弹脆性材料的损伤本构关系及应用

本文根据连续损伤力学方法,对弹脆性材料损伤的力学响应进行一般分析。理论分析中,材料与损伤都是各向异性的。还导出了计算损伤张量、有效弹性张量、真实应力张量以及损伤能耗率张量的实用表达式。     

A direct approach to fiber and membrane reinforced bodies. Part II. Membrane reinforced bodies

The paper deals with membrane reinforced bodies with the membrane treated as a two-dimensional surface with concentrated material properties. The bulk response of the matrix is treated separately in two cases: (a) as a coercive nonlinear material with convex stored energy function expressed in the small strain tensor, and (b) as a no-tension material (where the coercivity assumption is not satisfied). The membrane response is assumed to be nonlinear in the surface strain tensor. For the nonlinear bulk response in Case (a), the existence of states of minimum energy is proved. Under suitable growth conditions, the equilibrium states are proved to be exactly states of minimum energy. Then, under appropriate invertibility condition of the stress function, the principle of minimum complementary energy is proved for equilibrium states. For the no-tension material in Case (b), the principle of minimum complementary energy (in the absence of the invertibility assumption) is proved. Also, a theorem is proved stating that the total energy of the system is bounded from below if and only if the loads can be equilibrated by a stress field that is statically admissible and the bulk stress is negative semidefinite. Two examples are given. In the first, we consider the elastic semi-infinite plate with attached stiffener on the boundary (Melan’s problem). In the second example, we present a stress solution for a rectangular panel with membrane occupying the main diagonal plane.     

Relativistic equations of balance in continuum mechanics

In this paper we formulate the global expression of balance of energy starting from the relativistic definition of force, mechanical work, heat supply, and kinetic energy. We then prove that invariance under Lorentz transformation leads to the local expression for balance of momentum and to the equation of decomposition of the energy-momentum tensor.Previously, the principles of balance of momentum and balance of energy have been introduced as independent principles.     

Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables

In the vicinity of the glass transition, glass-forming materials exhibit pronounced frequency-dependent changes in the mechanical material properties, the thermal expansion behaviour and the specific heat. The frequency dependence becomes apparent under harmonic stress, strain or temperature excitations. The Prigogine-Defay ratio is a characteristic number which connects the changes in magnitude of these quantities at the glass transition. In order to represent the thermoviscoelastic properties of glass-forming materials in continuum mechanics, a three-dimensional approach which is based on the Gibbs free energy as thermodynamic potential is developed in this article. The Gibbs free energy depends on the stress tensor, the temperature and a set of internal variables which is introduced to take history-dependent phenomena into account. In the vicinity of an equilibrium reference state, the specific Gibbs free energy is approximated up to second order terms. Evaluating the Clausius-Duhem inequality, the constitutive relations for the strain tensor, the entropy and the internal variables are derived. In comparison with other approaches, the entropy, the strain tensor and the internal variables are functionals not only of the stress tensor but also of the temperature. Applying harmonic temperature- or stress-controlled excitations, complex frequency-dependent relations for the specific heat under constant stress, for the thermal expansion coefficients as well as for the dynamic mechanical compliance are obtained. The frequency-dependence of these quantities depicts the experimentally observed behaviour of glass-forming materials as published in literature. Under the assumption of isotropic material behaviour, it is shown that the developed theory is compatible with the Prigogine-Defay inequality for arbitrary values of the material parameters.     

On the Balance Equation for Stresses Concentrated on Curves

Conditions of equilibrium of forces are formulated for a force system in a continuous body which consists of a bulk stress and a stress concentrated on a curve. The system of forces is interpreted as a tensor valued measure and the equilibrium of forces is interpreted in a weak sense. The divergence theorem is proved for bulk stress fields with the corresponding singularity. Examples of such stress fields are given.      

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

Limit criteria and a model for inelastic deformation of anisotropic media

Limit criteria corresponding to the critical values of the specific strain energy and various eigenmoduli of anisotropic media were proposed based on a representation of the elastic tensor in terms of eigenmoduli and eigenstates. Examples of limit (plasticity) criteria independent of the safe stress state are given. Governing equations for the limiting strain of anisotropic materials were obtained using the concept of eigenstates. The corresponding equations for isotropic materials of classes {1, 5} and {5, 1} and materials of the cubic system of various classes are considered. Limit criteria can vary considerably depending on the class of anisotropic material.     

