U.S. patent application number 13/464461 was filed with the patent office on 2012-12-06 for compression programming of shape memory polymers below the glass transition temperature.
Invention is credited to Guoqiang Li, Wei Xu.
Application Number | 20120306120 13/464461 |
Document ID | / |
Family ID | 47219736 |
Filed Date | 2012-12-06 |
United States Patent
Application |
20120306120 |
Kind Code |
A1 |
Li; Guoqiang ; et
al. |
December 6, 2012 |
Compression Programming of Shape Memory Polymers Below the Glass
Transition Temperature
Abstract
Compression programming of a shape memory polymer without the
requirement of added heat, wherein the programming occurs at a
temperature below the glass transition of the shape memory polymer.
The shape memory polymer can be either a thermoset or a
thermoplastic shape memory polymer.
Inventors: |
Li; Guoqiang; (Baton Rouge,
LA) ; Xu; Wei; (Richland, WA) |
Family ID: |
47219736 |
Appl. No.: |
13/464461 |
Filed: |
May 4, 2012 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
61483196 |
May 6, 2011 |
|
|
|
Current U.S.
Class: |
264/320 |
Current CPC
Class: |
C08L 2312/00 20130101;
C08L 2203/02 20130101; C08L 2205/16 20130101; C08L 25/08 20130101;
C08L 25/08 20130101; C08L 67/02 20130101; C08K 3/40 20130101; C08L
2201/12 20130101 |
Class at
Publication: |
264/320 |
International
Class: |
B29C 59/02 20060101
B29C059/02 |
Goverment Interests
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
[0002] This invention was made with government support under grant
number CMMI 0946740 awarded by the National Science Foundation. The
Government has certain rights in the invention.
Claims
1. A method for compression programming of a shape memory polymer,
said method comprising: (a) applying a compressive force to a shape
memory polymer at a temperature less than the glass transition
temperature of the shape memory polymer, to deform the shape of the
shape memory polymer; and (b) releasing the compressive force,
while retaining a temporary shape deformation of the shape memory
polymer.
2. The method of claim 1, wherein the shape memory polymer is a
thermoset shape memory polymer.
3. The method of claim 1, wherein the shape memory polymer is a
thermoplastic shape memory polymer.
4. The method of claim 1, wherein the shape memory polymer
comprises a closed-cell foam.
5. The method of claim 1, wherein the compressing force step
applies a prestrain to the shape memory polymer, and wherein the
prestrain is larger than the yielding strain of the shape memory
polymer.
6. The method of claim 5, wherein the prestrain is less than 51%
strain.
7. The method of claim 6 wherein, the prestrain is less than 46%
strain.
8. The method of claim 1, wherein the compressing force step
applies a prestrain to the shape memory polymer, and wherein the
prestrain is at least 110% of the yielding strain of the shape
memory polymer, and wherein the prestrain is less than 100%
strain.
9. The method of claim 8, wherein the prestrain is at least 150% of
the yielding strain of the shape memory polymer.
10. The method of claim 1, wherein the compressing force step
applies a prestrain, and the prestrain is at least 7%.
11. The method of claim 10, wherein the prestrain is at least
10%.
12. The method of claim 1, wherein said releasing step comprises a
period of stress relaxation of at least 10 minutes.
13. The method of claim 1, wherein said compressing force step has
a strain rate in a range of 10.sup.-4/second to
10.sup.3/second.
14. The method of claim 1, additionally comprising the step of
heating the shape memory polymer above the glass transition
temperature, whereby the shape memory polymer returns from the
deformed shape to the shape memory polymer's memory shape.
15. A method for compression programming of a shape memory polymer,
said method comprising: applying prestrain force to a shape memory
polymer at a temperature less than the glass transition temperature
of the shape memory polymer, wherein the prestrain is greater than
the yielding strain of the shape memory polymer, and wherein a
temporary shape deformation of the shape memory polymer is
obtained.
16. The method of claim 15, additionally comprising a period of
stress relaxation.
17. The method of claim 16, wherein the period of stress relaxation
is at least 10 minutes.
18. The method of claim 15, wherein the force-applying step further
has a strain rate of 10.sup.-4/s to 10.sup.3/s.
19. The method of claim 15, additionally comprising the step of
heating the shape memory polymer above the glass transition
temperature, whereby the shape memory polymer returns from the
deformed shape to the shape memory polymer's memory shape.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority under 35 U.S.C.
.sctn.119(e) from U.S. Provisional Application Ser. No. 61/483,196,
filed 6 May 2011, entitled "Biomimetic Self-Healing Composite" the
contents of which are fully incorporated by reference herein. This
application is related to copending U.S. utility application Ser.
No. (to be assigned) entitled "Thermosetting Shape Memory Polymers
with Ability to Perform Repeated Molecular Scale Healing" in the
name of Guoqiang Li et al., the contents of which are fully
incorporated by reference herein.
BACKGROUND OF THE INVENTION
[0003] 1. Field of the Invention
[0004] This invention relates to polymeric materials, more
particularly to thermosetting polymers, and more particularly to
methods for programming a thermoset shape memory polymer at ambient
temperatures below the glass transition temperature of the
thermoset.
[0005] 2. Description of Related Art
[0006] Polymers:
[0007] Polymers are large molecules (macromolecules) composed of
repeating structural sub-units. These sub-units are typically
connected by covalent chemical bonds. The term polymer encompasses
a large class of compounds comprising both natural and synthetic
materials with a wide variety of properties. Because of the
extraordinary range of properties of polymeric materials, they play
essential and ubiquitous roles in everyday life. These roles range
from familiar synthetic plastics and elastomers to natural
biopolymers such as nucleic acids and proteins that are essential
for life.
[0008] Plastics, Thermoset and Thermoplastic:
[0009] A plastic is any of a wide range of synthetic or
semi-synthetic organic solids that are moldable. Plastics are
typically organic polymers of high molecular mass, but they often
contain other substances. There are two types of plastics:
thermoplastic polymers and thermosetting polymers. Thermoplastics
are the plastics that do not undergo chemical change in their
composition when heated and can be molded again and again. Examples
include polyethylene, polypropylene, polystyrene, polyvinyl
chloride, and polytetrafluoroethylene (PTFE). Common thermoplastics
range from 20,000 to 500,000 amu.
[0010] In contrast, thermosets are assumed to have an effectively
infinite molecular weight. These chains are made up of many
repeating molecular units, known as repeat units, derived from
monomers; each polymer chain will have several thousand repeating
units. Thermosets can take shape once; after they have solidified,
they stay solid. Thus, in a thermosetting process, a chemical
reaction occurs that is irreversible. In contrast to thermoplastic
polymers (discussed below), once hardened a thermoset resin cannot
be reheated and melted back to a liquid form.
[0011] Thermoplastic Polymers
[0012] A thermoplastic polymer, also known as a thermosoftening
plastic, is a polymer that turns to a viscous liquid when heated
and freezes to a rigid state when cooled sufficiently.
Thermoplastic polymers differ from thermosetting polymers (e.g.
phenolics, epoxies) in that they can be remelted and remolded.
[0013] Thermoplastics are elastic and flexible above a glass
transition temperature (T.sub.g) specific for each thermoplastic.
Between the T.sub.g and the higher melting temperature (T.sub.m)
some thermoplastics have crystalline regions alternating with
amorphous regions in which the chains approximate random coils. The
amorphous regions contribute elasticity and the crystalline regions
contribute strength and rigidity. Above the T.sub.m all crystalline
structure disappears and the chains become randomly interdispersed.
As the temperature increases above T.sub.m, viscosity gradually
decreases without any distinct phase change.
[0014] Thermoplastics can go through melting/freezing cycles
repeatedly and the fact that they can be reshaped upon reheating
gives them their name. However, this very characteristic of
reshapability also limits the applicability of thermoplastics for
many industrial applications, because a thermoplastic material will
begin to change shape upon being heated above its T.sub.g and
T.sub.m.
Thermosetting Polymers
[0015] According to an IUPAC-recommended definition, a
thermosetting polymer is a prepolymer in a soft solid or viscous
state that changes irreversibly into an infusible, insoluble
polymer network by curing. Thermoset materials are usually liquid
or malleable prior to curing and designed to be molded into their
final form, or used as adhesives. Others are solids like that of
the molding compound used in semiconductors and integrated circuits
(IC).
[0016] Curing of thermosetting polymers may be done, e.g., through
heat (generally above 200.degree. C. (392.degree. F.)), through a
chemical reaction (two-part epoxy, for example), or irradiation
such as electron beam processing. A cured thermosetting polymer is
often called a thermoset. The curing process transforms the
thermosetting resin into a plastic or rubber by a cross-linking
process. Energy and/or catalysts are added that cause the molecular
chains to react at chemically active sites (unsaturated or epoxy
sites, for example), linking into a rigid, 3-D structure. The
cross-linking process forms a molecule with a larger molecular
weight, resulting in a material with a heightened melting point.
During the curing reaction, the molecular weight increases to a
point so that the melting point is higher than the surrounding
ambient temperature, and the material solidifies.
[0017] However, uncontrolled heating of the material results in
reaching the decomposition temperature before the melting point is
obtained. Thermosets never melt. Therefore, a thermoset material
cannot be melted and re-shaped after it is cured. A consequence of
this is that thermosets generally cannot be recycled, except as
filler material.
[0018] Thermoset materials are generally stronger than
thermoplastic materials due to their three-dimensional network of
bonds. Thermosets are also better suited for high-temperature
applications (up to their decomposition temperature). However,
thermosets are generally more brittle than thermoplastics. Because
of their brittleness, thermosets are vulnerable to high strain rate
loading such as impact damage. Because many lightweight structures
use fiber reinforced thermoset composites, impact damage, if not
healed properly and timely, may lead to catastrophic structural
failure.
[0019] Smart Materials:
[0020] "Smart materials" or "designed materials" are materials that
have one or more properties that can be significantly changed in a
controlled fashion by external stimuli, such as stress,
temperature, moisture, pH, electric or magnetic fields. For
example, a shape memory polymer (SMP) is a material in which large
deformation can be induced and recovered through energy (often
thermal) changes or stress changes (pseudoelasticity). Shape memory
polymers have varying visual characteristics depending on their
formulation. Shape memory polymers may be epoxy-based, such as
those used for auto body and outdoor equipment repair;
cyanate-ester-based, which are used in space applications; and
acrylate-based, which can be used in very cold temperature
applications, such as for sensors that indicate whether perishable
goods have warmed above a certain maximum temperature.
[0021] Temperature-responsive shape memory polymers are materials
which undergo changes upon temperature change. There are also
several other types of shape memory polymers that undergo change
based on other than thermal energy. For example, pH-sensitive shape
memory polymers are materials that change in volume when the pH of
the surrounding medium changes. Photomechanical materials change
shape under exposure to light.
[0022] The shape of temperature-responsive SMPs can be repeatedly
changed by heating above their glass transition temperature
(T.sub.g). When heated, they become flexible and elastic, allowing
for easy configuration.
[0023] Once they are cooled, they will maintain their new shape.
However, the SMPs will return to their original shapes when they
are reheated above their T.sub.g. An advantage of shape memory
polymer resins is that they can be shaped and reshaped repeatedly
without losing their material properties, and these resins can be
used in fabricating shape memory composites.
[0024] Shape memory polymer composites are high-performance
composites, formulated using fiber or fabric reinforcement and
shape memory polymer resin as the matrix. Due to the shape memory
polymer matrix, these composites have the ability to be easily
manipulated into various configurations when they are heated above
their glass transition temperatures and exhibit high strength and
stiffness in their frozen or glassy state at temperatures lower
than their glass transition. SMPs can also be reheated and reshaped
repeatedly without losing their material properties.
[0025] Most SMPs are thermoplastics. However, a limited number of
thermoset SMPs have been identified. The thermoset SMPs have a
glass transition temperature above which the thermoset can be
molded. However, as thermosets, they do not have a melting
temperature, and after curing the polymer is set and can never be
re-molded. If a thermoset SMP continues to be heated beyond its
glass transition, it will never melt but will instead decompose
when it reaches its decomposition temperature.
[0026] Shape memory polymers have become increasingly used due to
their low cost, malleability, damage tolerance, and large ductility
(Lendlein et al., 2005; Otsuka and Wayman, 1998; Nakayama, 1991).
These advantages enable them to be active in various applications
such as micro-biomedical components, aerospace deployable equipment
and actuation devices (Tobushi et al., 1996; Liu et al., 2004;
Yakacki et al., 2007).
[0027] Lately, confined shape recovery of shape memory polymers has
been used for repeatedly sealing/closing structural-length scale
impact damages (Li and John, 2008; Nji and Li, 2010a; and John and
Li, 2010). A biomimetic two-step self-healing scheme,
close-then-heal (CTH), has been proposed by Li and Nettles (2010)
and Xu and Li (2010), and further detailed by Li and Uppu (2010),
for healing structural-length scale damage autonomously,
repeatedly, and molecularly. This concept has been validated by Nji
and Li (2010b). It is envisioned that SMPs will be used in
light-weight self-healing structures.
[0028] A thermally responsive shape memory polymer is not smart
without programming. A common programming cycle starts with a
deformation of the SMP at a temperature above the glass transition
temperature (T.sub.g). While maintaining the shape (strain) or
stress, the temperature is lowered below T.sub.g. With the
subsequent removal of the applied load, a temporary shape is
created and fixed. This completes the typical three-step
programming process. The original permanent shape can then be
recovered upon reheating above T.sub.g, which is the thermal
response aspect of a thermally responsive shape memory polymer.
[0029] The programming and shape recovery complete a
thermomechanical cycle. However, for practical applications such as
large structures, programming at very high temperature is not a
trivial task because it is a lengthy, labor-intensive, and
energy-consuming process. There is a need for alternative
programming approaches. Various types of programming have been
conducted on SMPs using the traditional
heating-loading-cooling-unloading method. If the applied load is a
tensile force or stretch, it is called tension or drawing
programming; if the applied load is a compressive force or shrink,
it is called compression programming. If either drawing or
compression programming is conducted at temperatures below T.sub.g,
it can be called cold-drawing programming or cold-compression
programming.
[0030] Several theories have been developed to explain the
thermomechanical profiles of SMPs. Earlier rheological models
(Tobushi et al., 1997; Bhattacharyya and Tobushi, 2000) were
capable of capturing the characteristic shape memory behavior of
SMPs but with limited prediction capability due to the loss of the
strain storage and release mechanisms. Later developments such as
mesoscale model (Kafka, 2001; Kafka, 2008) and molecular dynamic
simulation (Diani and Gall, 2007) propelled the understanding to a
rather detailed level. Recently, the phenomenological approach
(Morshedian et al., 2005; Gall et al., 2005; Liu et al., 2006;
Yakacki et al., 2007; Chen and Lagoudas, 2008a; Chen and Lagoudas,
2008b; Qi et al., 2008; Xu and Li, 2010) emerges to be an effective
tool to macroscopically investigate the thermomechanical mechanisms
of SMPs. The work by Liu et al. (2006) is a typical example of
these various phenomenological models, which proposed a continuum
mixture of a frozen and an active phase controlled by a sole
temperature dependent first-order phase transition concept for the
thermally activated SMPs. Although arguably treating the SMPs as a
special elastic problem without consideration of the time
dependence, the model reasonably captures the essential shape
memory responses to the temperature event. However, the involvement
of nonphysical parameters such as volume fraction of the frozen
phase and stored strain resulted in a controversial nonphysical
interpretation of the glass transition process. In order to address
such issues, Nguyen et al. (2008) presented a revolutionary concept
which attributes the shape memory effects to structural and stress
relaxation rather than the traditional phase transition hypothesis.
They proposed that the dramatic change in the temperature
dependence of the molecular chain mobility, which describes the
ability of the polymer chain segments to rearrange locally to bring
the macromolecular structure and stress response to equilibrium,
underpins the thermally activated shape memory phenomena of SMPs.
The fact that the structure relaxes instantaneously to equilibrium
at temperatures above T.sub.g but responds sluggishly at
temperature below T.sub.g, suggests that cooling macroscopically
freezes the structure into a non-equilibrium configuration below
T.sub.g, and thus allows the material to retain a temporary shape.
Reheating above T.sub.g reduces the viscosity, restores mobility
and allows the structure to relax to its equilibrium configuration,
which leads to shape recovery.
[0031] It is noted that cold-drawing programming of thermoplastic
SMPs has been conducted by several researchers. Lendlein and Kelch
(2002) indicated that shape memory polymer (SMP) can be programmed
by cold-drawing but did not give many details. Ping et al. (2005)
investigated a thermoplastic poly(.epsilon.-caprolactone) (PCL)
polyurethane for medical applications. In this polymer, PCL was the
soft segment, which could be stretched (tensioned) to several
hundred percent at room temperature (15-20.degree. C. below the
melting temperature of the PCL segment). They found that the
cold-drawing programmed SMP had a good shape memory capability.
