Chapter Three€¦ · (in a natural way) the material stability to the convexity requirements imposed on the integrand of the quasihyperelastic stress potential; (iii) it allows us

A mathematical description of damage anddeformation

S. Dimitrov & E. SchnackDepartment of Solid Mechanics,University of Karlsruhe, Germany.

Chapter Three

Abstract

The primary objective of this chapter is two-fold, first it aims to provide a con-cise and brief overview of the main physical phenomena behind the failure effectsattaining the evolution of deformation patterns; secondly, it aims to rationalize theinterplay between damage and deformation in terms of an incremental quasihyper-elastic constitutive description of the material response. This kind of constitutivedescription possesses plenty of advantages, among those: (i) it reveals a strong(meaning natural) conceptual correlation between the underlying physics and theanalytical framework that allows one to accommodate in a unified and very elegantform not only the intrinsic but the extrinsic type of failure modes also; (ii) it links(in a natural way) the material stability to the convexity requirements imposed onthe integrand of the quasihyperelastic stress potential; (iii) it allows us to formulatethe internal (material) dissipation in gradientless form on the basis of dissipationdistances measured between two inelastic states on the internal variables manifold.

In addition, the constitutive model of this type combines all the benefits of inter-nal variables and the functional formulations. Nevertheless, although this modellingparadigm is still in its youth it possesses the potential to become a strong and con-current field of research in continuum and computational solid mechanics in thenext decade.

1 Introduction

1.1 Background

Continuum damage mechanics (CDM) is a comparatively new research area in thetheory of response and reliability of engineering materials. It rests upon constitutiveequations defining material behavior weakened by many microdefects of irregular

WIT Transactions on State of the Art in Science and Engineering, Vol 1, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/1-85312-836-8/03

shape, randomly distributed in multiple scales and multiple dimensions. The phe-nomena of damage manifests itself either in surface (e.g. microcracks) or in bulk(e.g. microcavities) discontinuities being in permanent interaction and evolution. Itinvolves a rheological process that not only gives rise to a progressive net loss oflocal stiffness and integrity in the solid bodies, but also axiomatizes the kinematicalsplit of the total strain rate into elastic and inelastic counterparts.

The importance of the concept of damage strain rate can not be established,however, by studying the kinematics alone. Its relevance clearly in-creases whenone studies the restrictions placed on the constitutive assumptions by certain phe-nomenological rules. Those are implicitly postulated in the fundamental work [1]where the loss of stiffness is considered, always confined by three postulates: theprinciple of material frame-indifference, which asserts that the mechanical responseof a material is the same for all observers (excluding from consideration any rela-tivistic aspects), and the principles of determinism and local action, which assertthat the actual stress state at a particle is determined by the (arc-length) historyof an arbitrary small neighborhood of that particle. The corollary of these threeprinciples is that the change of macroscopic response, emerging by evolution ofmicro-defects is at least deterministic, and possibly gradual and on this basis thereduction of stiffness as a result of progressive loading, should be characterized bydeterministic macroscopic variables. In addition, the principle of local action allowsone to isolate, model and analyze a representative microdefect and after that, usingthe appropriate superposition, to employ the results for the case of periodic patternsarising from the reproduction of this microdefect in space.

It is beyond doubt, that the thermodynamics and the theory of internal vari-ables played the significant role for the confession of the phenomenological basis ofCDM. It was in the early 1960s when Walter Noll put to Bernard Coleman the ideathat the reduced Clausius–Duhem inequality (see p. 643 of [2]) is a restriction onall processes, admissible in the material (see also [3, 4]) and, because one defineseach material by giving a set of constitutive relations, this inequality must holdfor all processes compatible with those relations thus being a phenomenologicalrestriction on the constitutive equations. As a natural consequence seven years laterthe scientific world was pleased to read another fundamental work [5], which hasopened the door to a new stage before the mathematical modelling in continuummechanics, introducing the concept of internal variables. Later, it appeared to be acornerstone for the eloquent development of CDM since it asserts that the historyof material response and its change may be defined in terms of evolving internalvariables (see [6]) that depend on the mathematical expectation of the microdefectsdensity [7, 8].

The present chapter is written with the purpose of presenting the general princi-ples of inelastic damage accumulation in materials. It is predominantly analyticaland, as every work of such a nature, is based on traditions and history. The authorstry not to mix both, remembering that tradition evolves, but history extends. Themain approach adopted here focuses on one (very promising) concept, namely: theglobal functional representation of the incremental inelastic response based upon(non-convex) variational analysis. Being confined within the requirements of sim-plicity and clarity of exposition, the concern is only on materials whose responsecan be fully characterized using two scalar functions: first, reproducing the local

58 Advances in Fatigue, Fracture and Damage Assessment

WIT Transactions on State of the Art in Science and Engineering, Vol 1, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line)

storage of energy in the solid, and secondly rationalizing the energy dissipation.Such a conceptual way harks back to the works of Maurice Biot [9] and HansZiegler [10] (see also [11]). For this class of materials the non-isochoric processof damage accumulation relates to the constitutive response via an homogenizedCauchy stress tensor. It is shown that the Cauchy stress tensor can be obtained froma preliminary constructed incremental inelastic potential function that depends ontotal strain and an internal variables vector. This way is in principal analogy withthe well-known approach of the finite-deformations theory, where the second Piola-Kirchhoff stress tensor can be calculated as the first derivative of the strain-dependenthyperelastic stress potential function. The construction of the inelastic potential isperformed in the form of the incremental minimization problem written in terms ofthe above-mentioned stored energy function and dissipation function. In fact, theunderlying method for calculation of inelastic potential is crucially related to thedetermination of the geodesics on the manifold1 of internal variables with respectto the metric induced by the scalar dissipation function. This geodesics is called acurve of minimal dissipation length, and represents an extremal path in the spaceof internal variables. In simple words the material 'moves' in the space of internalvariables so that the dissipation path is extremal.

In writing this chapter the aim was to achieve two objectives:

(i) to present a retrospectively balanced treatment in as quantitative and rigor-ous terms as possible, of the mechanics, physics and mathematics of inelasticdeformation for generalized standard materials (see [11]),

(ii) to develop a balanced perspective of the incremental variational ap-proachto continuum damage mechanics and limitations before it.

Finally, in this analytic study it is discussed, as simply as possible, the foundationof the field viewpoint, without aiming to provide the reader with a full panoply ofCDM tools of research. This treatise is intended for advanced readers, becauseuntypically, it presents the foundations of damage mechanics not as those appearedin the well-renowned books and dwell on the textbooks.

1.2 Terminology and general scheme of notations

The departure here is not away from the customary notations found in classicalbooks in continuum mechanics. The hope is that the adopted form is most easilyunderstandable by someone not prejudiced by a past acquaintance with the sub-ject. Elsewhere, direct notations are used rather than Cartesian or indexed coordi-nates. The terminology, e.g. vector, linear transformation, functional space, tensor

1In mathematics, a manifold is a topological space that looks locally like the well-known Euclideanspace Rn. A frequently given example in this context is the surface of a sphere, which is not a plane, butsmall patches of it are topologically equivalent to patches of the Euclidean plane. In material modellingthe set of internal variables with all their admissible values form a manifold. Since every manifoldis a topological space, the notion of distance between two ”positions” on the manifold can be intro-duced. When considered further in the thermodynamical context this notion is equivalent to the distancebetween two thermodynamical states, characterized by different values of internal variables. Ergo, astate without bulk microcracks and a state with bulk microcracks can correspond to two values of theinternal damage variable d, d = 0 and d = 0.5, respectively.

Advances in Fatigue, Fracture and Damage Assessment 59


product, etc. used in this work is fairly standard in the modern books of contin-uum mechanics and functional analysis and follows the widely accepted scheme ofnotions.

General scheme of notations:

Roman bold-faced minuscules a, u, v, ...: vectors and vector fields;x, y, z, ...: points of space.

Greek bold-faced minuscules σ, ε, ξ, ...: second-order tensors or tensorfields.

