Principle Of Maximum Dissipation Research Articles

Stable discretization of the Boltzmann equation based on spherical harmonics, box integration, and a maximum entropy dissipation principle

The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.

Covariant description of contact interfaces considering anisotropy for adhesion and friction.: Part 2. Linearization, finite element implementation and numerical analysis of the model

A covariant description for contact problems including anisotropy for both adhesion and sliding domains is proposed. The principle of maximum dissipation is used to obtain a computational model in the case of quasi-static motions. This second part is first focusing on the linearization of the nonlinear equations necessary for the solution process. Then the finite element implementation for various contact elements is developed. In addition, a mechanical interpretation via a rheological model is discussed. Finally, different cases including curvilinear anisotropy on arbitrary surfaces are considered. The numerical examples are chosen to show the influence of the orthotropy type on the development of the sticking-sliding zone as well as on the kinematical behavior of the contact bodies.

Covariant description of contact interfaces considering anisotropy for adhesion and friction: Part 1. Formulation and analysis of the computational model

A covariant description for contact problems including anisotropy for both adhesion and sliding domains is proposed. The principle of maximum dissipation is used to obtain a computational model in the case of quasi-static motion. Various cases including curvilinear anisotropy on arbitrary surfaces illustrating the possibility to model machined surfaces are considered. The part is served to be a necessary preparation for further finite element implementations and numerical analysis.

Martensitic transformation in NiMnGa single crystals: Numerical simulation and experiments

In this article, we develop a continuum mechanical model to simulate deformation and phase transformation processes in shape memory alloys. The model is based on a detailed description of the stored energy. Furthermore, the energy dissipation due to phase transformations is taken into account via the maximum-dissipation principle. The results from the 3D numerical simulations of stress induced transformations from the cubic to the tetragonal phase and martensitic variant reorientations in NiMnGa are compared with laboratory experiments on NiMnGa [0 0 1]-oriented single crystals.

On multiscale significance of Rice’s normality structure

The normality structure proposed by [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455.] provides a minimal framework of multiscale thermodynamics. As shown in this paper, Rice’s multiscale thermodynamic formalism is exactly consistent with Ziegler’s essential notion [Ziegler, H., 1977. An Introduction to Thermomechanics, North-Holland, Amsterdam.] that the entire constitutive response is determined by the knowledge of two scalar potential functions: an energy function and a dissipation function. In Rice’s multiscale thermodynamic formulation, the variational equation relating macroscale and microscale thermodynamic fluxes and forces plays a central role and ensures the equality between microscale and macroscale dissipation rate. The variational equation can be further reformulated into a principle of maximum equivalent dissipation. Based on the variation equation, the transformation from microscale to macroscale is characterized by two linear transformations with the same corresponding matrix.

On the small and finite deformation thermo-elasto-viscoplasticity theory for strain localization problems

This work is focused on the numerical implementation of thermo-elasto viscoplastic constitutive equations in small and finite strain contexts using the FEM. A length-scale parameter is introduced implicitly through viscosity in order to address strain localization and material instability in the (initial) boundary value problems. According to the second law of thermodynamics a dissipation inequality described in the rotated material coordinate system is developed by Voyiadjis et al. (2004), which was used along with the principle of maximum dissipation to formulate a viscoinelastic model. In order to check the effectiveness of the present framework a set of numerical examples under strict deformation conditions are presented. These numerical examples prove the excellent performance of the present framework in describing the strain localization problem and in obtaining mesh-independent results.

Multiscale Thermodynamic Significance of the Scale Invariance Approach in Continuum Inelasticity

AbstractIn this paper, the scale invariance approach from micro- to macro-plasticity by Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach,” ASME J. Eng. Mater., 117, pp. 352–355) and Ning and Aifantis (1996, “Anisotropic Yield and Plastic Flow of Polycristalline Solids,” Int. J. Plasticity, 12, pp. 1221–1240) is investigated within Rice’s normality structure (1971, “Inelastic Constitutive Relations for Solids: An Integral Variable Theory and its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455; 1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms,” Constitutive Equations in Plasticity, A. S. Argon, ed., MIT Press, Cambridge, MA, pp. 23–79). The normality structure provides a minimal framework of multiscale thermodynamics, and the dissipation equivalence between the microscale and macroscale is ensured by a variational equation which can be further formulated into principle of maximum equivalent dissipation. It is revealed in this paper that within the framework of normality structure, the so-called hypothesis of generalized scale invariance holds for the kinetic rate laws, flow rules, and orthogonality conditions in the sense of Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach”). Stemming from Rice’s kinetic rate laws, the generalized scale invariance reflects the inherent self-consistent character of the normality structure. If the plastic work rate is assumed to be equal to the intrinsic dissipation rate, the kinematic hardening plasticity as a demonstration of the scale invariance approach by Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach”), can be well accommodated within the framework of normality structure. Therefore, the scale invariance approach is justified from a multiscale thermodynamic viewpoint. It is further shown that the maximization procedure in this approach just corresponds to the principle of maximum equivalent dissipation.