Rabani et al. (2006) also investigated the shape memory
functionality of two shape-memory polymers containing short aramid
hard segments and poly(c-caprolactone) (PCL) soft segments with
cold-drawing programming. As compared to the study by Ping et al.
(2005), the hard segment was different but the same soft segment
PCL was used. Wang et al. (2010) further studied the same SMP as
Ping et al. (2005). They used FTIR to characterize the
microstructure change during the cold-drawing programming and shape
recovery. They found that in cold drawing programming, the
amorphous PCL chains orient first at small extensions, whereas the
hard segments and the crystalline PCL largely maintain their
original state. When stretched further, the hard segments and the
crystalline PCL chains start to align along the stretching
direction and quickly reach a high degree of orientation; the
hydrogen bonds between the urethane units along the stretching
direction are weakened, and the PCL undergoes stress-induced
disaggregation and recrystallization while maintaining its overall
crystallinity. When the SMP recovers, the microstructure evolves by
reversing the sequence of the microstructure change during
programming. Zotzmann et al. (2010) emphasized that a requirement
for materials suitable for cold-drawing programming is their
ability to be deformed by cold-drawing. Based on their discussion,
it seems that an SMP with an elongation at break as high as 20% is
not suitable for cold-drawing programming.
BRIEF SUMMARY OF THE INVENTION
[0032] We have discovered that SMPs can gain the shape memory
capability, creating a non-equilibrium configuration at
temperatures below T.sub.g. We disclose a method for isothermal
compression programming of a shape memory polymer, said method
comprising: applying force to a shape memory polymer at a
temperature less than the glass transition temperature of the shape
memory polymer in a magnitude sufficient to produce a temporary
shape deformation of the shape memory polymer. The shape memory
polymer can be a thermoset or a thermoplastic shape memory polymer.
The shape memory polymer can optionally be a closed-celled foam. In
certain embodiments the applied force is a prestrain, and the
prestrain is larger than the yielding strain of the shape memory
polymer. In certain embodiments the applied force is a prestrain,
and the prestrain is less than 30, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, or 51% strain. When the applied
force is a prestrain, the prestrain can be at least 105%, 110%,
115%, 120%, 125%, 130%, 135%, 140%, 145%, 150%, 160%, 170%, 180%,
190%, 200%, 210%, 220%, 225%, 230%, 235%, 240%, 245%, 250%, 275%,
300%, 325%, 350%, 375%, 400%, 425%, 450%, 475%, 500%, 525%, 550%,
575%, 600%, 625%, or 650% of the yielding strain of the shape
memory polymer, with a proviso that the prestrain is never more
than a 100% strain. When the applied force is a prestrain, the
prestrain can be can be at least 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 25, 30, 35, 40, 45, 50 or 55%. In certain
embodiments, a method for isothermal compression programming of a
shape memory polymer further comprises a stress relaxation time of
at least 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 45,
60, 75, 90, 105, 120, 150, 180, 210, 240 or 260 min. Methods in
accordance with the invention comprise various non-mutually
exclusive combinations of the features set forth herein.
DEFINITIONS
[0033] "Decomposition Temperature (T.sub.D)" is the temperature at
which chemical bonds are broken or violent oxidation or fire
occurs.
[0034] "Fixed strain" is the difference between the prestrain and
the springback. At the end of programming, there is a rebound or
springback when the load is removed.
[0035] "Glass transition temperature (T.sub.8)": the temperature at
which amorphous polymers undergo a transition from a rubbery,
viscous amorphous liquid (T>T.sub.g), to a brittle, glassy
amorphous solid (T<T.sub.g). This liquid-to-glass transition (or
glass transition for short) is a reversible transition. The glass
transition temperature T.sub.g, if one exists, is always lower than
the melting temperature, T.sub.m, of the crystalline state of the
material. An amorphous solid that exhibits a glass transition is
called a glass. Supercooling a viscous liquid into the glass state
is called vitrification. Despite the massive change in the physical
properties of a material through its glass transition, the
transition is not itself a phase transition; rather it is a
phenomenon extending over a range of temperatures and is defined by
one of several conventions. Several definitions of T.sub.g have
been endorsed as accepted scientific standards. Nevertheless, all
such definitions are to some extent arbitrary, and they can yield
different numeric results. The various definitions of T.sub.g for a
given substance typically agree within a few degrees Kelvin.
[0036] "Healing Temperature (T.sub.H)": The healing temperature can
be defined functionally as a preferred temperature above the
melting temperature where thermoplastic molecules overcome
intermolecular barriers and are able to gain mobility and to more
effectively diffuse within a material.
[0037] "Melting point (T.sub.m)": The term melting point, when
applied to polymers, is not used to suggest a solid-liquid phase
transition but a transition from a solid crystalline (or
semi-crystalline) phase to a still solid but amorphous phase. The
phenomenon is more properly called the crystalline melting
temperature. Among synthetic polymers, crystalline melting is only
discussed with regards to thermoplastics, as thermosetting polymers
decompose at high temperatures rather than melt. Consequently,
thermosets do not melt and thus have no T.sub.m.
[0038] "Prestrain" is the maximum strain applied during
programming.
[0039] "Relaxation time" is the time elapsed during the stress
relaxation process.
[0040] "Shape fixity" is similar to strain fixity, suggesting that
a temporary shape is fixed.
[0041] "Shape fixity ratio" is the ratio of the strain after
programming over the prestrain.
[0042] "Strain recovery" is the amount of strain that is recovered
during shape recovery process.
[0043] "Stress relaxation" occurs when, after a material reaches a
certain deformation, the stress continuously reduces while the
strain remains constant.
[0044] "Yield strain" is the strain corresponding to yielding. In
the stress-strain curve, the change of slope signals the start of
yielding.
BRIEF DESCRIPTION OF THE DRAWINGS
[0045] FIG. 1 An illustration of a four-step thermomechanical cycle
in accordance with the present invention: Programming (Step 1-Step
3) and shape recovery (Step 4).
[0046] FIG. 2 DMA results as functions of temperature: (a) solid
line-storage modulus (b) dashed line-loss modulus.
[0047] FIG. 3 Shape fixity results at temperature below T.sub.g for
specimens programmed at different prestrain levels (5%, 10%, and
30%).
[0048] FIG. 4 Strain-time response during the entire
thermomechanical cycle for specimens programmed with (Panel a) 30%
and (Panel b) 10% prestrain. The four steps for the specimen with
120 min of stress relaxation time during programming are also
shown.
[0049] FIG. 5 The 3-D thermomechanical cycle in terms of
stress-strain-time for different stress relaxation times with
prestrain levels of 10% and 30%.
[0050] FIG. 6 An analogous decomposition scheme for the deformation
gradient.
[0051] FIG. 7 A linear rheological illustration for stress
response.
[0052] FIG. 8 Numerical simulation for samples with 30% prestrain
(FIGS. 8 a) and 10% prestrain (FIG. 8 b) during the entire
thermomechanical cycle. The four steps for the entire
thermomechanical cycle for the specimen with 120 min of stress
relaxation time during programming are also shown.
[0053] FIG. 9 Recovery strain as a function of temperature for
different heating rates.
[0054] FIG. 10 Recovery strain as a function of temperature for
different heating profiles.
[0055] FIG. 11 Thermomechanical cycle results for different
programming temperatures: (FIG. 11 a) programming followed with
immediate heating recovery, (FIG. 11 b) programming followed with
cooling then heating recovery.
[0056] FIG. 12: Flowchart of the MATLAB program.
[0057] FIG. 13 Thermal response to stress-free cooling.
[0058] FIG. 14 Stress-strain response of the SMP at different
temperatures.
[0059] FIG. 15 Stress-strain response of the SMP at different
strain rates.
[0060] FIG. 16. Thermal response for a stress-free, constant
heating rate (q=0.56.degree. C./min) test.
[0061] FIG. 17 DMA results for the SMP based syntactic foam and the
pure SMP
[0062] FIG. 18 XPS spectra of the pure SMP and the SMP based
syntactic foam for (FIG. 18a) the C 1s electron, and (b) the O 1s
electron
[0063] FIG. 19 Strain-time response during the entire
thermomechanical cycle for specimens programmed with (FIG. 19a) 30%
prestrain and (FIG. 19b) 20% prestrain (the four steps shown in the
figures are for the curve with 120 min of stress relaxation
time.)
[0064] FIG. 20 Viscoelastic behavior of the foam by creep test at
room temperature
[0065] FIG. 21 SEM observation of (a) pristine specimen and (b)
specimen after 30% cold-compression programming
[0066] FIG. 22 Thermo-mechanical cycle in terms of (FIG. 22a)
stress-strain-time and (FIG. 22b) stress-strain-temperature
responses for different stress relaxation time with a pre-strain
level of 30% and 20%
[0067] FIG. 23 Equivalent scheme for the SMP matrix
[0068] FIG. 24 An analogous decomposition scheme for the
deformation gradient FIG. 25 An arbitrary nonlinear damage model
with a linear equivalent FIG. 26 A linear rheological illustration
for stress response
[0069] FIG. 27 Comparison of numerical simulations with
experimental results for the full thermomechanical cycle (a) strain
evolution with 30% prestrain, (b) strain evolution with 20%
prestrain, and (c) thermomechanical cycle in terms of
stress-strain-time response
[0070] FIG. 28 Comparison of numerical simulation with test results
for a 2-D traditional thermomechanical cycle.
[0071] FIG. 29 Thermomechanical cycle results for specimen with
different .PHI.p.
[0072] FIG. 30 Thermomechanical cycle results for specimens with
different w.
[0073] FIG. 31 Thermal response to stress-free natural cooling.
[0074] FIG. 32 Stress-strain response of the SMP based syntactic
foam at various temperatures.
[0075] FIG. 33 Stress-strain response of the SMP based syntactic
foam at different strain rates.
[0076] FIG. 34 Thermal response for stress-free constant-rate
heating.
DETAILED DESCRIPTION OF THE INVENTION
[0077] Disclosed for the first time is a novel thermomechanical
programming process for thermally activated SMPs, either
thermoplastic or thermosetting SMPs. In accordance with the present
invention, a non-equilibrium configuration can be created and
maintained in shape memory polymers (SMPs) below T.sub.g. A new and
effective approach is set forth herein which programs glass
transition-activated SMPs directly at temperatures well below
T.sub.g. The 1-D compression programming below T.sub.g and free
shape recovery were extensively investigated both experimentally
(Example 1) and analytically (Example 2).
[0078] Example 3 applies the data and information from Example 1 to
a shape memory polymer (SMP)-based self-healing syntactic foam,
which was found to be capable of self-sealing structural scale
damage repeatedly, efficiently, and almost autonomously.
[0079] In Example 4, a structural-relaxation constitutive model
featuring damage-allowable thermoviscoplasticity was developed to
predict the nonlinear shape memory behavior of the SMP based
syntactic foam programmed at glassy temperatures. After validation
by both 1-D (compression) and 2-D (compression in longitudinal
direction and tension in transverse direction) tests, the
constitutive model was used to evaluate the effects of several
design parameters on the thermomechanical behavior of the SMP based
syntactic foam. It is concluded that the model is a useful tool for
designing and training this novel self-healing composite.
[0080] Thus, instead of the heating followed by cooling, the
programming was conducted at a constant temperature which was well
below the T.sub.g of the SMP. In one embodiment, this invention
comprises an approach to program thermoset or thermoplastic SMPs
directly at temperatures well below T.sub.g, which effectively
simplifies the shape fixing process. 1-D compression programming
below T.sub.g and free shape recovery of a thermoset SMP were
experimentally investigated. Functional stability of the shape
fixity under various environmental attacks was also experimentally
evaluated.
[0081] A mechanism-based thermoviscoelastic-thermoviscoplastic
constitutive model incorporating structural and stress relaxation
was developed to predict the nonlinear shape memory behavior of the
SMP trained below T.sub.g. Comparison between the prediction and
the experiment showed good agreement. The structure dependence of
the thermomechanical behavior of the SMP was further discussed
through a parametric study per the validated constitutive model.
This study validates that programming by cold-compression is a
viable alternative for thermally responsive thermoset SMPs.
[0082] In accordance with the present invention, a thermosetting
SMP was programmed by cold-compression. The elongation at break is
about 4% for this thermosetting SMP at temperature below T.sub.g,
which is not suitable for cold-drawing (tensioning)
programming.
[0083] The thermomechanical behavior of the thermally responsive
thermoset SMP with a unique programming process at glassy
temperature has been studied both experimentally and theoretically.
Among the results of this work are:
[0084] (1) The approach of cold-compression programming of a
thermosetting shape memory polymer was tested and modeled. The test
results show that this is an effective and efficient method which
achieves very large and durable shape fixity, and has similar shape
memory capability to specimens programmed by the more lengthy,
labor-intensive, and energy-consuming approach currently used.
[0085] (2) The concept that the shape memory effect in nature is a
transition between equilibrium and nonequilibrium configuration of
the SMP structure can explain the shape memory mechanism of a
thermoset SMP programmed by cold-compression.
[0086] (3) It was found that the prestrain level should be larger
than the yielding strain of the SMP in order to fix a temporary
shape at temperatures below T.sub.s.
[0087] (4) Longer stress relaxation time leads to larger shape
fixity ratio. The upper bound of the shape fixity is determined by
the difference between the prestrain and the spring-back, which is
the ratio of the relaxed stress over the relaxed modulus.
[0088] (5) A finite deformation theory and mechanism based
thermoviscoelastic constitutive model has been developed to study
the thermomechanical behavior of the SMP programmed by
cold-compression. Because the pseudo-plasticity and structure
evolution are incorporated, the model reasonably captures the
essential characteristics of the shape memory response. A fairly
good agreement has been reached between the testing and
modeling.
[0089] (6) The parametric simulation study reveals that the shape
memory behavior is highly dependent on the heating profile. A
faster heating rate shifts the onset of recovery to a higher
temperature.
[0090] (7) The effect of heating history further corroborates that
the shape recovery response is more a thermodynamic structure
evolution than a steady state variable-determined phase transition.
Beyond the glass transition temperature, even without further
heating to a higher temperature, an adequate time period of soaking
can still help achieve the full recovery.
[0091] (8) As long as the programming occurs in glassy state, the
programming at a higher temperature followed with an immediate
heating recovery leads to a higher shape fixity ratio and has
slight effect on the strain recovery. The recovery of the SMP
programmed at a higher temperature followed by a cooling process
initiates at a lower temperature and progresses at a faster
rate.
[0092] (9) It seems that the time-temperature equivalence principle
holds for the shape memory behavior. Similar shape recovery ratio
can be achieved at a higher temperature with a shorter time period
of soaking or a longer time period of soaking at a lower
temperature.
[0093] The programming of thermoset SMPs at glassy temperatures was
successfully applied to a SMP-based, self-healing syntactic foam. A
structure-evolving, damage-allowable thermoviscoplastic model has
been developed, which reasonably captured the most essential shape
memory response during this process. Results of this study
included:
[0094] (1) Cold programming was effective and efficient for
SMP-based self-healing syntactic foam. Considerable recoverability
was achieved, although some damage in glass hollow microsphere
inclusions was inevitable.
[0095] (2) A finite deformation, continuum constitutive model was
developed to study the thermomechanical behavior of the SMP-based
self-healing syntactic foam programmed at glassy temperature. As
thermoviscoplasticity, structural relaxation and inclusion damage
mechanism are considered in the model, the model plausibly captures
the essential elements of the shape memory response. A fairly good
agreement has been reached between the modeling results and the
experimental results.
[0096] (3) The parametric simulation study revealed preferred
embodiments for SMP-based syntactic foam: a high volume fraction of
microsphere inclusions leads to a low recovery ratio, and a high
wall thickness ratio of the glass microballoons leads to a larger
recovery strain. Particular optimized configurations are achieved
by adjusting and balancing these parameters.
[0097] The current model is based on closed-cell SMP based
syntactic foam. Preferred embodiments of the invention comprise
programming of closed-cell SMP foams, although open-cell foams may
also be used.
EXAMPLES
Example 1
Testing of Thermomechanical Behavior of Thermoset Shape Memory
Polymer Programmed by Cold-Compression
[0098] In this example the SMP specimens were isothermally and
uniaxially compressed to a certain strain level and then held for
relaxation while strain was maintained. It was found that
meaningful fixity ratios were achieved efficiently with an adequate
prestrain and various relaxation time periods.
[0099] The stability of the fixed temporary shape was then verified
under various environmental attacks such as water immersion and
ultraviolet light exposure. Subsequent free shape recovery tests
proved that the permanent shape was also recoverable upon heating,
similar to the specimens programmed using the traditional
approach.
