A. Needleman

Analysis of the cup-cone fracture in a round tensile bar

Abstract Necking and failure in a round tensile test specimen is analysed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of micro-voids. Final material failure by coalescence of voids, at a value of the void volume fraction in accord with experimental and computational results, is incorporated in this constitutive model via the dependence of the yield condition on the void volume fraction. In the analyses the material has no voids initially; but high voidage develops in the centre of the neck where the hydrostatic tension peaks, leading to the formation of a macroscopic crack as the material stress carrying capacity vanishes. The numerically computed crack is approximately plane in the central part of the neck, but closer to the free surface the crack propagates on a zig-zag path, finally forming the cone of the cup-cone fracture. The onset of macroscopic fracture is found to be associated with a sharp “knee” on the load deformation curve, as is also observed experimentally, and at this point the reduction in cross-sectional area stops.

Numerical simulations of fast crack growth in brittle solids

Dynamic crack growth is analysed numerically for a plane strain block with an initial central crack subject to tensile loading. The continuum is characterized by a material constitutive law that relates stress and strain, and by a relation between the tractions and displacement jumps across a specified set of cohesive surfaces. The material constitutive relation is that of an isotropic hyperelastic solid. The cohesive surface constitutive relation allows for the creation of new free surface and dimensional considerations introduce a characteristic length into the formulation. Full transient analyses are carried out. Crack branching emerges as a natural outcome of the initial-boundary value problem solution, without any ad hoc assumption regarding branching criteria. Coarse mesh calculations are used to explore various qualitative features such as the effect of impact velocity on crack branching, and the effect of an inhomogeneity in strength, as in crack growth along or up to an interface. The effect of cohesive surface orientation on crack path is also explored, and for a range of orientations zigzag crack growth precedes crack branching. Finer mesh calculations are carried out where crack growth is confined to the initial crack plane. The crack accelerates and then grows at a constant speed that, for high impact velocities, can exceed the Rayleigh wave speed. This is due to the finite strength of the cohesive surfaces. A fine mesh calculation is also carried out where the path of crack growth is not constrained. The crack speed reaches about 45% of the Rayleigh wave speed, then the crack speed begins to oscillate and crack branching at an angle of about 29° from the initial crack plane occurs. The numerical results are at least qualitatively in accord with a wide variety of experimental observations on fast crack growth in brittle solids.

Overview no. 42 Texture development and strain hardening in rate dependent polycrystals

Abstract A new rate dependent constitutive model is developed for polycrystals subjected to arbitrarily large strains. The model is used to predict deformation textures and large-strain strain hardening behavior following various stress-strain histories for single phase f.c.c. aggregates that deform by crystallographic slip. Examples involving uniaxial and plane strain tension and compression are presented which illustrate how texture influences polycrystalline strain hardening, in particular these examples demonstrate both textural strengthening and softening effects. Input to the model includes the description of single crystal strain hardening and latent hardening along with strain rate sensitivity, all properties described on the individual slip system level. The constitutive formulation used for the individual grains is essentially that developed by Peirce et al . [6, Acta metall . 31, 1951 (1983)] to solve rate dependent boundary value problems for finitely deformed single crystals. Inclusion of rate dependence is shown to overcome the long standing problem of nonuniqueness in the choice of active slip systems which is inherent in the rate independent theory. Because the slipping rates on all slip systems within each grain are unique in the rate dependent theory, the lattice rotations and thus the textures that develop are unique. In addition, the model makes it possible to study how strain rate sensitivity on the slip system, and single grain, levels is manifested in polycrystalline strain rate sensitivity. The model is also used to predict “constant offset plastic strain yield surfaces” for materials that are nearly rate insensitive—these calculations describe the development of rounded “yield surface vertices” and the resulting softening of material stiffness to a change in loading path that vertices imply. For our rate dependent solid this reduction in stiffness occurs after small but finite loading increments. Finally the model is used to carry out an imperfection-based sheet necking analysis both for isotropic and strongly textured sheets. The results show that larger strain hardening rates, and strain rate sensitivity, on the slip system level both increase the failure strains, as expected, but also demonstrate a strong influence of texture on localized necking.

Material rate dependence and localized deformation in crystalline solids

Abstract Nonuniform deformations of rate dependent single crystals subject to tensile loading are analyzed numerically. The crystal geometry is idealized in terms of a planar double slip model. In addition to allowing the effects of material rate sensitivity to be explored, the present rate dependent formulation permits the analysis of a range of material strain hardening properties and crystal geometries that could not be analyzed within a rate independent framework. Two crystal geometries are modeled. One is a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate slip. For this geometry, a direct comparison with a previous rate independent calculation shows that material rate sensitivity delays shear band development significantly. Our present rate dependent formulation also enables a more complete exploration of the effects of high (i.e. greater than Taylor) latent hardening ratios on “patchy” slip development. In particular we show that strong latent hardening and patchy slip can give rise to kinematical constraints that prevent shear bands from propagating completely across the gage section. The second geometry models a b.c.c. crystal oriented so that there is approximately a double mode of slip with the slip systems inclined by more than 45° to the tensile axis. This calculation displays the formation of a localized band of conjugate slip. The lattice rotations accompanying this mode eventually lead to a decrease in the resolved shear stress on the more active system in the band so that the bands do not accumulate large strains and catastrophic shear bands do not form. The implications of material rate sensitivity for uniqueness are also discussed with reference to implications for the prediction of mechanical properties of polycrystals.