Italic light-faced minuscules a, α, γ,...: scalars or scalar fields.

Sans-serif bold-faced majuscules C, D, M,...: fourth-order tensors ortensor fields.

Italic light-faced majuscules Ω, Γ, θ,...: regions in Euclidean space.

Blackboard majuscules W, R, M, ...: vector spaces.

2 Physical motivation

Any material body deforms under applied external loading. From a practical (andhenceforth slightly prejudiced) point of view the deformation is considered elasticwhen it is reversible and time independent, i.e. when it vanishes instantaneouslyas soon as the external loading is removed. A reversible but time-dependent defor-mation is known as viscoelastic. The latter increases with progress of time afterapplication of the load and decreases 'slowly' after the load is removed. The defor-mation is called inelastic if it is irreversible or permanent. To a certain extent thisclassification is biased, since it makes sense only after particular selection and com-plete functional formulation of the material model (see also [12]).

More reasonable and helpful taxonomy can be constructed taking into accounttwo invariant characteristics of material response: dependence on the velocity atwhich loading is applied and presence of hysteresis effects. Considered in a purelymechanical context the behavior of material during the course of (strain-controlled)loading is characterized with a temporarily constant state of stress. Since this stressis reached asymptotically and represents the equilibrium of internal forces in mate-rial it is termed equilibrium stress. The set of equilibrium states of stress attributedto the loading process can be summarized using the term- equilibrium relation. Thisis simply the stress–strain dependence well recognized in the classical infinitesimaltheory of elasticity with generalized Hooke’s law. The graphical representation ofthe equilibrium relation is given with the help of a so-called equilibrium curve. Ifone analyzes the equilibrium curve there could be noted the following four typesof response: rate-independent without hysteresis, rate-dependent with hysteresis,rate-dependent without equilibrium hysteresis, rate-independent with equilibriumhysteresis.

With respect to these four categories there are exactly four matching mathemat-ical models: theory of elasticity, describing the rate-independent material behav-ior without hysteresis; theory of plasticity, describing rate-independent materialbehavior with hysteresis, the theory of viscoelasticity, describing the rate-dependent



response without equilibrium hysteresis and the theory of viscoplasticity that for-malizes the rate-dependent material behavior with equilibrium hysteresis.

Our discussion in this section concerns the physical aspects of the rate-independentresponse with hysteresis and corresponding background for introduction of thenotion of damage and microstructural instability.

2.1 Characterization of inelastic deformation in ductile single crystals

When a well-annealed monocrystal suitably oriented for single slip is subjectedto cyclic (plastic) strain-controlled loading, a rapid hardening occurs. With con-tinued cyclic straining the rate of hardening progressively diminishes leading toa quasi-static deformation state known as saturation and characterized by a sta-ble configuration of the stress–strain, equilibrium curve. As the early experimentalevidence shows (see [13]), the cyclic hardening exhibits a gradual sensitivity tothe amplitude of applied plastic-strain control. Accordingly, there could be distin-guished three regions in the development of the equilibrium curve that correspondto three different stages of evolution in the representative material microstructure.These regions are denoted, for convenience, with Latin numbers I, II and III.

The region I accommodates hardening characteristics that emerge at compar-atively low values of the plastic-strain amplitude. In fact, the initial few cyclesof alternating strain produce dislocations that accumulate on the primary glideplane. This is a geometrical consequence accounting for the lack of rotation inthe monocrystalline slip system with respect to the loading axis. Thus the primaryslip system remains unchanged and the most highly stressed. It serves as an attrac-tor of positive and negative edge dislocations that encounter and trap each otherin small distances producing, in this way, so-called dislocation dipoles (see [14]).With continued cycling, accumulation of dislocation dipoles emerges in a well-structurized geometrical network referred to in the literature as veins, bundles orloop patches (see more representative results for copper single crystals in [15]).Microscopy results identify the building elements of loop patches as elongated for-mations with axis parallel to the primary dislocation lines. These formations areseparated by structures that are dislocation free and thus descriptively denoted aschannels. The dislocation patches contribute to the rapid hardening in the region Iof the cyclic stress–strain curve by partially constraining the dislocation motion onthe primary slip system. Increasing the number of cycles increases both the densityof edge dislocations per loop patch as well as the volumetric fraction of the looppatches in the material [16]. From the macrostructural point of view in region I,the specimen is able to withstand (practically) an infinite number of fatigue cyclesbecause the cyclic plastic strain does not cause progressively accumulated damage.This is true up to the upper boundary of the region characterized by the value ofthe plastic-strain amplitude above which the development of microstructural cracksstarted. The fine set of loop patches serves further as a kernel for development ofa macroscopically observable set of localization bands (so-called persistent slipbands).

This process nucleates at strain amplitudes corresponding to the region II in theequilibrium curve. A persistent slip band (PSB) is composed of a large number ofslip planes that form a flat lamellar structure very similar to that emerging in shape-



memory alloys. A particular PSB spans the entire cross section of the monocrys-talline specimen. The internal configuration of a PSB is characterized by a periodicarray of dislocation walls that divide the persistent lamellae into channels. Thisstructure is considerably different with respect to the 'amount' of the accommo-dated plastic strain from that of the matrix (i.e. the rest of the material). In fact, thematrix preserves the initial loop patches structure that is able to resolve uniaxialplastic strains of order 10−4, i.e. the matrix undergoes only microscopic yielding.On the other hand, PSBs support high plastic-shear strains of the order of 0.01 andundergo macroscopic yielding. As a result, the process of saturation in region II ismore complex and consists of contributions due to a deformation within the matrixveins and deformation within persistent slip bands. The logical consequence ofsuch multiscale localization of deformation is that fatigue cracks emerge primarilyat the interface between persistent slip bands and matrix.

The region III in the equilibrium curve is comparatively short and correspondsto a rapid decrease in material stiffness primarily due to increased nucleation andcoalescence of microdefects. It determines the unstable material behavior attainedwith softening.

2.2 Characterization of deformation in ductile polycrystalline materials

Following the discussion of microstructural effects underlying fatigue damage inductile monocrystals the attention is redirected now to polycrystalline ductile mate-rials. For these, the mechanisms of cyclic deformation and damage as observed inmonocrystals are generally known to be applicable for grains close to the boundaryof the specimen. Different rules govern the initiation and coalescence of microde-fects in the bulk of material where fatigue cracks occur under hardening regimesquite different from those for single crystals.

Again the plastic strain is localized in persistent slip bands that in variance tomonocrystals does not span the entire specimen but only the corresponding crosssection of a single grain. The formation of PSBs with wall structure identical to thatfound in monocrystals is no longer confined to a single slip system. In summary,two factors underly the difference between deformation mechanisms in polycrys-talline and single-crystal materials: (a) the grains in a given polycrystalline ensem-ble have rather different orientations of primary slip systems, which emerges in (b)incompatibility of deformations between adjacent grains and promotion of multiax-ial loading and multiple slip. On the macroscopic observation scale the cyclic defor-mation of polycrystalline conglomerates is associated with two important effects:shakedown and the Bauschinger effect. In particular, under constant amplitude, andfully reversed strain control the cyclic hardening or softening at a particular point inbulk of the specimen causes an increase or decrease, respectively, in the axial strainamplitude. In a strain-controlled outset the respective strain amplitude reaches astable saturation after a certain time known as 'shakedown' period. In fact, the ini-tial loading cycles initiate in the material microstructural plastic flow that generatesresidual stresses. The latter can be of such magnitude that a steady deformationstate is attained after a certain number of load reversals wherein a closed cycleof entirely elastic reversed deformation occurs. Since in this situation there is noaccumulation of plastic strain in subsequent cycles the mechanical system is said



to have undergone shakedown. When the residual stresses from the plastic flowinduce a steady-state cyclic deformation in material that is purely elastic, then theshakedown effect is called elastic.

