Thermomechanical Constitutive Modeling of Shape Memory Polymer Including
Continuum Functional and Mechanical Damage Effects
Amir Shojaei1, Guoqiang Li1,2*
1
Department of Mechanical & Industrial Engineering, Louisiana State University,
Baton Rouge, LA 70803, USA
2
Department of Mechanical Engineering, Southern University, Baton Rouge, LA 70813, USA
*Corresponding author. Tel.: 001-225-578-5302; Fax: 001-225-578-5924;
†
Emails: Ashoja1@lsu.edu, and A.Shojaei.Mech.Eng@gmail.com
lguoqi1@lsu.edu (G. Li)
1
(A.
Shojaei)
and
Appendix A. Typical thermomechanical cycle.
Figure A1 shows a typical thermomechanical cycle of an amorphous SMP and SMP based
syntactic foam (Li and Nettles 2010). The following four steps are included in each cycle: (1)
High temperature loading: the temperature is elevated to above the transition temperature, i.e. Tr,
where the mobility in the SMP molecular network is surged. The SMP molecular chains are
flexible in this stage and they can cope with the applied external traction field. (2) Cooling: The
SMP is cooled down to below Tr while the external traction field is maintained. In this step the
deformed molecular network maintains the induced shape in step 1. (3) Low temperature
unloading: The traction is then removed which results in elastically unloading the SMP and
completing the programming process, and (4) Recovery: During the shape recovery stage the
temperature is increased beyond the transition temperature where the locked molecular chains
are able to restore their original configuration, and in this step the SMP releases its memory.
Figure A1. Typical thermomechanical cycle for SMP and SMP based syntactic foam, figure
is after (Li and Nettles 2010)
Li, G. and D. Nettles (2010). "Thermomechanical characterization of a shape memory polymer
based self-repairing syntactic foam." Polymer 51(3)755-762.
2
Appendix B. Discussions on performance of “pnorm” and “dnorm” functions
In the case of the polystyrene based thermoset SMP system, used in this study, the glass
transition temperature is identified as Tg=67.7 oC and the standard deviation of the glass
transition loss event is obtained as σg=6.5 oC. Based upon these inputs the functions ̅̅̅̅̅̅̅̅̅
dnormg and
pnormg are plotted in Fig. B1(a). In the case of the shape recovery process, the transition
temperature is approximately centered at Tr=62.0 oC and its variance is σr=5.13 oC. The
associated functions ̅̅̅̅̅̅̅̅̅
dnormr and pnormr with these values are depicted in Fig. B1(b).
Figure B1 Responses of Eqs. (6) and (7) over the: (a) glass transition with Tg=67.7 oC and
σg=10 oC and (b) shape recovery with Tr=62.0 oC and σr=5.13 oC.
3
Appendix C. Parametric study of the viscoplastic model
The performance of these equations is parametrically studied in Fig. C1. In Fig. C1(a) the effect
of various central values of the strain hardening effect, i.e. 𝐼𝛼𝑝 , is studied, in which two different
𝐼𝛼𝑝 s result in two distinguishable responses. In Fig. C1(b) the role of different bandwidths values,
i.e. 𝜎𝛼 , are depicted.
Figure C1 Parametric study for the developed viscoplastic model: (a) effect of different
𝒑
mean values 𝑰𝜶 , and (b) effect of different bandwidth values, 𝝈𝜶 .
4
Appendix D. Model outline
To summarize the developed framework, the model for each aspect of SMP simulation is
presented in Tables D1 to D7.
