The Experts below are selected from a list of 66 Experts worldwide ranked by ideXlab platform
J. N. Reddy - One of the best experts on this subject based on the ideXlab platform.
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Ordered Rate Constitutive Theories for Non-classical Thermoviscoelastic Solids with Dissipation and Memory Incorporating Internal Rotations
Polytechnica, 2018Co-Authors: K. S. Surana, D. Mysore, J. N. ReddyAbstract:This paper presents constitutive theories for non-classical thermoviscoelastic solids with dissipation and memory using thermodynamic framework based on entirety of the displacement gradient Tensor. Thus, the conservation and the balance laws used in this work incorporate symmetric as well as antisymmetric parts of the displacement gradient Tensor. In this paper, we only consider small deformation small strain; hence, the constitutive theories are basis independent. The constitutive theories are derived in the presence as well as in the absence of balance of Moment of Moments balance law. It is shown that the energy storage, dissipation mechanism, and the fading memory in the non-classical thermoviscoelastic solids are due to strain rates, rotation rates, stress Tensor, Moment Tensor, and their rates. Constitutive theories are derived using the conditions resulting from the entropy inequality in conjunction with the representation theorem. The constitutive theories derived using integrity are followed by simplified constitutive theories. Material coefficients are derived and discussed for both cases. Constitutive models parallel to non-classical Maxwell, Oldroyd-B, and Giesekus models for thermoviscoelastic fluids are derived and shown to be a subset of a more generalized simplified constitutive theory presented in the paper. Retardation moduli are derived for stress Tensor as well as Moment Tensor and are compared with those in classical continuum theories for similar solids.
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Restrictions on the Material Coefficients in the Constitutive Theories for Non-Classical Viscous Fluent Continua
Applied Mathematics-a Journal of Chinese Universities Series B, 2018Co-Authors: Karan S. Surana, J. N. ReddyAbstract:This paper considers conservation and balance laws and the constitutive theories for non-classical viscous fluent continua without memory, in which internal rotation rates due to the velocity gradient Tensor are incorporated in the thermodynamic framework. The constitutive theories for the deviatoric part of the symmetric Cauchy stress Tensor and the Cauchy Moment Tensor are derived based on integrity. The constitutive theories for the Cauchy Moment Tensor are considered when the balance of Moments of Moments 1) is not a balance law and 2) is a balance law. The constitutive theory for heat vector based on integrity is also considered. Restrictions on the material coefficients in the constitutive theories for the stress Tensor, Moment Tensor, and heat vector are established using the conditions resulting from the entropy inequality, keeping in mind that the constitutive theories derived here based on integrity are in fact nonlinear constitutive theories. It is shown that in the case of the simplest linear constitutive theory for stress Tensor used predominantly for compressible viscous fluids, Stokes' hypothesis or Stokes' assumption has no thermodynamic basis, hence may be viewed incorrect. Thermodynamically consistent derivations of the restrictions on various material coefficients are presented for non-classical as well as classical theories that are applicable to nonlinear constitutive theories, which are inevitable if the constitutive theories are derived based on integrity.
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Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory
Applied Mathematics-a Journal of Chinese Universities Series B, 2016Co-Authors: Karan S. Surana, Stephen W. Long, J. N. ReddyAbstract:In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient Tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient Tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress Tensor and conjugate strain rate Tensors of up to orders n and Cauchy Moment Tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient Tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient Tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient Tensor is used as rotation rates to determine rate of rotation gradient Tensor. This is an approximation to true convected time derivatives of the rotation gradient Tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient Tensor; (2) first convected derivative of the symmetric part of the rotation gradient Tensor as used by Surana et al. is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy Moment Tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress Tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain Tensor, Moment Tensor and heat vector for thermofluids without memory in which convected time derivatives of strain Tensors up to order n are conjugate with the Cauchy stress Tensor and the convected time derivatives of the symmetric part of the rotation gradient Tensor up to orders 1n are conjugate with the Moment Tensor. Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress Tensor, Moment Tensor and heat vector. Stress Tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress Tensor and the Moment Tensor are further decomposed into equilibrium and deviatoric Tensors. It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress Tensor, Moment Tensor and heat vector. Density in the current configuration, convected time derivatives of the strain Tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient Tensor up to orders 1n, temperature gradient Tensor and temperature are considered as argument Tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence. The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates. The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al. for internal polar non-classical incompressible thermofluids.
Eliot Fried - One of the best experts on this subject based on the ideXlab platform.
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Statistical Foundations of Liquid-Crystal Theory II: Macroscopic Balance Laws
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Brian Seguin, Eliot FriedAbstract:Working on a state space determined by considering a discrete system of rigid rods, we use nonequilibrium statistical mechanics to derive macroscopic balance laws for liquid crystals. A probability function that satisfies the Liouville equation serves as the starting point for deriving each macroscopic balance. The terms appearing in the derived balances are interpreted as expected values and explicit formulas for these terms are obtained. Among the list of derived balances appear two, the Tensor Moment of inertia balance and the mesofluctuation balance, that are not standard in previously proposed macroscopic theories for liquid crystals but which have precedents in other theories for structured media.