Plastic shakedown refers to the steady-state cyclic deformation at which alter-nating plasticity occurs without any accumulation of plastic strain. The shakedowneffect at high degree is a synonym of material stability since its existence deter-mines the existence of a stable equilibrium curve without bifurcations. The kine-matically admissible plastic-strain cycle in which the shakedown will not occur canbe determined using the kinematical upper-bound theorem for elastoplastic stabilityas stated by Koiter (see [17]). This theorem certifies that if the internal plastic workper cycle is exceeded by the work done by external loading then shakedown doesnot occur. In this case the plastic strain accumulates from cycle to cycle and growsunbounded up to so-called incremental collapse of material. The limiting value ofthe plastic-strain amplitude above which the shakedown effect is not observed isknown as the shakedown limit.

The other very important effect was described in 1886 by Bauschinger [18].Bauschinger observed that after tensile loading into the plastic range and unload-ing followed by compression the absolute value of the yield stress is smaller thanthe initial yield stress. The origins of this reduction are related to the anisotropicchanges in dislocation substructure of the polycrystalline specimen induced byreversed loading and in changes in the local stresses. The presence of this effectcomplicates to a significant degree the modelling of plastic flow. Usually it is lin-earized and frequently considered in the context of an appropriate kinematic hard-ening model.

2.3 Concluding remarks to ductile deformation process

In conclusion, the following important characteristics pertaining to the processof cyclic loading and fatigue-induced damage accumulation in ductile crystallinematerials are summarized:

- the strain field on the microscopic observation scale possesses a stronglyoscillating character that is inherited primarily from the initial localization ofdeformation in loop patches,

- the strain field on the macroscopic observation scale possesses also a stronglyoscillating character, due to a set of evolving persistent slip bands as well asthe active microcracks nucleated at the interfaces between PSBs and matrix.

These observations can be suitably rationalized in a constitutive model of ahomogenized macro-continuum with locally attached microstructural constituents.On the basis of a general internal variable formulation of inelasticity the localmicroscopic constitutive response can be determined in terms of two functions gov-erning the energy storage and dissipation, respectively. These functions inducedincremental quasi-hyperelastic microstress potential (IQMP) on the microscopiclevel, which could be obtained as a result of minimization with respect to internalvariables. The IQMP attains a minimum in the standard Sobolev–Morrey2 solution

2A functional space is a collection of real-valued continuous functions defined on some interval.Roughly speaking, when this is a vector space of weakly differentiable functions it is usually referredto as a Sobolev space. The question of who is the founder of this class of specialized functional spaces



spaces, only when there is no microstructure developed.The development of microstructure (ergo texture or pattern of microcracks) as a

result of cyclic loading leads to the oscillating character of the strain field and inconsequence to a lack of sequential lower semicontinuity of the integrand in thecorresponding minimization problem. When a further attempt to minimize such anon-lower semicontinous functional is made, the result is more and more rapidlyoscillating sequences of solutions, where for every sequence the minimum of thefunctional is different. Hence the classical minimization procedure can not be per-formed. In this case it is said that the integrand needs a relaxation by convexifica-tion in order to obtain the corresponding minimizing displacement field. The nextsections are devoted to these and other very interesting questions.

2.4 Characterization of deformation in materials with inferior cohesivestrength (brittle solids)

The evolution of deformation in materials with inferior cohesive strength (ergo,brittle and quasibrittle solids) crucially depends on the type of mechanism of nucle-ation and growth of microstructural defects. The latter are influenced by the ther-modynamic state of the material and the ambient stimuli (i.e. the characteristics ofthe loading including counterparts, frequency, amplitude, etc.).

Local tensile stresses, high strain and temperature rates, oxidation and subse-quent corrosion microstructural homogeneity (note–omogeneity!) almost alwayspromote the rate of damage accumulation, microcrack growth and failure. In con-trast, the local compressive stresses low-temperature rates, and the energy barriersassociated with heterogeneity of microstructure lead to trapping of damage accu-mulation and hence increase the microstructural-fracture toughness. In this contextthe rate and the stability of damage evolution are merely reflecting the balancebetween the damage-promoting mechanisms and damage-hindering mechanisms.

The balance between the microcrack nucleation and growth is manifested by theoverall stability of the process and failure modes. In general, the damage evolutionand failure modes range from those that are dominated by the growth of existingmicrocracks to those dominated by microcrack nucleation.

In terms of analytical modelling this situation is conceptually rather straightfor-ward,

- If the damage evolution mode is dominated by microcrack growth the macro-failure has a negligible correlation with the accumulated damage and effec-tive stiffness. Since the macro-failure threshold depends only on the maxi-mal size and orientation (with respect to the loading direction) of the largest

is more or less a subject of long time polemics. In fact, the notion was introduced by S.L. Sobolev inhis famous paper 'Sur un theoreme danalyse fonctionnelle' (1938). However, the mathematical struc-ture and significance for variational calculus were investigated in detail, by Charles Morrey Jr. in thesecond part of his joint contribution with J. Calkin,'Functions of several variables and absolute con-tinuity II' (1940). In honor of both brilliant mathematicians this chapter uses the reference Sobolev–Morrey space, when discussion concerns the mathematical properties of a particular class of weaklydifferentiable functions.



pre-existing crack the failure threshold is not dependent on average quanti-ties (such as overall elasticity tensor, overall conductivity tensor and so on).Observe that the rupture threshold is not invariant with respect to the varia-tions of the radii of penny-shaped cracks and crack number. The formulationof a failure-threshold criterion then should be introduced in the form of alimit surface representing a locus of all points for which Griffith’s criterionis satisfied.

- Let now the damage-evolution mode be dominated by the nucleation of newmicrocracks that are randomly distributed within the specimen and approxi-mately equal in size (i.e. the specimen response remains statistically homo-geneous up to nucleation of a macroscopic crack). This type of failure mayoccur in brittle or quasibrittle solids that are either subjected to a compressivestate of stress characterized by a very high ratio between the averaged pres-sure and the differential stress or is strictly damage tolerant. The connectivitythreshold can not be easily discerned macroscopically during the progress ofloading. A specimen that is broken into very small pieces will still hardenwhen subjected to hydrostatic compression. The onset of the connectivitytransition can only be detected by repeated unloading and application of shorttensile tractions. For this kind of material response the failure threshold mustbe formulated in terms of critical microcrack densities. The failure thresholdthen will be invariant with respect to the radii of penny-shaped cracks andtheir number. In other words the failure threshold is related to the effectivespecimen properties obtained as volumetric averages.

- One of the most, let us say, intriguing phenomena that is placed somehowbetween the first two types of damage evolution is localization of failure. Thistakes place at intermediate confinement levels. Microcracks grow and nucle-ate and the growth is dependent on the interaction between microcracks. Thestrain field becomes rather inhomogeneous and a large part of the specimenis in fact unloaded to the elastic state. Depending on the boundary conditions(even involving a thermal counterpart) the specimen itself will not rupture atthe onset of localization and may not soften either when, for instance, it iswide and very short. What is crucial here is that the energy functional fails tobe convex any longer and hence the failure criterion must accommodate thismathematical effect.

- The initial phase of inelastic deformation in damage-tolerant materials (ergoceramic composites with elongated inclusions) is dominated by microcracknucleation regardless of the sign of the principle stresses. The preliminaryaccumulated damage can be large enough to reduce the effective tangen-tial stiffness to zero, causing specimen failure in load-controlled region. Thefailure threshold is related to the accumulated damage and the magnitude ofthe effective stiffness. In displacement-controlled conditions the final failurewill, in the majority of cases, occur when the largest microcrack or micro-crack cluster grows to a critical size. The failure threshold depends then onthe extreme statistics (large stress concentrations) and can not be predicted by



models in which the damage parameter is related to the microcrack density.In addition, the localization phenomena can be described also as a transition

between the hardening (local) and softening (non-local) regimes of macroscopicresponse.