Table D1 Temperature dependent properties for a non-programmed SMP (background
properties)
(𝑇 − 𝑇# )2
1
dnorm# (𝑇, 𝑇# , 𝜎# ) =
exp (−
)
2𝜎# 2
𝜎# √2𝜋
𝑇2
pnorm# (𝑇, 𝑇# , 𝜎# ) = ∫ dnorm# (𝑇, 𝑇# , 𝜎# )𝑑𝑇
𝑇1
̅̅̅̅̅̅̅̅̅# (𝑇, 𝑇# , 𝜎# ) =
dnorm
dnorm#
dnormmax
#
Loss tangent:
tan 𝛿𝑔 = tan 𝛿0 + (tan 𝛿𝑚𝑎𝑥 − tan 𝛿0 ) × ̅̅̅̅̅̅̅̅̅
dnormg
Dynamic tensile storage modulus:
1−pnormg
𝐸 ′ = 𝐸0′ 1+pnorm
g
Dynamic bulk storage modulus:
1−pnormg
𝐵 ′ = 𝐵0′ 1+pnorm
g
LTEC:
1.38
𝛽𝑏 = 3𝐸
𝐶
𝑐𝑜ℎ 𝑅
(1 + ℵ1 pnormg )
Table D2 Thermo-viscoplastic constitutive model
𝑛
|𝛾̇ 𝑝 |
|𝜏| 1
𝑝
|𝛾̇ | = 𝛾̇ 0 × {1 + 𝐶2 ln (
)} × ( )
𝛾̇ 0
𝜇
dnormα (𝐼
𝑝
, 𝐼𝛼𝑝 , 𝜎𝛼 )
pnormα (𝐼
𝑝
, 𝐼𝛼𝑝 , 𝜎𝛼 )
𝑝
2
(𝐼 𝑝 − 𝐼𝛼𝑝 )
=
exp (−
)
2𝜎𝛼 2
𝜎𝛼 √2𝜋
1
𝐼𝑝
= ∫ dnormα (𝐼 𝑝 , 𝐼𝛼𝑝 , 𝜎𝛼 ). d𝐼 𝑝
0
𝛼𝑖𝑗 = ℵ3 𝐼 (1 − pnormg )pnormα 𝛿𝑖𝑗
∗
𝑠𝑖𝑗
= 𝑠𝑖𝑗 − 𝑋𝑖𝑗
1 ∗ ∗
|𝜏| = √ 𝑠𝑖𝑗
𝑠𝑖𝑗
2
∗
𝑠𝑖𝑗
𝐷𝑖𝑗𝑝 = |𝛾̇ 𝑝 |
√2|𝜏|
5
1 ∗ ∗
√ 𝑠𝑖𝑗
𝑠𝑖𝑗 − 𝜏𝑦 ≤ 0
2
𝜏𝑦 = 𝜏0 × {1 + 𝐶 ln (
1 − pnormg
|𝛾̇ 𝑝 |
)} ×
𝛾̇ 0
1 + pnormg
Table D3 Shape memory simulation through shape memory affected properties
(𝑇)
𝜆𝑖
= [1 + (𝛽𝑏 + 𝛽𝑟𝑒𝑐 )Δ𝑇]𝐼𝑖
𝑚𝑖𝑛
Shape recovery parameter: 𝛽𝑟𝑒𝑐 = ±ℵ𝑡𝑟𝑒𝑙 × ϵ(prog) ×
SRR: ℵ
SRR: ℵ
𝑚𝑖𝑛
𝑡𝑟𝑒𝑙
𝑚𝑎𝑥
𝑡𝑟𝑒𝑙
𝑚𝑖𝑛
= 𝜖 (𝑟𝑒𝑐) /𝜖 (𝑝𝑟𝑜𝑔)
for 𝑡𝑟𝑒𝑙 = 𝑡𝑟𝑒𝑙
𝑚𝑎𝑥
= 𝜖 (𝑟𝑒𝑐) /𝜖 (𝑝𝑟𝑜𝑔) for 𝑡𝑟𝑒𝑙 = 𝑡𝑟𝑒𝑙
𝑚𝑎𝑥
Relaxation effect: 𝐹𝑡𝑟𝑒𝑙 = 1 + (
𝑞̇
𝐹𝑟𝑒𝑐
Heating rate effect: 𝑇𝑟′ =
𝑚𝑖𝑛
ℵ𝑡𝑟𝑒𝑙 −ℵ𝑡𝑟𝑒𝑙
𝑚𝑎𝑥
ℵ𝑡𝑟𝑒𝑙
× 𝑇𝑟 with
pnormr
2𝜎𝑟
× 𝐹𝑡𝑟𝑒𝑙
𝑛2
𝑡
𝑟𝑒𝑙
) (𝑡 𝑚𝑎𝑥
)
𝑟𝑒𝑙
𝑞̇
𝐹𝑟𝑒𝑐
𝑞̇
𝑛3
= (𝑞̇ (𝑚𝑖𝑛))
Table D4 Stress relaxation
𝜎𝑟𝑒𝑙 = 𝜎0 ×
𝑡
𝐹𝑟𝑒𝑙𝑟𝑒𝑙
𝑡
𝐹𝑟𝑒𝑙𝑟𝑒𝑙
−1
𝑡𝑟𝑒𝑙
= [exp (𝑎1 𝑚𝑎𝑥 )]
𝑡𝑟𝑒𝑙
Table D5 FD effect on SMP elastic modulus (for compression programming)
𝐷=𝑁
𝑁𝑖
with i=0,1,..,Nmax
𝐸𝑁
𝐸 = 𝐸0 (1 + ( 𝑚𝑎𝑥 − 1 ) 𝐷2 )
𝐸0
𝑚𝑎𝑥
Table D6 FD effect on the Shape Recovery Ratio (SRR)
ℵ#𝑁𝑚𝑎𝑥
#
#
ℵ𝐷 = ℵ (1 − (1 −
) 𝐷0.4 )
ℵ#
𝜖 (𝑟𝑒𝑐)
ℵ#𝑁𝑚𝑎𝑥 = 𝜖(𝑝𝑟𝑜𝑔)