Thermodynamics of magnetohydrodynamic flows with axial symmetry

We present strategies based upon optimization principles in the case of the axisymmetric equations of magnetohydrodynamics (MHD). We derive the equilibrium state by using a minimum energy principle under the constraints of the MHD axisymmetric equations. We also propose a numerical algorithm based on a maximum energy dissipation principle to compute in a consistent way the nonlinearly dynamically stable equilibrium states. Then, we develop the statistical mechanics of such flows and recover the same equilibrium states giving a justification of the minimum energy principle. We find that fluctuations obey a Gaussian shape and we make the link between the conservation of the Casimirs on the coarse-grained scale and the process of energy dissipation. We contrast these results with those of two-dimensional hydrodynamical turbulence where the equilibrium state maximizes a H function at fixed energy and circulation and where the fluctuations are nonuniversal.

Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading-unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah-Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.

A strain space gradient plasticity theory for finite strain

In this paper, an extension to the finite deformation regime of the infinitesimal theories of strain gradient plasticity discussed in a paper by R. Chambon, D. Caillerie and T. Matsushimas [Int. J. Solids Struct. 38 (2001) 8503–8527] is presented which extends and generalizes the previous works of R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (Série IIb) (2001) 797–802] and C. Tamagnini, R. Chambon and D. Caillerie [C.R. Acad. Sci. 329 (Série IIb) (2001) 735–739]. Central to the proposed theory are the kinematic assumptions concerning the decomposition of the assumed measures of strain and hyperstrain into an elastic and a plastic part. Following modern treatments of finite deformation plasticity, a multiplicative decomposition of the deformation gradient is postulated, while an additive decomposition is adopted for the second deformation gradient, as in the paper by R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (Série IIb) (2001) 797–802]. The elastic constitutive equations for Kirchhoff stress and double stress tensors are obtained by assuming the existence of a suitable free energy function in the spatial description . The requirements of (i) invariance for rigid body motions superimposed upon the intermediate configuration ; and, (ii) spatial covariance, see the book by J.E. Marsden and T.J.R. Hughes [Mathematical Foundations of Elasticity , Dover Publications Inc., New York, 1994], provide the corresponding material version of the hyperelastic constitutive equations. A fully covariant formulation of the evolution equations for the plastic strain and hyperstrain tensors, as well as for the internal variables is obtained by a straightforward application of the principle of maximum dissipation, after introducing a suitable yield condition in strain space. The resulting strain-space theory of second gradient plasticity can be considered an extension of the finite deformation plasticity theory proposed by J.C. Simo [Comput. Methods Appl. Mech. Engrg. 66 (1988) 199–219] to elastoplastic media with microstructure. As an example of application, a single-mechanism, isotropic hardening second gradient model for cohesive-frictional materials is proposed in which the cohesive component of the shear strength is assumed to increase with the magnitude of elastic hyperstrains, as advocated by, e.g. N.A. Fleck and J.W. Hutchinson [Strain gradient plasticity, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advances in Applied Mechanics, vol. 33 Academic Press, 1997] for metals, based on both experimental observations and micromechanical considerations. Due to the internal length scales provided by the microstructure, the model is ideally suited for the analysis of failure problems in which strain localization into shear band occurs.

