Nonlinear Viscoelastic Constitutive Model for Epoxy Polymers

Based on comprehensive experimental data (Hu et al. 2003, Shen et al. 2004) a new nonlinear viscoelastic model has been recently developed by the authors (Xia et al. 2003b, Xia et al. 2005). Here only a brief description will be given.

For the uniaxial stress state, the model can be represented by a finite number of nonlinear ‘Kelvin – Voigt type’ elements and a linear spring element, connected in series (Fig. 2). The constitutive equations, generalized to the multiaxial stress state, are sumarized below:

In the above, {fe}, {ee}, {fec}, {a} are the total strain-rate, elastic strain-rate, creep strain-rate, and stress-rate vectors (each contains six components, respectively). E is an elastic modulus which is assumed to be constant and [a] is a matrix related to the value of Poisson’s ratio, defined by

In Eqn. (14) Ti =njEt (i = 1, 2, •••, n) denotes the retardation time, Etis the spring stiffness and П is the dashpot viscosity for the i-th ‘Kelvin-Voigt type’ element, respectively. Based on Eqn. (14), the retardation time Ti has a damped exponential character as in an exponential-type function. Its value determines the time duration after which contribution from the individual ‘Kelvin-Voigt type’ element becomes negligible. Therefore, the number of the ‘Kelvin-Voigt type’ elements adopted in the constitutive equation depends on the required time range. For simplicity, we introduce a time scale factor a, and assume that

Ti =(a)i-1Ti (17)

The description of the nonlinear behaviour in the current model is achieved by letting Et’s be functions of the current equivalent stress, aeq. Furthermore, a single function form for all Et ’s is assumed, i. e.

where Ix = al + a2 +a3 is the first invariant of the stress tensor, J2 = sijsij /2 is the second invariant of the deviatoric stress and R is the ratio of compressive to tensile ‘yield stress’. Note that when R = 1, then Eqn. (19) reduces to the von Mises equivalent stress, aeq = -^3J2 .

The five constants (E, u, a, , R) and the functional form of E1(aeq) can be determined from

uniaxial creep curves tested at different stress levels by a simple procedure described in Xia et al. (2003b).

A distinct feature of this constitutive model is its capacity to distinguish between loading and unloading cases by introducing a stress memory surface and a corresponding switch rule. The stress memory surface is defined as:

3

fa (a ) – R2 = — s s – R2 = 0

J m v ij ‘ mem ^ ij ij mem

where stj = atj —ak±dij are the deviatoric stress components. The radius of the memory surface,

Rmem, is determined by the maximum von Mises stress level experienced by the material during its previous loading history, i. e. Rmem = ^(3stjstj /2)max. Therefore, for a monotonic loading from a stress free state, the memory surface will expand isotropically with the increasing stress level. If aj is

df

the current stress point, dais the stress increment at time t, and (——————- ) = , represents the

J dajj s‘j =s‘j

direction of the normal to the memory surface at the current stress point, then the criterion to distinguish the loading/unloading cases is defined as follows: [7]