A consistent algorithm for finite-strain visco-hyperelasticity and visco-plasticity of amorphous polymers

Abstract

Shared rheology elements of soft tissues and additive-manufactured amorphous thermoplastics benefit from a common visco-hyperelastic and visco-plastic constitutive framework. For finite-strain and general visco/elastic/plastic constitutive laws, allowing full anisotropy, we use the Kröner–Lee decomposition of the deformation gradient combined with Mandel stress-based yield function. Relatively weak conditions are necessary for specific laws to be incorporated in this framework. For example, stress-like variables present in visco-elasticity are removed from the list of constitutive unknowns. Constitutive iteration is performed for the elastic Cauchy–Green tensor Ce and the plastic multiplier increment Δλ. The source is here the right Cauchy–Green tensor provided by any discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential. The exact Jacobian of the second Piola–Kirchhoff stress is determined with respect to this source, consistent with the integrator. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function with each of the viscous terms added. The constitutive system is solved by a damped Newton–Raphson algorithm with the corresponding stress error being extensively assessed with stress error maps. A cellular beam structure is analyzed and results compared with experiments from our group. Some of the symbolic calculations and closed-form solutions are beyond what is manually achievable and therefore the symbolic sources for this work are made available in a source code repository.