𝑚𝑖𝑛
𝑚𝑎𝑥
for 𝑡𝑟𝑒𝑙 = 𝑡𝑟𝑒𝑙
and 𝑡𝑟𝑒𝑙 = 𝑡𝑟𝑒𝑙
Table D7 FD effect on the Stress Recovery Ratio
ΩD = Ω0 (1 − (1 −
𝜎 (𝑟𝑒𝑐)
ΩNmax
Ω0
) 𝐷0.4 )
Ω = 𝜎(𝑝𝑟𝑜𝑔)
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Appendix E. Computational implementation
The inelastic deformation in polymers is isochoric and the developed model in this work is based
on the incompressibility assumption. Simo et al. (Simo et al., 1985) introduced the isochoric
assumption into the kinematic of the large deformation by splitting the deformation gradient into
the volume preserving and non-preserving parts as shown in the following:
1
𝐹̂𝑖𝑗 = 𝐽−3 𝐹𝑖𝑗
where the hat symbol, ̂ , shows the isochoric components. Then 𝐹̂𝑖𝑗 is the volume-preserving
deformation gradient and 𝐽 = det(𝐹𝑖𝑗 ). This decomposition should be applied to the elastic
deformation gradients, i.e. 𝐹̂ 𝑒 = 𝐽−1/3 𝐹 𝑒 and it ensures that the computed inelastic deformation
is isochoric, i.e. det(𝐹𝑖𝑗𝑝 ) ≡ 1 (Simo and Ortiz 1985; Simo and Taylor 1986; Simo and Hughes
1997).
The return mapping algorithm solves for the plastic strain increments in an implicit integration
algorithm (Simo and Ortiz 1985; Simo and Hughes 1997; Shojaei et al. 2010; Shojaei et al. 2013;
Shojaei and Li 2013a; Shojaei et al. 2013b). The return mapping techniques in inelastic solutions
are natural consequence of splitting the total deformation gradient into the elastic and inelastic
components. The iterative solution algorithm for a finite deformation problem is outlined as
follows (Simo and Hughes 1997):
-
Introducing the loading condition such as rate of deformation tensor, 𝐷𝑖𝑗 , and calculating
(𝑛+1 )
the current deformation gradient, i.e. 𝐹𝑖𝑗
-
, where n indicates the last load step number.
