Relaxation Modulus Determination

For numerical analysis in ABAQUS, the power-law representation of the relaxa­tion modulus was fitted and replaced by a four-element Prony series according to Eq 9, as shown in Fig. 5(b). The fit with Prony series for a reference

temperature of 5 °C is shown in Fig. 5(b). The fit is not exactly identical to the power-law representation because, for simplicity, only four elements were used. However, for this investigation, four Maxwell elements are considered to be suf­ficient. The relaxation times of the four Maxwell elements and the Prony series parameters are presented in Table 3


m = біЄм


where E(t) is the relaxation modulus, Ei are Prony series spring-constant pa­rameters for the relaxation modulus master curve (spring constants or moduli), and tri are the relaxation times for each Maxwell element.

For 3D multi-axial stress state, it is convenient to describe the stress state with deviatoric and dilatational components. A detailed description is given in Ref 19, a generalized solid Maxwell model is used in ABAQUS to characterize these two stress-state components. The bulk relaxation modulus follows from Eq 10 and the shear relaxation modulus from Eq 11

K(t)=K«M Ki(1 – e*)j (10)

G(t)=G<^1 -£ Gi(1 – гЬ)j (11)

where G is shear modulus, K is bulk modulus, t is actual time, tri are relaxation times, Go and K0 are instantaneous shear and bulk elastic moduli, respectively, and Gi and Ki are Prony series parameters.

Assuming constant Poisson’s ratio in the analysis simplifies the problem. This assumption was made in spite of the fact that Poisson’s ratio in viscoelastic materials is generally time dependent. The time-dependent Poisson’s ratio of viscoelastic materials can increase or decrease depending on the bulk and shear relaxation with time [17]. However, because the modeling of blister in this study was assumed to remain in the small strain regime, this simplification was con­sidered acceptable. A time-independent Poisson’s ratio of 0.35 was assumed for MA, and introduced in Eqs 12 and 13 to relate time-dependent relaxation modu­lus to the time-dependent bulk and shear relaxation modulus

TABLE 3—Prony series parameters at reference temperature 5 °C.

Prony series relaxation time, s

Prony series spring constants, MPa

tr1 tr2 tr3




E3 E4

747.384 186.358 • 1003 9.183 • 1003




18.84 32.69


3(1 – 2v)

G(t) = 2(T+v) (13)

As shown in Fig. 5(b), the power function of the relaxation modules was used for determining the parameters of the Prony series for the generalized Maxwell model by curve fitting. Because the master curve is constructed for the reference temperature of 5 °C, the Prony series parameters represent the time-dependent shear and volumetric behavior of the material at this particular temperature. The Prony series parameters for 25 °C are determined by shifting the creep com­pliance data to the reference temperature of 25 °C and by applying a similar pro­cedure as mentioned above. Table 4 lists the Prony series parameters at 5 °C.