2.5 Concluding remarks to brittle deformation process

In view of the considerations discussed the inelastic failure modes can be dividedinto intrinsic (material) and extrinsic (structural) sub-classes. Besides,

- the failure threshold of the intrinsic failure modes depends on the density ofevolving microstructure (ergo microcrack pattern) and thus can be predictedfrom the constitutive properties derived by the methods of effective continua.At the incipient failure the material is still statistically homogeneous on themicroscale. The intrinsic failure does not necessarily represent a failure inthe common meaning of this word. A material can simply bifurcate into dif-ferent configurations that are still able to support the applied load;

- the failure threshold of the extrinsic failure modes does not depend exclu-sively on the effective constitutive properties and the microdefects density.The damage evolution and failure are dominated by the growth of existingmicrocracks. A specimen in a uniform state of stress ceases to be able tosupport the external load when a single microdefect of critical size becomesunstable in Griffith’s sense. The dependence on the specimen size, shape andboundary conditions are some of the recognized features of the extrinsic fail-ure modes.

3 Constitutive modelling

The seven axioms of continuum physics asserting the balance (conservation) of:mass, linear momentum, angular momentum, energy, electric charge, magnetic fluxand the principle of irreversibility expressed in terms of entropy, in themselves donot suffice to determine the deformation (motion) of a body subjected to given load-ing. Before this deterministic problem can be formulated, it is necessary to specifythe behavior of material of which the body is made. In every continuum-mechanicstreatise, such a specification is stated by constitutive laws, which relate the stresstensor to the deformation process. In this subsection the discussion is confined togeneralized standard materials (as referred to, in [11]), and corresponding constitu-tive laws all considered in an infinitesimal context.

3.1 Kinematics of deformation process

A body Ω is a set of elements called particles and labelled at their positions with X.The body is frequently referred to a system of coordinates that establishes one-to-one correspondence between material particles represented by material coordinatesX = X1,X2,X3 and triples x = x1,x2,x3 of real numbers. In this treatise,x is restricted to be a coordinate triple (point) in 3-dimensional Euclidean vectorspace E3 equipped with orthonormal basis ei, i = 1, 2, 3.



The one-to-one (homeomorphic) continuously differentiable correspondence χbetween material particles and coordinate triples determines a region χ(Ω) into E3

that is called the configuration of the body. In particular, x = χ(X) identifies theplace in E3 occupied by X, and X = χ−1(x) is the particle whose place is x. Theone-parameter family of regions χt(Ω)t∈[0;T ] in E3 that the body occupies whentime t changes between 0 and T determines the so-called deformation of the body.Thus,

x = χt(X) = χ(X, t), ∀X, t ∈ Ω × [0; T ]. (1)

The equation x = χ(X, t) can be regarded also as a transformation of the materialcoordinates X of the body to the new coordinates x, in the space equipped withthe Euclidean metric. If gij are the components of the Euclidean metric tensor,then the components Gkl of the metric tensor in the body are related to gij by thetransformation law

Gkl =∂xi

∂Xk

∂xj

∂Xlgij . (2)

Thus, different configurations will give rise to different metric tensors in thebody. Without lack of generality, one refers to the configuration at t = 0 as ini-tial (reference) and labels the places in the region χ0(Ω) with x, thus introduc-ing a homeomorphic identity map between X ∈ Ω and Euclidean points x ∈χ0(Ω). This corresponds to the assumption that at t = 0, the map χ0 preserves theEuclidean metric and hence the material body is initially non-deformed. Then, attime instant t the point χt(x) can be understood as the place occupied by x, in thedeformation χt, while

u(x, t) = χ(x, t) − x, (3)

is the displacement of x. For the sake of simplicity the explicit writing of the sym-bols x, and t is omitted. The tensor fields,

F = ∇χ, and ∇u = F − 1, (4)

are called, respectively the deformation gradient and displacement gradient, where1 is the second-order unity tensor. In order to exclude from consideration theself-penetration and annihilation of some regions in Ω, as physically meaninglessevents, it is assumed further that the second-order tensor F is non-singular, i.e.det [F] > 0. Non-singularity means that there is now such physically meaninglesseffects as annihilation or self-penetration of the material.

As far as the continuum mechanics of deformable bodies is concerned the atten-tion is naturally drawn to the kinematical behavior rather than rigid-body motion.The deviation from the latter is referred to as distortion or strain and is representedby a properly selected second-order tensor. Many different measures of strain havebeen proposed in the literature, as, however, was remarked by Truesdell (see p. 268of [2]), while in particular situations the specific choice of strain measure couldbe helpful, in general such contentions are mutually equivalent. The most usefulfor purposes hereafter is the finite-strain tensor introduced by Green (see p. 542 in[19])

F =12(FFT − 1), (5)



and its linearized equivalent formulated on the basis of

ε =12(∇u + (∇u)T), and hence E = ε +

12∇u∇uT. (6)

The equivalence between E and ε regarded in an infinitesimal context is easilyverified by just taking into account that within an error of O(‖∇u‖2), the contribu-tion ∇u∇uT is negligibly small in magnitude3.

The second-order tensor ε is called the total infinitesimal strain tensor. Consid-ered as a kinematic measure of deformation for generalized standard materials εis decomposed further into elastic and volume-preserving (isochoric) plastic parts,denoted by εe and εp, respectively, according to the relationship,

ε = εe + εp. (7)

The isochoric (volume-preserving) nature of the plastic flow is embodied into εp

by the assumption tr[εp] = 0. In the context of strain-driven deformation behaviorεe is regarded as an independent variable and the evolution of εp is defined throughthe flow rule (as described in the following). Thus, (7) can be viewed as the defini-tion of the elastic strain tensor as ε = ε − εp. In conclusion, note that in the localdecomposition (7) only the total strain tensor ε is considered to be the gradient ofthe displacement field u, the elastic-strain tensor is, in general, not a gradient of the”elastic” displacement field. Further, neither ε nor εp are skew-symmetric, they areassumed symmetric at the outset, and the phenomena of plastic spin, per se, shallbe excluded from consideration.

3.2 Internal variables representation of inelastic energetics

As is well known, the local constitutive response of material is constrained at pointχt(x) by the second principle of thermodynamics, which in a fully isothermal con-text is represented by the (reduced) Clausius-Plank inequality for the internal dis-sipation Dint,

Dint = σε − ψ ≥ 0, (8)

where, ψ is the stored (Helmholtz) free-energy function, σ is the Cauchy stress ten-sor and symbol ˙( ) denotes derivative with respect to the time t. The description inthe following is fairly general and is along the same conceptual line as the internalvariable formulation as outlined by Colleman and Noll [4].

The Helmholtz free-energy function has a meaning of thermodynamical poten-tial, and is defined as the Legendre conjugate4 as the total energy function e. For

3It is assumed the convention to denote contraction with respect to all indices of a particular tensorfield with a corresponding number of points in the inner product (for example, the inner product betweentwo second-order tensors a and b will be denoted as, a : b). The norm of the given symmetric second-order tensor a is then ‖a‖ =

√a : a.

4The notion of Legendre conjugacy is related to the so-called Legendre–Fenchel transformation. Thelatter allows one to describe a particular function using a different set of variables. In the context ofmaterial modelling, the basic is the set of internal variables. The evolution of internal variables is drivenby pseudo-forces that are conjugated to them in the sense of Legendre–Fenchel. In fact, without thenotion of conjugacy the mathematical formulation of evolution laws for internal variables would beimpossible.



isothermal processes, as considered here, it is more convenient to use ψ rather thane, since the physically more consistent analytic representation of the Clausius–Plankinequality. The potential ψ governs the local storage of mechanical energy at pointχt(x) and depends on the total strain tensor and generalized vector of internal vari-ables ζ. Without going into much detail, the Colleman method5 employed on (8)yields the following constitutive equation for stresses,

σ = ∂εψ(ε, ζ), (9)

and the so-called reduced dissipation inequality

ξ · ζ ≥ 0, where ξ = ∂ζψ(ε, ζ), (10)

where ξ is the generalized vector of internal forces conjugated to internal variablesζ. As usual, it is assumed that the initial configuration is undeformed and stressfree, which yields the normalization of the stored energy function according toψ(0, ζ) = 0 and ∂εψ(0, ζ) = 0.