[0100] Experimental Methods
[0101] Raw Materials, Curing, and Specimen Preparation
[0102] The shape memory polymer was a polystyrene-based thermoset
SMP resin system with a T.sub.g of 62.degree. C. commercially sold
by CRG Industries under the name of Vertex. A hardening agent
distributed by the same company was added to the SMP resin. The
mixture was blended for 10 min before it was poured into a
229.times.229.times.12 mm steel mold and placed into a vacuum
chamber at 40 kPa for 20 min for removal of any air pockets
introduced during the mixing process. The resin was then cured in
an oven at 79.degree. C. for 24 hours, followed by 6 hours at
107.degree. C. After curing, the SMP panel was de-molded and cut
into 30.times.30.times.12 mm block specimens for further
testing.
[0103] Dynamic Mechanical Analysis
[0104] In order to determine the glass transition zone of the SMP,
the dynamic mechanical analysis (DMA) test was conducted on a DMA
2980 tester from TA instruments per ASTM D 4092. A rectangular
sheet with dimensions of 17.5.times.11.9.times.1.20 mm was placed
into a DMA single cantilever clamping fixture. A small dynamic load
at 1 Hz was applied to a platen and the temperature was ramped from
room temperature to 120.degree. C. at a rate of 3.degree. C./min.
The amplitude was set to be 15 .mu.m.
[0105] Coefficient of Thermal Expansion
[0106] The linear thermal expansion coefficient was measured by
using a linear variable differential transducer (LVDT, Cooper
Instruments LDT 200 series) system to record the specimen surface
displacement and a Yokagawa DC100 data acquisition system to
collect the thermocouple measurement of the temperature change. The
temperature was ramped from room temperature to 100.degree. C. at
an average heating rate of 0.56.degree. C./min. After equilibration
for 30 minutes, the sample was naturally cooled down to room
temperature.
[0107] Programming by Isothermal Flat-Wise Uniaxial Compression
Test
[0108] Specimens were programmed at a temperature well below the
T.sub.g of the SMP, instead of the typical lengthy programming
process above T.sub.g. In this example room temperature (20.degree.
C.) was adopted for programming. The programming was conducted by a
uniaxial compression test. Uniaxial flat-wise compression was
performed with a MTS QTEST150 electromechanical frame outfitted
with a moveable furnace (ATS heating chamber) per the ASTM C 365
standard at a displacement rate of 1.3 mm/min to the test prestrain
level. Temperature control and monitoring were achieved through a
thermocouple placed in the chamber near the SMP specimen.
Stress-strain responses were generated for different prestrain
levels and stress relaxation time.
[0109] In this study, three prestrain levels (5%, 10%, and 30%),
corresponding to the elastic zone (5%) and post-yielding zone (10%
and 30%), respectively, were selected. The stress relaxation time
was determined at 0 min, 30 min, 120 min, and 260 min for the 5%
prestrain level, and 0 min, 5 min, 15 min, 30 min, and 120 min for
the 10% and 30% prestrain levels. At least three effective
specimens were tested for each prestrain level and stress
relaxation time. Based on the test results (1) the strain should be
greater than the yielding strain; (2) the strain is preferably as
high as about 40%, which starts to see significant strain
hardening; (3) strain rate affects the shape fixity, i.e., for the
same programming strain, the higher the strain rate, the lower the
shape fixity. For example, tests using a strain rate of about
1,000/s for cold-compression programming showed reduced shape
fixity, while shape memory capability was not affected, i.e.,
strain rate was reduced as compared to a lower strain rate such as
0.01/s.
[0110] Free Shape Recovery Test
[0111] Once the specimens were programmed, an unconstrained strain
recovery test was then implemented, where the compressed SMP
specimen was heated to T.sub.high=79.degree. C. at an average
heating rate of q=0.82.degree. C./min. The same LVDT system was
used to track the movement of the specimen during heating.
[0112] The thermomechanical cycle including programming and shape
recovery is schematically shown in FIG. 1. The programming
comprises three steps at a glassy temperature--typically (but not
necessarily) conducted at a fixed glassy temperature (room
temperature was used in this study): compression to the designed
prestrain (Step 1), stress relaxation (Step 2), and removal of
loading (Step 3).
[0113] Depending on the relaxation time, the entire programming
takes from minutes to a couple of hours, compared to prior
heat-based programming methods, which require refined temperature
control and typically over 10 hours of programming time (Li and
Nettles, 2010; Li and Uppu, 2010). Step 4, shape recovery, is
similar to what has been done in prior methods.
[0114] Environmental Conditioning Tests
[0115] The capability for the SMP to maintain its shape fixity has
been well established for specimens programmed by the prior
high-temperature programming approach. Prior to the present
invention, however there was no information about the ability to
achieve or the functional stability of SMP programmed at a
temperature below T.sub.g under various environmental attacks. The
stability of the temporary shape of the SMP specimens programmed in
accordance with the invention was investigated for water immersion,
ultraviolet light (UV) exposure and a combination of these two
conditions. For the water immersion test, one programmed specimen
was immersed in a cup of drinking water. The water level was about
2.5 cm above the surface of the specimen. For the UV exposure test,
one programmed specimen was put in the same plastic cup without
water. A 300-Watt Mog Base UV lamp, which had a wavelength ranging
from 280 to 340 nm (mixed UV-A and UV-B light), was placed about 30
cm away from the transparent plastic cup. For the combined water
immersion and UV exposure test, one programmed specimen was
immersed in the same transparent plastic cup containing the same
amount of drinking water. At the same time, the specimen was
exposed to the same UV source with the same intensity. The
specimens were monitored regularly for up to 3 months in order to
record any dimension changes. In the first two weeks, the dimension
of the specimens was measured every day and after that, the
dimension was recorded every week. After 3 months of environmental
attacks, the specimens were recovered using the same procedure as
the non-attacked specimens.
[0116] Experimental Results
[0117] DMA Test Results
[0118] The experimental results in FIG. 2 illustrate the storage
modulus and loss modulus of the SMP as functions of temperature.
The glass transition zone and T.sub.g can be found from the storage
modulus per ASTM D 4092. The intersection between the tangent at
the inflection point and the extrapolated tangent at the glassy
state defines the lower limit and the intersection between the
tangent at the inflection point and the extrapolated tangent at the
rubbery plateau defines the upper limit of the glass transition
zone. The average value in between them defines the glass
transition temperature T.sub.g=67.78.degree. C. The listed value of
T.sub.g=62.degree. C. by the distributor was determined by
differential scanning calorimetry (DSC), which was about 4.degree.
C. lower than their DMA results. Therefore, the T.sub.g provided by
the manufacturer is consistent with our test results.
[0119] Uniaxial Strain-Controlled Compression Programming
[0120] The strain evolution during the material programming
process, including the first three steps of the entire
thermomechanical cycle in FIG. 1, is presented in FIG. 3. It is
seen that shape fixity highly depends on the prestrain levels.
[0121] SMP specimens programmed at a 5% prestrain level could not
fix a temporary shape, regardless of the length of the stress
relaxation time. Upon removal of the load, immediate full
spring-back was observed. For specimens programmed at 30%
prestrain, however, a reasonable amount of strain was preserved,
even when the load was instantly removed (zero relaxation time).
With zero stress relaxation time, the shape fixity was still about
73%. Therefore, the level of prestrain does affect programming at
glassy temperatures.
[0122] As documented in a previous study (Li and Nettles, 2010),
the uniaxial compression yielding strain of the same thermosetting
SMP is about 7% at the same glassy temperature. A 5% prestrain
falls in the elastic region of the SMP. Therefore, immediate full
springback occurs regardless of the relaxation time held. At 30%
prestrain, the SMP specimen already yields and thus is able to
maintain a reasonable temporary fixed strain even without stress
relaxation. Therefore, a post-yielding prestrain level determines
the success of the programming at glassy temperature.
[0123] It can also be observed from FIG. 3 that, with 30%
prestrain, a longer stress relaxation time in Step 2 tends to
enhance the shape fixity ratio. As the relaxation time continuously
increased, the shape fixity asymptotically approached an upper
bound, which is equal to the difference between the prestrain and
elastic spring-back (ratio of the relaxed stress over the relaxed
modulus). Further increase in the relaxation time can hardly bring
up any significant increase in the shape fixity ratio.
[0124] With 10% prestrain, which is about 3% higher than the yield
strain, a tendency similar to 30% prestrain is observed. Therefore,
as long as the prestrain is above the yield strain, a certain
amount of shape fixity can be realized. Of course, as the prestrain
increases, the shape fixity also increases. For example, at zero
stress relaxation time, the shape fixity is about 62.5% for 10%
prestrain level, which is lower than the corresponding shape fixity
of 73% for 30% prestrain level. It is also observed that the shape
fixity with 10% prestrain plateaus earlier than that with 30%
prestrain as stress relaxation time increases, possibly due to less
viscoelastic and viscoplastic deformation with lower prestrain
level.
[0125] Environmental Conditioning Test
[0126] The environmental attack test detected no change in specimen
dimensions for any environmental conditions during the tests. Free
shape recovery test showed almost the same recovery ratio as those
non-attacked specimens. Since the observation time was up to 3
months and the environment conditions covered the most common
working conditions, the stability of the non-equilibrium
configuration created by cold-compression programming should be
well confirmed. Thus, the temporary shape of the thermosetting SMP
programmed at temperature below T.sub.g is stable.
[0127] Free Recovery Test
[0128] FIG. 19 shows the entire thermomechanical cycles, including
the unconstrained strain recovery during the heating process (Step
4 in FIG. 1). From FIG. 4 (a), which is programmed by 30%
prestrain, it is observed that initially the programmed specimen
only shows a slight and gradual thermal expansion. As the
temperature approaches T.sub.g, the influence of the entropy change
dominates, leading to a rapid strain recovery. At temperatures well
above T.sub.g, most of the prestrain has been released and the
strain converges to a stabilized value.
[0129] It is interesting to note that a similar sigmoidal-type
strain recovery path is shared by all the specimens with differing
relaxation times during programming, indicating that the strain
release mechanism is generally independent of the holding time
during programming. With 10% prestrain (FIG. 4 (b)), the shape
recovery follows a tendency similar to that with 30% prestrain. A
noticeable difference exists in the shape recovery ratio. With the
10% prestrain, the shape recovery ratio is about 100%, regardless
of the stress relaxation time during programming; with the 30%
prestrain, there is a small amount of strain that cannot be
recovered. A possible reason is that with the 30% prestrain, some
damage may have been created within the SMP specimen, which cannot
be recovered during free shape recovery.
[0130] Overall, the shape memory capability of the thermosetting
SMP programmed by cold-compression is considerable. The approach of
programming at a glassy temperature is much simpler and easier to
implement, and exhibits a considerable shape memory capability.
[0131] The 3-D stress-strain-time behaviors for the entire
thermomechanical cycle, which include the three-step
cold-compression programming process and the one step heating
recovery, are shown in FIG. 5, for both the 10% and 30% prestrain
levels. An extremely nonlinear, time- and temperature-dependent
behavior is revealed. In-depth understanding of this complex
thermomechanical behavior is elucidated by comprehensive
constitutive modeling, which is developed in the following
example.
Example 2
Constitutive Modeling of Thermomechanical Behavior of Thermoset
Shape Memory Polymer Programmed by Cold-Compression
[0132] A continuum finite deformation based thermoviscoelastic
model was developed to further elucidate the finding obtained in
Example 1. The concept presented by Nauyen et al. (2008) that the
shape memory effect reflects the transition between equilibrium and
nonequilibrium configuration of the SMP structure was adopted and
extended to the isothermal shape fixity process below T.sub.g. The
Narayanaswamy-Moynihan model (Narayanaswamy, 1971; Moynihan et al.,
1976) was incorporated to represent the structure relaxation.
Comparisons with experiments showed that the model could fairly
well reproduce the general thermomechanical behavior of the
thermoset SMP. Subsequent parametric studies were conducted to
explore the shape memory responses to different stimuli and
different programming temperatures per the validated constitutive
model.
[0133] Constitutive Modeling
[0134] General Consideration
[0135] The molecular resistance to inelastic deformation for
amorphous thermoset SMPs below the glass transition temperature
(T.sub.g) mainly originates from two sources: the intermolecular
resistance to segmental rotation and the entropic resistance to
molecular alignment (Boyce et al., 1989, 2001).
[0136] The four-step thermomechanical cycle shown in FIG. 5 can be
analyzed as follows: It is assumed that the plastic flow does not
commence until the stressed material completely overcomes the free
energy barrier to the molecular chain mobility, a restriction
imposed on molecular chain motion from neighboring chains.
Following the initial yield, molecular alignment occurs and
subsequently alters the configurational entropy of the material
(Step 1). Since the plastic strain develops in a rate-dependent
manner, the length of relaxation time physically indicates the
degree of the nonequilibrium configuration (Step 2). A relaxed
configuration is then obtained after elastically unloading to a
stress free state (Step 3). Due to the high material viscosity and
vanishing chain mobility at the glassy programming temperature, the
nonequilibrium structure is prevented from relaxing to the
equilibrium state during the observed time frame, resulting in a
retained temporary shape at the end of Step 3. Upon heating above
T.sub.g the viscosity decreases and chain mobility increases. The
thermodynamically favorable tendency of increasing entropy allows
the material to restore its equilibrium configuration and thus
achieve shape recovery (Step 4).
[0137] Based on this understanding, a mechanism-based constitutive
model was developed by incorporating the nonlinear structural
relaxation model into the continuum finite-deformation
thermoviscoelastic theory. The aim of this effort was to establish
a quantitative understanding of the shape memory behavior of the
thermally responsive thermoset SMP programmed at temperatures below
T.sub.g. To keep the model simple, several basic assumptions were
made for purposes of the modeling:
[0138] 1) The SMP system is assumed to be macroscopically isotropic
and homogeneous. The stress field is assumed to be uniform.
[0139] 2) Heat transfer in the material is not considered. The
temperature is treated as uniform throughout the entire body.
[0140] 3) The structural relaxation and inelastic behavior of the
material is assumed to be solely dependent on the temperature, time
and stress.
[0141] 4) The material is assumed to undergo no damage during the
thermomechanical cycle.
[0142] Deformation Response
[0143] As illustrated in FIG. 6, any arbitrary thermomechanical
path can be considered as a transition of the material between an
initial reference configuration of an undeformed and unheated
continuum body denoted by .OMEGA..sub.0 and a spatial configuration
.OMEGA. of the deformed body which may have also experienced a
certain temperature change. It is assumed that the configuration
.OMEGA..sub.0 is either in thermodynamic equilibrium in rubbery
state or in a stress-free glassy configuration originated from
mechanically unconstrained cooling from high temperature. A
deformation gradient
F = .differential. x .differential. X ##EQU00001##
represents the tangent of a general nonlinear mapping
x=x(X(t),T(t),t) of a material point from .OMEGA..sub.0 to .OMEGA..
This deformation mapping is then considered to be a combination of
a thermal deformation and a mechanical deformation, which can be
separated through a multiplicative decomposition scheme (Lu and
Pister, 1975; Lion, 1997):
F.sub.T=F.sub.MF.sub.T (1)
[0144] Here, F.sub.M defines the mechanical deformation gradient;
F.sub.T defines the mapping path from .OMEGA..sub.0 to
.OMEGA..sub.T, an intermediate heated configuration. Because the
material is assumed to be isotropic, the thermal deformation
gradient can be expressed as
F.sub.T=J.sub.T.sup.1/3I (2)
where J.sub.T=det (F.sub.T) is the determinant of the thermal
deformation gradient, representing the volumetric thermal
deformation.
[0145] To separate the elastic and viscous responses, we introduce
a multiplicative split of the mechanical deformation gradient into
elastic and viscous components (Sidoroff, 1974; Lion, 1997):
F.sub.M=F.sub.eF.sub.v (3)
[0146] Although a discrete spectrum of nonequilibrium processes
F.sub.M.sup.i=F.sub.e.sup.iF.sub.v.sup.i (i=1, . . . N) (Govindjee
and Reese, 1997) would be more appropriate to describe the general
behavior of the real solid materials, only single stress relaxation
is considered in the following derivation for the sake of
convenience. The viscous part of the velocity gradient is then
defined as:
L.sub.v={dot over (F)}.sub.vF.sub.v.sup.-1=D.sub.v+W.sub.v (4)
where D.sub.v is the symmetric part of L.sub.v, representing the
plastic stretch of the velocity gradient and W.sub.v is the
asymmetric component, representing the plastic spin. By applying
the polar decomposition, we can also split F.sub.e into a stretch
(V.sub.e) and a rotation (R.sub.e) as:
F.sub.e=V.sub.eR.sub.e (5)
[0147] Structural Relaxation Response
[0148] A fictive temperature T.sub.f based approach firstly
introduced by Tool (1946) has been proved to be extremely
successful in supplying the information about the free volume or
the structure in the formulation of the free energy density. The
fictive temperature T.sub.f is an internal variable to characterize
the actual thermodynamic state during the glass transition, defined
as the temperature at which the temporary nonequilibrium structure
at T is in equilibrium (Nguyen et al., 2008). It was assumed that
the rate change of the fictive temperature is proportional to its
deviation from the actual temperature and the proportionality
factor depends on both T and T.sub.f (Narayanaswamy, 1971), as
indicated in the evolution equation (Tool, 1946):
T f t = K ( T , T f ) ( T - T f ) ( 6 ) ##EQU00002##
[0149] The Narayanaswamy-Moynihan model (NMM), discussed in detail
by Donth and Hempel (2002), is an improvement for this approach.
