This paper presents a kinematic assumption free and thermodynamically consistent non-linear formulation incorporating finite strain and finite deformation for thermoviscoelastic plates/shells based on the conservation and balance laws of the classical continuum mechanics (CCM) in ${\mathbb{R}}^3$ (see Surana and Mathi, (2020) for linear theory). The conservation and balance laws in Lagrangian description with finite strain measure and the conjugate stress measure in ${\mathbb{R}}^3$ are considered. The conjugate pairs in the second law of thermodynamics (SLT), the consideration of additional physics and the principle of equipresence are utilized to determine the constitutive variables and their argument tensors. The constitutive theory for the contravariant second Piola–Kirchhoff stress tensor is derived using Green’s strain tensor and its convected time derivatives of up to order $n$ as its argument tensors using representation theorem with complete basis (i.e. using integrity). The convected time derivatives of the Green’s strain tensor up to order $n$ provide ordered rate constitutive theory for the dissipation mechanism that is naturally non-linear due to Green’s strain measure. The constitutive theory for heat vector derived using representation theorem and integrity is also a non-linear constitutive theory. Simplified linear forms of these constitutive theories are also presented. The solution methods for the mathematical model for the BVPs as well as the IVPs using $p$-version hierarchical higher degree and higher order finite element method are presented. Due to dissipation, the energy equation is integral part of the mathematical model that accounts for rate of mechanical work resulting in rate of entropy production, hence heat.

The plate/shell geometry (flat or curved) is described by its middle surface containing the nodal vectors (at each of the nine nodes), the ends of which define bottom and top surfaces of the plate/shell (Surana and Mathi, 2020). The geometry is mapped into $\xi \eta \zeta$ natural coordinate space in a two unit cube. The $p$-version hierarchical local approximations in $\xi \eta \zeta$ are constructed to describe the deformation of the plate/shell middle surface as well as its faces that is controlled by $p$-levels in $\xi ,\eta$ and $\zeta$ directions (element local approximation). The formulation presented here is accurate for very thin as well as very thick plate/shells and for small as well as finite deformation and finite strains. Non-linear constitutive theories based on integrity allow more complex constitutive behavior in term of strains as well as strain rates, hence dissipation mechanism. Dissipation mechanism is described by an ordered rate theory based on SLT and additional physics. The formulation presented here always ensures thermodynamic equilibrium during the evolution as it is derived using the conservation and balance laws of CCM in ${\mathbb{R}}^3$. The formulation always preserves three dimensional nature of the deformation physics regardless of the plate/shell thickness and is free of locking problems and shear correction factors that plague most of the currently used plate/shell formulations.

本文基于经典连续力学（CCM）的守恒和平衡定律，提出了热力学粘弹性板/壳的运动学假设自由且热力学一致的非线性公式，其中包含有限应变和有限变形。 ${\mathbb{[R}}^3$（有关线性理论，请参见Surana和Mathi，（2020年））。拉格朗日描述中的守恒和平衡规律的有限应变度量和共轭应力度量。${\mathbb{[R}}^3$被考虑。利用热力学第二定律（SLT）中的共轭对，对附加物理学的考虑和等价原理来确定本构变量及其自变量张量。使用格林的应变张量及其对流时间导数高达2阶，推导出协变的第二Piola–Kirchhoff应力张量的本构理论$ñ$作为其参数张量，使用具有完全基础的表示定理（即使用完整性）。格林应变张量的对流时间导数直至阶$ñ$为耗散机制提供了有序的速率本构理论，由于格林的应变测度，其自然是非线性的。利用表示定理和完整性推导的热矢量本构理论也是非线性本构理论。还提出了这些本构理论的简化线性形式。BVP和IVP的数学模型的求解方法$p$提出了高阶高阶有限元方法。由于耗散，能量方程式是数学模型的组成部分，该数学模型考虑了导致熵产生率和热量产生率的机械功率。

板/壳的几何形状（平坦或弯曲）由包含节点矢量的中间表面（在九个节点中的每个节点）描述，节点矢量的端部定义了板/壳的底部和顶部表面（Surana和Mathi，2020年） 。几何映射到$\xi \eta \zeta$两个单位立方体中的自然坐标空间。的$p$版本的层次局部近似 $\xi \eta \zeta$ 用于描述板/壳中间表面的变形及其受控制的面 $p$级别 $\xi ，\eta$ 和 $\zeta$方向（元素局部近似）。这里介绍的公式对于非常薄和非常厚的板/壳以及小的变形和有限的变形以及有限的应变都是准确的。基于完整性的非线性本构理论允许在应变以及应变率方面实现更复杂的本构行为，从而实现耗散机制。耗散机制由基于SLT和其他物理学的有序速率理论来描述。这里介绍的公式始终确保在演化过程中的热力学平衡，因为它是使用CCM的守恒和平衡定律推导出来的。${\mathbb{[R}}^3$。不管板/壳的厚度如何，该配方始终保持变形物理的三维性质，并且没有困扰大多数当前使用的板/壳配方的锁定问题和剪切校正因子。