The requirement is ζ ∈ M, where the manifold M is embedded in the set R3×3×

Rm. Thus the internal variables vector can have different constituents depending on

the physicality behind the microstructural changes. For example, in the case of vonMises plastic flow with isotropic hardening, accompanied by isotropic evolution ofmicrodefects, ζ can have the following representation,

ζ = εp, α, d, (11)

where d is the scalar damage variable and α is the equivalent plastic strain accumu-lated up to the moment t and calculated according to,

α =∫ t

0

‖εp‖dt. (12)

Changes in internal variables during the process of loading and unloading giverise to dissipation of energy. The dissipation could be a result of internal friction ordue to nucleation of internal surfaces etc. It can be rationalized for a broad spec-trum of ductile materials with the scalar dissipation function φ : ζ → R+, whichdepends on the flux of internal variables ζ only. The derivative of function φ withrespect to the rate ζ is proportional to the internal dissipative forces arising fromthe change in ζ. The function φ is convex on M and positively homogenous ofdegree-1, in the sense that φ(ηζ) = ηφ(ζ) for every positive scalar η. The posi-tive homogeneity is an elegant way to model the rate independence. Indeed, if φ ishomogenous of degree-1 then for η > 0, and rate ζ,

|φ(ηζ)|‖ηζ‖ =

|φ(ζ)|‖ζ‖ . (13)

In other words, any transformation of the time scale (induced by 'scale factor' η),which can also be interpreted as a change in the rate of process ζ, cancels out the

5See more in [5].



dissipation function. Using ψ and φ the evolution of ζ in time is given with the Biotdifferential equation [9],

∂ζψ(ε, ζ) + ∂ζφ(ζ) = 0. (14)

Because freezing the flow of internal variables leads to zero dissipation, the Biotequation is supplemented with condition φ(0) = 0. Finally, observe that the above-given properties of the dissipation function allow us to regard φ as an infinitesi-mal metric on the manifold M. Consider now a smooth evolution process (i.e. ζwell defined in space and time) of internal variables in the time interval [ti; tj ].The integral taken from the dissipation metric φ over [ti; tj ] determines the local(pointwise) dissipation distance, Dloc between the two states of evolution ζi, ζj ofinternal variables at ti and tj , respectively, i.e.

Dloc =∫ tj

ti

φ(ζ)dt. (15)

From this expression the total energy that the body dissipates between states ζi, ζj

could be obtained. This energy is determined simply as the integral over the wholebody Ω, from the local dissipation distance, i.e.

D(ζi, ζj) =∫

Ω

DlocdΩ. (16)

It measures the global dissipation of energy between internal states, characterizedby vectors ζi, ζj , respectively.

Bear in mind that it was never assumed that metric φ (and hence distance D) issymmetric (which would mean φ(−ζ) = φ(ζ)). In other words the order on theset of internal states will matter, and thus D(ζi, ζj) = D(ζj , ζi)6. In conclusion,

observe that D(ζi, ζj) has the physical meaning of energy, while∫Ω

φ(ζ)dΩ issimply the globally dissipated power. For the phenomenology developed in thefollowing it is necessary to introduce the notion of dissipation distance Dloc(ζi, ζ)between actual state ζi and any other admissible state ζ in the same time interval.To this end let us introduce the substitution,

ζ = ζi + ηv = ζj , η ∈ [0, 1], (17)

in (15). Here the state ζ is interpreted as an admissible linear variation of the stateζi in a sufficiently smooth ”direction” v. Then, after substituting (17) and exploringdegree-1 homogeneity of the dissipation function, one obtains,

Dloc = φ(ζi)(tj − ti) + η

∫ tj

ti

φ(v)dt. (18)

The total dissipation is calculated accordingly,

D =∫

Ω

DlocdΩ. (19)

6To this extent hence, the function φ violates the classical definition of metric. The violation, how-ever, is conceptually insignificant.



3.3 Global energetic formulation of the model

For generalized standard materials, the local energetics (and hence the mechanicalcharacterization of the deformation process in terms of strains and stresses) is com-pletely determined by the Helmholtz free-energy function ψ, dissipation functionφ, displacement boundary conditions u, tractions t and energy of external loading,

〈b, u〉 + 〈t, u〉 =∫

Ω

ρb · udΩ +∫

Γ

t · udΓ, (20)

where one made use of the symbol 〈·, ·〉 to define the L2(Ω) inner product of eitherfunctions, or tensors depending on the particular context. In (20) bt = b(x, t) isthe dead load, a function of position and time; ρ = ρ(x) is the time-independentmaterial density function.

The function ψ represents the local storage of (strain) energy. For the purposesof discussion, however, it is more suitable to define this function in integral form,additionally taking into account the energy of external loads. Accordingly, the totalreduced Helmholtz free energy of the solid body is calculated as

Ψ =∫

Ω

ψdΩ + [〈b, u〉 + 〈t, u〉], (21)

for any pair u, ζ ∈ Ut × M. The expression [〈b, u〉 + 〈t, u〉] corresponds to thework done by external forces at t for displacement u. Further, using this expressionone is able to formulate the following problem of evolution in the case of rate-independent elastoplasticity coupled to damage,

For all t ∈ [0, T ] find u, ζ ∈ Ut × M associated to ψ(ε, ζ), φ(ζ) suchthat the following inequalities hold,

Ψ(ε, ζ) ≤ Ψ(ε, ζ) + D(ζ, ζ), (22)

for all admissible u, ζ ∈ Ut × M, and

Ψ(εi+1, ζi+1) + D(ζi, ζi+1) ≤ Ψ(εi, ζi)−

−∫ ti+1

ti

[〈b, u〉 + 〈t, u〉]dt, (23)

for all ti, ti+1 ∈ [0, T ], with 0 ≤ ti ≤ ti+1 ≤ T .

It was assumed that at every fixed but otherwise arbitrary time instant t, ε =∇Su. Inequalities (22) and (23) are referred to as the stability and energy inequality,respectively. Both have a simple mechanical interpretation. The stability inequalityrepresents the fact that, while holding time t fixed, the system is stable in the sensethat any admissible change from ζ to ζ does not release more stored energy thanhas to be paid by dissipation.

In other words, at fixed t the energy state characterized in terms of actual inter-nal variables vector differs from any other energy state characterized in terms of



the admissible internal variables vector, at least with the dissipation distance cal-culated between actual and admissible states. The energy inequality says that thetotal reduced energy stored at ti+1 plus the energy dissipated in the time interval[ti, ti+1] has to be less than the total reduced energy at ti plus the work of externalforces7. The major importance of the above-given formulation is that neither Ψ northe dissipation distance D must be differentiable.

3.4 Flow rules

It is possible to show that the energy and stability inequalities lead to classicalflow rules for damage and plasticity. To this end let us introduce the conjugatedthermodynamical variables,

σ = ∂εψ(ε, ζ), ξ = −∂ζψ(ε, ζ), (24)

where σ is the (ergo, microscale) Cauchy stress tensor corresponding to total strainε, and ξ is the vector of thermodynamic variables conjugated to ζ. To be moreillustrative let us consider an extension of the example discussed in Section 3.2.Given a two-dimensional plate with unit thickness, under the plane-strain condition.The plate occupies the domain Ω ∈ E2. The internal variables vector is chosento account for plasticity combined with linear kinematic-isotropic hardening andisotropic damage. Accordingly,

ζT =

⎛⎜⎜⎜⎜⎝

εp

α

Ξ

d

⎞⎟⎟⎟⎟⎠ ∈ M × R

3×3 × R+ × R3×3 × R+, (25)

where Ξ denotes the traceless (i.e. tr[Ξ] = 0) kinematic hardening variable associ-ated to the back-stress tensor q, α is the isotropic hardening variable, and d is thescalar damage variable. Introducing further, the bulk κ ∈ R+ and shear µ ∈ R+

moduli, the fourth-order 'virgin' stiffness tensor C can be written in the form,

C = κ1 ⊗ 1 + 2µ

[I − 1

31 ⊗ 1

], (26)

with I, the fourth-order unit tensor. For convenience the matrix H of generalizedplasticity moduli is also introduced

