Vlado A. Lubarda

THE ONSET OF TWINNING IN METALS: A CONSTITUTIVE DESCRIPTION

A constitutive approach is developed that predicts the critical stress for twinning as a function of external (temperature, strain rate) and internal (grain size, stacking-fault energy) parameters. Plastic defor- mation by slip and twinning are considered as competitive mechanisms. The twinning stress is equated to the slip stress based on the plastic flow by thermally assisted movement of dislocations over obstacles, which leads to successful prediction of the slip-twinning transition. The model is applied to body centered cubic, face centered cubic, and hexagonal metals and alloys: Fe, Cu, brasses, and Ti, respectively. A constitutive expression for the twinning stress in BCC metals is developed using dislocation emission from a source and the formation of pile-ups, as rate-controlling mechanism. Employing an Eshelby-type analysis, the critical size of twin nucleus and twinning stress are correlated to the twin-boundary energy, which is directly related to the stacking-fault energy (SFE) for FCC metals. The effects of grain size and SFE are examined and the results indicate that the grain-scale pile-ups are not the source of the stress concentrations giving rise to twinning in FCC metals. The constitutive description of the slip-twinning transition are incorporated into the Weertman-Ashby deformation mechanism maps, thereby enabling the introduction of a twinning domain. This is illustrated for titanium with a grain size of 100 µm.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Damage tensors and the crack density distribution

Abstract The paper presents an algorithm for the derivation of damage tensors emphasizing its relationship with the actual and approximate crack density distributions. The proposed model is illustrated using scalar, second and fourth order continuous tensor approximations of some typical two and three dimensional crack distributions. It is also shown that the occurrence of regions with negative crack density (anticrack regions) is in many cases a common and as yet unexplored feature of the approximate solutions.

On the mechanics of solids with a growing mass

A general constitutive theory of the stress-modulated growth of biomaterials is presented with a particular accent given to pseudo-elastic soft living tissues. The governing equations of the mechanics of solids with a growing mass are revisited within the framework of finite deformation continuum thermodynamics. The multiplicative decomposition of the deformation gradient into its elastic and growth parts is employed to study the growth of isotropic, transversely isotropic, and orthotropic biomaterials. An explicit representation of the growth part of the deformation gradient is given in each case, which leads to an effective incremental formulation in the analysis of the stress-modulated growth process. The rectangular components of the instantaneous elastic moduli tensor are derived corresponding to selected forms of the elastic strain energy function. Physically appealing structures of the stress-dependent evolution equations for the growth induced stretch ratios are proposed.

On the effective lattice parameter of binary alloys

Abstract A study of the effective lattice parameter in binary alloys is given based on an analysis of the volume change produced by dilute resolution of the solute atoms in the solvent matrix. An apparent size of the solute atom is incorporated in the analysis to approximately account for the electronic interactions between the outermost quantum shells of the solute and solvent atoms. The comparison with experimental data for various alloy systems and deviations from Vegard’s law are analyzed. The free energy of binary alloys and the ordering of solute atoms at higher solute concentrations are then discussed.

Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics

Some fundamental issues in the formulation of constitutive theories of material response based on the multiplicative decomposition of the deformation gradient are reviewed, with focus on finite deformation thermoelasticity, elastoplasticity, and biomechanics. The constitutive theory of isotropic thermoelasticity is first considered. The stress response and the entropy expression are derived in the case of quadratic dependence of the elastic strain energy on the finite elastic strain. Basic kinematic and kinetic aspects of the phenomenological and single crystal elastoplasticity within the framework of the multiplicative decomposition are presented. Attention is given to additive decompositions of the stress and strain rates into their elastic and plastic parts. The constitutive analysis of the stress-modulated growth of pseudo-elastic soft tissues is then presented. The elastic and growth parts of the deformation gradient and the rate of deformation tensor are defined and used to construct the corresponding rate-type biomechanic theory. The structure of the evolution equation for growth-induced stretch ratio is discussed. There are 112 references cited in this review article. DOI: 10.1115/1.1591000 The objective of this survey is to give an overview of the application of the multiplicative decomposition of the deformation gradient in constitutive theories of finite deformation thermoelasticity, elastoplasticity, and biomechanics. The multiplicative decomposition of the deformation gradient is based on an intermediate material configuration, which is obtained by a conceptual destressing of the currently deformed material configuration to zero stress. The significance of such configuration for material modeling was pointed out ˙

Damage model for brittle elastic solids with unequal tensile and compressive strengths

