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thermodynamic consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a thermodynamically Consistent New Thermoelastic Formulation

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American Journal of Computational Mathematics 2020年第2期10卷 167-220页

作者： Karan S. Surana Sri Sai Charan Mathi Department of Mechanical Engineering University of Kansas Lawrence Kansas

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathema

关键词： Plate and Shell Mathematical Models Energy Functional thermodynamic consistency Classical Continuum Mechanics Non-Classical Continuum Mechanics Internal Rotations Cosserat Rotations Principle of Virtual Work

thermodynamic consistency and Thermomechanical Dynamics (TMD) for Nonequilibrium Irreversible Mechanism of Heat Engines

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Journal of Applied Mathematics and Physics 2021年第6期9卷 1364-1390页

作者： Hiroshi Uechi Lisa Uechi Schun T. Uechi Osaka Gakuin University Suita Japan University of California Los Angeles USA KPMG Ignition Tokyo Data Tech. Tokyo Japan

The irreversible mechanism of heat engines is studied in terms of thermodynamic consistency and thermomechanical dynamics (TMD) which is proposed for a method to study nonequilibrium irreversible thermodynamic systems. As an example, a water drinking bird (DB) known as one of the heat engines is specifically examined. The DB system suffices a rigorous experimental device for the theory of nonequilibrium irreversible thermodynamics. The DB nonlinear equation of motion proves explicitly that nonlinear differential equations with time-dependent coefficients must be classified as independent equations different from those of constant coefficients. The solutions of nonlinear differential equations with time-dependent coefficients can express emergent phenomena: nonequilibrium irreversible states. The couplings among mechanics, thermodynamics and time-evolution to nonequilibrium irreversible state are defined when the internal energy, thermodynamic work, temperature and entropy are integrated as a spontaneous thermodynamic process in the DB system. The physical meanings of the time-dependent entropy, T(t)dS(t), , internal energy, dƐ(t), and thermodynamic work, dW(t), are defined by the progress of time-dependent Gibbs relation to thermodynamic equilibrium. The thermomechanical dynamics (TMD) approach constitutes a method for the nonequilibrium irreversible thermodynamics and transport processes.

关键词： thermodynamic consistency Thermomechanical Dynamics (TMD) Nonlinear Differential Equation with Time-Dependent Coefficients Nonequilibrium Irreversible States

Magnetized strange quark matter in a mass-density-dependent model

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Chinese Physics C 2015年第1期39卷 43-49页

作者：侯佳荀彭光雄夏铖君徐建峰 Theor. Physics Center for Science Facilities Institute of High Energy PhysicsChinese Academy of Sciences School of Physics University of Chinese Academy of Sciences

We investigate the properties of strange quark matter （SQM） in a strong magnetic field with quark confinement by the density dependence of quark masses considering the total baryon number conservation, charge neutrality and chemical equilibrium. It is found that an additional term should appear in the pressure expression to maintain thermodynamic consistency. At fixed density, the energy density of magnetized SQM varies with the magnetic field strength. By increasing the field strength an energy minimum exists located at about 6×10^19 Gauss when the density is fixed at two times the normal nuclear saturation density.

关键词： quark matter thermodynamic consistency mass-density dependence strange stars

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

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Applied Mathematics 2023年第12期14卷 773-838页

作者： Karan S. Surana Sri Sai Charan Mathi Department of Mechanical Engineering Univeristy of Kansas Lawrence KS USA

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor

关键词： thermodynamic consistency Dynamic Bifurcation Static Bifurcation Nonlinear Formulation Finite Strain Finite Deformation Thermoviscoelastic Classical Continuum Mechanics Conservation and Balance Laws Nonlinear Damping

Deformable Permanent Ferroelectric or Ferromagnetic Media

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Journal of Applied Mathematics and Physics 2023年第10期11卷 3061-3074页

作者： Michael Grinfeld Pavel Grinfeld The U.S. Army Research Laboratory WMRD Aberdeen Proving Ground USA Mathematics Department Drexel University Philadelphia USA

In the framework of continuum mechanics, one of possible consistent definitions of deformable permanent magnets is introduced and explored. Similar model can be used for ferroelectric substances. Based on the suggested definition, we establish the key kinematic relationship for the deformable permanent magnet and suggest the simplest master system, allowing to analyze behavior of such substances.

关键词： Ferroelectric and Ferromagnetic Media thermodynamic consistency The Aleph Tensor Boundary Value Problems of Electro- and Magnetostatics and Quasi-Statics

A thermodynamic Approach to Rate Equations in Continuum Physics

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Journal of Physical Science and Application 2017年第6期7卷 15-23页

作者： Angelo Morro DIBRIS University of Genoa Genova 16145 Italy

The paper addresses the formulation of rate equations, via objective time derivatives, within continuum physics. The concept of objectivity is reviewed and distinction is made with material frame-indifference whose meaning is restricted to the invariance of the balance equations relative to Galilean frames. Objective time derivatives are defined to leave the tensor character of the appropriate field invariant within the set of Euclidean frames. Rate equations are required to involve objective time derivatives and to be consistent with the second law of thermodynamics. Here the general structure of objective time derivatives is established and the known derivatives of the physical literature are shown to be particular cases. Next, to fix ideas, a rate equation is considered for the model of heat conduction via a generalization of the Maxwell-Cattaneo equation with higher-order gradients as in the Guyer-Krumhansl equation. The thermodynamic restrictions are investigated and the expected effects, of the selected derivative of the heat flux, on the stress tensor are derived.

关键词： Objective derivatives rate equations thermodynamic consistency