H =

[12K 0T

0 13H1

], (27)

where K and H denote the isotropic and kinematic hardening moduli, respectively.With this notation at hand the following uncoupled form of the Helmholtz free-energy function can be considered,

ψ(ε, ζ) = ψe(ε ⊗ ε, εp ⊗ εp, d) + ψp(α,Ξ), (28)

7Observe that the work of external forces is negative.



where the elastic counterpart ψe(ε ⊗ ε, εp ⊗ εp, d) depends on the scalar damagevariable d and the two thermodynamic fluxes ε⊗ ε and εp ⊗ εp. The plastic coun-terpart ψp(α,Ξ) depends on the isotropic hardening variable α and the kinematichardening variable Ξ. Using,

ψe = d

[32κ1 ⊗ 1 + µ

[I − 1

31 ⊗ 1

]]:: [ε ⊗ ε − εp ⊗ εp]

= d

[32κ(tr[ε])2 + µ‖dev[ε]‖2 + ‖dev[εp]‖2

], (29)

and

ψp = α,ΞH

(α

Ξ

)=

12Kα2 +

13H‖Ξ‖2. (30)

For ψ one finally obtains,

ψ = d

[32κ(tr[ε])2 + µ‖dev[ε]‖2 + ‖dev[εp]‖2

]+

12Kα2 +

13H‖Ξ‖2. (31)

Let us now consider the concept of plastic relaxation. It can be employed by thefollowing additive decomposition of actual (effective) stress tensor σ,

σ = σ − σp. (32)

Here, σ is the elastic stress tensor related to the total strain ε through the consti-tutive laws of linearized elasticity. Substituting (32) in the Clausius–Plank form ofthe second law one has

σ : ε − ψ ≥ 0, (33)

i.e.[σ + σp] : ε − ψ ≥ 0. (34)

Further differentiating ψ in time one obtains,

[σ + σp] : ε − ∂εψ : ε − ∂εpψ : εp − ∂dψd − ∂αψα − ∂Ξψ : Ξ ≥ 0, (35)

and hence

[σ + σp − ∂εψ] : ε − ∂εpψ : εp − ∂dψd − ∂αψα − ∂Ξψ : Ξ ≥ 0. (36)

By imaging independent processes and requiring each one of them to be thermo-dynamically admissible one finally arrives at the following system of constitutiveequations for conjugated variables,

σ = ∂εψ − σp, σp = −∂εpψ, Y = −∂dψ, β = −∂αψ, q = −∂Ξψ. (37)

The latter could be determined more precisely using the above expression for ψ,i.e. ⎛

⎜⎜⎜⎜⎝σp

β

q

Y

⎞⎟⎟⎟⎟⎠ = ∂ζψ =

⎛⎜⎜⎜⎜⎝

2dµdev[εp]−Kα

− 23HΞ

− 12C :: [ε ⊗ ε − εp ⊗ εp]

⎞⎟⎟⎟⎟⎠ . (38)



From (37)1 it follows,

σ = ∂εψ + ∂εpψ. (39)

The assumed form of ψ shows the additive composition of thermodynamic fluxesε ⊗ ε and εp ⊗ εp in the damage-elastic counterpart of the Helmholtz free-energyfunction induces the classical kinematic decomposition (7) of the total strain tensor.Indeed, since the plastic strain tensor is traceless from the equation,

2dµ

[I − 1

31 ⊗ 1

]: εp = d

[κ1 ⊗ 1 + 2µ

[I − 1

31 ⊗ 1

]]: εp, (40)

in combination with the constitutive law (37)1 one has

σ = ∂εψ − σp = d

[κ1 ⊗ 1 + 2µ

[I − 1

31 ⊗ 1

]]: [ε − εp], (41)

which explicitly demonstrates that the actual (effective) stress state can be deter-mined from the additive split of total strain into elastic and plastic parts. In additionto these results one could remark that

dC = ∂ε⊗εψ = −∂εp⊗εpψ. (42)

Let us now return again to the analysis of plastic and damage evolution using stabil-ity and energy inequalities. First, the stability inequality will be redefined locally.To this end one needs some preliminary evaluations. Using (21) at the fixed time t,and performing local (Gateaux) variations ε = ε+ η∇Su, ζ = ζ + ηv on the totalstrain tensor ε and the vector of internal variables ζ, respectively,

δΨ(ε;∇Su, ζ; v) =∂

∂ηΨ(ε, ζ)

∣∣∣∣η=0

=∂

∂ηΨ(ε + η∇Su, ζ + ηv)

∣∣∣∣η=0

, (43)

for the Gateaux derivative at η = 0 one obtains,

δΨ =∫

Ω

[∂εψ : ∇Su + ∂ζψ · v]dΩ + [〈b,u〉 + 〈t, u〉Γσ ]. (44)

The Gateaux variation δD of the dissipation distance D can be obtained in the sameway. Taking into account (18), δD reads as

δD(ζ, ζ) =∂

∂ηδD(ζ, ζ + ηv)

∣∣∣∣η=0

=∫

Ω

∫ tj

ti

φ(v)dΩdt. (45)

Introducing the linear variations of ε and ζ in stability inequality (23) and takinginto account (44) and (45) as well as (16) leads to



∫Ω

[σ : ∇Su − ξ · v + Dloc(ζ, v)]dΩ − [〈b,u〉 + 〈t, u〉Γσ] ≥ 0, (46)

where, for the sake of notational simplicity the vector ξ of thermodynamic variablesconjugated to ζ was used. The inequality (46) emerges in the following unilaterallyconstrained local stability criterion, ∫

Ωσ : ∇Su = 〈b,u〉 + 〈t, u〉Γσ ,

ξ · v ≤ ∫ tj

tiφ(v)dt, for all admissible v.

(47)

The first equality is classical and represents in weak form the balance of linearmomentum. The second inequality furnishes the fact that in a locally stable materialthe ”work” done by internal fluxes ξ, conjugated to internal variables ζ does notovercome the dissipation threshold set by Dloc.

Let us now investigate the features of energy inequality. With the observationthat

Ψ(εi+1, ζi+1) − Ψ(εi, ζi) =∫ ti+1

ti

Ψdt

=∫ ti+1

ti

∫Ω

ψdΩ − [〈b, u〉 + 〈 ˙t, u〉Γσ] − [〈b, ˙u〉 + 〈t, ˙u〉Γσ

]

dt, (48)

the energy inequality is transformed in the following equivalent form∫ ti+1

ti

∫Ω

[ψ + φ(ζ)]dΩdt ≤∫ ti+1

ti

[〈b, ˙u〉 + 〈t, ˙u〉Γσ ]dt. (49)

Employing the chain rule for the time derivative of ψ one first obtains∫ ti+1

ti

∫Ω

[∂εψ : ε + ∂ζψ · ζ + φ(ζ)]dΩdt ≤∫ ti+1

ti

[〈b, ˙u〉 + 〈t, ˙u〉Γσ]dt, (50)

and after differentiating (47)2 in time and substituting the result in (49) finallyarrives at∫ ti+1

ti

∫Ω

φ(ζ)dΩdt ≤∫ ti+1

ti

∫Ω

ξ · ζdΩdt, ⇒∫

Ω

φ(ζ)dΩ ≤∫

Ω

ξ · ζdΩ. (51)

In conclusion, the following time-local form of energy and stability inequalities areobtained

Local stability criterion, ∫Ω

σ : ∇Su = 〈b, u〉 + 〈t, u〉Γσ ,

ξ · v ≤ ∫ tj

tiφ(v)dt, for all admissible v.