The paper presents a rate-type constitutive analysis of damage, applicable to brittle materials whose elastic properties degrade during a deformation process. Different tensile and compressive material responses are modeled incorporating positive and negative projections of the stress or strain tensors. Proposed evolution laws for the rate of compliance tensors are consistent with some of the prominent features of brittle material response. A new structure of the damage surface is introduced for a more accurate account of the effects of the hydrostatic states of stress on the overall response. Derived rate constitutive equations provide the explicit representation of the tangent compliance tensor. The proposed model is applied to uniaxial tension and compression to illustrate nonlinear relationships between stress and longitudinal, lateral, and volumetric strains. The proposed model is compared with some of the existing theories. DEGRADATION of elastic properties reflecting accumulating damage in brittle materials is primarily a consequence of the evolution of internal microcrack structure. Depending on the material microstructure and the current state of stress and strain and their rates, instantaneous material response and further evolution of damage may involve activation of different microcrack mechanisms. The tensile hoop stress generated at the surface of relatively large pores represents a preferential crack nucleation mechanism in very porous rocks. In low porosity, compact rocks frictional sliding of crack surfaces can destabilize original shear cracks, causing them to kink and develop wing cracks. Other mechanisms of microcracking are also possible. Some of these mechanisms were implemented in the failure analysis of brittle solids by Ashby and Sammis (I), and others. Progressive degradation of mechanical properties is an inherent feature of brittle material behavior. Several analytical models were developed to estimate the effective elastic properties of a solid weakened by a given distribution of microcracks or other defects. A comprehensive review of these models can be found in a recent treatise by Nemat-Nasser and Hori (2). An important issue in the formulation of continuum damage models is related to the appropriate choice of the mathematical form for the damage variable. This has been recently studied by Lubarda and Krajcinovic (3), who considered several frequently encountered two and three dimensional distri- butions of microcracks. Their analysis demonstrated the shortcomings of the scalar and second- order tensor damage variables, and the accuracy gained by using the fourth-order tensor variable. The mode and stability of crack growth and, therefore, behavior of brittle material strongly depends on the sign and magnitude of applied stresses. For instance? the response of brittle material subjected to a compressive loading is strongly dependent on the magnitude of lateral confinement. An unconfined specimen fails by axial splitting, attributed to unstable growth of a single crack, at relatively small microcrack density. As the confinement is increased, axial splitting is suppressed and at large levels of confinement homogeneous microcracking prevails throughout the sample, resulting in a quasi-ductile overall response (Horii and Nemat-Nasser (4)). The analysis presented in this paper addresses the important issue of modeling different tensile and compressive responses of brittle materials. The concept of positive and negative projections of stress and strain tensors is used to account approximately for two basic, tensile and compressive, damage evolution modes. This idea was independently introduced by Ladeveze and Lemaitre (5), Ortiz (6) and Mazars (7), and was subsequently utilized by Simo and Ju (8), Yazdani and Schreyer (9), Ju (IO), Stevens and Liu (l 11, Yazdani (12), and others. The positive parts of the stress

Some fundamental issues in rate theory of damage-elastoplasticity

Abstract The paper elaborates on some fundamental constitutive issues in the rate theory of damage-elastoplasticity. The analysis combines the constitutive theories of elastoplastic and progressively damaged solids. After defining needed kinematic and kinetic preliminaries, the anisotropic elastic response is analyzed by introducing a set of damage tensors which represent material degradation and induced elastic anisotropy. Decomposition of the rate of stress and deformation tensors into their elastic and inelastic parts is then defined in a manner analogous to the corresponding decomposition used in large-deformation elastoplasdcity theory. The procedure is further developed to partition the inelastic stress and strain rates into the damage and plastic parts, which takes into account the physics of these deformation processes. The energy dissipation rate is derived and the thermodynamic forces conjugate to elastic stiffness and compliance tensors are identified, based on a thermodynamic analysis of isothermal deformation process. The damage potentials for the corresponding fluxes are introduced and the constitutive expressions for the damage stress and strain rates are established. The concept of a damage surface is used to define the onset and evolution of damage. A constitutive analysis for inelastic stress and strain rates is then presented. The inelastic potential function and the yield surface are introduced. A dual formulation is constructed in both the stress and strain spaces. The two limiting cases, one involving plasticity without damage, and the other involving damage without plasticity, are deduced from the developed and more general constitutive framework of damage-elastoplasticity.

AN ANALYSIS OF EQUILIBRIUM DISLOCATION DISTRIBUTIONS

Equilibrium distributions of collections of discrete dislocations are analyzed, with the dislocations modelled as line defects in a linear elastic medium. The dislocated equilibrium configuration is determined by finding a minimum potential energy configuration, with respect to variations in the dislocation positions, for a fixed number and type of dislocations. Numerical results are presented for finite and infinite bodies with distributions of edge dislocations under plane strain conditions. Calculations involving doubly periodic arrays of cells, within which there is a single set of parallel slip planes, show a strong tendency for sharp dislocation walls to form. Perturbations of the wall structure due to the presence of pinned dislocations, vacant slip planes and free surfaces are illustrated. The stress fields due to the dislocation walls are calculated and large shear stress values are found away from any dislocation core. Pileups involving dislocations on two sets of intersecting slip planes are found to give rise to equilibrium configurations involving dislocation free regions. The response of dislocation patterns in an infinite medium to an imposed shear stress is also analyzed.

Conservation integrals in couple stress elasticity

Abstract Noether’s theorem on invariant variational principles is applied in the case of infinitesimal couple stress elasticity, thereby extending the analysis of Knowles and Sternberg (1972. On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44, 187–211) beyond the range of classical elasticity. Two conserved integral quantities are deduced which generalize the J -integral and L -integral in the notation of Budiansky and Rice (1973: Budiansky, B. and Rice, J. R. (1973) Conservation laws and energy-release rates. J. Appl. Mech. 40, 201–203 ). An expression for an M -integral is also obtained, but it is demonstrated that there is no corresponding conservation law for this integral. Relationships of the derived path integrals to other similar quantities for couple stress elasticity which have appeared in the literature are discussed.

Anisotropy in the compressive mechanical properties of bovine cortical bone and the mineral and protein constituents

The mechanical properties of fully demineralized, fully deproteinized and untreated cortical bovine femur bone were investigated by compression testing in three anatomical directions (longitudinal, radial and transverse). The weighted sum of the stress-strain curves of the treated bones was far lower than that of the untreated bone, indicating a strong molecular and/or mechanical interaction between the collagen matrix and the mineral phase. Demineralization and deproteinization of the bone demonstrated that contiguous, stand-alone structures result, showing that bone can be considered an interpenetrating composite material. Structural features of the samples from all groups were studied by optical and scanning electron microscopy. Anisotropic mechanical properties were observed: the radial direction was found to be the strongest for untreated bone, while the longitudinal one was found to be the strongest for deproteinized and demineralized bones. A possible explanation for this phenomenon is the difference in bone microstructure in the radial and longitudinal directions.