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 ...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 mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.展开更多
This paper presents two specific thermodynamically consistent nonclassical continuum theories for solid and fluent continua.The first non-classical continuum theory for solid continua incorporates Jacobian of deformat...This paper presents two specific thermodynamically consistent nonclassical continuum theories for solid and fluent continua.The first non-classical continuum theory for solid continua incorporates Jacobian of deformation in its entirety in the conservation and the balance laws and the derivation of the constitutive theories.The second non-classical continuum theory for solid continua considers Jacobian of deformation in its entirety as well as the Cosserat rotations in the conservation and balance laws as well as the constitutive theories.The first non-classical continuum theory for fluent continua presented here considers velocity gradient tensor in its entirety.The second non-classical continuum theory for fluent continua considers velocity gradient tensor in its entirety as well as Cosserat rotation rates in the derivation of the conservation and balance laws and the constitutive theories.Since the non-classical continuum theories for solid and fluent continua considered here incorporate additional physics of deformation due to rotations and rotation rates compared to classical continuum mechanics,the conservation and balance laws of classical continuum mechanics are shown to require modification as well as a new balance law balance of moment of moments is required to accommodate the new physics due to rotations and rotation rates.Eringen’s micropolar,micromorphic and microstretch theories,couple stress theories and nonlocal theories are also discussed within the context of the non-classical theories presented here for solid and fluent continua.Some applications of these theories are also discussed.展开更多
文摘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 mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.
文摘This paper presents two specific thermodynamically consistent nonclassical continuum theories for solid and fluent continua.The first non-classical continuum theory for solid continua incorporates Jacobian of deformation in its entirety in the conservation and the balance laws and the derivation of the constitutive theories.The second non-classical continuum theory for solid continua considers Jacobian of deformation in its entirety as well as the Cosserat rotations in the conservation and balance laws as well as the constitutive theories.The first non-classical continuum theory for fluent continua presented here considers velocity gradient tensor in its entirety.The second non-classical continuum theory for fluent continua considers velocity gradient tensor in its entirety as well as Cosserat rotation rates in the derivation of the conservation and balance laws and the constitutive theories.Since the non-classical continuum theories for solid and fluent continua considered here incorporate additional physics of deformation due to rotations and rotation rates compared to classical continuum mechanics,the conservation and balance laws of classical continuum mechanics are shown to require modification as well as a new balance law balance of moment of moments is required to accommodate the new physics due to rotations and rotation rates.Eringen’s micropolar,micromorphic and microstretch theories,couple stress theories and nonlocal theories are also discussed within the context of the non-classical theories presented here for solid and fluent continua.Some applications of these theories are also discussed.
