Egidio Rizzi

On the Portevin¿Le Chatelier effect: theoretical modeling and numerical results

Abstract A new model is presented for a physically-consistent description of plastic material instabilities referred to as Portevin–Le Chatelier (PLC) effect, namely the oscillatory plastic flow that may be observed in metallic alloys subjected to load-or displacement-controlled plastic deformation in a certain range of strain, strain rate and temperature. The model is conceived to describe the kinetics of Dynamic Strain Ageing (DSA), that is the dynamic interaction between mobile dislocations and diffusing solute atoms which is known to be the primary mechanism inducing the PLC effect. The model is coupled in time and (one-dimensional) space and introduces two intrinsic time scales in the evolution equations and a characteristic length scale through a diffusion-like term with spatial second-order gradient. Approximate analytical solutions are first derived for the boundaries of the PLC range and for the strain localization characteristics defining the kinematics of PLC deformation bands. Numerical results are then obtained through Finite Differences solutions of the space–time coupled equations: a considerable wealth of features is discovered, including the appearance of various PLC band types depending on the applied strain rate. Phenomenological characteristics (i.e. stress–strain curves) are presented together with the corresponding space–time patterns of strain localization. The obtained results are in agreement with the analytical solutions developed here and with the available experimental observations on the PLC effect.

Mechanical behavior of a syntactic foam: experiments and modeling

This paper reports the results of a research activity concerning the mechanical behavior of a syntactic foam employed as core material for sandwich composite panels. Following a purely phenomenological approach, experimental and numerical results are presented and compared at the macroscopic scale. The main features observed in the uniaxial, biaxial and Three Point Bending (TPB) tests are highlighted. A bimodulus constitutive model of the Drucker-Prager type is chosen for modeling biaxial stress states with diffused damage. An alternative discrete crack approach is devised for the computer simulation of the (TPB) three point bending tests: the best matching is achieved for a rectangular Dugdale-type cohesive law. Though not proposing novel experimental or numerical methodologies, the present engineering approach should interest readers generally involved in computational composite mechanics and, specifically, in modeling particulate composites of the type considered here.

On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo-logarithmic damage tensor rate

Abstract In spite of its appeal, anisotropic damage is being introduced in the constitutive equations of engineering materials at a slow pace. One of the main reasons is the difficulty of establishing general evolution laws. This originates from the lack of physical meaning of the thermodynamic forces conjugate to the damage variables, which finally constitute the space in which loading functions and `damage rules are defined. In this article, the authors propose a new `pseudo-logarithmic rate of damage, which has the advantage of exhibiting a simple and meaningful conjugate force with very convenient properties. A main advantage is the physical interpretation of the corresponding “damage rule”, which clearly separates the effects of its volumetric part, responsible for isotropic degradation, from its deviatoric part, responsible for anisotropic effects. This new concept is applied to a second-order tensor secant formulation, which is developed using traditional concepts of continuum damage mechanics within the general theoretical framework of elastic degradation and damage recently proposed by the authors. A first example of anisotropic damage formulation based on these concepts, the `generalized pseudo-Rankine model, is presented and verified with analytical and numerical examples in a companion `Part II paper.

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.

Experimental characterization and numerical simulations of a syntactic-foam/glass-fibre composite sandwich

This note presents the main results of an experimental and numerical investigation on the mechanical behaviour of a composite sandwich primarily designed for naval engineering applications. The skins of the sandwich are made of glass-fibre/polymer-matrix composites; their interior layers are connected with interwoven threads called piles which cross the sandwich core. Such core consists of a syntactic foam made by hollow glass microspheres embedded in an epoxy matrix. Experimental tests and numerical finite element (FE) simulations on both the sandwich composite and its separate components have been performed in order to characterise fully the complex mechanical behaviour of such a highly heterogeneous material.