A strain space gradient plasticity theory for finite strain

In this paper, an extension to the finite deformation regime of the infinitesimal theories of strain gradient plasticity discussed in a paper by R. Chambon, D. Caillerie and T. Matsushimas [Int. J. Solids Struct. 38 (2001) 8503–8527] is presented which extends and generalizes the previous works of R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (Série IIb) (2001) 797–802] and C. Tamagnini, R. Chambon and D. Caillerie [C.R. Acad. Sci. 329 (Série IIb) (2001) 735–739]. Central to the proposed theory are the kinematic assumptions concerning the decomposition of the assumed measures of strain and hyperstrain into an elastic and a plastic part. Following modern treatments of finite deformation plasticity, a multiplicative decomposition of the deformation gradient is postulated, while an additive decomposition is adopted for the second deformation gradient, as in the paper by R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (Série IIb) (2001) 797–802]. The elastic constitutive equations for Kirchhoff stress and double stress tensors are obtained by assuming the existence of a suitable free energy function in the spatial description. The requirements of (i) invariance for rigid body motions superimposed upon the intermediate configuration; and, (ii) spatial covariance, see the book by J.E. Marsden and T.J.R. Hughes [Mathematical Foundations of Elasticity, Dover Publications Inc., New York, 1994], provide the corresponding material version of the hyperelastic constitutive equations. A fully covariant formulation of the evolution equations for the plastic strain and hyperstrain tensors, as well as for the internal variables is obtained by a straightforward application of the principle of maximum dissipation, after introducing a suitable yield condition in strain space. The resulting strain-space theory of second gradient plasticity can be considered an extension of the finite deformation plasticity theory proposed by J.C. Simo [Comput. Methods Appl. Mech. Engrg. 66 (1988) 199–219] to elastoplastic media with microstructure. As an example of application, a single-mechanism, isotropic hardening second gradient model for cohesive-frictional materials is proposed in which the cohesive component of the shear strength is assumed to increase with the magnitude of elastic hyperstrains, as advocated by, e.g. N.A. Fleck and J.W. Hutchinson [Strain gradient plasticity, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advances in Applied Mechanics, vol. 33 Academic Press, 1997] for metals, based on both experimental observations and micromechanical considerations. Due to the internal length scales provided by the microstructure, the model is ideally suited for the analysis of failure problems in which strain localization into shear band occurs.

Concepts for modeling impacts without friction

There are three basic equations in mechanics for treating collisions: the law of impact, kinematic compatibility, and energetic consistency. In this paper, the conditions are examined under which a natural extension of the dynamics at an impact is possible without taking additional impact laws, and which additional assumptions have to be made to solve the impact for different classes of systems. It will be shown that Newton’s law of impact for two colliding point masses can be derived from the concept of energy conservation and the principle of maximum dissipation, and has therefore not to be regarded as an independent equation. Moreover, it can be assigned to single-contact impacts in multibody systems as soon as the classical definition of perfect constraints is being extended to impulsive dynamics and unilateral contacts. It will further be shown that the principle of maximum dissipation leads to a unique post-impact velocity in the case of multi-contact collisions. In all other cases, however, the velocities remain undetermined, and laws of impact have to be postulated as additional and independent equations, whereas the classic definition of the restitution coefficient as a dissipation parameter can still be kept.

Implicit plane stress algorithm for multilayer J2‐plasticity using the Prager–Ziegler translation rule

AbstractA totally implicit algorithm for plane stress multilayer plasticity is presented. The algorithm presents the plane stress version of the 3D/plane strain model for multilayer plasticity presented in a previous work. As in the 3D/plane strain case, the model is consistent with the principle of maximum dissipation and it may be considered as an extension of classical J2‐plasticity for anisotropic non‐linear kinematic behaviour preserving Masing's rules. In order to obtain the asymptotic second‐order convergence of the Newton algorithm, the consistent tangent moduli are also given. Copyright © 2003 John Wiley & Sons, Ltd.

On a finite strain viscoplastic theory based on a new internal dissipation inequality

This work is focused on the theoretical development and numerical implementation of a viscoplastic law. According to the second law of thermodynamics a dissipation inequality described in the rotated material coordinate system is developed. Based on this dissipation inequality and the principle of maximum dissipation a finite strain viscoplastic model described also in the rotated material coordinate system is formulated. The evolution equations are expressed in terms of the material time derivatives of the rotated elastic logarithmic strain, the accumulated plastic strain and the strain-like tensor conjugate to the rotated back stress. The mathematical structure of this theory is concise and similar to that of the infinitesimal viscoplastic theory. These characteristics make the numerical implementation of this theory easy. The stress integration algorithm and the algorithmic tangent moduli for the infinitesimal theory can be applied to the numerical implementation of the present finite strain theory with a little reformulation. The complicated algorithmic formulations for most of other finite plastic laws can be therefore circumvented. In order to check the effectivity of the present finite strain theory a set of numerical examples under strict deformation conditions are presented. These numerical examples prove the excellent performance of the present viscoplastic material law at describing the finite strain elastoplastic and viscoplastic problems.