Karan S. Surana - One of the best experts on this subject based on the ideXlab platform.
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Restrictions on the Material Coefficients in the Constitutive Theories for Non-Classical Viscous Fluent Continua
Applied Mathematics-a Journal of Chinese Universities Series B, 2018Co-Authors: Karan S. Surana, J. N. ReddyAbstract:This paper considers conservation and balance laws and the constitutive theories for non-classical viscous fluent continua without memory, in which internal rotation rates due to the velocity gradient Tensor are incorporated in the thermodynamic framework. The constitutive theories for the deviatoric part of the symmetric Cauchy stress Tensor and the Cauchy Moment Tensor are derived based on integrity. The constitutive theories for the Cauchy Moment Tensor are considered when the balance of Moments of Moments 1) is not a balance law and 2) is a balance law. The constitutive theory for heat vector based on integrity is also considered. Restrictions on the material coefficients in the constitutive theories for the stress Tensor, Moment Tensor, and heat vector are established using the conditions resulting from the entropy inequality, keeping in mind that the constitutive theories derived here based on integrity are in fact nonlinear constitutive theories. It is shown that in the case of the simplest linear constitutive theory for stress Tensor used predominantly for compressible viscous fluids, Stokes' hypothesis or Stokes' assumption has no thermodynamic basis, hence may be viewed incorrect. Thermodynamically consistent derivations of the restrictions on various material coefficients are presented for non-classical as well as classical theories that are applicable to nonlinear constitutive theories, which are inevitable if the constitutive theories are derived based on integrity.
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Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory
Applied Mathematics-a Journal of Chinese Universities Series B, 2016Co-Authors: Karan S. Surana, Stephen W. Long, J. N. ReddyAbstract:In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient Tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient Tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress Tensor and conjugate strain rate Tensors of up to orders n and Cauchy Moment Tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient Tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient Tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient Tensor is used as rotation rates to determine rate of rotation gradient Tensor. This is an approximation to true convected time derivatives of the rotation gradient Tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient Tensor; (2) first convected derivative of the symmetric part of the rotation gradient Tensor as used by Surana et al. is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy Moment Tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress Tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain Tensor, Moment Tensor and heat vector for thermofluids without memory in which convected time derivatives of strain Tensors up to order n are conjugate with the Cauchy stress Tensor and the convected time derivatives of the symmetric part of the rotation gradient Tensor up to orders 1n are conjugate with the Moment Tensor. Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress Tensor, Moment Tensor and heat vector. Stress Tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress Tensor and the Moment Tensor are further decomposed into equilibrium and deviatoric Tensors. It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress Tensor, Moment Tensor and heat vector. Density in the current configuration, convected time derivatives of the strain Tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient Tensor up to orders 1n, temperature gradient Tensor and temperature are considered as argument Tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence. The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates. The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al. for internal polar non-classical incompressible thermofluids.
Brian Seguin - One of the best experts on this subject based on the ideXlab platform.
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Statistical Foundations of Liquid-Crystal Theory II: Macroscopic Balance Laws
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Brian Seguin, Eliot FriedAbstract:Working on a state space determined by considering a discrete system of rigid rods, we use nonequilibrium statistical mechanics to derive macroscopic balance laws for liquid crystals. A probability function that satisfies the Liouville equation serves as the starting point for deriving each macroscopic balance. The terms appearing in the derived balances are interpreted as expected values and explicit formulas for these terms are obtained. Among the list of derived balances appear two, the Tensor Moment of inertia balance and the mesofluctuation balance, that are not standard in previously proposed macroscopic theories for liquid crystals but which have precedents in other theories for structured media.
Herbert Kimura - One of the best experts on this subject based on the ideXlab platform.
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Monte Carlo Approximate Tensor Moment Simulations
Numerical Linear Algebra With Applications, 2014Co-Authors: Juan Arismendi Zambrano, Herbert KimuraAbstract:An algorithm to generate samples with approximate first-, second-, and third-order Moments is presented extending the Cholesky matrix decomposition to a Cholesky Tensor decomposition of an arbitrary order. The Tensor decomposition of the first-, second-, and third-order objective Moments generates a non-linear system of equations. The algorithm solves these equations by numerical methods. The results show that the optimisation algorithm delivers samples with an approximate error of 0.1%-4% between the components of the objective and the sample Moments. An application for sensitivity analysis of portfolio risk assessment with Value-at-Risk (VaR) is provided. A comparison with previous methods available in the literature suggests that methodology proposed reduces the error of the objective Moments in the generated samples.