Elastically stretching the material by a “Trial Elastic” deformation gradient given by:
)
(𝑛+1 ) 𝑝(𝑛 ) −1
𝐹𝑖𝑗
where
𝐹𝑖𝑗𝑒(𝑛+1 = 𝐹𝑖𝑗
)
𝐹𝑖𝑗𝑝(𝑛 shows the frozen inelastic deformation gradient
-
from the last load step.
Relaxing the elastically stretched configuration until the state of the corrected stress
satisfies the yield condition, i.e. 𝜏𝑦 ≤ 0.
-
The converged values are then utilized to update the inelastic stretch tensor and 𝐹𝑖𝑗𝑝(𝑛+1) .
At the end of each thermomechanical cycle the damage parameter D is updated.
The details of the coupled elastic, plastic and damage solution algorithms are discussed in
(Shojaei, et al. 2013).
The developed MD and FD framework can be coupled in an FEA code, in which the MD
damage is utilized to incrementally update the material elastic moduli during the course of
deformation. This can be accomplished with introducing a USDFLD to incrementally update the
elastic modulus, E. On the other hand the material properties are updated after each round of the
7
thermomechanical cycle. This provides a mathematically simple and computationally efficient
approach for capturing full damage spectrum in SMP systems.
The viscoplastic, damage and thermomechanical models are well suited for the FEA
implementation through user defined materials. The authors are developing a virtual SMP design
lab by introducing these models into ABAQUS through USDFLD and UMAT subroutines. The
verified FEA model will then be utilized to predict the final product properties based on very
basic inputs about the SMP system performance such as Tg, SRR, SFR, rate dependency, MD
and FD damage responses, etc. In this way the SMP material properties are found based upon
optimum response of the final product to a specific type of loading or service condition. For
example a mechanical component, made of SMP, is subjected to its worst case scenario loading
conditions in the FEA code to obtain the required SMP system material properties.
8
Appendix F. Simulation of stress relaxation:
Fig. F1 shows the performance of the established stress relaxation formula, Eq. (24), in
capturing the experimental data. The stress relaxation experiments concern the compressed SMP
up to 𝜖 =30% under three different temperature levels are also depicted. Due to the fact that we
have not considered a direct term for the temperature effect in the stress relaxation model, and
the predicted SMP properties in Figs. 6 and 7 are utilized in the simulations, the captured results
in the mid-range temperature show some deviations with respect to the experimental data.
Figure F1 Experimental and simulation results for the stress relaxation process in SMP
sample
9
Appendix G. Simulating 2D TM cycles:
The performance of the proposed framework is examined in a more general loading
condition that is the thermomechanical cycle of a 2D traditional programming process, reported
by (Li and Xu 2011). As shown in Fig. G1, 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
o
C. Afterwards the sample is subjected to the conventional training method to achieve shape
fixity (cooling to room temperature in about ten hours while holding the load, and then removing
the load completely and instantly). The free recovery is then carried out by reheating up the
sample to 79 oC at a heating rate of 0.3 C/min and letting the sample to be equilibrated for 30
min. The strain evolution in the horizontal and vertical directions during the entire
thermomechanical cycle is depicted in Fig. G1 (𝜖 (𝑝𝑟𝑜𝑔) = 𝟎. 𝟒). The simulation results in Fig.
G1 agree well with the experimental data for the 3 steps of programming and 4th step of the free
recovery process. The same elastic properties are assumed for the tension and compression cases.
Figure G1 Comparison of numerical simulation with test results for a 2D
thermomechanical cycle, the hashed lines are the experimental data which are constructed
after (Li and Xu 2011).
Li, G. and T. Xu (2011). "Thermomechanical Characterization of Shape Memory Polymer–
Based Self-Healing Syntactic Foam Sealant for Expansion Joints." Journal of Transportation
Engineering 137(11): 805-814.