Eddy Heat Diffusivity at Maximum Dissipation in a Radiative-convective One-dimensional Climate Model

The hypothesis that the steady state of the climate is constrained to maximize its dissipation is investigated by using a one-dimensional radiative-convective model where the convective fluxes are parameterized following the mixing length theory. The eddy heat diffusivity is chosen to produce the maximum energy dissipation caused by convection (maximum dissipation principle). The state of maximum dissipation due to heat diffusion in a model atmosphere with a fixed vertical profile of specific humidity is obtained for a value of the eddy heat diffusivity, which is in close agreement with that expected for air in stirred conditions. By including the temperature—opacity feedback (i.e., condensable absorbing gas in the infrared with a fixed vertical profile of relative humidity), the state of maximum dissipation is attained for a value of the eddy heat diffusivity that lies within the range of the expected values. In this case, the vertical temperature profile is convectively stable and does not differ in excess from the observed one. Thus, it is suggested that the results here reported represent an ‘empirical’ support to the maximum dissipation principle.

A minimization principle for finite strain plasticity: incremental objectivity and immediate implementation

AbstractA finite strain plasticity formulation is proposed which meets several requirements that often appear contradictory. On a physical ground, it is based on a multiplicative split of the deformation, hyperelasticity for the reversible part of the behaviour and the maximal dissipation principle to define the evolution laws. On a numerical ground, it is incrementally objective and the integration over a time increment can be expressed as a minimization problem, a proper framework to examine the questions of existence and uniqueness of the solutions. Last but not least, the implementation is immediate since it relies on the same equations for finite and infinitesimal transformations. Finally, the formulationis applied to von Mises plasticity with isotropic linear hardening and introduced in the finite element software Code_Aster®. The numerical computation of a cantilever beam shows that it leads to results in good agreement with those obtain with common approaches. Copyright © 2002 John Wiley & Sons, Ltd.

Evolution model for martensitic phase transformation in shape-memory alloys

A mesoscopical-level model for the evolution of microstructure in simple-laminate martensite undergoing an isothermal phase-transformation process within the context of a uniaxial deformation is proposed using a Hamiltonian approach to a relaxed problem involving a Young-measure-valued deformation gradient and Hill's maximum-dissipation principle involving positive homogeneous dissipation potential which reflects the energy needed for (and dissipated by) a phase transformation. A regularization by adding a (modified) volume-fraction gradient, which can be understood as a limit Ericksen-Timoshenko beam-like construction, is considered to ensure existence of a weak solution for a slow-process model. A numerical algorithm and computational experiments are also presented.

A meso–macro finite element modelling of laminate structures: Part I: time-independent behaviour

A meso–macro modelling is proposed for laminates made of unidirectional layers of a polymer matrix reinforced with long fibres. The behaviour of a layer is expressed through elasticity coupled to damage and plasticity. The anisotropic damage is completely described by a single scalar variable and its evolution law is specified from the principle of maximum dissipation. The effective undamaged space concept associated with the strain equivalence principle is used for the plastic analysis. The flow rules are defined within the context of the generalised standard materials assuming the existence of a dissipation pseudo-potential and a plastic criterion expressed as scalar valued functions of the effective stresses and the thermodynamic force associated with a non-linear kinematic hardening. The integration of the layer behaviour through the thickness is obtained within a Kirchhoff shell element. The constitutive equations of the layer are numerically integrated using an unconditionally stable algorithm consisting of a multi-level iterative scheme. A consistent integration of the equilibrium equations is performed to assure a quadratic convergence of the global solution scheme. A methodology is also proposed for the evaluation of the inter-laminar shear stresses in a consistent manner with the stresses and internal variables computation algorithm. The proposed formulation has been implemented in the finite element code CASTEM2000 in order to test its validity. The obtained results have been compared with the available experimental ones in the case of progressive repeated loading tests by applying pure traction, pure internal pressure and internal pressure with end effect on a tube.

A non-equilibrium thermodynamic geometric structure for thermoviscoplasticity with maximum dissipation

Abstract A maximum dissipation principle induces a class of viscoplastic evolution equations within a non-linear, non-equilibrium thermodynamic structure which extends Gibbs thermostatics. The evolution of a relaxation process is determined through non-linear affinities, which generalize the linear Onsager construct, along with the maximum dissipation principle and the long term states. The relaxation modulus is the norm of the plastic strain rate. Forced processes, in which the control variables change with time, are assumed to be relaxation processes at each instant. This class of thermoviscoplastic models includes the three-dimensional Freed–Chaboche–Walker model in which the internal state variables are the back stress, the drag strength and the elastic yield strength for creep initiation.

A finite deformation second gradient theory of plasticity

Extending the previous work by Chambon et al. [2] to the finite deformation regime, a local second gradient theory of plasticity for isotropic materials with microstructure is developed based on the multiplicative decomposition of the deformation gradient, the additive decomposition of the second deformation gradient and the principle of maximum dissipation.