Crack driving force and energy-momentum tensor in electroelastodynamic fracture

The energy flux integral and the energy-momentum tensor for studying the crack driving force in electroelastodynamic fracture are formulated within the framework of the nonlinear theory of coupled electric, thermal and mechanical fields based on fundamental principles of thermodynamics. This formulation lays a foundation for in-depth understanding of the fracture behavior of piezoelectric materials. Remarkably, the dynamic energy release rate thus obtained has an odd dependence on the electric displacement intensity factor for steady-state propagation of a conventional (unelectroded) crack with exact, electrically permeable, semi-permeable, or impermeable crack surface condition, which is in agreement with experimental evidence.     

Second gradient viscoelastic fluids: dissipation principle and free energies

We consider a generalization of the constitutive equation for an incompressible second order fluid, by including thermal and viscoelastic effects in the expression for the stress tensor. The presence of the histories of the strain rate tensor and its gradient yields a non-simple material, for which the laws of thermodynamics assume a appropriate modified form. These laws are expressed in terms of the internal mechanical power which is evaluated, using the dynamical equation for the fluid. Generalized thermodynamic constraints on the constitutive equation are presented. The required properties of free energy functionals are discussed. In particular, it is shown that they differ from the standard Graffi conditions. Various free energy functionals, which are well-known in relation to simple materials, are generalized so that they apply to this fluid. In particular, expressions for the minimum free energy and a more recently introduced explicit functional of the minimal state are proposed. Derivations of various formulae are abbreviated if closely analogous proofs already exist in the literature.     

Method of molecular dynamics in mechanics of deformable solids

The basic principles of the method of molecular dynamics are analyzed. Symplectic difference schemes for the numerical solution of molecular dynamics equations are considered. Stability is studied, and the errors in the energy conservation law, which are induced by using these schemes, are estimated. Equations of mechanics of continuous media are derived by means of averaging over the volume of an atomic system. Expressions for the stress tensor are obtained by using the virial principle and the method of averaging over the volume. The principles of construction of EAM and MEAM potentials of atomic interaction in crystals are analyzed. Two problems of fracture of copper-molybdenum composites are solved by the method of molecular dynamics.     

A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals

A thermodynamical theory of gradient elastoplasticity, including kinematic hardening, is developed by introducing the concept of dislocation density tensor. The theory is self-consistent and is based on two fundamental principles, the principle of increase of entropy and the maximal entropy production rate. Thermodynamically consistent constitutive equations for plastic stretching, plastic spin and back stress are rigorously derived. Also, an expression for the plastic spin is obtained from the constitutive equation of dislocation drift rate and an expression for the back stress is given as a balance equation expressing equilibrium between internal stress and microstress conjugate to the dislocation density tensor. Moreover, it is shown that the present gradient theory yields a symmetric stress tensor. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given.     

A linearized theory of elastic dielectrics which couples electrically to flexural vibrations

Assuming that the free energy depends on the deformation gradient and the spatial electric field, we derive the expressions for the Cauchy stress tensor and the spatial electric displacement from an observer invariant quadratic form of the free energy via the strict definitions of these quantities. Specific forms of the Piola-Kirchhoff stress tensor and the material electric displacement are then deduced and linearized in a particular sense. As an application of the resulting theory, we formulate the problem of an electrically driven disc within the context of the classical bending theory of thin plates. The material of the disc is assumed to have at most the symmetry of a hexagonal system of classC _6v.The resulting coupled differential equations for the axial mechanical displacement of the middle surface and the material electric potential indicate that the problem is not empty. This result is of particular interest in view of the fact that it is generally held that the classical theory of piezoelectricity does not permit such couplings to occur.     

Interfacial excess energy, excess stress and excess strain in elastic solids: Planar interfaces

In this paper, interfacial excess energy and interfacial excess stress for coherent interfaces in an elastic solid are reformulated within the framework of continuum mechanics. It is shown that the well-known Shuttleworth relationship between the interfacial excess energy and interfacial excess stress is valid only when the interface is free of transverse stresses. To account for the transverse stress, a new relationship is derived between the interfacial excess energy and interfacial excess stress. Dually, the concept of transverse interfacial excess strain is also introduced, and the complementary Shuttleworth equation is derived that relates the interfacial excess energy to the newly introduced transverse interfacial excess strain. This new formulation of interfacial excess stress and excess strain naturally leads to the definition of an in-plane interfacial stiffness tensor, a transverse interfacial compliance tensor and a coupling tensor that accounts for the Poisson's effect of the interface. These tensors fully describe the elastic behavior of a coherent interface upon deformation.     

