Thermodynamic constitutive modeling for analyzing the consistencies and differences of intelligent responses induced by thermo-elastic-viscoplastic couplings

doi:10.1007/s10483-026-3409-7

Abstract

Abstract:

Through appropriate thermo-mechanical stimulation, shape memory polymers (SMPs) can exhibit autonomous shape-morphing capabilities, which are referred to as entropic elastic (EE) or reversible plastic (RP) intelligent responses. Currently, few studies in the literature address unified constitutive modeling for these two types of intelligent responses, and thermodynamic consistency is lacking. Here, we develop a unified thermodynamically consistent constitutive model that captures hyperelastic-viscoelastic and elasto-viscoplastic couplings, thereby reproducing both the EE and RP intelligent responses. Through verification against experimental data and results from subsequent mechanistic and parametric studies, the constitutive model can not only integrate the theoretical representations of these two phenomena into a mathematical framework, but also reveal the consistencies and differences between the two types of intelligent responses.

Key words: intelligent response, hyperelasticity-viscoelasticity, elasto-viscoplasticity, thermodynamic constitutive modeling

2010 MSC Number:

O345

Hao DUAN, W. M. HUANG, Hao ZENG, Jianping GU, Huiyu SUN. Thermodynamic constitutive modeling for analyzing the consistencies and differences of intelligent responses induced by thermo-elastic-viscoplastic couplings. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1533-1548.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3409-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I7/1533

Figures/Tables 11

Table 1

Table 2

Model parameters from Refs. [28], [50], and [51]"

Parameter	Value	Parameter	Value
$T g$	42.1 °C	$T ref$	27 °C
$μ eq$	2.53 MPa	$τ s0$	0.5 s
Q	2 325.79 K	$K B$	$1.380 69 × 10 − 23$ J/K
$α g$	$6.0 × 10 − 5 (∘ C) − 1$	$τ R0$	0.5 s
$B R$	15 900 J/m³	$η ref$	6 J·m^-3·K
C	50 J·m^-3·K	$s s$	13.9 MPa
$C s$	1	$E g$	453 MPa
$Δ G$	$0.28 × 10 − 19$ J	A	3.7
$T r$	42.1 °C	h₀	499 MPa
s₀	16.7 MPa

Table 2

Table 3

Model parameters from Ref. [52]"

Parameter	Value	Parameter	Value
$T g$	59 °C	$E g$	1 450 MPa
$μ eq$	2.93 MPa	$τ s0$	$4.0 × 10 − 2$ s
$T 1, T 2, T 3$	299.15 K, 312.15 K, 353.15 K	$T ref$	35 °C
$Q / s y$	101 K/MPa	$K B$	$1.380 69 × 10 − 23$ J/K
$α r$	$1.45 × 10 − 4 (∘ C) − 1$	$Δ G$	$3.3 × 10 − 20$ J
$C s$	1	$α 1, α 2, α 3$	41, 55, 99
$α g$	$1.1 × 10 − 4 (∘ C) − 1$	$τ R0$	$3.5 × 10 − 1$ s
$B R$	25 000 J/m³	$η ref$	12 J·m^-3·K
C	65 J·m^-3·K	$E 1, E 2, E 3$	824.7 MPa, 911.95 MPa, 33.05 MPa

Table 3

Fig. 1

Fig. 2

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Fig. 7

Fig. 8

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Physical quantity	Significance
η; C	Entropy per unit reference volume; heat capacity
T; T_ref	Temperature; thermodynamic reference temperature
λ_L; λ_eff	Number of Kuhn segments; effective stretch
μ_neq; μ_ne	Neo-Hookean characteristic stiffness; rubbery shear modulus
q; k	Heat flux vector; heat conduction coefficient
α_r and α_g	Coefficients of thermal expansion (CTE) in the rubbery and glassy states
τ_R	Structure relaxation time
τ_s	Stress relaxation time
η_ref	Reference entropy at the thermodynamic reference temperature
ΔG; s_y	Activation parameter of stress relaxation; yield strength
Q; K_B	Activation parameter; Boltzmann’s constant
B_R; C_s	Activation parameter of structure relaxation; coupling factor