# Viscoelastic Model

Elastic materials having the capacity to dissipate the mechanical energy due to viscous effects are characterized as viscoelastic materials. For multi-axial stress state, the constitutive relation can be written as:

where e and f are the deviatoric and volumetric strains; G(t - t ) and K(t - t ) are shear and bulk relaxation functions. The relaxation functions can then be represented by the mechanical model, (shown in this figure ) which is usually referred to as a Generalized Maxwell Model having the expressions as the following:

where G 0 and K 0 are the initial shear and bulk moduli (t = 0) given by: G 0   = E/2(1+v) and K 0   = E/3(1-2v).

g i, k i, t i G, and t i K are the i-th shear and bulk moduli and corresponding times.

The effect of temperature on the material behavior is introduced through the time-temperature correspondence principle. The mathematical form of the principle is:

where g t is the reduced time and g is the shift function. The WLF (Williams-Landel-Ferry) equation is used to approximate the function:

where TO is the reference temperature which is usually picked as the Glass transition temperature; C1 and C2 are material dependent constants.

The required parameters include the following:

 Parameter Symbol Description Linear Elastic Parameters EX Elastic modulus NUxy Poisson's ratio GXY (optional) Shear modulus Relaxation Function Parameters G1, G2, G3,..., G8 represent g1, g2, ...,g8 in the Generalized Maxwell Model equations TAUG1, TAUG2, ....., TAUG8 represent t1g, t2g,..., t8g in the Generalized Maxwell Model equations K1, K2, ..., K8 represent k1, k2, ...,k8 in the Generalized Maxwell Model equations TAUK1, TAUK2, ..., TAUK8 represent t1k, t2k,..., t8k in the Generalized Maxwell Model equations WLF Equation Parameters REFTEMP represents T0 in the WLF equation VC1 represents C1 in the WLF equation VC2 represents C2 in the WLF equation

When defining a shear or bulk relaxation curve under the Tables & Curves tab, the first point of the curve is the G 1   or K 1   moduli at time t 1. At time t = 0, the program automatically computes G 0   or K 0 from the Elastic modulus and Poisson's ratio.

The viscoelastic material model can be used with the draft and high quality solid and thick shell elements.

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