Instead of postulating a simple exponential relaxation mechanism
governed by a single relaxation time (Tool, 1946), the
non-exponential structural relaxation behavior as well as the
spectrum effect were studied. It is assumed that the whole thermal
history T(t) starts from a thermodynamic equilibrium state where
T(t.sub.0)=T.sub.f(t.sub.0). And Tool's fictive temperature is
defined by:
T.sub.f(t)=T(t)-.intg..sub.t.sub.0.sup.t.phi.(.DELTA..zeta.)dT(t)
(7)
[0150] The response function is chosen, according to Moynihan et
al. (1976), in the manner of a Kohlrausch function (Kohlrausch,
1847), in which the value of .beta. describes the non-exponential
characteristic of the relaxation process:
.phi.=exp[-(.DELTA..zeta.).sup..beta.], 0<.beta..ltoreq.1
(8)
[0151] The dimensionless material time difference .DELTA..zeta. is
introduced to linearize the relaxation process:
.DELTA. = ( t ) - ( t ' ) = .intg. t ' t t .tau. s . ( 9 )
##EQU00003##
[0152] where the structural relaxation time .tau..sub.s, a
macroscopic measurement of the molecular mobility of the polymer,
accounts for the characteristic retardation time of the volume
creep (Hempel et al., 1999; Nguyen et al., 2008). As presented
earlier, the structural relaxation in terms of .tau..sub.s is
controlled by both the actual temperature T and the fictive
temperature T.sub.f. A Narayanaswamy mixing parameter x was
introduced to weigh the individual influence (Narayanaswamy,
1971):
.tau. s = .tau. 0 exp [ B ( T g - T .infin. ) 2 ( x T - T .infin. +
1 - x T f - T .infin. ) ] , 0 < x .ltoreq. 1 ( 10 )
##EQU00004##
[0153] It can be observed that the term of (1-x) describes the
contribution of T.sub.f. Here, T.sub.g is the glass transition
temperature. T.sub..infin. denotes the Vogel temperature, defined
as (T.sub.g-50) (.degree. C.). .tau..sub.0 corresponds to the
reference relaxation time at T.sub.g. B is the local slope at
T.sub.g of the trace of time-temperature superposition shift factor
in the global William-Landel-Ferry (WLF) equation (William et al.,
1955).
[0154] After obtaining the evolution profile of T.sub.f, we can
then evaluate the isobaric volumetric thermal deformation
corresponding to a temperature change from T.sub.0 to T
(Narayanaswamy, 1971; Scherer, 1990; Nguyen et al., 2008):
J.sub.T(T,T.sub.f)=1+.alpha..sub.r(T.sub.f-T.sub.0)+.alpha..sub.g(T-T.su-
b.f) (11)
where .alpha..sub.r and .alpha..sub.g represent the long-time
volumetric thermal expansion coefficients of the material in the
rubbery state and the short-time response in the glassy state,
respectively.
[0155] Stress Response
[0156] The mechanical behavior of amorphous glassy polymers under
various temperature conditions has been extensively studied by
numerous researchers (Boyce et al., 1988a, b; Treloar, 1958; Boyce
et al., 1989; Govindjee and Simo, 1991; Arruda and Boyce, 1993;
Bergstrom et al., 1998; Miehe and Keck, 2000; Boyce et al., 2001;
Qi and Boyce, 2005). Although other approaches can still
accommodate the present constitutive framework, the method of Boyce
and co-workers was adopted in this study to model the general
stress-strain behavior of the SMPs.
[0157] The overall mechanical resistance to the strain of a polymer
mainly comes from two distinct sources: the temperature
rat-dependent intermolecular resistance and the entropy-driven
molecular network orientation resistance. It is possible to capture
this nonlinear behavior by decomposing the stress response into an
equilibrium time-dependent component .sigma..sub.ve representing
the viscoplastic behavior and an equilibrium time-independent
component .sigma..sub.n representing the rubber-like behavior. The
two stress components can be represented by a three-element
conceptual model as schematically illustrated in FIG. 7 for a
one-dimensional analog. An elastic-viscoplastic component consists
of an Eyring dashpot monitoring an isotropic resistance to chain
segment rotation and a linear spring used to characterize the
initial elastic response, while a parallel nonlinear hyperelastic
element accounts for the orientation strain hardening behavior.
[0158] If we further denote the deformation gradient acting on the
elastic-viscoplastic component by F.sub.ve and the deformation
gradient acting on the network orientation spring by F.sub.n, the
following constitutive relations are revealed:
.sigma.=.sigma..sub.ve+.sigma..sub.n (12)
.sigma..sub.ve=.sigma..sub.e=.sigma..sub.v (13)
F.sub.ve=F.sub.n=F.sub.m (14)
F.sub.ve=F.sub.eF.sub.v (15)
[0159] The equilibrium response on the network orientation element
can be defined following the Arruda-Boyce eight chain model (Arruda
and Boyce, 1993) as:
.sigma. n = 1 J n .mu. r .lamda. L .lamda. chain L - 1 ( .lamda.
chain .lamda. L ) B _ ' + k b ( J - 1 ) I ( 16 ) ##EQU00005##
[0160] where .mu..sub.r is the initial hardening modulus, and
k.sub.b denotes the bulk modulus to account for the
incompressibility of rubbery behavior. Because most amorphous
polymers exhibit vastly different volumetric and deviational
behavior, the volumetric and deviational contributions are
considered separately by taking out the volumetric strain through
the split formulation (Flory, 1961; Simo et al., 1985):
F.sub.n-J.sub.n.sup.-1/3F.sub.n (17)
where J.sub.n=det(F.sub.n). B= F.sub.n F.sub.n.sup.T, is the
isochoric left Cauchy-Green tensor, and B'= B-1/3 .sub.n1I
represents the deviational component of B .sub.n1=tr( B) is the
first invariant of B. .lamda..sub.chain= {square root over (
.sub.n1/3)} is the effective stretch on each chain in the
eight-chain network. .lamda..sub.L is the locking stretch
representing the rigidity between entanglements. The Langevin
function is defined by:
L ( .beta. ) = coth ( .beta. ) - 1 .beta. ( 18 ) ##EQU00006##
[0161] whose inverse leads to the feature that the stress increases
dramatically as the chain stretch approaches its limiting
extensibility .lamda..sub.L.
[0162] The nonequilibrium stress response acting on the
elastic-viscoplastic component can be determined through the
elastic contribution F.sub.e:
.sigma. ve = .sigma. e = 1 J e L e ( ln V e ) ( 19 )
##EQU00007##
[0163] where J.sub.e=det(F.sub.e), and L.sup.e=2G+.lamda.II is the
fourth order isotropic elasticity tensor. G and .lamda. are Lame
constants, is the fourth order identity tensor and I is the second
order identity tensor.
[0164] The Viscous Flow
[0165] As proposed earlier, the molecular process of a viscous flow
is to overcome the shear resistance of the material for local
rearrangement. Therefore, a plastic shear strain rate {dot over
(.gamma.)}.sub.v is given to help constitutively prescribe the
viscous stretch rate D.sub.v as:
D.sub.v={dot over (.gamma.)}.sub.vn (20)
[0166] where
n = .sigma. ve ' .sigma. ve ' .sigma. ve ' = .sigma. ve ' .sigma.
ve ' ##EQU00008##
is the normalized deviational portion of the nonequilibrium stress.
This shows that the viscous stretch rate scales with the plastic
shear strain rate and evolves in the direction of the flow
stress.
[0167] Taking into account that the non-Newtonian fluid
relationship must be valid for the dashpot of the mechanical model,
the shear strain rate {dot over (.gamma.)}.sub.v can be formulated
in an Eyring model (Eyring, 1936) with the temperature dependence
in a WLF kinetics manner:
.gamma. . v = s .eta. g T Q exp ( c 1 ( T - T g ) c 2 + T - T g )
sinh ( Q T .tau. _ s ) ( 21 ) ##EQU00009##
[0168] here
.tau. _ = .sigma. ve ' 2 ##EQU00010##
is defined as the equivalent shear stress; c.sub.1, c.sub.2 are the
two WLF constants; Q is the activation parameter; s represents the
a thermal shear strength; and .eta..sub.g denotes the reference
shear viscosity at T.sub.g. The evolution Eq. (21) reveals the
nature of the viscoplastic flow to be temperature-dependent and
stress-activated.
[0169] More recently, Nguyen et al. (2008) further extended the
viscous flow rule to a structure-dependent glass transition region
by introducing the fictive temperature T.sub.j into the temperature
dependence:
.gamma. . v = s .eta. g T Q exp ( c 1 ( c 2 ( T - T f ) + T ( T f -
T g ) T ( c 2 + T - T g ) ) ) sinh ( Q T .tau. _ s ) ( 22 )
##EQU00011##
[0170] It can be observed that once the material reaches
equilibrium where T.sub.f=T, Eq. (22) will reduce to Eq. (21) for a
structure independent time-temperature shift factor.
[0171] Following yielding, the initial rearrangement of the chain
segments alters the local structure configuration, resulting in a
decrease in the shear resistance. To further account for the
macroscopic post-yield strain softening behavior, the
phenomenological evolution rule for athermal shear strength s
proposed by Boyce et al. (1989) is implemented,
s . = h ( 1 - s s s ) .gamma. . v ( 23 ) ##EQU00012##
[0172] The initial condition s=s.sub.0 applies. Here s.sub.0
denotes the initial shear strength, while s.sub.s denotes the
saturation value. h is the slope of the yield drop with respect to
plastic strain. It should be noted that a softening characteristic
can only be captured when s.sub.0>s.sub.s holds.
[0173] The constitutive relations for the sophisticated
temperature- and time-dependent thermo-mechanical behavior of the
thermally activated thermoset SMP are summarized in Table 1. The
comprehensive model considers the material mechanical response in
the manner of structure dependent thermoviscoelasticity. It is
capable of capturing the important features of polymer behavior
such as yielding, strain softening and strain hardening. Since our
aim is to establish a thermomechanic framework for the
extraordinary characteristics of SMPs programmed at glassy
temperature, the present constitutive model does somewhat simplify
real SMP behavior. Several factors such as heat conduction and
pressure on the structure relaxation response are not taken into
account. A single nonequilibrium stress relaxation process is also
assumed for the sake of convenience, yet multiple relaxation
mechanism (i.e., more separate Maxwell elements in FIG. 7) are
required to distinguish the long-range entropic stiffening process
and the short-range viscoplastic flow induced strain-hardening
behavior.
[0174] Results
[0175] Model Validation
[0176] The constitutive relations were coded and implemented into a
MATLAB program, for which a flowchart is illustrated in FIG. 12 to
simulate the corresponding experimental data. The model parameters
were obtained through various mechanical testing measurements.
Detailed parameter identification procedures are briefly described
below. The final values of these parameters are listed in Table 2.
The mathematical formulation for 1-D compression is demonstrated
below.
[0177] Based on the parameters in Table 2, the numerical simulation
results, which cover the entire thermomechanical profile of the SMP
programmed at 30% prestrain for different relaxation histories in a
strain-time scope, is shown in FIG. 8 (a). The material was
initially stressed to the pre-defined strain level after overcoming
the yielding point and experiencing a slight strain-softening,
followed by significant strain hardening (Step 1). Afterwards it
was held with different time periods of relaxation for plastic
strain development (Step 2). Finally the remaining stress was
instantly removed, leading to a stress-free state (Step 3). Lengthy
relaxation seemingly enhanced the level of the strain fixity. The
stored deformation was then released and the original shape
recovered during a subsequent heating process (Step 4).
TABLE-US-00001 TABLE 1 Summary of the thermoviscoelastic model
deformation F = F.sub.eF.sub.vF.sub.T response F.sub.T =
J.sub.T.sup.-1/3I structure relaxation T f ( t ) = T ( t ) - .intg.
t 0 t .PHI. ( .DELTA..zeta. ) dT ( t ) ##EQU00013## .phi. =
exp[-(.DELTA..zeta.).sup..beta.] .DELTA..zeta. = .zeta. ( t ) -
.zeta. ( t ' ) = .intg. t ' t dt .tau. s ##EQU00014## .tau. s =
.tau. 0 exp [ B ( T g - T .infin. ) 2 ( x T - T .infin. + 1 - x T f
- T .infin. ) ] ##EQU00015## stress .sigma. = .sigma..sub.ve +
.sigma..sub.n response .sigma. n = 1 J n .mu. T .lamda. L .lamda.
chain L - 1 ( .lamda. chain .lamda. L ) B _ ' + k b ( J - 1 ) I
##EQU00016## .sigma. ve = 1 J e L e ( ln V e ) ##EQU00017## viscous
flow D.sub.v = {dot over (.gamma.)}.sub.vn rule .gamma. . v = s
.eta. g T Q exp ( c 1 ( c 2 ( T - T f ) + T ( T f - T g ) T ( c 2 +
T - T g ) ) ) sinh ( Q T .tau. _ s ) ##EQU00018##
TABLE-US-00002 TABLE 2 Material parameters of the preliminary
constitutive model Model parameters Values T.sub.g (.degree. C.)
glass transition temperature 62 T.sub.0 (.degree. C.) programming
temperature 20 .DELTA.t (minute) relaxation time 0/5/15/30/120
.alpha..sub.g (10.sup.-4 .degree. C..sup.-1) volumetric CTE of
glassy state 5.462 .alpha..sub.r (10.sup.-4 .degree. C..sup.-1)
volumetric CTE of rubbery state 8.441 G (MPa) glassy shear modulus
196.4 .lamda. (MPa) Lame constant for glassy state 785.7 .mu..sub.r
(MPa) rubbery modulus 1.2 k.sub.b (MPa) bulk modulus 1000
.lamda..sub.L locking stretch 0.95 .mu..sub.g (MPa s.sup.-1)
reference shear viscosity at T.sub.g 1550 s.sub.0 (MPa) initial
shear strength. 35 s.sub.s (MPa) steady-state shear strength 33
Q/s.sub.0 (.degree. K/MPa) flow activation ratio 380 h (MPa) flow
softening constant 250 c.sub.1 first WLF constant 25.8 c.sub.2
(.degree. C.) second WLF constant 90 .tau. (s) structure relaxation
characteristic time 200 x NMM constant 0.95 .beta. Kohlrausch index
0.95
[0178] From FIG. 8 (a), the model simulation generally has a
reasonable agreement with the test results. It proves that the
model is capable of capturing the basic nonlinear material behavior
of the SMP during a thermomechanical cycle. The real SMP samples
did not achieve the full predicted recovery; this discrepancy may
come from a couple of sources. Considering the large peak
compressive stress applied during programming (about 40 MPa in FIG.
5 and FIG. 8(a)), some irreversible damage may have been induced in
the SMP specimen. Also the deficiency of the single relaxation
assumption appears evident in the discrepancies between the
simulation and experiments when the relaxation time is
insufficient. This can be validated by FIG. 8(a) that when the
relaxation time is short, the discrepancy is large; when the
relaxation time is long enough (120 min), the discrepancy becomes
comparatively small. Therefore, a spectrum of multiple
nonequilibrium processes would be required to describe the actual
stress relaxation process of a real thermosetting SMP.
[0179] In this study, the same parameters calibrated in modeling
the constitutive behavior of the SMP programmed by 30% prestrain
level were also used to predict the thermomechanical behavior of
the same SMP programmed by 10% prestrain level; see FIG. 8 (b). It
is clear that, with the same set of parameters, the model predicted
well the constitutive behavior of the SMP programmed by 10%
prestrain. This further validated the developed model.
[0180] Prediction and Discussion
[0181] To demonstrate that the shape memory response of the SMP has
a strong dependence on the structural evolution, the influence of
the temperature profile has been investigated through the
unconstrained recovery simulations.
[0182] Dependence on the Heating Rate
[0183] FIG. 9 exhibits the free recovery prediction results for two
different heating rates q=0.6.degree. C./min and q=3.degree.
C./min. It is observed that a faster heating rate shifts the
initiation of the recovery process to a higher temperature and
leads to a more gradual temperature dependence at the start of the
strain release, but hardly affects the final recovery ratio.
[0184] Dependence on the Heating History
[0185] Besides the heating rate, the heating profile also
influences the structure evolution. The calculation results for two
types of heating profiles are shown in FIG. 10. Heating profile #1
represents a heating profile from 22.degree. C. to 79.degree. C.
with a constant heating rate of q=1.degree. C./min; while heating
profile #2 represents a heating profile from 22.degree. C. to
68.degree. C. with a constant heating rate of q=1.degree. C./min
followed by a 50 minute soaking period. It shows that although
heating profile #2 does not reach the same high temperature of
79.degree. C. as that of the heating profile #1, it still reaches
the same recovery strain level after adequate soaking. This is an
indication of time-temperature equivalence.
