Shyam Keralavarma

The growth and coalescence of micro-voids in a material undergoing ductile fracture depends strongly on the loading path. Void growth occurs by diffuse plasticity in the material and is sensitive to the hydrostatic stress, while void coalescence occurs by the localization of plastic deformation in the inter-void ligaments under a combination of normal and shear stresses on the localization plane. In this paper, a micromechanics-based plasticity model is developed for an isotropic porous material, accounting for both diffuse and localized modes of plasticity at the micro-scale. A multi-surface approach is adopted, and two existing plasticity models that separately account for the two modes of yielding, above, are synthesized to propose an effective isotropic yield criterion and associated state evolution equations. The yield criterion is validated by comparison with quasi-exact numerical yield loci computed using a finite elements based limit analysis procedure. It is shown that the new criterion is in better agreement with the numerical loci than the Gurson model, particularly for large values of the porosity for which the loading path dependence of the yield stress is well predicted by the new model. Even at small porosities, it is shown that the new model predicts marginally lower yield stresses under low triaxiality shear dominated loadings compared to the Gurson model, in agreement with the numerical limit analysis data. Predictions for the strains to the onset of coalescence under proportional loading, obtained by numerically integrating the model, indicate that void coalescence tends to occur at relatively small plastic strain and porosity levels under shear dominated loadings. Implications on the prediction of ductility using the new model in fracture simulations are discussed.

Ductile failure under plane stress conditions is analyzed at the meso-scale using periodic unit cell model simulations of void growth in a plastically deforming matrix. Equivalent strains to failure by the onset of plastic instability at the macro-scale are estimated using the loss of ellipticity criterion for the equilibrium equations. Failure loci obtained from the cell model simulations are compared with the predictions of an instability-based ductile failure model and the Hosford–Coulomb damage indicator model, under both proportional and non-proportional loading conditions. The instability-based model is shown to quantitatively predict the shape of the failure locus under proportional loading, including the presence of a cusp at uniaxial tension and a ductility minimum under plane strain tension, in the absence of heuristic adjustable parameters in the failure criterion. It is shown that the characteristic shape of the plane stress failure locus is primarily due to the Lode dependence of the failure criterion, and not the damage growth law as assumed in the damage indicator models. Under non-proportional loading involving a step change in loading direction at an intermediate strain, the instability-based model correctly predicts the non-linear variation of the failure strain as a function of the intermediate strain; unlike a linear variation predicted by the damage indicator models, which is not in agreement with the cell model simulations. Forming limit curves showing the strains to the onset of localized necking in thin sheets are also obtained from the cell model simulations using an appropriate modification of the macroscopic instability criterion.

The growth of sub-grain voids in crystalline materials is affected by plastic anisotropy as well as void size effects. In this paper, an effective yield criterion is derived for a porous single crystal using homogenization theory and limit analysis. A two dimensional planar model of a single crystal containing a random distribution of cylindrical voids is assumed. The effective yield criterion is derived using plastic limit load analysis of a hollow cylindrical representative volume element, containing a concentric cylindrical void in a single crystalline matrix. A conventional strain gradient plasticity model with an embedded material length scale is assumed for the matrix, in order to account for the void size dependence of yielding. The yield loci and void growth rates predicted by the model under plane strain conditions are validated by comparison with numerical results obtained using finite elements, as well as prior analytical results for the special case of size independent matrix behavior. The model predictions for the size dependence of the yield stress as well as void growth under proportional loading conditions are discussed with reference of known results from experiments and lower scale dislocation dynamics simulations.