An analysis of nonuniform and localized deformation in ductile single crystals

Abstract The nonuniform and localized deformations of ductile single crystals subject to tensile loading are analyzed numerically. The crystal is modelled by a rate independent, elastic-plastic relation based on Schmids law which precisely accounts for lattice rotations. Both self hardening and latent hardening of the slip systems are included in the model. The crystal geometry is idealized in terms of a planar double slip model. Initial imperfections are specified in the form of slight thickness inhomogeneities and the calculations follow the crystal deformation through diffuse necking and the formation of shear bands. The pattern of shear bands depends on the initial imperfection, but, independent of the particular small imperfection, the material planes of the bands are inclined at a characteristic angle to the slip planes. Also, the lattice misorientation across the shear band, which is such as to cause geometrical softening of the bands, is not sensitive to the imperfection form. For high strength, low hardening crystals a comparison with existing experimental data shows remarkably good qualitative and quantitative agreement between the calculations and observations. We also model a relatively soft high hardening crystal which undergoes more diffuse necking than the strong high hardening crystal. Diffuse necking leads to lattice rotations which produce geometrical softening and hence promote shear band formation. Furthermore, we carry out a calculation for a high strength low hardening crystal with the latent hardening rate prescribed somewhat larger than for isotropic hardening. In this case a ‘patchy’ pattern of slip emerges. However, the course of shear band development is unaffected.

An experimental and numerical study of deformation in metal-ceramic composites

The deformation characteristics of ceramic whisker- and particulate-reinforced metal-matrix composites were studied experimentally and numerically with the objective of investigating the dependence of tensile properties on the matrix microstructure and on the size, shape, and distribution of the reinforcement phase. The model systems chosen for comparison with the numerical simulations included SiC whisker-reinforced 2124 aluminum alloys with well-characterized microstructures and 1100-o aluminum reinforced with different amounts of SiC particulates. The overall constitutive response of the composite and the evolution of stress and strain field quantities in the matrix of the composite were computed using finite element models within the context of axisymmetric and plane strain unit cell formulations. The results indicated that the development of significant triaxial stresses within the composite matrix, due to the constraint imposed by the reinforcements, provides an important contribution to strengthening. Systematic calculations of the alterations in matrix field quantities in response to controlled changes in reinforcement distribution give valuable insights into the effects of particle clustering on the tensile properties. The numerical results also deliver a mechanistic rationale for experimentally observed trends on: (i) the effects of reinforcement morphology and volume fraction on yield and strain hardening behavior of the composite, (ii) the pronounced influence of reinforcement clustering on the overall constitutive response, (iii) ductile failure by void growth within the composite matrix, (iv) the insensitivity of the yield strength of the composite to changes in matrix microstructure, and (v) the dependence of ductility on the microstructure of the matrix and on the morphology and distribution of the reinforcement. The predictions of the present analyses are compared and contrasted with current theories of elastic and plastic response in multi-phase materials in an attempt to develop an overall perspective on the mechanisms of composite strengthening and of matrix and interfacial failure.

Material rate dependence and mesh sensitivity in localization problems

Abstract The role of material rate dependence in setting the character of governing equations is illustrated in the context of a simple one-dimensional problem. For rate-dependent solids, the incremental equilibrium equations for quasi-static problems remain elliptic and wave speeds for dynamic problems remain real, even in the presence of strain-softening. The pathological mesh sensitivity associated with numerical solutions of localization problems for rate-independent solids is eliminated. In effect, material rate dependence implicity introduces a length scale into the governing equations, although the constitutive description does not contain a parameter with the dimensions of length. Numerical results are presented that illustrate the localization behavior of slightly rate-dependent solids under both quasi-static and dynamic loading conditions.

Discrete dislocation plasticity: a simple planar model

A method for solving small-strain plasticity problems with plastic flow represented by the collective motion of a large number of discrete dislocations is presented. The dislocations are modelled as line defects in a linear elastic medium. At each instant, superposition is used to represent the solution in terms of the infinite-medium solution for the discrete dislocations and a complementary solution that enforces the boundary conditions on the finite body. The complementary solution is nonsingular and is obtained from a finite-element solution of a linear elastic boundary value problem. The lattice resistance to dislocation motion, dislocation nucleation and annihilation are incorporated into the formulation through a set of constitutive rules. Obstacles leading to possible dislocation pile-ups are also accounted for. The deformation history is calculated in a linear incremental manner. Plane-strain boundary value problems are solved for a solid having edge dislocations on parallel slip planes. Monophase and composite materials subject to simple shear parallel to the slip plane are analysed. Typically, a peak in the shear stress versus shear strain curve is found, after which the stress falls to a plateau at which the material deforms steadily. The plateau is associated with the localization of dislocation activity on more or less isolated systems. The results for composite materials are compared with solutions for a phenomenological continuum slip characterization of plastic flow.

Void growth and coalescence in porous plastic solids

Abstract A boundary value problem simulating a periodic array of spherical voids in an isotropically hardening elastic-viscoplastic matrix is analyzed. The calculations show a shift from a general axisymmetric deformation state to a mode of uniaxial straining at which point the plastic deformation localizes to the ligament between neighboring voids. This event is associated with the accelerated void growth accompanying coalescence. The numerical results are related to the description of void growth and coalescence within a phenomenological constitutive framework for progressively cavitating solids.

A tangent modulus method for rate dependent solids

Abstract A one step forward gradient time integration scheme is developed which leads to a tangent stiffness type method for rate dependent solids. Within the context of small strain theory numerical examples are presented showing application of the method to material behaviors ranging from elasticnonlinearly viscous to nearly rate independent. The adaptability of this rate dependent tangent modulus method to complex constitutive relations and to finite deformation analyses is also illustrated.