[0186] Dependence on the Programming Temperature T.sub.0
[0187] The effect of the programming temperature T.sub.0 is shown
in FIG. 11. The SMP samples are considered to be programmed at
20.degree. C. and 40.degree. C. respectively for the same
relaxation time period of 20 minutes. Two cases are considered. For
Case (a), shape recovery immediately follows the programming at a
heating rate of 3.degree. C./min, which means that the starting
temperature for recovery is different (20.degree. C. and 40.degree.
C., respectively). It can be seen that a higher T.sub.0
significantly increases the shape fixity ratio due to the decrease
of molecular segmental resistance during the plastic flow, and
shortens the recovery time period. As the temperature-recovery
strain subfigure in FIG. 11 (a) shows, the two programmed SMPs
generally follow a similar recovery path except for the small
deviation caused by the structure relaxation and thermal expansion.
For Case (b), the sample programmed at 40.degree. C. is first
cooled to 20.degree. C. before being heated to recover, which means
that the starting temperature for recovery is the same (20.degree.
C.). It can be seen from FIG. 11 (b) that for the sample programmed
at 40.degree. C., it takes a longer time for completion of Step 4.
The major recovery was completed at a lower temperature, again
showing a time-temperature equivalency.
[0188] Detailed Parameter Identification Procedures
[0189] Although the final values of the material parameters used
for demonstration, as listed in Table 2, were mainly obtained from
curve fitting various testing results shown in FIG. 13 through FIG.
16, several basic guidelines were followed to assist in the
estimates:
[0190] (1) A cooling profile of the thermal deformation is plotted
versus the temperature in FIG. 13. The reference height L.sub.0
denotes the initial sample height. It can be observed that the
thermal response is not linear as the temperature traverses through
the glass transition region. Linear .alpha..sub.r and .alpha..sub.g
were computed from the slopes above and below the T.sub.g.
Volumetric CTE is three times the value of the linear CTE.
[0191] (2) .mu..sub.r and .lamda..sub.L are the parameters
characterizing the rubbery behavior of the material, and can be
determined from the stress-strain response at temperatures above
T.sub.g (FIG. 14). Lame constants G and A can be related to the
initial slope of the isothermal uniaxial compression stress-strain
curve in glassy state by assuming a typical polymer Poisson's ratio
of 0.4 (Qi et al., 2008). Although it has been suggested that
different sets of parameters .mu..sub.r and .mu..sub.L are
preferred to capture the fundamentally different response of the
rubbery state and the glassy state (Anand and Ames, 2006; Qi et
al., 2008), they are treated as being temperature-independent for
the sake of convenience in parameter identification and
computational simplicity.
[0192] (3) As suggested in previous efforts (Boyce et al., 1989;
Nguyen et al., 2008; Qi et al., 2008), the viscoplastic parameters
such as Q, s, s, and h can be roughly determined from curve fitting
of the compression tests at different strain rates (FIG. 15). The
ratio Qls determines the strain rate dependence of the yield
strength, and s/s.sub.s indicates the drop of the shear strength. h
characterizes the strain-softening rate after yielding.
[0193] (4) The structure relaxation parameters x and .beta. are
fitted to a stress-free, constant heating profile of the thermal
deformation (FIG. 16).
[0194] Mathematical Formulation for 1-D Compression
[0195] For uniaxial compression, if we consider that the load is
applied in the n.sub.1 direction, the mathematical formula can be
further reduced as follows:
[0196] Because of the assumption of isotropic material and uniform
stress field,
F = [ .lamda. 1 .lamda. 2 .lamda. 2 ] ( C .1 ) ##EQU00019##
[0197] Here .lamda..sub.1 represents the stretch in the n.sub.1
direction and .lamda..sub.2 is the stretch in the other two
directions.
[0198] The isochoric left Cauchy strain tensor can be specified
as:
B _ = ( J n ) - 2 / 3 [ .lamda. 1 2 .lamda. 2 2 .lamda. 2 2 ] , J n
= .lamda. 1 ( .lamda. 2 ) 2 / J .tau. ( C .2 ) ##EQU00020##
[0199] Hence the effective stretch .lamda..sub.chain is defined
as:
.lamda. chain = ( J n ) - 1 / 3 .lamda. 1 2 + 2 .lamda. 2 2 3 ( C
.3 ) ##EQU00021##
[0200] If .lamda..sub.1.sup.e and .lamda..sub.2.sup.e denote the
elastic stretches,
J.sub.e=.lamda..sub.1.sup.e(.lamda..sub.2.sup.e).sup.2 then the
equilibrium and the non-equilibrium stresses can be identified
by:
.sigma. n = .lamda. 1 2 - .lamda. 2 2 3 J n s / e .mu. r .lamda. L
.lamda. chain L - 1 ( .lamda. chain .lamda. L ) [ 2 - 1 - 1 ] + k b
( J - 1 ) I ( C .4 ) .sigma. ve = 1 J e [ ln ( .lamda. 1 e ( 2 G +
.lamda. ) .lamda. 2 e 2 .lamda. ) ln ( .lamda. 1 e .lamda. .lamda.
2 e 2 ( G + .lamda. ) ) ln ( .lamda. 1 e .lamda. .lamda. 2 e 2 ( G
+ .lamda. ) ) ] ( C .5 ) ##EQU00022##
.tau. _ = 2 3 3 J e G | ln .lamda. 2 e .lamda. 2 e | .
##EQU00023##
[0201] As a result, the equivalent shear stress
Example 3
Testing of Shape Memory Polymer Based Self-Healing Syntactic Foam
Programmed at Glassy Temperature
[0202] The novel process of programming at glassy temperatures has
been set forth herein, and the recoverability and functional
stability of thermosetting SMP programmed according to this "cold
compression" programming method have been confirmed. In this
example, the work is extended to SMP-based syntactic foams. Also,
because of the composite nature and the damage tendency of the
microballoons in the foam, a constitutive model underpinning the
imperfect shape memory behavior developed and set forth in Example
4.
[0203] As set forth in Example 1, it was shown that, as long as a
nonequilibrium configuration can be created for a glass-transition
activated SMP, a temporary shape can be fixed, even if the
temperature creating this nonequilibrium configuration is below the
glass transition temperature. In other words, programming of SMPs
can be conducted at glassy temperatures. A systematic experimental
testing and constitutive modeling have validated this concept (also
see [1]). We found that SMPs can be programmed at glassy
temperature as long as the prestrain is greater than the yielding
strain of the SMPs.
[0204] In this example, the three-step programming process set
forth in Example 1 was applied to the SMP based syntactic foam at
glassy temperatures. In laboratory testing the foam specimens were
first programmed at glassy temperature with various stress
relaxation time periods. Free shape recovery was then conducted.
The shape fixity ratio and shape recovery ratio were determined.
These test results were used as baseline data for the constitutive
modeling set forth in Example 4.
[0205] Experimental Methods
[0206] Specimen Preparation
[0207] The SMP based syntactic foam was formulated through the
dispersion of 40% by volume of glass hollow microspheres into the
SMP matrix. The SMP named Veriflex from CRG Industries was used, a
styrene-based thermoset SMP resin system (T.sub.g=62.degree. C.).
The glass hollow microspheres were from Potters Industries (Q-CEL
6014) with an average outer diameter of 85 .mu.m, an effective
density of 0.14 g/cm.sup.3, and a wall thickness of 0.8 .mu.m. The
microspheres were incrementally added into the SMP resin, allowing
several minutes for blending. A hardening agent was then added and
the solution was blended for another 10 minutes before it was
poured into a 229.times.229.times.12.7 mm steel mold. It was then
placed in a vacuum chamber at 40 kPa for 20 minutes to remove any
entrapped air bubbles. The curing process initiated at 79.degree.
C. for 24 hours, and then 107.degree. C. for 3 hours, followed by
121.degree. C. for 9 hours in an industrial oven, as recommended by
Li and Nettles [7]. After curing, the foam panel was de-molded and
was machined into different dimensions for various testing:
30.times.30.times.12.5 mm.sup.3 block specimens, which were
determined per ASTM C365 standard [28], were used for thermal
expansion, uniaxial compression, thermomechanical programming and
shape recovery tests; and 17.5.times.11.9.times.1.20 mm.sup.3 plate
specimens, which were determined per ASTM E1640-04 standard [29],
were used for DMA tests. In this study, 40% by volume of
microballoons was chosen for several reasons. (1) For most
polymeric syntactic foams, the volume fraction of microballoons is
around 40-60% [30]. (2) For this specific SMP, 40% was the volume
fraction that maintained workability without the use of diluents.
Diluents were not a preferred choice because they might affect the
curing as well as the shape memory functionality of the foam. (3)
This was the volume fraction we have used previously for the same
foam [7]. Maintaining the same volume fraction facilitated
comparisons.
[0208] Dynamic Mechanical Analysis
[0209] In order to determine the T.sub.g of the foam, the single
cantilever mode dynamic mechanical analysis (DMA) test was
conducted on a DMA 2980 tester from TA instruments per ASTM E
1640-04 [29]. The specimen had a dimension of
17.5.times.11.9.times.1.20 mm.sup.3. The dynamic load frequency was
set to be 1 Hz and the amplitude was 15 .mu.m. The temperature
ramped from room temperature to 120.degree. C. at a rate of
3.degree. C./min.
[0210] X-Ray Photoelectron Spectroscopy
[0211] The X-ray photoelectron spectroscopy (XPS) spectra of the
pure SMP and the foam specimen were collected on a Kratos AXIS 165
high performance multi-technique surface analysis system with an
information depth of 10 nm and a scan area of 700.times.300
.mu.m.sup.2. This was performed to qualitatively evaluate the
interface between the SMP matrix and the glass hollow
microspheres.
[0212] Thermal Expansion Measurement
[0213] A linear variable differential transducer (LVDT, Cooper
Instruments LDT 200 series) system was used to measure the thermal
expansion and a Yokagawa DC100 data acquisition system was used to
monitor the temperature. The specimen was heated from room
temperature to 100.degree. C. at 0.4.degree. C./min and naturally
cooled down after thermally equilibrated for 30 minutes.
[0214] Programming of the Foam Below Glass Transition
Temperature
[0215] The thermomechanical cycle including the new programming
method and shape recovery was as schematically shown in FIG. 1. The
programming comprised three steps at a fixed glassy temperature
(e.g., room temperature in the present study): compression to the
designed pre-strain (Step 1), stress relaxation (Step 2), and load
removal (Step 3). Step 4 is the shape recovery step, which was
conducted the same as in the traditional approach. Isothermal
uniaxial flat-wise compression programming was performed on a MTS
QTEST150 electromechanical frame outfitted with a moveable furnace
(ATS heating chamber) per the ASTM C 365 standard [29]. The
displacement rate was set to be 1.3 mm/min. A thermocouple placed
in the chamber near the SMP specimen was used to control the
environmental temperature.
[0216] As set forth in Example 1, successful shape fixity at glassy
temperatures should have a post-yield pre-strain (i.e., a strain
greater than yield strain). We tested prestrains below yielding
strain, slightly above yielding strain, and well away from yielding
but below fracture or significant strain hardening. Thus, two
prestrain levels, 30% and 20%, which were above the yield strain of
7% for the same foam at room temperature [7], were selected with
stress relaxation times of 0 min, 5 min, 15 min, 30 min, and 120
min. At least three effective specimens were tested for each stress
relaxation time period.
[0217] Free Shape Recovery Tests
[0218] Unconstrained strain recovery tests were performed on the
programmed specimens. During the test, the programmed foam specimen
was reheated to T.sub.high=80.degree. C. at an average heating rate
of q=0.4.degree. C./min. The displacement at the specimen surface
was tracked by the same LVDT system.
[0219] Experimental Results
[0220] DMA Test Results
[0221] The experimental results in FIG. 17 illustrate the loss
modulus and storage modulus of the pure SMP and the SMP based
syntactic foam as a function of temperature. It was found that the
peak of the loss modulus of the foam had been shifted to a higher
temperature as compared to that of the pure SMP. From FIG. 17, the
difference in the T.sub.g temperature was estimated to be
2.3.degree. C. The T.sub.g of 62.degree. C. for the pure SMP
provided by the manufacturer was determined by differential
scanning calorimetry (DSC), which was about 6.degree. C. lower than
the DMA result from FIG. 17. To maintain consistency, we used the
T.sub.g of the pure SMP as 62.degree. C. Therefore, the T.sub.g of
the foam was estimated to be 62.degree. C.+2.3.degree.
C.=64.3.degree. C.
[0222] XPS Test Results
[0223] The XPS results shown in FIG. 18 reveal that different
binding energies exist in the pure SMP and the foam sample for the
same emitted electrons (C (1s) and O (1s)). It indicated that some
chemical shifts may have occurred at the glass hollow
microsphere/SMP matrix interface. The mobility of the SMP polymer
chains in the vicinity of the interface has probably been reduced,
leading to an increase in glass transition temperature of the foam,
which echoes the DMA test results.
[0224] Uniaxial Strain-Controlled Compression Programming
[0225] The strain evolution during the material programming process
(Step 1-3) can be observed in FIG. 19. A reasonable shape fixity
ratio (70.5% for 20% pre-strain and 72.6% for 30% pre-strain) was
reached even when the constraint was instantly removed (zero
relaxation time). Similar to the pure SMPs, it was found that
longer stress relaxation times tend to increase the shape fixity
ratio. However, an upper limit of the shape fixity ratio could be
reached as the relaxation time continually increases. Further
lengthening the relaxation time barely produced a noticeable
increase in the shape fixity ratio.
[0226] The strain evolution with time (i.e., the change of strain
with time) is further highlighted in FIG. 20 for viscoelastic
tests. One is a creep test with a constant stress and the other
with zero stress. It is clear that, even at room temperature, the
foam showed creep. This is direct evidence that viscoelastic
deformation can occur in the glassy state.
[0227] Therefore, a viscoelastic component was added in our
modeling of Example 4. With zero stress, however, there is no
change of strain with time, suggesting stability of the fixed level
of strain.
[0228] Free Shape Recovery Test
[0229] FIG. 19 also shows the unconstrained heating recovery (Step
4). The programmed specimen initially showed slight thermal
expansion. As the temperature further approached T.sub.g, the
entropy increase led to a rapid strain recovery. At temperatures
well above T.sub.g, the strain appeared to stabilize. A typical
recovery path was shared by all the specimens with different
relaxation times during programming, indicating a universal strain
release mechanism. It was observed that the irrecoverable strain
for all the specimens programmed by the same prestrain appeared to
be at nearly the same level (about 8% for 20% pre-strain and 10%
for 30% pre-strain), indicating a similar irrecoverable amount of
damage occurred regardless of the relaxation time period. Therefore
it is assumed that the damage occurred primarily in the compression
process (Step 1). Since the damage in the SMP matrix itself under
30% prestrain can be neglected [1], the damage presumably came
entirely from crushing and implosion of the glass hollow
microspheres.
[0230] A Hitachi S-3600N VP-Scanning Electron Microscope was used
to examine the microstructure change due to programming; see FIG.
21. From FIG. 21 (b), some of the microballoons have been crushed
after cold-compression programming at 30% prestrain, which
contributed to the irreversible strain after free shape
recovery.
[0231] The extremely nonlinear behaviors for the entire
thermomechanical cycle including a three-step glassy temperature
programming process and a one-step heating recovery in the
stress-strain-time view and the stress-strain-temperature view are
shown in FIG. 22 (a) and FIG. 22 (b), respectively.
[0232] In-depth understanding of this complex thermomechanical
behavior could be better elucidated by the constitutive modeling
set forth in Example 4. It is noted that, as instant unloading
occurs at the end of the programming, straight lines were used to
connect the final loading point of Step 2 and the initial point of
the free-recovery path in Step 4 in FIG. 22. These straight lines
are not actual physical unloading curves, because the sudden
removal of the load could not be recorded by the MTS machine.
Therefore, the slopes of these straight lines do not represent the
true unloading modulus.
Example 4
Thermoviscoplastic Modeling of Shape Memory Polymer Based
Self-Healing Syntactic Foam Programmed at Glassy Temperature
[0233] As shown by the material characterization test results (DMA
and XPS results), the incorporation of glass microballoons altered
the chemical bonds at the interface between the SMP matrix and
glass hollow microsphere inclusions. Earlier studies [31,32]
reported that there exists a long-range gradient (over 100.degree.
K difference) for the polymer matrix glass transition temperature
in the vicinity of the particles. Therefore, it was believed that
an interfacial transition zone (ITZ) layer similar to the
phenomenon in cement-based materials [33-35] also occurs in the SMP
based syntactic foam. To consider the influence of such a layer on
the performance of the foam, a unit cell of the SMP based syntactic
foam was treated as a three-phase composite with ITZ-coated glass
hollow microspheres embedded in the pure SMP matrix, as illustrated
in FIG. 23. However, since current techniques have difficulties in
characterizing the ITZ layer in details, a convenient approach of
integrating the ITZ and pure SMP as a new equivalent SMP medium
[25] was adopted. The equivalent scheme is also shown in FIG. 23 on
the right.