颗粒材料三维应力路径下的接触组构特性

颗粒材料的宏观应力变形特征与其微观接触力、组构等紧密相关.一般而言,强接触系统属于颗粒内部体系的传力结构,其对应的组构张量是影响宏观应力性质的重要因素.细观数值方法(如离散单元法)能够反映物理试验的基本规律,并且可以方便地提取宏微观数据来研究颗粒体系的应力变形机制.采用离散单元法(discrete element method,DEM)进行一系列等$p$等$b$应力路径下颗粒材料的真三轴试验,在此基础上研究了三维应力路径下颗粒材料的宏微观力学参数的演化过程、三维组构张量与应力张量多重联系以及强接触体系反映的宏观应力特征.研究表明:颗粒体系偏应力峰值状态和临界状态均存在与加载路径无关的宏微观特征;三维应力路径下组构张量与应力张量存在非共轴性,但其联合不变量演化过程表现出加载路径无关的特征;与弱接触系统的组构张量相比,强接触系统的组构张量更能反映宏观应力张量的特征;强弱接触体系的组构张量对颗粒体系宏观响应的贡献不同,其分界点存在一定取值范围,但采用平均接触力较为简单合理.      

Contemporary perspectives in finite strain plasticity

From an internally consistent theory of finite strain plasticity in which are introduced a new stress tensor that includes rigid body rotation and a compatible finite strain tensor amenable to simple geometrical interpretation, a close correlation with experiment is obtained for stress paths of arbitrary composition and direction. Measured material constants are proportional to the elastic shear modulus, as indicated for dislocation theory. At large strain there exists a quantum mechanical structure for the continuum, in which are defined a series of relative reference configurations and second order transitions.     

Thermomechanical material modelling based on a hybrid free energy density depending on pressure,isochoric deformation and temperature

In order to represent temperature-dependent mechanical material properties in a thermomechanical consistent manner it is common practice to start with the definition of a model for the specific Helmholtz free energy. Its canonical independent variables are the Green strain tensor and the temperature. But to represent calorimetric material properties under isobaric conditions, for example the exothermal behaviour of a curing process or the dependence of the specific heat on the temperature history, the temperature and the pressure should be taken as independent variables. Thus, in the field of calorimetry the Gibbs free energy is usually used as thermodynamic potential whereas in continuum mechanics the Helmholtz free energy is normally applied. In order to simplify the representation of calorimetric phenomena in continuum mechanics a hybrid free energy density is introduced. Its canonical independent variables are the isochoric Green strain tensor, the pressure and the temperature. It is related to the Helmholtz free energy density by a Legendre transformation. In combination with the additive split of the stress power into the sum of isochoric and volumetric terms this approach leads to thermomechanical consistent constitutive models for large deformations. The article closes with applications of this approach to finite thermoelasticity, curing adhesives and the glass transition.     

An Area Modulus of Elasticity: Definition and Properties

A new modulus of elasticity is defined to be the ratio of an equibiaxial stress to the relative area change in the planes in which the stress acts. This area modulus of elasticity is intermediate in properties between Young's modulus and the bulk modulus. Expressions for the area modulus are computed in isotropic elasticity. A simple, convenient expression for the compliance tensor of transverse isotropy is found in terms of, amongst others, the longitudinal (axial) area modulus and this leads to a new, concise condition for positive definiteness of the compliance tensor. The limits of incompressibility, inextensibility and constant area are briefly considered. This revised version was published online in August 2006 with corrections to the Cover Date.     

Approche globale,minima relatifs et Critère d'Amorçage en Mécanique de la RuptureGlobal approach,relative minima and yield criterion in Fracture Mechanics

We show that it is possible to predict the onset of cracking by searching relative minima of the total energy of a body, the yield criterion depending on the choice of the energies. In particular in the case where the bulk energy is elastic and the surface energy depends on the jump of the displacement, we prove that the yield criterion is formulated in terms of the stress tensor and of intrinsic curve type. To cite this article: J. Laverne, J.-J. Marigo, C. R. Mecanique 332 (2004).