On the formulation of anisotropic elastic degradation.: II. Generalized pseudo-Rankine model for tensile damage

Abstract In the companion `Part I article, the theoretical aspects of anisotropic damage based on second-order tensors were discussed, and the concept of pseudo-logarithmic rate of damage was introduced. The thermodynamic forces conjugate to this damage rate exhibit physical meaning, which greatly simplifies the task of defining loading surfaces and evolution laws. In this second part, a formulation for anisotropic tensile damage which takes advantage of those concepts is developed and verified: the `generalized pseudo-Rankine model. Depending on the value of a single parameter, the loading surface in pseudo-log space may assume shapes which vary gradually between a π-plane and a Rankine-type criterion. This corresponds to a transition from a purely isotropic to a highly anisotropic tensile degradation model. In spite of the relative complexity of anisotropy, one of the important advantages of the model is that closed-form solutions are possible for a number of simple loading cases. The first one developed is uniaxial tension, which makes it possible to interpret the remaining two material parameters in terms of the tensile strength σt and fracture energy per unit volume gf. Adding the two isotropic elastic constants, this makes a total of only five material parameters. Additional closed-form solutions are developed for the simple loading cases of pure shear, pure distortion, and uniaxial tension after tensile loading–unloading in a perpendicular direction. The behavior of the new model under complex loading histories is illustrated with a numerical tension/shear test with a significant rotation of principal strains.

On the kinematics of Portevin-Le Chatelier bands: theoretical and numerical modelling

A model is presented for the description of the Portevin–Le Chatelier (PLC) effect, namely the repetitive plastic yielding that may be observed in metallic alloys for certain ranges of temperature and applied stress or strain rates. The model reflects the underlying microstructural dynamic strain ageing (DSA), i.e. the dynamic interaction between mobile dislocations and diffusing solute atoms. Focus is made on PLC instabilities of Type A, that is PLC bands that nucleate and propagate continuously throughout the tensile specimen as solitary plastic waves. The kinematics of these PLC bands is studied analytically and validated numerically by a Finite Differences integration of the model equations. The characteristics of the PLC bands, that is band speed, band width and band plastic strain, are determined from the space–time fields of plastic activity. The band parameters exhibit very good matching with the theoretical results and order-of-magnitude agreement with the experimental observation of the PLC effect.

On failure indicators in multidissipative materials

Abstract Multi-dissipative constitutive descriptions of irreversible material degradation result in tangent operators that are made up of multiple rank-one updates of the elasticity tensors: multisurface elastoplasticity, plastic yielding combined with elastic degradation and multicrack models are representative examples. The spectral properties of these tangent operators determine failure conditions at the material level in terms of loss of uniqueness and discontinuous bifurcation of the incremental response. These failure properties are analysed herein by studying the eigensolution of the sum of n rank-one ( m × m ) matrices, and the eigensolution of n rank-one updates of the ( m × m ) identity matrix, whereby the tangent material tensors are written in matrix form. Analytical eigensolutions are presented and interpreted mechanically in terms of continuous and discontinuous failure indicators. In the light of these spectral properties, the failure indicators of single-dissipative materials are revisited and explicit results are presented for double-dissipative models in the form of plasticity combined with elastic-damage. In particular, it is shown that the activation and interaction of two dissipation processes may destabilize the tangent operators beyond the state resulting from a single active mechanism.

Relations between drained and undrained moduli in anisotropic poroelasticity

Although the knowledge of the drained moduli is often assumed to define the material coefficients of elastic fluid-saturated porous media, it is not sufficient. Resorting to the properties of the constituents is possible but may not be satisfactory due to lack of accuracy. On the other hand, the mechanical information contained in the undrained moduli is complementary to that provided by the drained moduli but is also overabundant. The compatibility relations between these two types of moduli are examined for several classes of anisotropic solid skeletons and the information required from the undrained moduli is exactly defined through a spectral analysis of the dyadic difference in tensor compliances. A switch of the results is possible if the undrained moduli are given instead of the drained moduli. An incomplete data set of material coefficients for a transverse isotropic shale is treated as an example. Considerable simplifications arise for a particular form of anisotropy defined by a second order fabric tensor.

A Formulation of Anisotropic Elastic Damage Using Compact Tensor Formalism

An anisotropic elastic-damage model for initially-isotropic materials is presented. The model is based on a pseudo-logarithmic second-order damage tensor rate. To derive the complete expression of the tangent stiffness entering the rate constitutive law, various tensor operations and derivatives of tensor functions must be developed. Such derivations have been performed in compact form. Some useful tensor derivatives and a table of tensor algebra operations are given in Appendix. This note should interest engineering researchers involved in the development of constitutive models through tensor formalism.