[0234] Since the aim of this work was to establish a theoretical
framework for the shape memory behavior of a damage-allowable SMP
based syntactic foam programmed at glassy temperatures, several
fundamental assumptions were made for further model derivation:
[0235] 1) The material is considered to be isotropic, homogeneous
and uniformly stressed.
[0236] 2) The temperature is assumed to be spatially uniform.
[0237] 3) Structural and stress relaxation are considered to be
solely temperature, time and stress dependent.
[0238] 4) The equivalent SMP matrix is considered to be thoroughly
perfect. All the damage originates from the crushing and implosion
of the glass hollow microspheres.
[0239] Kinematics
[0240] As documented previously, an arbitrary thermomechanical
deformation mapping from an initial undeformed and unheated
configuration .OMEGA..sub.0 to a spatial configuration .OMEGA. can
be considered as a combination of a thermal deformation and a
mechanical response; see FIG. 24. The scheme is expressed as a
multiplicative decomposition of the deformation gradient
[36,37]:
F=F.sub.MF.sub.TF=F.sub.MF.sub.T (1)
where F.sub.M defines the mechanical deformation gradient and
F.sub.T defines the mapping path from .OMEGA..sub.0 to
.OMEGA..sub.T, an intermediate heated configuration. Because the
material is assumed to be macroscopically isotropic, the thermal
deformation gradient is:
F.sub.T=J.sub.T.sup.1/3IF.sub.T=J.sub.T.sup.1/3I (2)
[0241] where J.sub.T=det(F.sub.T) is the determinant of the thermal
deformation gradient, representing the volumetric thermal
deformation and I is the second order identity tensor.
[0242] To consider the composition of the syntactic foam, the rule
of mixtures applies:
F.sub.M=.phi..sub.pF.sub.p(1-.phi..sub.p)F.sub.iF.sub.M=.phi..sub.pF.sub-
.p+(1-.phi..sub.p)F.sub.i (3)
where F.sub.p represents the deformation of the SMP matrix and
F.sub.i represents the deformation of the glass microsphere
inclusions. .PHI..sub.p is the volume fraction of the polymer
matrix.
[0243] Usually glass microspheres crush during the loading step;
therefore, a damage allowable constitutive model of the microsphere
inclusions is used. If an internal stress and time dependent
evolution parameter .PHI..sub.d(.sigma.,t) is introduced to
represent the volume fraction of the damaged microspheres out of
the total microsphere volume, the deformation of the inclusions
could be expressed as:
F.sub.i=(1-.PHI..sub.d)F.sub.i.sup.ud=.PHI..sub.d
F.sub.i=(1.phi..sub.d)F.sub.i.sup.ud+.phi..sub.dF.sub.i.sup.d
(4)
where F.sub.i.sup.ud refers to undamaged microspheres while
F.sub.i.sup.d refers to damaged microspheres.
[0244] To separate the elastic and viscous response of the SMP
matrix, the multiplicative split scheme can be operated on the
polymer deformation gradient [37,38]:
F.sub.p=F.sub.p.sup.eF.sub.p.sup.vF.sub.p=F.sub.p.sup.eF.sub.p.sup.v
(5)
[0245] where F.sub.p.sup.e represents the elastic component and
represents the viscous component.
[0246] Further polar decomposition of F.sub.p.sup.e leads to a left
stretch tensor and a rotation tensor
F.sub.p=V.sub.p.sup.eR.sub.p.sup.e (6)
[0247] The viscous velocity gradient is then defined as:
L.sub.p.sup.v={dot over
(F)}.sub.p.sup.vF.sub.p.sup.v-1=D.sub.p.sup.vW.sub.p.sup.v (7)
[0248] where D.sub.p.sup.v=1/2(L.sub.p.sup.v+L.sub.p.sup.eT)
represents the plastic stretch of the velocity gradient and is the
spin. tensor.
[0249] 4.3 Structural Relaxation and Thermal Deformation
[0250] The concept of fictive temperature T.sub.f was first
introduced by Tool [39] to explain the nonlinearity of structural
relaxation. As defined, T.sub.f is the temperature at which the
temporary nonequilibrium structure at T is in equilibrium [26].
Considering that there exists an equilibrium configuration at a
different temperature T.sub.f, which is equivalent to the current
nonequilibrium configuration at the current temperature T, T.sub.f
serves as a measurement of the actual nonequilibrium structure
state. The rate change of the fictive temperature is assumed to be
proportionally dependent on its deviation from the actual
temperature [40]. Its evolution was proposed as follows [39], where
the temperature and structure dependent K represents the
proportionality factor:
T f t = K ( T , T f ) ( T - T f ) ( 8 ) ##EQU00024##
[0251] The Narayanaswamy-Moynihan model (NMM) [40,41] further
improved this approach by taking into account the non-exponential
structural relaxation behavior as well as the spectrum effect. As
discussed in detail by Donth and Hempel [42], with the assumption
that the whole thermal history T(t) starts from a thermodynamic
equilibrium state where T(t.sub.0)=T.sub.f(t.sub.0), Tool's fictive
temperature is given by:
T.sub.f(t)=T(t)-.intg..sub.t.sub.0.sup.t.phi.(.DELTA..zeta.)dT(t)T.sub.f-
(t)=T(t)-.intg..sub.t.sub.0.sup.t.phi.(.DELTA..zeta.)dT(t) (9)
[0252] where .phi. is the response function and is expressed as a
Kohlrausch function [43]:
.phi.=exp(-(.DELTA..zeta.).sup..beta.).phi.=exp[-(.DELTA..zeta.).sup..be-
ta.], 0<.beta..ltoreq.1 (10)
[0253] It is found from the equation above that, for very small
departures from equilibrium is not constant [44]. Therefore .beta.
describes the non-exponential characteristic of the relaxation
process.
[0254] .DELTA..zeta. is introduced as the dimensionless material
time difference to linearize the relaxation process, roughly
measuring the time in units of a mean structural relaxation time
[45]:
.DELTA. = ( t ) - ( t ' ) = .intg. t ' t t .tau. s .DELTA. = ( t )
- ( t ' ) = .intg. t ' t t .tau. s ( 11 ) ##EQU00025##
[0255] where the parameter .tau..sub.s, commonly referred to be the
structural relaxation time, is a macroscopic measurement of the
molecular mobility of the polymer [26,46]. As elaborated earlier
that the structural relaxation is dependent on both T and T.sub.f,
a Narayanaswamy parameter x was introduced to weigh their
individual influence [40]:
.tau. s = .tau. 0 exp [ B ( T g - T .infin. ) 2 ( x T - T .infin. +
1 - x T f - T .infin. ) ] .tau. s = .tau. 0 exp [ B ( T g - T
.varies. ) 2 ( x T - T .infin. + 1 - x T f - T .infin. ) ] , , 0
< x .ltoreq. 1 ( 12 ) ##EQU00026##
[0256] It is understood that (1-x) describes the effect of the
nonequilibrium state. T.sub.g is the glass transition temperature
and T.sub..varies.=T.sub.g-50(.degree. C.) denotes the Vogel
temperature. T.sub.0 corresponds to the reference relaxation time.
B is the local slope at T.sub.g of the trace of time-temperature
superposition shift factor [47].
[0257] Since the material has been assumed to be statistically
homogeneous and heat transfer is not considered, the global
isobaric volumetric thermal deformation corresponding to a
temperature change from T.sub.0 to T can then be evaluated as
follows [26,40,48]:
J.sub.T(T,T.sub.f)=1+.alpha..sub.r(T.sub.f-T.sub.0)+.alpha..sub.g(T-T.su-
b.f) (13)
.alpha..sub.r and .alpha..sub.g respectively represent the
long-term volumetric thermal expansion coefficients of the material
in the rubbery state and the short-term response in the glassy
state.
[0258] Constitutive Behavior of Glass Microsphere Inclusions
[0259] Since the glass hollow microspheres are brittle and have a
high Young's modulus, the constitutive behavior of the undamaged
portion can be considered to be purely elastic:
.sigma.=.sigma..sub.i=L.sub.i.sup.e(ln F.sup.ud) (14)
where L.sub.i.sup.e=2G.sub.i+.lamda..sub.iII is the fourth order
isotropic elasticity tensor of the glass microspheres. G.sub.i and
.lamda..sub.i are Lame constants, is the fourth order identity
tensor and I is the second order identity tensor.
[0260] Physically, the evolution of the crushing and implosion of
the hollow microspheres can be extremely complex. Since our focus
is just on establishing a thermomechanical framework for the SMP
based syntactic foam, for simplicity we assume an instant and
complete damage mechanism occurring to the hollow microspheres
partly because the glass hollow microspheres are brittle and thus
the crack propagation speed is high. So
.PHI..sub.d(.sigma.,t)=.PHI..sub.d(.sigma.).
[0261] .PHI..sub.d(.sigma.) normally evolves nonlinearly. If a
normal statistical distribution applies, then an arbitrary
nonlinear curve of the volume fraction of the damaged microballoons
should start slowly when the applied load initially overcomes the
bearing stress .sigma..sub.b and then should accelerate as the load
further increases, and finally slow down gradually as damage
proceeds and reaches a complete failure of all the microsphere
inclusions, as illustrated in FIG. 25. Since it is difficult to
capture the actual nonlinear damage profile, a linear equivalent
damage model was considered. As the irrecoverable strain is assumed
to fully come from the damage and volume reduction of the hollow
microspheres, the total damage volume fraction
(.PHI..sub.d.sup.total) of the microspheres can be calculated based
on its relation to the final irrecoverable strain
(.epsilon..sub.ir) as:
1+.epsilon..sub.ir=.PHI..sub.p+(1-.phi..sub.p)((1-.PHI..sub.d.sup.total)+-
.PHI..sub.d.sup.total(1+(1-w).sup.3).sup.1/8), where w is the wall
thickness ratio for the glass hollow microspheres. The
proportionate factor k for the linear equivalent damage model is
given by
k = .phi. d total ( .sigma. m - .sigma. b ) , ##EQU00027##
where .sigma..sub.m is the maximum stress during the programming
process. Because the maximum stress is achieved at the end of
loading in Step 1 of the programming process, the peak stress at
the corresponding prestrain (30% or 20%) is used. .sigma..sub.b
corresponds to the initial damage stress, which is the crushing
pressure of the glass microspheres as provided by the manufacturer
(1.72 MPa). It is noted that the microballoons are not completely
crushed (damaged) in the first programming cycle; see FIG. 21 (b).
The damage should accumulate as the programming-recovery cycles
increase and stabilize after several cycles, which may lead to a
decrease in the shape recovery ratio in the first several cycles
and an increase in the shape recovery ratio thereafter. For
simplicity, however, the dependence of damage on the number of
programming-recovery cycles was not considered in this study; this
simplification could be a potential source of discrepancy between
the model prediction and the test results.
[0262] If we additionally consider the glass microspheres to be
isotropic, the damage gradient can be given by:
F.sub.d=J.sub.d.sup.1/3I (15)
[0263] where J.sub.d represents the ratio of the volume reduction
during the damage, which can be determined as:
I d = v a d v b d = ? .pi. ( r B - ( r - t ) B ) ? .pi. r B = 1 - (
1 - w ) 3 ? indicates text missing or illegible when filed ( 16 )
##EQU00028##
[0264] where V.sub.bd and V.sub.ad represent the volume of the
hollow microsphere before and after damage, respectively; r is the
outer radius of the microsphere; i is the wall thickness; and w=t/r
is the wall thickness ratio.
[0265] It should be noted that even if completely crushed, the
fractured pieces of the glass microspheres should still behave
elastically. Hence, the deformation gradient of the damaged portion
of the microspheres can be expressed as:
F.sub.i.sup.d=F.sub.t.sup.udF.sub.d (17)
[0266] Constitutive Behavior of the Equivalent Shape Memory Polymer
Matrix
[0267] Many efforts have been made to detail the constitutive
relations of the highly nonlinear mechanical behavior of amorphous
glassy polymers [49-59]. As the time-dependent mechanical behavior
of the equivalent shape memory polymer involves equilibrium and
nonequilibrium responses, a three-element conceptual model proposed
by Boyce and co-workers, as illustrated in FIG. 26, were adopted to
capture the stress response. A Maxwell element paralleling with a
hyperelastic rubbery spring represents the stress split scheme:
.sigma.=.sigma..sub.p=.sigma..sub.p.sup.ve.sigma..sub.p.sup.n
(18)
here .sigma..sub.p.sup.ve and are the stresses on the viscoplastic
component and the rubbery spring.
[0268] The scheme indicates that the overall mechanical response to
the straining can be expressed as the sum of the intermolecular
segmental rotation resistance and the entropy driven molecular
network orientation resistance. By further applying Hooke's Law to
the linear elastic spring which characterizes the initial elastic
response and Arruda-Boyce eight chain model [54] to the nonlinear
rubbery spring which monitors the molecular network hardening, we
can express the Cauchy stress as:
.sigma. p = [ 1 J p e L p e ( ln V p e ) ] + [ 1 J n .mu. r .lamda.
L .lamda. chain L - 1 ( .lamda. chain .lamda. L ) B _ + k b ( J n -
1 ) I ] ( 19 ) ##EQU00029##
[0269] where the first part is and the second part is,
J.sub.p.sup.e=det(F.sub.p.sup.e), and L.sub.p.sup.e is the
elasticity tensor; J.sub.n=det(F.sub.p.sup.n), and
B=J.sub.n.sup.-2/3F.sub.p.sup.nF.sub.p.sup.nT is the isochoric left
Cauchy-Green tensor to consider the vastly different volumetric and
deviational behavior exhibited by most amorphous polymers [60,61];
B' is its deviational part; .lamda..sub.chain= {square root over (
.sub.n1/3)} is the effective stretch; and =tr( B) represents the
first invariant. .lamda..sub.L is the locking stretch representing
the rigidity between entanglements. The Langevin function L is
given by
L ( .beta. ) = coth ( .beta. ) - 1 .beta. . ##EQU00030##
[0270] The Eyring dashpot accounts for the isotropic resistance to
the local molecular rearrangement such as chain rotation. A
structure dependent viscous flow rule [26] was used to help
describe its constitutive behavior:
.gamma. . v = s .eta. g T Q exp ( C 1 ( c 2 ( T - T f ) + T ( T f -
T g ) T ( c 2 + T - T g ) ) ) sinh ( Q T .tau. _ s ) ( 20 )
##EQU00031##
[0271] here
.tau. _ = .sigma. P ' ve 2 ##EQU00032##
is the equivalent shear stress; c.sub.1, c.sub.2 are WLF constants;
Q is the activation parameter; .eta..sub.g denotes the shear
viscosity at T.sub.g; s represents the a thermal shear strength,
and a phenomenological evolution rule
s . = h ( 1 - s s S ) .gamma. . v ( s = s 0 , t = t 0 )
##EQU00033##
proposed by Boyce et al. [52] can be adopted to further feature the
post-yield strain softening, where s.sub.0 denotes the initial
shear strength, s.sub.s is the saturation value, and h describes
the yield drop with respect to plastic strain; and {dot over
(.gamma.)}.sub.v is the plastic shear strain rate. It is related to
the viscous stretch rate D.sub.p.sup.v as
.gamma. . v = .sigma. P ' ve .sigma. P ' ve = D P v ,
##EQU00034##
indicating that the viscous stretch rate scales with the plastic
shear strain rate and evolves in the direction of the flow stress.
It is also noted that Eq. (20) will be reduced to the standard
Eyring equation [62] upon thermal equilibrium where T.sub.f=T.
[0272] Model Summary
[0273] The temperature- and time-dependent, damage-allowable
thermo-mechanical constitutive relations for the SMP based
syntactic foam are summarized in Table 3. The preliminary model
considers the novel composite material in a structure-evolving
manner. It was capable of capturing the essential mechanical
behavior such as yielding, strain softening and strain hardening.
The influence of the crushing and implosion of the glass hollow
microspheres is also taken into account. However, since the focus
is on developing a theoretical thermo-mechanical framework for the
SMP based syntactic foam programmed at glassy temperature, the
proposed constitutive model is rough as compared to the actual
material behavior. Factors such as heat conduction,
deformation-induced entropy change and pressure effects on the
structure relaxation are excluded. A comparatively simple instant
and complete-damage process is also assumed for the glass hollow
microspheres. Detailed modeling efforts on the interaction between
the matrix and inclusions would help capture the more vivid
physical phenomenon.
