A hygro-thermo-mechanical constitutive model for shape memory polymers filled with nano-carbon powder

Figures & data

Figure 1. A rheological analog illustration for the constitutive relations

Figure 2. Parameter fitting for the effect of filler on the glass transition temperature

Figure 3. Parameter fitting for the moisture diffusion of CB0. The test data are plotted as geometrical shapes, while the model results are shown as solid lines

Figure 4. Parameter fitting for the effect of moisture on the glass transition temperature of CB0

Figure 5. Parameter fitting and phase decomposition for the moisture diffusion of (a) CB4, (b) CB7, (c) CB10, (d) CB13, and (e) CB15

Figure 6. Parameter fitting for the effect of moisture on the glass transition temperature of (a) CB4, (b) CB7, (c) CB10, (d) CB13, and (e) CB15

Figure 7. Parameter fitting for the young’s modulus of the carbon powder

Figure 8. Comparison of the model results and test data for the stress-strain response of the first 3 steps for (a) CB0 and (b) CB4

Table 1. Parameters of the model

Parameter	Description	Value
N	Kuhn segments	1.7
Kb	Bulk modulus	1000 MPa
θref	Temperature of the equilibrium state	33/37°C (CB0/CB4)
sref	Specific entropy of the equilibrium state	2.2 J/kg·K
T1ref	Axial stress of the equilibrium state	−5 MPa
cp0	Specific heat	80 J/kg·K
k0	Coefficient of thermal expansion	1.4 × 10^−4 /^oC
d	Parameter of internal variable perturbations	1.5 × 10^−16 J/kg
e	Coupling parameter for internal variables and temperature	4.5 × 10^−9 J/kg·K
w1t	Coupling parameter for internal variables and stress	1 × 10^−16 m^3/kg
v1t	Coupling parameter for internal variables and stress	9 × 10^−15 m^3/kg
B0	Activation parameter	20000 J/kg
τ0	Reference structural relaxation time	3 × 10^−2 s
p	Equivalent coefficient of the moisture	200
ρ	Density	1050 kg/m^3
Q/sy	Activation parameter for viscous flow	1550 K/MPa
ηref	Reference shear viscosity	4 MPa/s
Δ	Half the thickness	1 mm
w∞	Moisture weight fraction at the equilibrium state	3.8/3.2/3.2/3.15/3/3.05% (CB0/4/7/10/13/15)
β	Probability per unit time for bound molecules to become free	2.63 day^−1
γ	Probability per unit time for free molecules to become bound	9.55 day^−1
Dγ	Diffusion coefficient	6.3/8/12/7/4/3.5 × 10^−3cm^2/day (CB0/4/7/10/13/15)
θg0 ξ	Glass transition temperature Coefficient of the filler for the free volume	52.2/50.4/48.5/47.3/46.2/45.8°C (CB0/4/7/10/ 13/15) 6.4 × 10^−3
ξN	Coefficient of the bound water for the free volume	1.6/1.7/1.8/1.8/2.1/2.1 × 10^−3(CB0/4/7/10/13/15)
ξn	Coefficient of the free water for the free volume	0
Em	Young’s modulus of the matrix at room temperature	1648 MPa
Ef	Young’s modulus of the carbon powder	17000 MPa
Er	Young’s modulus of the matrix at the rubbery state	3.0 MPa
θrefs	Reference temperature of material stiffness	28°C
m1	Parameter denoting the moisture effect on modulus	5.0
m2	Parameter denoting the temperature effect on modulus	0.25
urm	Shear modulus of the matrix	0.25 MPa
urf	Shear modulus of the carbon powder	10 MPa
μg	Poisson’s ratio of the glassy phase	0.35
μr	Poisson’s ratio of the rubbery phase	0.499
Z	Width of the transition zone	3
θm	Reference temperature of phase transition	37.5°C

Figure 9. Comparison of the model results with the test data for the free recovery of (a) CB0 and (b) CB4 during heating

Figure 10. Comparison of the model results with the test data for the free recovery of (a) CB0 and (b) CB4 during moisture absorption