cover page

Multi-mechanism Modeling of Inelastic Material Behavior

Georges Cailletaud

Kacem Saï

Lakhdar Taleb

image

Preface

It is common to say that each industrial sector relies on the performance of its materials. Examples of this type are multiple, think of the railways (development of rail steels at the end of the 19th Century), civil construction (development of advanced concrete formulations), the space shuttle (composites, carbon–carbon tiles) or aeronautics, where the performance of reactors depends on the maximum temperature supported by the materials in the hottest areas. But in fact, it would be more accurate to say that the performance obtained also depends on knowledge of the material used. An improved knowledge of the material paves the way for a structure design that, in addition to its elegance, has two important aspects: (1) safety improvement, insofar as having a good knowledge of the physical phenomena is better than applying large safety parameters, which often sound like coefficients to hide ignorance; (2) better ecological performance, since weight reduction decreases the fuel consumption of cars or aircrafts.

This modern approach has been greatly facilitated by the tremendous increase in the power of computers and the robustness of the numerical algorithms. Materials are too often forgotten in this process, and engineers’ fear is then to apply the “garbage in, garbage out” proverb when there is such a weak point in the calculation chain. Yet researchers have made considerable progress in the field of material modeling. The time has now come for efforts to popularize the models obtained and to encourage their use by providing examples on materials of current use.

About 50 years after Mandel’s paper on the “Généralisation de la théorie de la plasticité de W. T. Koiter” in Int. J. Solids Struct. (1965), the authors of the present book decided that it was time to gather the most recent results on the field of the so-called multi-mechanism models (MM). After Zarka and his co-workers, they have been active developers of this model class, which was reformulated in a thermodynamic framework, introduced in finite element codes, and adapted for a large number of materials and loading conditions. As a result, they have reached a good level of maturity. They offer a versatile toolbox to develop new constitutive equations for metals, polymers and geomaterials under monotonic or cyclic loading paths.

The implementation of the ideas was rather fast, and it was then time to start writing. This was spread over several years. It gave rise to several discoveries and developments of the models that had remained unexploited in the original versions. It has been enriched by ongoing research, which ensures that the document is fully up to date. Being able to resume the work begun every time a niche was released in the schedule was only possible, thanks to the mutual encouragements that the authors gave each other and to the sincere friendship that now crowns their efforts.

The hope is that the text now meets its readers, that it allows the sharing of results and a certain know-how, and that it is useful for students, researchers and engineers who will have it in hand.

The authors extend their sincere thanks to Vladislav A. Yastrebov, who designed the cover illustration from the map of a plastic deformation field in a cobalt matrix tungsten carbide composite. They would also like to thank Odile Adam for the careful attention she payed to the review of the bibliographies.

Georges CAILLETAUD
Kacem SAÏ
Lakhdar TALEB
October 2017

Introduction

“Plasticity” is a word with various meanings that change from brain reconstruction to metal deformation processes depending on the context and speaker. As far as materials are concerned, the aim of researchers is to develop not only models with a physical basis, but also models that are able to deliver results in minutes rather than hours. This is why the literature contains two types of approaches, namely (poly-)crystal plasticity and macroscopic models. The former are inherited from the early work of Taylor (1938), then Koiter (1953). It is clearly acknowledged in this case that plasticity has a lot of sources at the microstructure scale, and that plastic flow has to be built by collecting the contribution of each slip system. After von Mises (1911), the latter have been clearly formulated by Hill (1950), then Rice (1972) and Mandel (1972), with the introduction of internal variables to describe hardening, and Germain (1973) who elaborated the thermodynamic formulation. In the 1950s, a few authors attempted to bridge the gap between the two model classes. What remains from crystal plasticity when embedded in a macroscopic approach is the concept of “mechanism” and the fact that plastic flow may have various origins, possibly at different scales in the material. Accordingly, there is a loss of information, as the morphology is no longer explicit, and the variables represent averages more than the precise physical mechanisms. Extensions of the corresponding models can be made by considering that a mechanism does not directly refer to a slip system, but is the result of a more complex, combined plastic flow.

The purpose of the book is to provide readers with a rigorous framework showing the limits of the validity of the multi-mechanism (MM) models, and also to exhibit a series of “success stories” where the models correctly represent complex material behavior. Depending on the context, the mechanisms can be associated with different phases or follow various deformation regimes. Each mechanism will be represented by a (pseudo-) potential, from which plastic or viscoplastic flow can be derived. In all chapters, precise explanations are given, such that the reader can reproduce the calculations and apply the equations to his own experimental data base.

After a short state of the art (Chapter 1), the model formulation shown in Chapter 2 first mentions the thermodynamic framework, then derives two sets of models, namely (i) where each mechanism has its own plasticity criterion or (ii) all the mechanisms contribute to an unique plasticity criterion. In both cases, isotropic and kinematic hardening can be introduced. Chapter 3 contains a series of typical mechanical responses. It describes how time-dependent and time-independent flow can be combined to produce relevant histories in plasticity and creep. It also discusses the behavior of the various versions for the case of specific phenomena such as ratcheting or static recovery, and in the presence of non-proportional loading paths. Practical examples are provided in Chapter 4. A number of applications are devoted to metallic alloys (Ni-based, steels, aluminum alloys, zirconium alloys), but polymers (semi-crystalline, glassy) are also considered. The identification process is detailed, and the material parameters are available. Chapter 5 is devoted to damage. A few applications demonstrated that MM models may appear as good companions for expressions that introduce damage in metallic materials or mortar–rubber mixtures, under monotonic or cyclic loadings. The goal of Chapter 6 is to demonstrate that the models can be easily implemented in a finite element code in order to be used in structural calculations. After showing the framework for explicit and implicit integration, a few applications are shown, together with the full open source code.