TABLE-US-00003 TABLE 3 Summary of the constitutive model
deformation F = F.sub.MF.sub.T; F.sub.M = .phi..sub.pF.sub.p + (1 -
.phi..sub.p)F.sub.i response F.sub.i = (1 -
.phi..sub.d)F.sub.i.sup.ud + .phi..sub.dF.sub.i.sup.d;
F.sub.i.sup.d = F.sub.i.sup.udF.sub.d; F.sub.d = J.sub.d.sup.1/3I;
J.sub.d = 1 - (1 - w).sup.3 F.sub.p = F.sub.p.sup.eF.sub.p.sup.v;
F.sub.T = J.sub.T.sup.1/3I J.sub.T = 1 + .alpha..sub.r(T.sub.f -
T.sub.0) + .alpha..sub.g(T - T.sub.f) structure T f ( t ) = T ( t )
- .intg. t 0 t .PHI. ( .DELTA..zeta. ) dT ( t ) ##EQU00035##
relaxation .PHI. = exp ( - ( .DELTA..zeta. ) .beta. ) ;
.DELTA..zeta. = .zeta. ( t ) - .zeta. ( t ' ) = .intg. t ' t dt
.tau. s ##EQU00036## .tau. s = .tau. 0 exp [ B ( T g - T .infin. )
2 ( x T - T .infin. + 1 - x T f - T .infin. ) ] ##EQU00037## stress
response .sigma. = .sigma..sub.i = .sigma..sub.p =
.sigma..sub.p.sup.ve + .sigma..sub.p.sup.n .sigma..sub.i =
L.sub.i.sup.e(lnF.sub.i.sup.ud) .sigma. p = 1 J p e L p e ( ln V p
e ) + [ 1 J n .mu. r .lamda. L .lamda. chain L - 1 ( .lamda. chain
.lamda. L ) B _ ' + k b ( J n - 1 ) I ] ##EQU00038## viscous flow D
p v = .gamma. . v .sigma. p ve ' .sigma. p ve ' ##EQU00039##
.gamma. . v = s .eta. g T Q exp ( c 1 ( c 2 ( T - T f ) + T ( T f -
T g ) T ( c 2 + T - T g ) ) ) sinh ( Q T .tau. _ s )
##EQU00040##
TABLE-US-00004 TABLE 4 Material parameters of the preliminary
thermoviscoplastic constitutive model Model parameters Values
T.sub.g (.degree. C.) glass transition temperature 64.3 T.sub.0
(.degree. C.) programming temperature 20 .DELTA.t (minute)
relaxation time 0/5/15/30/120 .PHI..sub.p .PHI..sub.p volume
fraction of SMP matrix 0.6 .alpha..sub.g (10.sup.-4 .degree.
C..sup.-1) volumetric CTE of glassy state 5.062 .alpha..sub.r
(10.sup.-4 .degree. C..sup.-1) volumetric CTE of rubbery state
6.841 G.sub.i (GPa) Shear modulus of glass hollow microspheres 27.7
.lamda..sub.i (GPa) Lame constant for glass hollow microspheres
41.5 k (MPa.sup.-1) damage rate for glass hollow microspheres 0.02
w wall thickness ratio for glass hollow microspheres 0.019 G.sub.P
(MPa) glassy shear modulus of SMP 96.4 .lamda..sub.P(MPa) Lame
constant for glassy state of SMP 385.7 .mu..sub.r (MPa) rubbery
modulus of SMP 0.3 k.sub.b (MPa) bulk modulus of SMP 1000
.lamda..sub.L locking stretch 1.4 .eta..sub.g (MPa s.sup.-1)
reference shear viscosity at T.sub.g 4050 s.sub.0 (MPa) initial
shear strength. 20 s.sub.s (MPa) steady-state shear strength 18
Q/s.sub.0 (.degree. K/MPa) flow activation ratio 800 h (MPa) flow
softening constant 200 c.sub.1 first WLF constant 17.3 c.sub.2
(.degree. C.) second WLF constant 70 .tau. (s) structure relaxation
characteristic time 20 x NMM constant 0.95 .beta. Kohlrausch index
0.95
[0274] Results
[0275] Model Validation
[0276] The structure-evolving, damage-allowable thermoviscoplastic
constitutive model was computed in MATLAB. The corresponding model
parameters were mainly obtained by curve-fitting various thermal
and mechanical testing results. The mechanical and material
parameter values are listed in Table 4.
[0277] The numerical simulation results shown in FIG. 27(a), (b),
and (c) cover the full thermomechanical cycle of the SMP based
syntactic foam programmed at room temperature with the pre-strain
of both 20% and 30% in both strain-time scale and
stress-strain-time scale. All of the five different relaxation
histories (0 min, 5 min, 15 min, 30 min, and 120 min) for both
pre-strains are included. The material was initially compressed to
the pre-defined strain level, which was beyond the yielding point.
A slight strain-softening behavior appears followed by the strain
hardening (Step 1). After different periods of relaxation for
viscoplastic strain development (Step 2), the remaining stress
constraint is instantly removed, leading to an externally
stress-free state (Step 3). The temporary shape is fixed and
lengthy relaxation apparently promotes the strain fixity. During
the subsequent heating recovery (Step 4), the stored deformation is
released, although there is a considerable amount of irrecoverable
strain due to the damage of the glass hollow microspheres.
[0278] The simulation generally showed a reasonable agreement with
the experimental results and captured most of the essential
nonlinear material behavior, although less agreement on final
recovery strain was found for samples programmed to 20% pre-strain
than those programmed to 30% pre-strain. This may be because under
20% pre-strain, damage in the microballoons was considerably less
than that under 30% prestrain and was below the linear
interpolation prediction. In other words, the linear damage
evaluation assumption is more appropriate for heavily damaged
microballoons than for slightly damaged counterparts. It is also
noted that the approximate nature of the single relaxation
assumption appears evident. When the relaxation time is
insufficient, such as 0 minutes, the discrepancy is particularly
apparent. As the relaxation time further increases, the discrepancy
becomes comparatively less significant. Multiple non-equilibrium
relaxation processes would be required to more closely describe an
actual stress relaxation.
[0279] The thermomechanical cycle for a 2-D traditional programming
process as reported by Li and Xu [27] was also compared. The
cruciform specimen was initially subjected to a constant load of
54.3 N (168.3 kPa) vertically in compression and horizontally in
tension at 79.degree. C., after which the conventional training
method was followed to achieve shape fixity (cooling to room
temperature for about ten hours while holding the load, and then
removing the load completely and instantly). After that it was
reheated to 79.degree. C. at a heating rate of 0.3.degree. C./min
and equilibrated for 30 minutes for free recovery. The simulation
results in FIG. 28 show the strain evolution in the horizontal and
vertical directions during the entire thermomechanical cycle.
Again, good agreement was found between the testing and modeling
results.
[0280] Prediction and Discussion
[0281] The effects of the material composition on the
thermomechanical behavior were numerically investigated.
[0282] Volume fraction of the SMP matrix Error! Objects cannot be
created from editing field codes. .PHI..sub.p
[0283] The thermo-mechanical cycle prediction results of two
specimens with different volume fractions of SMP matrix
(.PHI..sub.p=0.5 and .PHI..sub.p=0.6) experiencing 40-minute
relaxation period are shown in FIG. 29. The recovery heating rate
was 0.4.degree. C./min.
[0284] It is found that less SMP appeared to slightly increase the
shape fixity ratio, which seems anomalous. Further observation of
the heating recovery revealed that the seeming enhancement in shape
fixity originated from an increase in glass hollow microsphere
damage. This is because the specimen with less SMP experienced
greater irreversible strain, and the loss of recoverability was
noticeably greater than the gain in the shape fixity. Therefore, it
is believed that lower .PHI..sub.p should lead to more damage and a
lower recovery ratio.
[0285] Wall Thickness Ratio w
[0286] Further consideration was given to the wall thickness ratio
of the hollow glass microspheres. FIG. 30 shows the full
thermomechanical cycle prediction for two specimens with different
w. The corresponding variation in microsphere strength was assumed
to be negligible.
[0287] The specimen with a higher w was found to be able to achieve
a larger recovery ratio (lower permanent strain), as it contained
fewer voids and hence suffered less damage during programming. It
is also interesting to notice that the shape fixity seemed to be
hardly affected by the variation in w, because the same crushing
strength was assumed. Although the irreversible deformation of
microballoons with lower w may tend to increase the shape fixity
ratio, the reduction in the reversible viscous deformation in SMP
counterbalanced that tendency.
[0288] The final values of the model parameters, as listed in Table
4, were mainly obtained from curve fitting various testing results
shown in FIG. 31 through FIG. 34. Several basic guidelines were
used to assist the initial estimations:
[0289] (1) A cooling history for the SMP based syntactic foam is
plotted as thermal deformation versus the temperature in FIG. 31.
L.sub.0 denotes the initial reference sample height. Because the
cooling rate is extremely slow, an average of 0.17.degree. C./min,
the thermal shrinkage can be perceived as the structural response.
Linear CTEs .alpha..sub.r and a.sub.g were computed from the slopes
at temperatures above and below T.sub.g. Volumetric CTE is three
times the values of the linear CTE.
[0290] (2) .mu..sub.r and .lamda..sub.L characterize the rubbery
behavior of the material, and can be determined from the
stress-strain response at temperatures above T.sub.g. The initial
slope of the isothermal uniaxial compression stress-strain curve in
glassy state gives an estimate for the Lame constant if a typical
polymer Poisson ratio of 0.4 is assumed [22]. The final values for
all these polymer mechanical parameters are fitted against the
stress-strain curves at various temperatures, as shown in FIG.
32.
[0291] (3) The viscoplastic parameters such as Q, s, s.sub.5, and h
can be roughly fitted from the compression tests at different
strain rates (FIG. 33). The ratio Q/s determines the strain rate
dependence of the yield strength, and s/s.sub.s represents the
shear strength drop. h characterizes the post-yield
strain-softening rate. It is found that noticeable discrepancies
appear between the modeling prediction and test results in FIG. 33,
especially at large strain. It is believed that more detailed
consideration of the interaction between matrix and inclusions and
a more realistic anisotropic flow model could be able to achieve a
better agreement.
[0292] (4) The structural relaxation parameters x and .beta. are
fitted to a stress-free, constant heating profile of the thermal
deformation (FIG. 34).
CITATIONS
[0293] Anand, L., Ames, N. M., 2006. On modeling the
micro-indentation response of amorphous polymer. Int. J. Plasticity
22, 1123-1170. [0294] Arruda, E. M., Boyce, M. C., 1993. A
three-dimensional constitutive model for the large stretch behavior
of rubber elastic materials. J. Mech. Phys. Solids 41, 389-412.
[0295] Behl, M., Lendlein, A., 2007. Shape-memory polymers.
Materials Today. 10 (4), 20-28. [0296] Bergstrom, J. S., Boyce, M.
C., 1998. Constitutive modeling of the large strain time-dependence
behavior of elastomers. J. Mech. Phys. Solids 46 (5), 931-954.
[0297] Bhattacharyya, A., Tobushi, H., 2000. Analysis of the
isothermal mechanical response of a shape memory polymer
rheological model. Polym. Eng. Sci. 40 (12), 2498-2510. [0298]
Boyce, M. C., Parks, D. M., Argon, A. S., 1988a. Large inelastic
deformation of glassy-polymers. 1: Rate dependent constitutive
model. Mech. Mater. 7 (1), 15-33. [0299] Boyce, M. C., Park, D. M.,
Argon, A. S., 1988b. Large inelastic deformation of
glassy-polymers. 2: Numerical-simulation of hydrostatic extrusion.
Mech. Mater. 7 (1), 35-47. [0300] Boyce, M. C., Weber, G. G.,
Parks, D. M., 1989. On the kinematics of finite strain plasticity.
J. Mech. Phys. Solids 37 (5), 647-665. [0301] Boyce, M. C., Kear,
K., Socrate, S., Shaw, K., 2001. Deformation of thermoplastic
vulcanizates. J. Mech. Phys. Solids 49 (5), 1073-1098. [0302] Chen,
Y. H., Lagoudas, D. C., 2008a. A constitutive theory for shape
memory polymers. Part 1-large deformations. J. Mech. Phys. Solids
56, 1752-1765. [0303] Chen, Y. H., Lagoudas, D. C., 2008b. A
constitutive theory for shape memory polymers. Part II-A linearized
model for small deformations. J. Mech. Phys. Solids 56, 1766-1778.
[0304] Diani, J., Gall, K., 2007. Molecular dynamics simulations of
the shape-memory behaviour of polyisoprene. Smart Mater. Struct.
16, 1575-1583. [0305] Donth, E., Hempel, E., 2002. Structural
relaxation above the glass temperature: pulse response simulation
with the Narayanaswamy Moynihan model for glass transition. J.
Non-Cryst. Solids 306, 76-89. [0306] Eyring, H., 1936. Viscosity,
plasticity, and diffusion as examples of absolute reaction rates.
J. Comput. Phys. 28, 373-383. [0307] Flory, P. J., 1961.
Thermodynamic relations for highly elastic materials. Trans.
Faraday Soc. 57, 829-838. [0308] Gall, K., Yakacki, C. M., Liu, Y.,
Shandas, R., Willett, N., Anseth, K. S., 2005. Thermomechanics of
the shape memory effect in polymers for biomedical applications. J.
Biomed. Mater. Res. A 73, 339-348. [0309] Govindjee, S., Reese, S.,
1997. A presentation and comparison of two large deformation
viscoelasticity models. Trans. ASME J. Eng. Mater. Technol. 119,
251-255. [0310] Govindjee, S., Simo, J., 1991. A micro-mechanically
based continuum damage model for carbon black-filled rubbers
incorporating Mullins effect. J. Mech. Phys. Solids 39 (1), 87-112.
[0311] Hempel, E., Kahle, S., Unger, R., Donth, E., 1999.
Systematic calorimetric study of glass transition in the homologous
series of poly(n-alkyl methacrylate)s: Narayanaswamy parameters in
the crossover region. Thermochimica Acta. 329, 97-108. [0312] John,
M., Li, G., 2010. Self-healing of sandwich structures with a grid
stiffened shape memory polymer syntactic foam core. Smart Mater.
Struct. 19(7) (paper No. 075013), 1-12. [0313] Kafka, V., 2001.
Mesomechanical constitutive modeling. World Scientific, Singapore.
[0314] Kafka, V., 2008. Shape memory polymers: a mesoscale model of
the internal mechanism leading to the SM phenomena. Int. J. Plast.
24, 1533-1548. [0315] Kohlrausch, F., 1847. Pogg. Ann. Phys. 12,
393-399. [0316] Lendlein, A., Langer, R., 2002. Shape memory
polymers. Angew. Chem. Int. Ed. 41, 2034-2057. [0317] Lendlein, A.
S., Kelch, S., Kratz, K., Schulte J., 2005. Shape-memory polymers.
In: Encyclopedia of Materials. Elsevier, Amsterdam, 1-9. [0318] Li,
G., John, M., 2008. A self-healing smart syntactic foam under
multiple impacts. Compos. Sci. Technol. 68(15-16), 3337-3343.
[0319] Li, G., Nettles, D., 2010. Thermomechanical characterization
of a shape memory polymer based self-repairing syntactic foam.
Polymer 51 (3), 755-762. [0320] Li, G., Uppu, N., 2010. Shape
memory polymer based self-healing syntactic foam: 3-D confined
thermomechanical characterization. Comp. Sci. Technol. 40 (9),
1419-1427. [0321] Lion, A., 1997. On the large deformation behavior
of reinforced rubber at different temperatures. J. Mech. Phys.
Solids 45, 1805-1834. [0322] Liu, Y. P., Gall, K., Dunn, M. L.,
McCluskey P., 2004. Thermomechanics of shape memory polymer
nanocomposites. Mech. Mater. 36 (10), 929-940. [0323] Liu, Y.,
Gall, K., Dunn, M. L., Greenberg, A. R., Diani, J., 2006.
Thermomechanics of shape memory polymers: uniaxial experiments and
constitutive modeling. Int. J. Plast. 22, 279-313. [0324] Lu, S. C.
H., Pister, K. S., 1975. Decomposition of deformation and
representation of the free energy function for isotropic
thermoelastic solids. Int. J. Solids Struct. 11, 927-934. [0325]
Miehe, C., Keck, J., 2000. Superimposed finite
elastic-viscoelastic-plastoelastic stress response with damage in
filled rubbery polymers. Experiments, modelling and algorithmic
implementation. J. Mech. Phys. Solids 48 (2), 323-365. [0326]
Morshedian, J., Khonakdar, H. A., Rasouli, S., 2005. Modeling of
shape memory induction and recovery in heatshrinkable polymer.
Macromol. Theory Simulat. 14, 428-434. [0327] Moynihan, C. T.,
Easteal, A. E., Debolt, M. A., Tucker, J., 1976. J. Am. Ceram. Soc.
59, 12-16. [0328] Nakayama, K., 1991. Properties and application of
shape-memory polymers. Int. J. Polym. Sci. Technol. 19, T43-T48.
