Ignacio Carol

Damage and plasticity in microplane theory

The paper deals with the microplane model, in which the stress-strain relations are defined independently on planes of all possible orientations in the microstructure, and the microplane stresses or strains are then constrained kinematically or statically to the macroscopic stress or strain tensor. The existing formulations of the microplane constitutive model for concrete are mainly based on the kinematic constraint. They have been shown capable of reproducing satisfactorily most experimental results available for concrete specimens, with the advantages of great conceptual simplicity, convenient numerical explicitness, intrinsic induced anisotropy and microcrack opening- closure conditions, etc. However, from the theoretical viewpoint little has been said about how these formulations relate to classical constitutive models of elasto-plasticity or continuum damage mechanics. In this paper, a new apemu of microplane theory is achieved by systematically intro- ducing damage and plasticity concepts into the microplane framework. New insight is provided on the role played by the split of the normal components, and on the role of the different possible types of micro-macro constraint. Specific formulations are developed and discussed within the new theoretical framework, which can be easily related to von Mises plasticity and to the existing models based on the second and fourth-order damage tensors. 0 1997 Elsevier Science Ltd.

AN INTERFACE ELEMENT FORMULATION FOR THE ANALYSIS OF SOIL-REINFORCEMENT INTERACTION

Abstract The formulation of a family interface elements belonging to the zero-thickness class is briefly described. The formulation is isoparametric and can be applied to 2- D and 3- D analysis. A flexible constitutive law, based on elastoplasticity, describing the mechanical behaviour of the interface is presented. In order to assess the suitability of the elements for the analysis of soil-reinforcement interaction, they are employed in the simulation of a pull-out test in two and three dimensions. It is shown that the performance of the elements can depend strongly on the type of integration rule adopted. The use of integration schemes, such as that of Newton-Cotes, which result in the uncoupling of the degrees of freedom of the element is shown to be very advantegeous. An application to the analysis of a problem involving softening behaviour of the interface is also included.

A unified theory of elastic degradation and damage based on a loading surface

A number of new models with stiffness degradation have been proposd in recent literature in the small strain regime. However, most of these works represent specific formulations, each using its own terminology, notation and assumptions, and relatively little effort has been spent so far towards achieving a common theoretical framework similar for instance to the theory of elastoplasticity. Moreover, most of the existing damage models are presented with intensive recourse to abstract thermodynamics concepts, and they combine stiffness degradation with plasticity, which (though being ultimately necessary to represent the actual material behavior) makes it much more difficult to isolate, analyse and understand the properties of the formulation for elastic stiffness degradation. As a contribution in this field, this paper presents a unifying theoretical framework to describe a class of models for elastic stiffness degradation based on the concept of loading surface. The derivation includes two consecutive steps: first, the constitutive framework for elastic-degrading models with evolution laws which are expressed directly in terms of the secant stiffness (or compliance) tensor, and second the elastic-damage models, in which the scant stiffness (or compliance) is assumed to depend on a reduced set of damage variables with clearer physical meaning and simpler evolution laws. Whenever possible, terminology is borrowed from the classical formulation of elastoplasticity, and thermodynamic concepts are introduced only as needed. Both stress-based and strain-based developments are compared throughout the paper, and the concept of associativity is reanalysed and generalized within the new unified framework of elastic degradation. The most significant scalar damage models found in the literature are reinterpreted in the context of this unified theory. Finally, a general expression is obtained for the tangential stiffness operator of associated scalar models (stress- and strain-based) of the (1 —D) type, that includes all the models considered as particular cases. More general damage formulations [scalar non-(1 — D), vectorial, tensorial] are reviewed and discussed systematically in a sequel paper.

A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses

Microplane models are based on the assumption that the constitutive laws of the material may be established between normal and shear components of stress and strain on planes of generic orientation (so-called microplanes), rather than between tensor components or their invariants. In the kinematically constrained version of the model, it is assumed that the microplane strains are projections of the strain tensor, and the stress tensor is obtained by integrating stresses on microplanes of all orientations at a point. Traditionally, microplane variables were defined intuitively, and the integral relation for stresses was derived by application of the principle of virtual work. In this paper, a new thermodynamic framework is proposed. A free-energy potential is defined at the microplane level, such that its integral over all orientations gives the standard macroscopic free energy. From this simple assumption, it is possible to introduce consistent microplane stresses and their corresponding integral relation to the macroscopic stress tensor. Based on this, it is shown that, in spite of the excellent data fits achieved, many existing formulations of microplane model were not guaranteed to be fully thermodynamically compliant. A consequence is the lack of work conjugacy between some of the microplane stress and strain variables used, and the danger of spurious energy dissipation/generation under certain load cycles. The possibilities open by the new theoretical framework are developed further in Part II companion paper.

New explicit microplane model for concrete: Theoretical aspects and numerical implementation

Abstract The microplane model is a powerful approach for the representation of the complex triaxial behavior of concrete and other similar materials. However, most efforts in previous formulations were devoted to the development of the model itself and to the experimental data fitting, rather than to a comprehensive theoretical description or to attainment of a modular and computationally efficient implementation in a computer code. In this paper, these objectives are pursued. The formulation of the model has been modified to rationalise the structure of the basic hypotheses, simplify the equations and generalise the concepts whenever possible. The result is a new formulation which, while retaining the favorable properties achieved previously, is also easier to understand, and convenient for computer implementation and large-scale calculations. A computational scheme is presented with the unified structure of a general code serving the double purpose of lest specimen analysis and finite element analysis. In practice, this structure includes two different main programs which call the same set of constitutive subroutines. A salient feature of the new version of the model is that the compulation of the stress corresponding to a prescribed strain increment of finite si/e is fully explicit. Step-by-step numerical integration, usually necessary for the practical use of constitutive models, can be avoided. Consequently, the complexity of the code and the cost of computations can be dramatically reduced. Some examples of applications, used to verify the previous version of the model, are also presented. They demonstrate that this new formulation gives a much better numerical efficiency for code implementation while keeping the same desirable features and accuracy in experimental data fitting.