(52)

Local energy criterion, ∫Ω

φ(ζ)dΩ ≤∫

Ω

ξ · ζdΩ. (53)



To obtain further the analytic form of the flow rules it is necessary to employthe generalization of the notion of differential operator in the context of the above-derived time local stability and energy criteria (see Sect. 1.3.3). Let us observefirstly that differentiating the second inequality from the local stability criterion intime one obtains,

ξ · v + ξ · v ≤ φ(v), and thus ξ · v ≤ φ(v), (54)

or equivalently,

ξ · ˙ζ ≤ φ( ˙

ζ), (55)

for particular ζ. Observe that when ξ, ζ = ξ, ζ from (52)2 and (53) it follows,ξ · ζ ≤ φ(ζ), i.e. φ(ζ) has a meaning of reduced dissipation. With (55) in mindthe convex set E of admissible thermodynamic forces ξ calculated at ζ as a sub-differential,

E = ∂ζφ(ζ = 0) = ξ|ξ · ˙ζ ≤ φ( ˙

ζ), for all ζ ∈ M. (56)

is introduced. It is underlined that ∂ζφ is calculated for ζ = 0 (i.e. for fixed, non-

evolving microstructure at time instant t). Then the vector ξ ∈ E is the correspond-ing sub-gradient of the function φ. The interpretation of the set E is well recognizedin plasticity theory. This is the classical elastic domain. When plasticity is coupledto damage this set determines an effective elastic domain. On the basis of this con-cept we formally understand the evolution (flow) of vector ξ conjugated to ζ as,

ξ ∈ ∂ζφ, ⇔ 0 ∈ −ξ + ∂ζφ. (57)

In other words, the 'actual' value of ξ induced by the rate ζ is understood as an ele-ment of the set furnished by the set-valued function ∂ζφ. In smooth, single-surfaceinelasticity the sub-differential ∂ζφ is single-valued and thus ξ is unique. The alter-native definition of the flow rule could be obtained by employing the Legendre–Fenchel transformation on φ and introducing the notion of dual dissipation func-tion, φ∗, i.e.

φ∗(ξ) = supζ

[ζ · ζ − φ(ζ)]. (58)

Further, taking the sub-differential of φ∗(ξ) we arrive at,

ζ ∈ ∂ξφ∗(ξ) = NξE, ⇔ 0 ∈ −ζ + NξE, (59)

where NξE denotes the normal cone to E at ξ. The reduced dissipation inequalitythen takes the form,

∂ζφ · ζ ≥ 0. (60)

Inequality (60) serves as a phenomenological constraint on the dissipation func-tion. It is satisfied by assuming for φ the properties of a gauge, i.e. convex, andpositive definite. In this contribution the discussion is restricted to the case whenthe convex set E has a smooth boundary (see more details in Sect. 3.6). The smoothboundary is analytically determined with the so-called inelastic potential (in anal-ogy with the already discussed yield surface in classical plasticity). In general, the



inelastic potential depends on spatial position, internal variables, and the vector ofconjugated variables, i.e. f = f(x, ξ, ζ). It allows a simple and very useful repre-sentation of the normal cone in the form NξE = λ∂ξf(x, ξ, ζ) for λ > 0. Uti-lizing this representation in (94) one arrives at the classical Karush–Kuhn–Tuckerformulation of the flow rules,

0 ∈ −ζ + NξE = −ζ + λ∂ξf(x, ξ, ζ) (61)

λ > 0, f(x, ξ, ζ) ≤ 0, λf(x, ξ, ζ) = 0. (62)

For smooth single-surface inelasticity defined only in the space of conjugatedthermodynamics variables these transform into,

ζ = λ∂ξf(ξ) (63)

λ ≥ 0, f(ξ) ≤ 0, λf(ξ) = 0, (64)

where the spatial position x was omitted for notational convenience.

3.5 Incremental variational representation of constitutive response

In this section we will describe a consistent approximation of constitutive responsenot in terms of classical Biot equations but in the form of an incremental stresspotential function. This approximation solves the stability and energy inequali-ties and allows us to cast the inelasticity due to damage and plasticity in termsof classical finite strain-theory. In this context the incremental variational formu-lation outlined in the following is along the same conceptual line as the so-calleddeformation theories of plasticity [20] and is based on the definition of the inelasticpotential function W , that under prescribed (i.e. known) ε, depends on the strainεi+1 at ti+1. Using this function the stress at ti+1 can be easily determined by thequasi-hyperelastic constitutive relation,

σi+1 = ∂εW (εi+1). (65)

To construct W let us first analyze the energy inequality in representation (50).The latter clearly defines the global balance of energy in the material. The left sideof (50) has the meaning of stored energy, while the right side represents the energyof external loading. The nontypical introduction of dissipation function in the leftexpression aims to accommodate the inelastic reaction of the solid body to externalloading. As was explained in Sect. 2, this inelastic reaction demonstrates in thedevelopment of representative multiscale microstructure, ergo on the nanoscopicobservation scale, loop patches and dislocation walls followed on the microscopiclevel by persistent slip bands and cavities as well as interface microcracks. Hencewe are motivated to discuss the following local form of incremental inelastic strainenergy function,

W (εi+1) = infζ

∫ ti+1

ti

[ψ(ε, ζ) + φ(ζ)]dt, with ζi = ζ(ti). (66)

For prescribed strains, this problem leads to determination of the incremental inelas-tic potential W as a minimum of incremental work

∫ ti+1

ti[ψ(ε, ζ) + φ(ζ)]dt done



from 'internal' thermodynamic forces on the material. Starting from the given ini-tial condition for internal variables this problem defines an incremental extremalpath in the space M. The problem (66) is consistent with the Biot equation definedabove. To demonstrate this let us first recast (66) into the form

W (εi+1) = infζ

[ψ(ε, ζ)|t=ti+1

t=ti+

∫ ti+1

ti

φ(ζ)dt

], (67)

and then, find the corresponding Gateaux variations with respect to the linear vari-ation of vector ζ = ζ + ηv. Accordingly,

δψ(ε, ζ; v) =d

dηψ(ε, ζ + ηv)

∣∣∣∣η=0

= ∂ζψ · v, (68)

δφ(ζ; v) =d

dηφ(ζ + ηv)

∣∣∣∣η=0

= ∂ζφ · v. (69)

Substituting this result into (66) gives,

∂ζψ · v|t=ti+1t=ti

+∫ ti+1

ti

∂ζφ · vdt = 0. (70)

Then, after integration of the second term by parts one obtains

(∂ζψ + ∂ζφ) · v∣∣∣t=ti+1

t=ti

−∫ ti+1

ti

[∂

∂t∂ζφ

]· vdt = 0. (71)

Thus setting ‖v‖ = 1 the incremental variational problem leads to two Euler–Lagrange equations,

∂ζψ + ∂ζφ = 0 (72)

− ∂

∂t∂ζφ = 0. (73)

Equation (72) is exactly the Biot equation at the discrete right boundary of thetime increment [ti; ti+1]. The second equation determines the curve of minimaldissipation length or so-called geodesic curve on M with respect to the dissipationmetric induced by φ. As was mentioned at the beginning of this chapter the intro-duction of incremental inelastic potential allows at least 'straightforward' appli-cation of the existence results from finite-strain elasticity to initial-boundary valueproblems of small-strain incremental inelasticity. This is achieved by applying thenotion of weak convexity on the integrand of (67). Indeed, as was obtained first byTonelli [21], the existence of regular minimizing functions (minimizers) is ensuredif the potential W is a quasi-convex function with respect to strain εi+1 in the sensethat

W (εi+1) ≤∫

Ω

W (εi+1 + ∇Sw)dΩ, (74)

for every superimposed displacement field w w. In other words, for all displace-ment fluctuations w defined on the solid body Ω, with support on its boundary, the



homogeneous strain field εi+1 provides an absolute minimizer of the incrementalstress potential.

Since the incremental potential depends on the developed micsrostructure, thiscondition could be understood as a rule for microstructural instabilities in mate-rial. What is important for calculations, is that quasi-convexity (and hence rank-1convexity) implies strong ellipticity of the incremental stress potential, i.e.

[a ⊗ b] : ∂2εεW [a ⊗ b] ≥ 0, (75)

for every two mutually orthogonal vectors a, b such that, rank [a − b] = 1. In thiscontext the strong elipticity condition is equivalent to rank-1 convexity of W .

