Modèles standards généralisés à convexe fixe en endommagementDamage standard models with a fixed convex domain

A thermodynamic framework is introduced for damage models. This framework, which consists in adding internal variables called complementary variables, lies within the framework of standard materials whose convex domain of admissible forces does not depend on the present state of internal variables. Thus its advantages are kept. Marigo's model is put back in this framework and another example combining isotropic and kinematic hardening is given. To cite this article: A. Cimetière et al., C. R. Mecanique 331 (2003).     

Continuous symmetries and constitutive laws of dissipative materials within a thermodynamic framework of relaxation: Part I: Formal aspects

An analysis of the continuous symmetries of the constitutive laws of inelastic materials written within a thermodynamical framework of relaxation is performed. This framework relies on the generalization of Gibb’s relationship outside the equilibrium of a uniform system, and the use of the fluctuation theory to model the material dissipation due to its internal microstructure change [Cunat, C., 2001. The DNLR approach and relaxation phenomena. Part I – Historical account and DNLR formalism. Mech. Time-depend. Mater. 5, 39–65]. The approach leads to a viscoelastic like formulation for small deformations, and changes gradually for finite strains towards elastoviscoplasticity (with or without damage) via a dependence of characteristic times with the loading path, in a way similar to the endochronic approach developed by Valanis [Valanis, K.C., 1975. On the fundations of the endochronic theory of viscoplasticity. Arch. Mech. 27, 857–868]. The present thermodynamic framework has been previously applied to elastoviscoplastic materials under cyclic and non-proportional loadings [Dieng, L., Abdul-Latif, A., Haboussi, M., Cunat, C., 2005b. Cyclic plasticity modeling with the distribution of non-linear relaxations approach. Int. J. Plasticity 21, 353–379]. The constitutive laws split into the state laws relating intensive variables (thermodynamics forces) to extensive-like variables, and the complementary evolution laws of the internal variables associated to the dissipative mechanisms. An interpretation of a non-equilibrium thermodynamic approach of irreversible processes in terms of an extremum principle is proposed, associated to a Lagrangian functional. It is shown that one possible choice for the Lagrangian kernel is the material derivative of the internal energy density, augmented by a complementary term that accounts for the evolution laws of the internal variables. Interpreting the material behavior during the non-equilibrium evolution as the Euler–Lagrange equations of the resulting action integral, a differential condition expressing both the local and variational symmetries encapsulated into the Lagrangian formulation is formulated. It is further shown that both symmetry conditions are fully equivalent along the optimal path corresponding to the satisfaction of the constitutive laws. In terms of both practical and methodological aspects, the predictive nature of the symmetry analysis is highlighted, as a systematic tool for the exploitation of the constitutive response. Its performance and utility are exemplified by the construction of a time–temperature equivalence principle for a dry viscous polymer (PA66); the calculated shift factor is shown to well agree with the empirical shift factor given by Williams–Landel–Ferry (WLF) expression. A systematic interpretation of the calculated symmetry groups of the constitutive laws in terms of master curves for various plastic and viscoplastic materials shall be presented in a forthcoming contribution.     

A thermomechanical framework for rate-independent dissipative materials with internal functions

This paper builds on previous work by Houlsby and Puzrin (Int. J. Plasticity 16 (2000) 1017) in which a framework was set out for the derivation of rate-independent plasticity theory from thermodynamic considerations. A key feature of the formalism is that the entire constitutive response is determined by knowledge of two scalar functions. The loading history is effectively captured through the use of internal variables. In this paper, we extend the concept of internal variables to that of internal functions, which represent infinite numbers of internal variables. In this case the thermodynamic functions are replaced by functionals. We set out the formalism necessary to derive constitutive behaviour within this approach. The principal advantages of this development is that it can provide realistic modelling of kinematic hardening effects and smooth transitions between elastic and elastic–plastic behaviour.     

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.     

Il'iushin's postulate and resulting thermodynamic conditions on elasto-plastic coupling

Il'iushin's postulate is restated within a general thermodynamic strain space formulation of rate independent plasticity by means of plastic internal variables. This yields a general expression in terms of appropriate thermodynamic potentials. A combination of a thermodynamic condition, derived from the general development, with the results of Il'iushin's postulate, furnishes explicit conditions on elasto-plastic coupling. A specific example is presented, with the plastic work being the only plastic internal variable. Necessary and sufficient consitbns on the elastic moduli and their change with plastic deformation are derived, for the thermodynamic condition to be satisfied.     

Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a linear elastic transversely isotropic bi-material full-space is presented. Three-dimensional point-load Green’s functions for stresses and displacements are given in complex-plane line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, within the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration. As illustrations, the present Green’s functions are analytically degenerated to the special cases such as half-space, surface and full-space Green’s functions. Some typical numerical examples are also given to show the general features of the bi-material Green’s functions.     

A thermodynamic approach with internal variables using Lagrange formalism. Part II. Continuous symmetries in the case of the time–temperature equivalence

In the first part of this contribution, the Lie-symmetries of the principle of least action associated to the constitutive equations of the DNLR formalism of relaxation have been presented. We examine in this second part the continuous symmetries corresponding to the simple case of stress relaxation under isothermal conditions. The well-known principle of time/temperature equivalence is discussed in terms of variational symmetry for the Jacobi’s action functional, and connected to the Onsager’s relation near the thermodynamic equilibrium.     

Chemical potential tensor for a two-phase continuous medium model

Relations for jumps of thermodynamic variables with allowance for inertial terms are derived under conditions of thermal equilibrium and in the absence of dissipation on the interphase surface. The notion of the chemical potential tensor is generalized for this case within the framework of the elastic continuous medium model. A thermodynamically well-posed definition of the chemical potential tensor is proposed for a class of two-phase models of deformable solids. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 12–22, May–June, 2005.     

On the role of material dissipation for the crack-driving force

The thermodynamic setting for the formulation of the “crack-driving force” for a singular crack in conjunction with rate-independent material response is discussed. One key ingredient is the introduction of a fixed (absolute) configuration, relative to which both physical and (virtual) configurational and spatial changes can be described. Only quasistatic and isothermal conditions are considered in this paper. A variational framework is established for the rate of global energy dissipation (integrated over the whole material domain) due to the combined action of a (discrete) crack extension and continuum inelasticity, whereby the material time derivative of internal variables and the rate of crack extension are coupled. The classical assumption (previously adopted in the literature) is that there is no coupling, i.e. the internal variables are considered as fixed (material) fields just like an inhomogeneous material property. The other (extreme) assumption is that the internal variables fields are convected with the configurational motion due to the virtual crack extension. Both cases are investigated in this paper for a simple 2D example of an edge crack in a plate in a setting of small strains and hardening plasticity. In particular, we consider convergence issues from mesh refinement.     

A Generalised Multiple Mapping Conditioning Approach for Turbulent Combustion

This paper follows the evolution in understanding of the multiple mapping conditioning (MMC) approach for turbulent combustion and reviews different implementations of MMC models. As the MMC name suggests, the original version represents a consistent combination of CMC-type conditional equations (conditional moment closure) and generalised mapping closure. It seems that the strength of the MMC model, and especially that of its stochastic version, lies in a more general (and much more transparent) interpretation. In this new generalised interpretation, we can replace complicated derivations by physical reasoning and the model appears to be a natural extension of modelling approaches developed in recent decades. MMC can be seen as a methodology for enforcing certain known characteristics of turbulence on a conventional mixing model. This is achieved by localising the mixing operation in a reference space. The reference space variables are selected to emulate the properties of a turbulent flow which have a strong effect on reactive quantities. The best and simplest example is an MMC model which has a single reference variable emulating the mixture fraction. In diffusion flames turbulent fluctuations of reacting quantities are strongly correlated with fluctuations of the mixture fraction. By making mixing local in the reference mixture fraction space a CMC-type mixing closure is enforced. In the original interpretation of MMC the reference variables are modelled as Markov processes. Since the reference variables should emulate properties of turbulent flows as realistically as possible the next step, and the basis of generalised MMC, is to remove the Markovian restriction and set reference variables equal to traced Lagrangian quantities within DNS or LES flow fields. Indeed, no Markov value can emulate the mixture fraction better than the mixture fraction itself. (Using a Markov vector process of dimension higher than the number of conditioning variables represents a more economical alternative for producing reference variables in generalised MMC.) The generalised MMC approach effectively incorporates the mixture fraction-based models, the PDF methods and LES/DNS techniques into a single methodology with possibility of blending useful features developed previously for conventional models. The generalised approach to MMC stimulates a more flexible understanding of simulations using sparsely placed Lagrangian particles as tools that may provide accurate joint distributions of reactive scalars at relatively low computational cost. The physical reasoning behind the new interpretation of MMC is supported by example computations for a partially premixed methane/air diffusion flame (Sandia Flame D). The scheme utilises LES for the dynamic field and a sparse-Lagrangian filtered density function method with MMC mixing for the scalar field. Two different particle mixing schemes are tested. Simulations are performed using only 35,000 Lagrangian particles (of these only 10,000 are chemically active) on a single workstation. The relatively low computational cost allows the use of realistic chemical kinetics containing 34 reactive species and 219 reactions. Intended for publication in the special issue of Flow, Turbulence and Combustion arising from the 2nd ECCOMAS Thematic Conference on Computational Combustion held at Delft in mid-2007.     

Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048–1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.     

Fatigue based three-dimensional structural design optimisation studies implementing the generalised Frost–Dugdale crack growth law

Experimental studies of fatigue crack growth in aluminium alloys have shown that, at the low-to-mid stress intensity factor range, there is often a log-linear relationship between the crack length and the fatigue life. These observations have led to the development of the generalised Frost–Dugdale crack growth law, which allowed the accurate prediction of fatigue crack growth from Region I. For this research paper the ‘generalised Frost–Dugdale’ law was used to perform an optimisation study of 7050-T7451 Aluminium structures. The structural optimisation procedure proposed integrates geometrical modelling, structural analysis and optimization into one complete and automated computer-aided design process. The results from the structural optimisation study compared the ‘generalised Frost–Dugdale’ law and the traditional Paris law. Gradient-less, gradient-based optimisation algorithm and an enumeration scheme were considered in this investigation. The enumeration scheme takes advantage of a cluster computer architecture which enables a visualisation of the solution space allowing verification and validation of the optimisation algorithm. The results indicated that the optimal geometrical shape and predicted fatigue life depended on the crack sizes, structural geometry, boundary conditions and fatigue crack growth law. As a result, this procedure illustrates that for the design of light weight structures, a fatigue based optimisation used in conjunction with visualisation of the solution space may provide a viable design methodology. The importance of non-destructive inspection (NDI) and its role in determining optimal structural geometries is also revealed. Furthermore, the possibility of the application of the generalised Frost–Dugdale model in design optimisation has been demonstrated. This procedure has the potential to be applied to structures with complex structural configurations taking into account crack propagation in Region I.     

Evolution Equations for Non-Simple Viscoelastic Solids

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.     

Hyperviscosity for unstructured ALE meshes

An artificial viscosity, originally designed for Eulerian schemes, is adapted for use in arbitrary Lagrangian–Eulerian simulations. Changes to the Eulerian model (dubbed ‘hyperviscosity’) are discussed, which enable it to work within a Lagrangian framework. New features include a velocity-weighted grid scale and a generalised filtering procedure, applicable to either structured or unstructured grids. The model employs an artificial shear viscosity for treating small-scale vorticity and an artificial bulk viscosity for shock capturing. The model is based on the Navier–Stokes form of the viscous stress tensor, including the diagonal rate-of-expansion tensor. A second-order version of the model is presented, in which Laplacian operators act on the velocity divergence and the grid-weighted strain-rate magnitude to ensure that the velocity field remains smooth at the grid scale. Unlike sound-speed-based artificial viscosities, the hyperviscosity model is compatible with the low Mach number limit. The new model outperforms a commonly used Lagrangian artificial viscosity on a variety of test problems.     

