In non-classical thermoelastic solids incorporating internal rotation and conjugate Cauchy moment tensor the mechanical deformation is reversible. This suggests that within the realm of linear mathematical models that...In non-classical thermoelastic solids incorporating internal rotation and conjugate Cauchy moment tensor the mechanical deformation is reversible. This suggests that within the realm of linear mathematical models that only consider small strains and small deformation the mechanical deformation is reversible. Hence, it is possible to recast the conservation and balance laws along with constitutive theories in a form that adjoint A* of the differential operator A in mathematical model is same as the differential operator A. This holds regardless of whether we consider an initial value problem (IVP) (when the integrals over open boundary are neglected) or boundary value problem (BVP). Thus, in such cases Galerkin method with weak form (GM/WF) for BVPs and space-time Galerkin method with weak form (STGM/WF) for IVPs are highly meritorious due to the fact that: 1) the integral form for BVPs is variationally consistent (VC) and 2) the space-time integral forms for IVP are space time variationally consistent (STVC). The consequence of VC and STVC integral forms is that the resulting coefficient matrices are symmetric and positive definite ensuring unconditionally stable computational processes for both BVPs and IVPs. Other benefits of GM/WF and space-time GM/WF are simplicity of specifying boundary conditions and initial conditions, especially traction boundary conditions and initial conditions on curved boundaries due to self-equilibrating nature of the sum of secondary variables that only exist in GM/WF due to concomitant. In fact, zero traction conditions are automatically satisfied in GM/WF, hence need not be specified at all. While VC and STVC feature also exists in least squares process (LSP) and space-time least squares finite element processes (STLSP) for BVPs and IVPs, the ease of specifying traction boundary conditions feature in GM/WF and STGM/WF is highly meritorious compared to LSP and STLSP in which zero traction conditions need to be explicitly specified. A disadvantage of GM/WF and STGM/ WF is that the mathematical models (momentum equations) needed in the desired form contain higher order derivatives of displacements (upto fourth order), hence necessitate use of higher order spaces in their solution. As well known, this problem can be easily overcome in LSP and STLSP by introduction of auxiliary equations and auxiliary variables, thus keeping the highest orders of the derivatives of the dependent variables to one or any other desired order. A serious disadvantage of this approach in LSP is the significant increase in the number of dependent variables, hence poor computational efficiency. In this paper we consider non-classical continuum models for internally polar linear elastic solids in which internal rotations due to displacement gradient tensor (hence internal polar physics) are considered in the conservation and the balance laws and the constitutive theories. For simplicity, we only consider isothermal case;hence energy equation is not part of mathematical model. When using mathematical models derived in displacements in GM/WF and LSP in constructing integral forms, we note that in GM/WF the number of dependent variables is reduced drastically (only three in R3), whereas in case of first order systems used in LSP and STLSP we may have as many as 22 dependent variables for isothermal case. Thus, GM/WF results in dramatic improvement in computational efficiency as well as accuracy when minimally conforming spaces are used for approximations. In this paper we only consider mathematical model in R2 for BVPs (for simplicity). Mathematical models for IVP and BVP in R3 will be considered in subsequent paper. The integral form is derived in R2 using GM/WF. Numerical examples are presented using GM/WF and LSP to demonstrate advantages of finite element process derived using integral form based on GM/WF for non-classical linear theories for solids incorporating internal rotations due to displacement gradient tensor.展开更多
文摘In non-classical thermoelastic solids incorporating internal rotation and conjugate Cauchy moment tensor the mechanical deformation is reversible. This suggests that within the realm of linear mathematical models that only consider small strains and small deformation the mechanical deformation is reversible. Hence, it is possible to recast the conservation and balance laws along with constitutive theories in a form that adjoint A* of the differential operator A in mathematical model is same as the differential operator A. This holds regardless of whether we consider an initial value problem (IVP) (when the integrals over open boundary are neglected) or boundary value problem (BVP). Thus, in such cases Galerkin method with weak form (GM/WF) for BVPs and space-time Galerkin method with weak form (STGM/WF) for IVPs are highly meritorious due to the fact that: 1) the integral form for BVPs is variationally consistent (VC) and 2) the space-time integral forms for IVP are space time variationally consistent (STVC). The consequence of VC and STVC integral forms is that the resulting coefficient matrices are symmetric and positive definite ensuring unconditionally stable computational processes for both BVPs and IVPs. Other benefits of GM/WF and space-time GM/WF are simplicity of specifying boundary conditions and initial conditions, especially traction boundary conditions and initial conditions on curved boundaries due to self-equilibrating nature of the sum of secondary variables that only exist in GM/WF due to concomitant. In fact, zero traction conditions are automatically satisfied in GM/WF, hence need not be specified at all. While VC and STVC feature also exists in least squares process (LSP) and space-time least squares finite element processes (STLSP) for BVPs and IVPs, the ease of specifying traction boundary conditions feature in GM/WF and STGM/WF is highly meritorious compared to LSP and STLSP in which zero traction conditions need to be explicitly specified. A disadvantage of GM/WF and STGM/ WF is that the mathematical models (momentum equations) needed in the desired form contain higher order derivatives of displacements (upto fourth order), hence necessitate use of higher order spaces in their solution. As well known, this problem can be easily overcome in LSP and STLSP by introduction of auxiliary equations and auxiliary variables, thus keeping the highest orders of the derivatives of the dependent variables to one or any other desired order. A serious disadvantage of this approach in LSP is the significant increase in the number of dependent variables, hence poor computational efficiency. In this paper we consider non-classical continuum models for internally polar linear elastic solids in which internal rotations due to displacement gradient tensor (hence internal polar physics) are considered in the conservation and the balance laws and the constitutive theories. For simplicity, we only consider isothermal case;hence energy equation is not part of mathematical model. When using mathematical models derived in displacements in GM/WF and LSP in constructing integral forms, we note that in GM/WF the number of dependent variables is reduced drastically (only three in R3), whereas in case of first order systems used in LSP and STLSP we may have as many as 22 dependent variables for isothermal case. Thus, GM/WF results in dramatic improvement in computational efficiency as well as accuracy when minimally conforming spaces are used for approximations. In this paper we only consider mathematical model in R2 for BVPs (for simplicity). Mathematical models for IVP and BVP in R3 will be considered in subsequent paper. The integral form is derived in R2 using GM/WF. Numerical examples are presented using GM/WF and LSP to demonstrate advantages of finite element process derived using integral form based on GM/WF for non-classical linear theories for solids incorporating internal rotations due to displacement gradient tensor.