3.6 Analysis of the smooth-single surface models of inelasticity

In this section we focus attention on the smooth models of inelasticity and partic-ulary on the smooth models of elastoplasticity coupled to damage. In the classicaltreatment these describe the effective elastic domain E in terms of the smooth func-tion f called the inelastic potential. As was mentioned in Sect. 3.4. the inelasticpotential depends on the thermodynamic forces ξ conjugated to internal variablesζ. The boundary of the effective elastic domain E is described by the level surfacef(ξ) = r, with threshold r ∈ R+, while the elastic domain is described throughthe level-set function,

E = levrf = ξ|f(ξ) ≤ r. (76)

The level-set function f is assumed to be always positive and homogeneous ofdegree one. In contrast to [22] we do not require f to be a gauge, since this shallautomatically constrain us to the case of isotropic hardening only. It is required thatφ be a gauge function. Further, for the known elastic domain the dissipation func-tion φ for rate-independent models of elastoplasticity coupled to damage may bedefined as a generalization of the principle of maximum plastic dissipation. In fact,two principles govern this particular type of inelastic response. Assuming no statecoupling between plasticity and damage, those two central postulates are stated asfollows: (i) the principle of maximum plastic dissipation [23] in terms of the effec-tive stress tensor [24, 25]; and (ii) the principle of maximum damage dissipation[26]. In combination, both allow one to determine the dissipation function φ inclosed form. Here we will follow a different route to obtain the closed form of φ.Let us first observe that the second inequality in the local stability criterion has themeaning of an extremal problem. In fact, it motivates us to consider the dissipationfunction φ as a solution of the following constrained maximization problem,

φ(ζ) = supξ · ζ for all ξ ∈ E. (77)

This can be solved by introducing the positive Lagrangian multiplier λ and redefin-ing the constrained optimization problem (77) as an unconstrained one, i.e.

φ(ζ) = supξξ · ζ − λ(f(ξ) − r). (78)

As was already discussed, this problem results in the classical form of flow rules.Substituting ζ = λ∂ξf(ξ) in the reduced dissipation inequality one obtains,

ξ · [λ∂ξf(ξ)] ≥ 0, (79)



and after employing the Euler theorem for derivative of positive definite functionsof degree one,

λf(ξ) ≥ 0. (80)

Thus, the image of the function f can be considered as a scalar force that drivesthe amount λ of inelastic flow [22]. An important observation then is that the dissi-pation function φ and its conjugated φ∗ can be expressed from simple geometricalconsiderations in terms of those new forces and fluxes. Accordingly, substitutingthe flow rule ζ = λ∂ξf(ξ) in the unconstrained minimization problem (78) andtaking into account once again the Euler theorem, we arrive at

φ(ζ) = supξλf(ξ) − λ(f(ξ) − r). (81)

This result gives us a very simple and attractive, from a computational pointof view, representation of the dissipation function φ . For vigor of treatment thedissipation function already recast in terms of λ is defined as an extended real-valued function in the following form

φ(λ) =

λr, λ ≤ 0−∞, λ ≥ 0

. (82)

An alternative way to obtain φ is via geometric interpretation of the notion ofgauge function. Since the effective elastic domain is a convex set then the gaugefunction φ centered at its origin is defined by

φ(λ) = infλ > 0|ξ ∈ λE, (83)

where we set φ(λ) = +∞ if ξ ∈ λE but λ < 0. Obviously in the present case thisinterpretation transforms simply into (82).

The finite-element-based discretization of the constitutive framework outlinedhere can be found in [22].

References

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[2] Truesdell, C. & Toupin, R., The classical field theories. Vol. III/1 of Handbuchder Physik, ed. S. Fluegge, Springer-Verlag, 1960.

[3] Coleman, B., Thermodynamics of materials with memory. Archive for Ratio-nal Mechanics and Analysis, 17, pp. 1–46, 1964.

[4] Coleman, B. & Noll, W., Thermodynamics of elastic materials with heat con-duction and viscosity. Archive for Rational Mechanics and Analysis, 13, pp.167–178, 1963.

[5] Coleman, B. & Gurtin, M., Thermodynamics with internal state variables.Journal of Chemical Physics, 47, pp. 597–613, 1967.

[6] Maugin, G., The Thermomechanics of Plasticity and Fracture. CambridgeUniversity Press, 1992.



[7] Krajcinovic, D. & Fonseka, G., The continuum damage theory of brittle mate-rials: 1. general theory. Transactions of the ASME-Journal of applied mechan-ics, 48(4), pp. 809–815, 1981.

[8] Krajcinovic, D., Selection of damage parameter - art of science? Mechanicsof Materials, 28, pp. 165–179, 1998.

[9] Biot, M., Mechanics of Incremental Deformations. John Wiley and Sons,1965.

[10] Ziegler, H., Some extremum principles in irreversible thermodynamics withapplication to continuum mechanics. Vol. IV of Progress in Solid Mechanics,eds. I. Sneddon & R. Hill, North-Holland Publishing Company, 1963.

[11] Halphen, B. & Nguyen, Q., Sur les materiaux standards generalises. Journalde Mecanique, 14, pp. 39–63, 1975.

[12] Betten, J., Net-stress analysis in creep mechanics. Archive of Applied Mechan-ics (Ingenieur-Archiv), 52, pp. 405–419, 1982.

[13] Mughrabi, H., The cyclic hardening and saturation behaviour of copper singlecrystals. Materials Science and Engineering, 33, pp. 207–223, 1973.

[14] Mughrabi, H., Microscopic mechanisms of metal fatigue. Vol. 3 of TheStrength of Metals and Alloys, eds. V.G. P. Haasen & G. Kostorz, Oxford:Pergamon Press, pp. 1615–1639, 1980.

[15] Z. Basinski, A.K. & Basinski, S., The temperature dependence of the satu-ration and stress and dislocation structure in fatigued copper single crystals.Acta Metalurgica, 28, pp. 191–207, 1980.

[16] Mughrabi, H.K., H. & Stark, X., Cyclic deformation and fatigue behaviourof α-iron mono- and polycrystals. International Journal of Fracture, 17, pp.193–220, 1981.

[17] Koiter, W., A new general theorem on shake-down of elastic-plastic structures.Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, SeriesB 59, eds. C.A. Brebbia, C. Maksimovic & M. Radojkowic, Elsevier, pp. 24–34, 1956.

[18] Bauchinger, J., Ueber die veranderungen der elastizitatsgrenze erwaermanabkuhlen und durch oftmals wiederholte belastung. Mitteilung aus dem Mech-anisch, Technischen Laboratorium der Koenigliche Technische Hochschule zuMuenchen, 13(5), 1886.

[19] Truesdell, C. & Noll, W., The Nonlinear Field Theories of Mechanics.Springer-Verlag, 1992.

[20] Martin, J., Plasticity. The MIT Press, 1975.[21] Tonelli, L., Criteri per lesistenza della soluzione in problemi di calcolo delle

variazioni. Annali di matematica pura ed applicata, 3(30), 1921.[22] Miehe, C., Schotte, J. & Lambrecht, M., Homogenization of inelastic solid

materials at finite strains based on incremental minimization principles. appli-cation to the texture analysis of polycrystals. Journal of the Mechanics andPhysics of Solids, 50(10), pp. 2123–2167, 2002.

[23] Lubliner, J., A maximum dissipation principle in generalized plasticity. ActaMechanica, 52, pp. 225–237, 1984.

[24] Lemaitre, J. & Chaboche, J.L., Mechanics of Solid Materials. Cambridge Uni-versity Press, 1990.



[25] Lemaitre, J. & Plumtree, A., Application of damage concepts to predict creep-fatigue failures. ASME, Journal of Engineering Materials and Technology,101, pp. 284–292, 1979.

[26] Simo, J. & Ju, J., Strain and stress based continuum damage models, part 1:Formulation. International Journal of Solids and Structures, 23(7), pp. 821–840, 1987.



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Chapter Three€¦ · (in a natural way) the material stability to the convexity requirements imposed on the integrand of the quasihyperelastic stress potential; (iii) it allows us