Abstract

In this work, we demonstrate the existence of rotational waves in deforming thermoelastic non-classical solid continua in which the conservation and balance laws consider internal rotations due to the deformation gradient tensor (Jacobian of deformation) as well as their time varying rates. In this non-classical continuum theory, time dependent deformation of solid continua results in time dependent varying rotations, angular velocities and angular accelerations at material points. Resistance to these by deforming continua results in additional moments due to rotations, angular momenta due to rotation rates and rotational inertial effects due to angular accelerations at the material points. Currently this physics is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM) based on internal and/or Cosserat rotations. In this paper, we present derivation of conservation and balance laws in Lagrangian description: conservation of mass, balance of linear momentum, balance of angular momentum, balance of moment of moments, first and second laws of thermodynamics that include: (i) physics due to internal rotations resulting from the displacement gradient tensor (ii) new physics associated with rotation rates (angular velocities) and angular accelerations resulting from the varying, time dependent internal rotations at the material points. The balance laws derived here are compared with those that only consider internal rotations and their rates in the absence of rotational inertial effects (Surana et al. in J Therm Eng 1(6):446–459, 2015; Int J Eng Res Ind Appl 8(2):77–106, 2015) to demonstrate the influence of new physics. Constitutive variables and their argument tensors are established using the conjugate pairs in the entropy inequality, additional desired physics and the principle of equipresence. The constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity. It is shown that the mathematical model consisting of the conservation and balance laws and the constitutive theories has closure. Existence of rotational waves is demonstrated due to the presence of angular velocities and angular accelerations arising from the time varying rotations and their rates when deforming solid continua offer rotational inertial resistance. In this paper, we only consider isotropic and homogeneous solid continua with small strain, small deformation physics and reversible mechanical deformation. Model problem studies are presented in a follow up paper.

中文翻译：

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包含内部旋转的非经典热弹性固体连续体中旋转波的存在

摘要

在这项工作中，我们证明了在变形热弹性非经典固体连续体中存在旋转波，其中守恒和平衡定律考虑了由于变形梯度张量（变形雅可比）引起的内部旋转及其时变率。在这种非经典连续体理论中，固体连续体的时间相关变形会导致材料点处随时间变化的旋转，角速度和角加速度。通过使连续体变形来抵抗这些阻力会导致由于旋转而产生的额外力矩，由于旋转速度而产生的角力矩以及由于材料点处的角加速度而产生的旋转惯性效应。当前，基于内部和/或Cosserat旋转的经典连续体力学（CCM）和非经典连续体力学（NCCM）均未考虑该物理学。在本文中，我们在拉格朗日描述中给出了守恒和平衡定律的推导：质量守恒，线性动量平衡，角动量平衡，矩矩平衡，热力学第一定律和第二定律，包括：（i）物理力学位移梯度张量产生的内部旋转（ii）与物质点处随时间变化的内部旋转而产生的旋转速率（角速度）和角加速度相关的新物理学。将此处得出的平衡定律与仅考虑内部旋转及其速率而没有旋转惯性效应的平衡定律进行比较（Surana等人，在J Therm Eng 1（6）：446–459，2015; Int J Eng Res Ind Appl 8（2）：77–106，2015）以展示新物理学的影响。本构变量及其自变量张量是使用熵不等式中的共轭对，附加的期望物理原理和等价原理建立的。本构理论是使用亥姆霍兹自由能密度以及表示定理和完整性推导出来的。结果表明，由守恒律和平衡律以及本构理论组成的数学模型是闭合的。由于存在时变旋转引起的角速度和角加速度的存在，证明了旋转波的存在，并且当固体连续体变形时，它们的速度会提供旋转惯性阻力。在本文中，我们仅考虑具有小应变，小变形物理学和可逆机械变形的各向同性和均质固体连续体。模型问题研究在后续论文中介绍。