Spurious energy dissipation/generation in stiffness recovery models for elastic degradation and damage

Abstract A number of constitutive models have been proposed in recent years for elastic degradation and damage, many of which include procedures for the recovery of stiffness upon closure of tensile raicrocracks. Most of these recovery procedures are based on the decomposition of stress or strain into positive (tensile) and negative (compressive) components, which are incorporated in the elastic formulation taking recourse to fourth-order positive and negative projection operators. Due to the non-dissipative nature of microcrack closure-reopening for a certain fixed state of degradation, the recovery formulation should possess a well-defined energy potential along the line of hyperelasticity, which conserves energy upon closed-loop load histories. This condition seems to have escaped the rapidly expanding literature on damage mechanics, i.e. closure formulations have not been verified in this regards. In the paper, the (lack of) energy conservation is examined in terms of the spurious dissipation rate, which is developed for a relatively general class of recovery models. They include the positive-negative projection operators and the bimodular formulations with different stiffnesses for tension and compression. It is shown that under proportional loading in strain or stress, all these formulations are energy conservative. Under non-proportional loading, however, they are only conservative in conjunction with isotropic degradation, and they exhibit spurious dissipation-generation when anisotropic degradation is considered and the load history involves rotation of principal directions.

A constitutive model for rock joints formulation and numerical implementation

Abstract An elastoplastic constitutive law for describing the three-dimensional mechanical behaviour of rock joints is presented. The model is intended for use in numerical analysis and is formulated with sufficient flexibility so that it can reproduce a wide range of observed joint stress-strain behaviour. An hyperbolic failure criterion is shown to fit well reported joint strength data and the same type of function is adopted to define the family of yield surfaces. The evolution of hardening/softening is controlled by the total length of the plastic tangential strain path. Locking behaviour under normal loading and dilatancy varying with stress and strain level are also accounted for. Examples of application of the model to the reproduction of test results are presented. The model can be generalized to include anisotropy effects. Finally, some aspects of the procedures used in the numerical implementation of the constituive law are described.

A thermodynamically consistent approach to microplane theory. Part II. Dissipation and inelastic constitutive modeling

The main objective of the present work is to provide a general framework for constitutive laws based on the microplane theory applicable to any kind of rheological behavior. Therefore, a thermodynamically consistent concept of deriving microplane based constitutive equations is presented. Microscopic constitutive laws are formulated on characteristic material planes, the so-called microplanes, resulting in an overall anisotropic macroscopic material characterization. The microscopic strain components of one plane are derived by the projection of the macroscopic strain tensor, leading to a kinematically constrained model. As proposed in the first part of this paper, the introduction of individual potentials on each microplane yields thermodynamically consistent microplane laws. They can be related to the macroscopic material description through an integration over the hemisphere. The microplane laws are chosen such that the macroscopic version of the Clausius-Duhem inequality is satisfied. This generic concept will be applied to the classical models of elasticity, elasto-damage and elasto-plasticity. The results are documented by the analysis of pointwise texture evolution for the model problems of uniaxial tension and simple shear.

A framework for geometrically nonlinear continuum damage mechanics

The objective of this work is the derivation of a framework for geometrically nonlinear continuum damage mechanics which allows for the description of damage by second order tensors. To this end, so called fictitious undamaged or rather microscopic configurations are considered which are related to the macroscopic configurations by linear tangent maps which allow for the interpretation as damage deformation gradients. The corresponding Finger tensors defined in the macroscopic configuration are then understood as damaged metrics for measuring the free energy for given strains. Thereby, the underlying motivation is provided by the hypothesis of strain energy equivalence between microscopic and macroscopic configurations. Based on the standard dissipative material approach the constitutive framework is completed by stresses, damage stresses, a damage condition and the associated evolution laws for the damage metric and the internal hardening variable. The relationship of the present damage metric based theory to the classical understanding of damage as an area reduction is highlighted. Finally, a simple prototype model problem is presented which allows for an efficient and accurate algorithmic treatment. Numerical examples demonstrate the applicability of the proposed framework.

A framework for microplane models at large strain, with application to hyperelasticity

A general framework is proposed for the formulation of microplane models at large strain. It is based on the thermodynamic approach to microplane formulation recently presented by the authors, which defines the macroscopic free energy of the material as an integral of a microplane free-energy potential over all possible orientations. By simple differentiation with respect to strain, it is possible to obtain the consistent definition of microplane stresses and integral expressions for evaluation of the macroscopic stress tensor. To apply this approach to large strains, new microplane strain measures need to be defined, including volume change, stretch of fibers, ‘‘thickening’’ of planes, deviatoric parts of the stretch and thickening, and distortion (shear) angles. Based on these, various microplane formulations are developed. Each formulation starts with the definition of microplane stresses and the derivation of the integral expressions which are valid for the general case of dissipative materials. Then, these expressions are particularized to specific forms of hyperelastic potentials leading to various hyperelastic models. The simplest model, with a quadratic microplane potential in terms of the fiber stretch, corresponds to the classical Gaussian statistical theory of long-chain molecules and leads to the neo-Hookean type of macroscopic free-energy potential. Many other, more complex forms of the microplane potential are investigated and their relation to existing models for rubber elasticity is analyzed. It is shown that, in the small-strain limit, they collapse into well-known small-strain microplane formulations, either with restricted or with unrestricted values of Poissons ratio. 2003 Elsevier Ltd. All rights reserved.