[0329] Narayanaswamy, O. S., 1971. A model of structural relaxation
in glass. J. Am. Ceramics Soc. 54 (10), 491-498. [0330] Nguyen T.
D., Qi, H., Castro, F., Long, K. N., 2008. A thermoviscoelastic
model for amorphous shape memory polymers: Incorporating structural
and stress relaxation. J. Mech. Phys. Solids 56(9), 2792-2814.
[0331] Nji, J., Li, G., 2010a. A self-healing 3D woven fabric
reinforced shape memory polymer composite for impact mitigation.
Smart Mater. Struct. 19(3) (paper No. 035007), 1-9. [0332] Nji, J.
and Li, G., 2010b. A biomimic shape memory polymer based
self-healing particulate composite. Polymer 51, 6021-6029. [0333]
Otsuka, K., Wayman, C. M., 1998. Shape memory materials. Cambridge
University Press, New York. [0334] Qi, H. J., Boyce, M. C., 2005.
Stress-strain behavior of thermoplastic polyurethanes. Mech. Mater.
37 (8), 817-839. [0335] Qi, H. J., Nguyen, T. D., Castro, F.,
Yakacki, C. M., Shandas, R., 2008. Finite deformation
thermo-mechanical behavior of thermally induced shape memory
polymers. J. Mech. Phys. Solids 56, 1730-1751. [0336] Ping, P.,
Wang, W., Chen, X., and Jing, X., 2005.
Poly(.epsilon.-caprolactone) polyurethane and its shape-memory
property. Biomacromol. 6, 587-592. [0337] Rabani, G., Luftmann, H.,
Kraft, A., 2006. Synthesis and characterization of two shape-memory
polymers containing short aramid hard segaments and
poly(c-caprolactone) soft segments. Polymer 47, 4251-4260. [0338]
Scherer, G. W., 1990. Theories of relaxation. J. Non-Cryst. Solids
123, 75-89. [0339] Sidoroff, F., 1974. Un modele viscoelastique non
lineaire avec configuration intermediare. J. Mec. 13, 679-713.
[0340] Simo, J. C., Taylor, R. L., Pister, K. S., 1985. Variational
and projection methods for the volume constraint in finite
deformation elasto-plasticity. Comput. Methods Appl. Mech. Eng. 51,
177-208. [0341] Tobushi, H., Hara, H., Yamada, E., Hayashi, S.,
1996. Thermomechanical properties in a thin film of shape memory
polymer of polyurethane series. Smart Mater. Struct. 5 (4),
483-491. [0342] Tobushi, H., Hashimoto, T., Hayashi, S., Yamada,
E., 1997. Thermomechanical constitutive modeling in shape memory
polymer of polyurethane series. J. Intel'. Materl. Syst. Struct. 8,
711-718. [0343] Tool, A. Q., 1946. Relation between inelastic
deformability and thermal expansion of glass in its annealing
range. J. Amer. Ceram. Soc. 29 (9), 240-253. [0344] Treloar, L. R.
G., 1958. The physics of Rubber Elasticity. Clarendon Press,
Oxford. [0345] Wang, W., Jin, Y., Ping, P., Chen, X., Jing, X., Su,
Z., 2010. Structure evolution in segmented poly(ester urethane) in
shape-memory process. Macromolecules 43, 2942-2947. [0346] William
M. L., Landel R. F., Ferry J. D., 1955. The temperature dependence
of relaxation mechanisms in amorphous polymers and other
glass-forming liquids. J. Amer. Chem. Soc. 77, 3701-3707. [0347]
Xu, W., Li, G., 2010. Constitutive modeling of shape memory polymer
based self-healing syntactic foam. Int. J. Solids Structs. 47 (9),
1306-1316. [0348] Yakacki, C. M., Shandas, R., Lanning, C., Rech,
B., Eckstein A., Gall K., 2007. Unconstrained recovery
characterization of shape-memory polymer networks for
cardiovascular applications. Biomaterials 28 (14), 2255-2263.
[0349] Zotzmann, J., Behl, M., Feng, Y., Lendlein, A., 2010.
Copolymer networks based on poly(o-pentadecalactone) and
poly(.epsilon.-caprolactone) segments as a versatile triple-shape
polymer system. Adv. Funct. Mater. 20, 3583-3594. [0350] [1] Li,
G., Xu, W., 2011. Thermomechanical behavior of shape memory polymer
programmed at glassy temperature: testing and constitutive
modeling. J. Mech. Phys. Solids, (Available on-line Mar. 9, 2011),
doi: 10.1016/j.jmps.2011.03.001. [0351] [2] Li, G., John, M., 2008.
A self-healing smart syntactic foam under multiple impacts. Compos.
Sci. Technol. 68(15-16), 3337-3343. [0352] [3] Lendlein, A. S.,
Kelch, S., Kratz, K., Schulte J., 2005. Shape-memory polymers. In:
Encyclopedia of Materials. Elsevier, Amsterdam, 1-9. [0353] [4]
Behl, M., Lendlein, A., 2007. Shape-memoy polymers. Materials
Today. 10 (4), 20-28. [0354] [5] Anderson, T. F., Walters, H. A.,
Glesner, C. W., 1970. Castable, sprayable, low density foam and
composites for furniture, marble, marine. J. Cell. Plast. 6,
171-178. [0355] [6] Gupta, N., and Woldesenbet, E., 2005.
Characterization of flexural properties of syntactic foam core
sandwich composites and effect of density variation. J. Compos.
Mater. 39, 2197-2212. [0356] [7] Li, G., Nettles, D., 2010.
Thermomechanical characterization of a shape memory polymer based
self-repairing syntactic foam. Polymer 51 (3), 755-762. [0357] [8]
Li, G., Uppu, N., 2010. Shape memory polymer based self-healing
syntactic foam: 3-D confined thermomechanical characterization.
Comp. Sci. Technol. 40 (9), 1419-1427. [0358] [9] Nji, J., Li, G.,
2010. A biomimic shape memory polymer based self-healing
particulate composite. Polymer 51, 6021-6029. [0359] [10] Nji, J.,
Li, G., 2010. A self-healing 3D woven fabric reinforced shape
memory polymer composite for impact mitigation. Smart Mater.
Struct. 19(3), 035007. [0360] [11] John, M., Li, G., 2010.
Self-healing of sandwich structures with a grid stiffened shape
memory polymer syntactic foam core. Smart Mater. Struct. 19(7)
075013. [0361] [12] Tobushi, H., Hara, H., Yamada, E., Hayashi, S.,
1996. Thermomechanical properties in a thin film of shape memory
polymer of polyurethane series. Smart Mater. Struct. 5 (4),
483-491. [0362] [13] Tobushi, H., Hashimoto, T., Hayashi, S.,
Yamada, E., 1997. Thermomechanical constitutive modeling in shape
memory polymer of polyurethane series. J. Intell. Materl. Syst.
Struct. 8, 711-718. [0363] [14] Bhattacharyya, A., Tobushi, H.,
2000. Analysis of the isothermal mechanical response of a shape
memory polymer rheological model. Polym. Eng. Sci. 40 (12),
2498-2510. [0364] [15] Kafka, V., 2001. Mesomechanical constitutive
modeling. World Scientific, Singapore. [0365] [16] Kafka, V., 2008.
Shape memory polymers: a mesoscale model of the internal mechanism
leading to the SM phenomena. Int. J. Plast. 24, 1533-1548. [0366]
[17] Diani, J., Gall, K., 2007. Molecular dynamics simulations of
the shape-memory behaviour of polyisoprene. Smart Mater. Struct.
16, 1575-1583. [0367] [18] Morshedian, J., Khonakdar, H. A.,
Rasouli, S., 2005. Modeling of shape memory induction and recovery
in heatshrinkable polymer. Macromol. Theory Simulat. 14, 428-434.
[0368] [19] Gall, K., Yakacki, C. M., Liu, Y., Shandas, R.,
Willett, N., Anseth, K. S., 2005. Thermomechanics of the shape
memory effect in polymers for biomedical applications. J. Biomed.
Mater. Res. A 73, 339-348. [0369] [20] Liu, Y., Gall, K., Dunn, M.
L., Greenberg, A. R., Diani, J., 2006. Thermomechanics of shape
memory polymers: uniaxial experiments and constitutive modeling.
Int. J. Plast. 22, 279-313. [0370] [21] Yakacki, C. M., Shandas,
R., Lanning, C., Rech, B., Eckstein A., Gall K., 2007.
Unconstrained recovery characterization of shape-memory polymer
networks for cardiovascular applications. Biomaterials 28 (14),
2255-2263. [0371] [22] Qi, H. J., Nguyen, T. D., Castro, F.,
Yakacki, C. M., Shandas, R., 2008. Finite deformation
thermo-mechanical behavior of thermally induced shape memory
polymers. J. Mech. Phys. Solids 56, 1730-1751. [0372] [23] Chen, Y.
H., Lagoudas, D. C., 2008. A constitutive theory for shape memory
polymers. Part I-large deformations. J. Mech. Phys. Solids 56,
1752-1765. [0373] [24] Chen, Y. H., Lagoudas, D. C., 2008. A
constitutive theory for shape memory polymers. Part II-A linearized
model for small deformations. J. Mech. Phys. Solids 56, 1766-1778.
[0374] [25] Xu, W., Li, G., 2010. Constitutive modeling of shape
memory polymer based self-healing syntactic foam. Int. J. Solids
Structs. 47 (9), 1306-1316. [0375] [26] Nguyen T. D., Qi, H.,
Castro, F., Long, K. N., 2008. A thermoviscoelastic model for
amorphous shape memory polymers: Incorporating structural and
stress relaxation. J. Mech. Phys. Solids 56(9), 2792-2814. [0376]
[27] Li, G., Xu, T., 2011. Thermomechanical characterization of
shape memory polymer based self-healing syntactic foam sealant for
expansion joint. ASCE J. Mater. Civ. Eng., (Available on-line Mar.
23, 2011), doi:10.1061/(ASCE)TE.1943-5436.0000279. [0377] [28] ASTM
C365-Stardard Test Method for Flatwise Compressive Properties of
Sandwich Cores. [0378] [29] ASTM E1640-04-Standard Test Method for
Assignment of Glass Transition. [0379] [30] Li, G., Nji, J., 2007.
Development of rubberized syntactic foam. Compos. Part A: App. Sci.
Manuf 38, 1483-1492. [0380] [31] Berriot, J., Montes, H., Lequeux,
F., Long, D., Sotta, P., 2002. Evidence for the shift of the glass
transition near the particles in silica-filled elastomers.
Macromolecules. 35(26), 9756-9762. [0381] [32] Berriot, J., Montes,
H., Lequeux, F., Long, D., Sotta, P., 2003. Gradient of glass
transition temperature in filled elastomers. Europhys. Lett. 64(1),
50-56. [0382] [33] Oliver, J. P., Maso, J. C., Bourdette, B., 1995.
Interfacial transition zone in concrete. J. Adv. Cem. Based Mater.
2(1), 30-38. [0383] [34] Li, G., Zhao, Y., and Pang S. S., 1998. A
three-layer built-in analytical modeling of concrete. Cem. Concr.
Res. 28, 1057-1070. [0384] [35] Li, G., Zhao, Y., and Pang S. S.,
1999. Four-phase sphere modeling of effective bulk modulus of
concrete. Cem. Concr. Res. 29, 839
l-845. [0385] [36] Lu, S. C. H., Pister, K. S., 1975. Decomposition
of deformation and representation of the free energy function for
isotropic thermoelastic solids. Int. J. Solids Struct. 11, 927-934.
[0386] [37] Lion, A., 1997. On the large deformation behavior of
reinforced rubber at different temperatures. J. Mech. Phys. Solids
45, 1805-1834. [0387] [38] Sidoroff, F., 1974. Un modele
viscoelastique non lineaire avec configuration intermediare. J.
Mec. 13, 679-713. [0388] [39] Tool, A. Q., 1946. Relation between
inelastic deformability and thermal expansion of glass in its
annealing range. J. Amer. Ceram. Soc. 29 (9), 240-253. [0389] [40]
Narayanaswamy, O. S., 1971. A model of structural relaxation in
glass. J. Am. Ceramics Soc. 54 (10), 491-498. [0390] [41] Moynihan,
C. T., Easteal, A. E., Debolt, M. A., Tucker, J., 1976. J. Am.
Ceram. Soc. 59, 12-16. [0391] [42] Donth, E., Hempel, E., 2002.
Structural relaxation above the glass temperature: pulse response
simulation with the Narayanaswamy Moynihan model for glass
transition. J. Non-Cryst. Solids 306, 76-89. [0392] [43]
Kohlrausch, F., 1847. Pogg. Ann. Phys. 12, 393-399. [0393] [44]
DeBolt M A., Easteal A J., Macedo P B., Moyhinan C T., 1976.
Analysis of structural relaxation in glass using rate heating data.
J. Am. Ceramics Soc. 59 (1-2), 16-21. [0394] [45] Donth E., 1982.
Analysis of thermoluminescence curves of polymers using current
methods of relaxation phenomenology. Polymer Bulletin 8, 211-217.
[0395] [46] Hempel, E., Kahle, S., Unger, R., Donth, E., 1999.
Systematic calorimetric study of glass transition in the homologous
series of poly(n-alkyl methacrylate)s: Narayanaswamy parameters in
the crossover region. Thermochimica Acta. 329, 97-108. [0396] [47]
William M. L., Landel R. F., Ferry J. D., 1955. The temperature
dependence of relaxation mechanisms in amorphous polymers and other
glass-forming liquids. J. Amer. Chem. Soc. 77, 3701-3707. [0397]
[48] Scherer, G. W., 1990. Theories of relaxation. J. Non-Cryst.
Solids 123, 75-89. [0398] [49] Treloar, L. R. G., 1958. The physics
of Rubber Elasticity. Clarendon Press, Oxford. [0399] [50] Boyce,
M. C., Parks, D. M., Argon, A. S., 1988. Large inelastic
deformation of glassy-polymers. 1: Rate dependent constitutive
model. Mech. Mater. 7 (1), 15-33. [0400] [51] Boyce, M. C., Park,
D. M., Argon, A. S., 1988. Large inelastic deformation of
glassy-polymers. 2: Numerical-simulation of hydrostatic extrusion.
Mech. Mater. 7 (1), 35-47. [0401] [52] Boyce, M. C., Weber, G. G.,
Parks, D. M., 1989. On the kinematics of finite strain plasticity.
J. Mech. Phys. Solids 37 (5), 647-665. [0402] [53] Govindjee, S.,
Simo, J., 1991. A micro-mechanically based continuum damage model
for carbon black-filled rubbers incorporating Mullins effect. J.
Mech. Phys. Solids 39 (1), 87-112. [0403] [54] Arruda, E. M.,
Boyce, M. C., 1993. A three-dimensional constitutive model for the
large stretch behavior of rubber elastic materials. J. Mech. Phys.
Solids 41, 389-412. [0404] [55] Miehe, C., Keck, J., 2000.
Superimposed finite elastic-viscoelastic-plastoelastic stress
response with damage in filled rubbery polymers. Experiments,
modelling and algorithmic implementation. J. Mech. Phys. Solids 48
(2), 323-365. [0405] [56] Bergstrom, J. S., Boyce, M. C., 1998.
Constitutive modeling of the large strain time-dependence behavior
of elastomers. J. Mech. Phys. Solids 46 (5), 931-954. [0406] [57]
Govindjee, S., Reese, S., 1997. A presentation and comparison of
two large deformation viscoelasticity models. Trans. ASME J. Eng.
Mater. Technol. 119, 251-255. [0407] [58] Boyce, M. C., Kear, K.,
Socrate, S., Shaw, K., 2001. Deformation of thermoplastic
vulcanizates. J. Mech. Phys. Solids 49 (5), 1073-1098. [0408] [59]
Qi, H. J., Boyce, M. C., 2005. Stress-strain behavior of
thermoplastic polyurethanes. Mech. Mater. 37 (8), 817-839. [0409]
[60] Flory, P. J., 1961. Thermodynamic relations for highly elastic
materials. Trans. Faraday Soc. 57, 829-838. [0410] [61] Simo, J.
C., Taylor, R. L., Pister, K. S., 1985. Variational and projection
methods for the volume constraint in finite deformation
elasto-plasticity. Comput. Methods Appl. Mech. Eng. 51, 177-208.
[0411] [62] Eyring, H., 1936. Viscosity, plasticity, and diffusion
as examples of absolute reaction rates. J. Comput. Phys. 28,
373-383. [0412] [63] Li H. X. and Buckley C. P., 2010. Necking in
glassy polymers: Effects of intrinsic anisotropy and structural
evolution kinetics in their viscoplastic flow. Int. J. Plast 26,
1726-1745.
[0413] All documents, including patents or published applications,
journal papers, and other documents either cited in this
specification, or relied upon for priority, are fully incorporated
by reference herein. In the event of an otherwise irreconcilable
conflict, the present specification shall control.
* * * * *