Viscoelastic model with complex rheological behavior (VisCoR): incremental formulation

Abstract

A thermo-rheologically complex linear viscoelastic material model, accounting for temperature and degree of cure (DoC), is developed starting with series expansion of the Helmholtz free energy and systematically implementing simplifying assumptions regarding the material behavior. In addition to the temperature and DoC dependent shift factor present in rheologically simple models, the derived novel model contains three cure and temperature dependent functions. The first function is identified as the rubbery modulus. The second is a weight factor to the transient integral term in the model and reflects the current temperature and cure state, whereas the third function is under the sign of the convolution integral, thus affecting the "memory" of the material. An incremental form of this model is presented which, due to improved approximation inside the time increment, has better numerical convergence than most of the similar forms. Parametric analysis is performed simulating stress development in a polymer, geometrically constrained in the mold during curing and cool-down. The importance of using proper viscoelastic model is shown, and the role of parameters in the model is revealed and discussed.

Stress relaxation and strain recovery phenomena during curing and thermomechanical loading: Thermorheologically simple viscoelastic analysis

“…The decomposition is coherent with assumptions used in developing the viscoelastic model. Details of the derivation of the model are given in Saseendran et al 24 Here, we will briefly discuss the main steps and assumptions when deriving the model. The derivation follows the routine used by Schapery 23 for nonlinear VE materials: the first terms of the Helmholtz free energy function expansion is written (assumption 1) in terms of applied strain " and viscoelastic internal state variables (…”

“…In equation ( 9) and ( 10), H is the Heaviside step function. The linear approximation used in each increment leads to calculation scheme, which converges much faster, 24 than common schemes where values of parameters in the starting point of the increment are used. The time increment does not have to be constant.…”

“…In the current paper, we present and use a thermodynamically consistent, thermorheologically complex VE model. The complete derivation of it, which is based on expansion of Helmholtz energy, similar as in Schapery, 23 is given in Saseendran et al 24 The obtained material model has the rubbery modulus and the glassy modulus dependent on DoC and T. Three DoC-and T-dependent functions in the constitutive law ensure that. One is the rubbery modulus, another one is a weight factor in front of the convolution integral and the last one is a multiplier to the strain in the derivative under the sign of the integral, defining the effect of strains during the whole loading history.…”

On Temperature-Related Shift Factors and Master Curves in Viscoelastic Constitutive Models for Thermoset Polymers

“…For example, in the branch of thermodynamically consistent linear viscoelasticity models derived using Schapery's approach [1,2], temperature ( T )-dependent parameters appear in front of the heredity integral and under its sign. A temperature dependent function a T ( ) is also present in exponents of the Prony series as a multiplier to relaxation times [2,3,4].…”

“…For example, in the branch of thermodynamically consistent linear viscoelasticity models derived using Schapery's approach [1,2], temperature ( T )-dependent parameters appear in front of the heredity integral and under its sign. A temperature dependent function a T ( ) is also present in exponents of the Prony series as a multiplier to relaxation times [2,3,4]. The latter, called the "shift factor," may also depend on the degree of cure a DoC ( ) (in thermoset composites) [3,5,6] and on the load or strain level in the case of nonlinear viscoelasticity, a T DoC ,� ,� ε ( ) [7,8].…”