A 3D staggered Lagrangian scheme for ideal magnetohydrodynamics on unstructured meshes

In this paper, we propose a 3D staggered Lagrangian scheme for the ideal magnetohydrodynamics (MHD) on unstructured meshes. All the thermal variables and the magnetic induction are defined in the cell centers while the fluid velocity is located at the nodes. The meshes are compatibly discretized to ensure the geometric conservation laws in Lagrangian computation by the classical subcell method, then the momentum equation is discretized using the subcell forces and the specific internal energy equation is obtained by the total energy conservation. Invoking the Galilean invariance, magnetic flux conservation, and the thermodynamic consistency, the expressions of subcell force as well as the cell-centered velocity are derived. Besides, the magnetic divergence-free constraint is fulfilled by a projection method after each time step. Various numerical tests are presented to assert the robustness and accuracy of our scheme.     

On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions

Expressions for thermodynamic potentials (internal energy, Helmholtz energy, Gibbs energy and enthalpy) of a thermoelastic material are developed under the assumption of small strains and finite changes in the thermal variable (temperature or entropy). The literature provides expressions for the Helmholtz energy in terms of strain and temperature, most often as expansions to the second order in strain and to a higher order in temperature changes, which ensures an affine stress–strain relation and a certain temperature dependence of the moduli of the material. Expressions are here developed for the four potentials in terms of all four possible pairs of independent variables. First, an expression is obtained for each potential as a quadratic function of its natural mechanical variable with coefficients depending on its natural thermal variable that are identified in terms of the moduli of the material. The form of the coefficients’ dependence on the thermal variable is not specified beforehand so as to obtain the most general expressions compatible with an affine stress–strain relation. Then, from each potential expressed in terms of its natural variables, expressions are derived for the other three potentials in terms of these same variables using the Gibbs–Helmholtz equations. The paper provides a thermodynamic framework for the constitutive modeling of thermoelastic materials undergoing small strains but finite changes in the thermal variables, the properties of which are liable to depend on the thermal variables.     

A theory of thermo-visco-elastic-plastic materials: thermomechanical coupling in simple shear

The constitutive relations of a theory of thermo-visco-elastic-plastic continuum have been formulated in Lagrangian form. The Lagrangian strains, strain rates, temperature, temperature rate and temperature gradients are considered as the independent constitutive variables. Three internal state variables (plastic strain tensor, back strain tensor and a scalar hardening parameter) are also incorporated. The axioms of objectivity and equipresence are followed. The Clausius–Duhem inequality is taken as the second law of thermodynamics. Several special theories are deduced based on material symmetries and/or conventionally adopted assumptions. The applications to the formation of shear bands and dynamic crack propagation are discussed.     

Analysis of non-local models through energetic formulations

Non-local formulations have been intensively applied for the past 20 years to control the strain localisation resulting from strain-softening constitutive laws. We propose a framework that encompasses most of these non-local approaches. It relies on an energetic interpretation of both equilibrium equations and constitutive relations for generalised standard materials. It allows a systematic exploration of the properties of these models. We focus here on three aspects: the existence of solutions when applied to strain-hardening laws, the effective localisation control when applied to strain-softening laws and the thermodynamic bases.A quite extended analysis is led under some assumptions, the more restrictive being that all the internal variables are dealt with the same regularising operator. It appears that only few approaches bring effective regularisation. Unfortunately, they do not rely on thermodynamic bases so that the Clausius–Duhem inequality is not automatically fulfilled. An alternative consists in giving up the assumptions of the analysis. In that case, it seems that no general result can be stated. Moreover, an illustration in the context of brittle damage shows that slightly different formulations can result in strongly differentiated behaviour, even though some specific choices lead nevertheless to a suitable model.     

Crystallization under external pressure

The influence of external pressure on the dynamics of crystallization is examined by considering a solid–liquid two-phase system in a cylinder closed by a piston. The dynamic equations are derived using three methods, namely, Rational Thermodynamics (Liu’s procedure), the Matrix Model, and the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) formalism. The constitutive relation for the multiphase pressure on the piston is identical for all three methods, whereas some aspects of the result for the phase change dynamics differ. The rational thermodynamics treatment constrains the phase change dynamics of only those structural variables that enter into the dissipation inequality, whereas the other two formalisms make statements about the phase change of all structural variables. Nevertheless, all three methods show, first, that the phase change happens instantaneously at constant volume and, second, where morphological detail can be built into the model without violating thermodynamic principles. It is discussed how an appropriate choice of the morphological variables allows one to incorporate impingement of crystals and depletion zone effects, as well as to distinguish crystal shapes.