A thermodynamically consistent formulation for bending of thermoviscoelastic beams for small deformation, small strain based on classical continuum mechanics

Abstract

The work presented in this paper extends the kinematic assumption free and thermodynamically consistent formulation for bending of thermoelastic beams presented by Surana et al. for bending of thermoviscoelastic beams with dissipation mechanism without memory. We consider small strain, small deformation physics in Lagrangian description. Conservation and balance laws of classical continuum mechanics (CCM) constitute the mathematical model for the physics considered in this paper. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and the principle of equipresence. Cauchy stress tensor is decomposed into equilibrium stress tensor (${}_e\mathit{\sigma }$) and deviatoric stress tensor (${}_d\mathit{\sigma }$). Constitutive theory for ${}_e\mathit{\sigma }$ is derived using Helmholtz free energy density in conjunction with incompressibility condition. The constitutive theory for ${}_d\mathit{\sigma }$ is derived in by first establishing its argument tensors using conjugate pairs in the entropy inequality and other desired physics and then using the representation theorem with complete basis (integrity). The constitutive theory for ${}_d\mathit{\sigma }$ is a nonlinear constitutive theory in terms of strain tensor containing up to fifth degree terms in the components of the strain tensor and an ordered rate theory in the strain rate tensors up to order n. Simplified linear ordered rate theory for ${}_d\mathit{\sigma }$ is also presented. The formulation presented here for thermoviscoelastic beams is based on the conservation and balance laws of classical continuum mechanics and construction of beam finite element formulation using hpk framework with variationally consistent integral form. In this approach the mathematical model consist of true conservation and balance laws and the choice of local approximations for the beam finite elements facilitates incorporation of required kinematic description based on the application. This approach is free of the a priori assumptions of kinematic relations, computations are unconditionally stable and the local approximations can be of higher degree (p) and of higher order (k). This approach addresses slender as well as deep beam physics and can be used to measure error in the computed solution through residual functional. The rate constitutive theory is based on the second law thermodynamics consistent with physics of deformation. Mathematical details of new formulation and the model problem studies and comparisons with currently used beam models are presented for slender as well as deep beams. This approach permits consideration of reversible (thermoelastic) as well as irreversible (thermoviscoelastic) processes. It is shown that Rayleigh damping used currently to derive modal damping has no physical basis and can lead to spurious solution. The dissipation mechanism presented here has physical basis and yields non-spurious and valid solutions Model problem studies are presented for undamped natural modes of vibration, damped and undamped transient dynamic response. The results obtained from the formulation presented here are compared with published works.

Keywords:

• Thermoviscoelastic
• beams
• p-version
• hierarchical
• strain rates
• ordered rate theory
• classical continuum mechanics
• conservation and balance laws

Acknowledgements

The computational facilities provided by the Computational Mechanics Laboratory of the mechanical engineering department are acknowledged.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

First author is grateful for his endowed professorship and the department of mechanical engineering of the University of Kansas for providing financial support to the second and third authors.