A Thermodynamically Consistent Formulation for Dynamic Response of Thermoviscoelastic Plate/Shell Based on Classical Continuum Mechanics (CCM)

Abstract

This paper presents a thermodynamically consistent and kinematic assumption free formulation for dynamics of thermoviscoelastic plates/shells based on the conservation and balance laws of classical continuum mechanics (CCM) in which dissipation mechanism has been incorporated through ordered rate constitutive theory for deviatoric stress tensor. In this paper, we consider small deformation, small strain. The conservation and balance laws of CCM in [Formula: see text] in Lagrangian description using Cauchy stress tensor ([Formula: see text]) and linearized Green strain tensor ([Formula: see text]) constitute the mathematical model. The constitutive theory for the deviatoric Cauchy stress ([Formula: see text]) is derived using conjugate pairs in entropy inequality in conjunction with representation theorem. The argument tensors of [Formula: see text] are [Formula: see text] and rates of [Formula: see text] up to order [Formula: see text]. This yields a constitutive theory with dissipation mechanism based on rates of strain up to order [Formula: see text]. Constitutive theory for heat vector is also derived using the conjugate pairs in the entropy inequality and representation theorem. Finite element method is used to obtain solutions of the initial value problems descried by the balance of linear momenta (BLM), energy equation and the constitutive theories. The shell element geometry is described by the middle surface and the nodal vectors at the middle surface defining bottom and top surfaces of the element. The local approximation for the displacement field is [Formula: see text] - version hierarchical in the plane of the element as well as in the transverse direction. A space-time decoupled finite element formulation using Galerkin Method with Weak Form (GM/WF) in space is constructed for BLM as well as energy equation, both resulting in ordinary differential equations (ODEs) in time. The ordinary differential equations (ODEs) in time resulting from the finite element formulation of BLM are used to study: (i) natural undamped modes of vibration (ii) the transient dynamic response using the ODEs in time recast in modal basis: (a) using Rayleigh damping (b) using the ordered rate damping proposed in this paper. Time response is calculated using modal damping based on Rayleigh damping as well as using the proposed ordered rate damping mechanism. Model problem studies are presented to demonstrate: (1) accuracy of the natural frequencies obtained from the present formulation for thin and thick plates/shells (in which shear deformation is significant) and the results are compared with the currently used plate formulations (2) accuracy of damped transient response using proposed damping mechanism is compared with time response using Rayleigh damping (3) it is shown that Rayleigh damping has no physical basis and leads to spurious stationary states. The proposed damping yields accurate stationary states that are in exact agreement with the solution of corresponding BVP. A single formulation presented in this paper remains valid and accurate for very thin as well as very thick plates/shells and correctly simulates 3D state of deformation regardless of plate/shell thickness and is free of shear locking problems as well as need for shear corrections. When obtaining the time response, solution for an increment of time alternates between the solution of BLM followed by the solution of the energy equation. Details are presented in the paper.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

“…3) Definition of force F using (18) and its use in (23) both are erroneous. We remark that in BLM (force balance) gradients of stresses appear, not forces defined using stresses based on some area (or unit area).…”

“…These cannot be transformed using a modal basis determined from the linear form of ODEs (1) in which [ ] K is constant. Authors in ref [14] presented details in the paper to show that forcing this transformation on (1) followed by the use of Rayleigh damping (see reference [18] [19] for issues with Rayleigh damping) leads to a system of decoupled ODEs that neither describes linear dynamics nor nonlinear dynamics. Hence, our view is that all nonlinear dynamic studies based on this approach are not supported by the mathematical models based on CBL of CCM followed by rather straightforward space-time coupled or space-time decoupled methods of obtaining their solutions, like we have presented in this paper and ref [13] [14].…”

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

“…Some of these are summarized in the following. We have presented details in this paper to show that forcing this transformation on (1) followed by use of Rayleigh damping (see reference [33] [34]) for issues with Rayleigh damping) leads to a system of decoupled ODEs that neither describe linear dynamics nor nonlinear dynamics. Hence, our view is that all nonlinear dynamic studies based on this approach are not supported by the mathematical models based on CBL of CCM followed by rather straight forward space-time coupled or space-time decoupled methods of obtaining their solutions, like we have presented in this paper.…”