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Appendix H. List of material parameters
To summarize the material parameters in this work, they are listed below in Table H1 based
upon their functionality in the model. Next to each parameter the method of its identification is
also identified to facilitate tracking the work. The term “from exp.” means the parameter is given
by the experimental data, and “curve fit.” denotes that the material parameter is found through
curve fitting techniques.
Table H1 List of model parameters based on their functionality and method of
identification
Temperature dependent SMP properties
initial value of tanδ (from exp.)
tan𝛅0
maximum value of tanδ (from exp.)
tan𝛅max
𝐸0′ (MPa) reference tensile storage moduli (from exp.)
𝑩′𝟎 (MPa) reference bulk storage moduli (from exp.)
Glass transition (from exp.)
𝑻𝒈 (oC)
o
𝝈𝒈 ( C)
standard deviation of 𝑇𝑔 (from exp.)
Parameter given analytically after Eq. (12) (from exp.)
ℵ𝟏
Parameter given analytically after Eq. (13) (from exp.)
ℵ𝟐
Viscoplastic model
reference shearing strain rate (from exp.)
𝛾̇ 0 (sec-1)
rate sensitivity exponent (curve fit)
𝑛1
shear modulus (from exp.)
𝜇 (MPa)
𝒑
plastic stretch (from exp.)
𝑰
𝒑
limit of plastic stretch in a polymer network (from exp.)
𝑰𝑳
Material Parameter (curve fit)
ℵ𝟑
Shape recovery
recovery strain (from exp.)
𝜖 (𝑟𝑒𝑐)
(prog)
programming strain level (from exp.)
ϵ
standard deviation of recovery temperature 𝑇𝑟 (from exp.)
𝜎𝑟
𝑚𝑖𝑛
𝑡𝑟𝑒𝑙 (min) Minimum relaxation time in a set of experiments (from exp.)
𝒕𝒎𝒂𝒙
𝒓𝒆𝒍 (𝐦𝐚𝐱) Maximum relaxation time in a set of experiments (from exp.)
𝒎𝒂𝒙
shape recovery ratio for 𝒕𝒎𝒂𝒙
ℵ𝒕𝒓𝒆𝒍
𝒓𝒆𝒍 case (from exp.)
𝑚𝑖𝑛
𝒕𝒎𝒊𝒏
shape recovery ratio for 𝑡𝑟𝑒𝑙
case (from exp.)
ℵ 𝒓𝒆𝒍
Heat rate (from exp.)
𝑞̇
(𝑚𝑖𝑛)
minimum heat rate in a set of experiments (from exp.)
𝑞̇
n2
relaxation time exponent (curve fit)
n3
heating rate exponent (curve fit)
Stress Relaxation
Rate parameter (curve fit)
𝑎1
Continuum Damage Model
𝐸𝑁𝑚𝑎𝑥 (MP experimentally measured elastic modulus after the 𝑁𝑚𝑎𝑥 ’th
11
a)
𝑬𝟎 (𝐌𝐏𝐚)
𝑵𝒎𝒂𝒙
𝒕𝒎𝒊𝒏
ℵ𝑵𝒓𝒆𝒍
𝒎𝒂𝒙
𝑡 𝑚𝑎𝑥
ℵ𝑁𝑟𝑒𝑙
𝑚𝑎𝑥
ΩNmax
Ω0
thermomechanical cycle (from exp.)
elastic moduli of the intact bulk SMP (from exp.)
maximum number of the thermomechanical cycles(from exp.)
shape recovery ratio for 𝒕𝒎𝒂𝒙
𝒓𝒆𝒍 case after the 𝑁𝑚𝑎𝑥 ’th thermomechanical
cycle (from exp.)
𝑚𝑖𝑛
shape recovery ratio for 𝑡𝑟𝑒𝑙
case after the 𝑁𝑚𝑎𝑥 ’th thermomechanical
cycle (from exp.)
stress recovery ratio in the 𝑁𝑚𝑎𝑥 TM cycle (from exp.)
stress recovery ratio in the first TM cycle (from exp.)
12