In ductile materials containing micro-voids, diffuse plasticity or localisation of plastic strain in narrow bands bridging the ligaments between voids can both occur as precursors to failure. In particular, localisation of plastic strain leads to coalescence through the formation of void sheets or plastic collapse of the ligament. If yielding is defined as the point in macro stress space at which the macroscopic plastic dissipation becomes large, either diffuse plasticity or localisation can cause yielding. Appropriate combinations of triaxiality T and Lode parameter L can cause localisation to occur earlier, and thereby modify the yield surface significantly. This is more likely to happen at high values of porosity. The competition between the two modes of yielding has been captured by a recently proposed multi-surface yield framework (Keralavarma, S. M., 2017, “A multi-surface plasticity model for ductile fracture simulations,” Journal of the Mechanics and Physics of Solids, 103, pp. 100–120), where the competition between the Gurson criterion for yielding by diffuse plastic flow and a criterion for localized yielding within discrete coalescence bands leads to a piecewise smooth yield locus with sharp vertices. In the present work, we generate yield surfaces computationally by using a voided, cuboidal unit cell and a computational homogenisation framework that allows for both macro deformation gradient and macro Cauchy stress control. The basic aim is to see how the multi-surface framework of yield compares with macro yield loci generated computationally using a formulation where both finite deformations and void shape changes are allowed. We show, for spherical, prolate and oblate initial voids, that localisation inevitably hastens macro yield and adds sharp vertices to the yield locus, for a wide range of L and T. The multi-surface framework, at least for spherical initial voids, is remarkably successful in capturing this competition.

Plastic anisotropy in the form of texture development and void shape evolution can significantly affect damage growth rates and overall strain to failure in ductile materials. A criterion for the onset of coalescence, which is the transition from void growth by diffuse plastic deformation to localized plasticity in the ligament connecting neighboring voids, is a critical component of any predictive model for ductile fracture. In this paper, a new micromechanics-based criterion for void coalescence, combining both forms of anisotropy above, is developed using homogenization and limit analysis of a hollow cylindrical representative volume element made of an orthotropic material of the Hill type. Two possible modes of coalescence, corresponding to necking instability and shear strain localization in the transverse inter-void ligament, are accounted for in the analysis. The final form of the coalescence criterion has an interesting symmetry with Gurson-type yield criteria for porous materials and is shown to be an improvement over existing models for the special case of isotropic matrix behavior. For validation of the analytical model, quasi-exact numerical coalescence loci are computed using a finite elements based limit analysis method for the special case of transversely isotropic materials. The analytical model is shown to be in good agreement with the numerical data, except for highly oblate void shapes approaching penny shape cracks. A heuristic modification for the model is proposed, which significantly improves the model predictions in that limit.

A framework for solving problems of dislocation-mediated plasticity coupled with point-defect diffusion is presented. The dislocations are modeled as line singularities embedded in a linear elastic medium while the point defects are represented by a concentration field as in continuum diffusion theory. Plastic flow arises due to the collective motion of a large number of dislocations. Both conservative (glide) and nonconservative (diffusion-mediated climb) motions are accounted for. Time scale separation is contingent upon the existence of quasi-equilibrium dislocation configurations. A variational principle is used to derive the coupled governing equations for point-defect diffusion and dislocation climb. Superposition is used to obtain the mechanical fields in terms of the infinite-medium discrete dislocation fields and an image field that enforces the boundary conditions while the point-defect concentration is obtained by solving the stress-dependent diffusion equations on the same finite-element grid. Core-level boundary conditions for the concentration field are avoided by invoking an approximate, yet robust kinetic law. Aspects of the formulation are general but its implementation in a simple plane strain model enables the modeling of high-temperature phenomena such as creep, recovery and relaxation in crystalline materials. With emphasis laid on lattice vacancies, the creep response of planar single crystals in simple tension emerges as a natural outcome in the simulations. A large number of boundary-value problem solutions are obtained which depict transitions from diffusional to power-law creep, in keeping with long-standing phenomenological theories of creep. In addition, some unique experimental aspects of creep in small scale specimens are also reproduced in the simulations.

In this paper, the effect of porosity on the ductility of as-cast AA7075 (a commercial highstrength aluminum alloy) was investigated. The as-cast material was processed through hot upsetting, and specimens with different porosity content were achieved. These were then subjected to tensile and compression tests. It was shown that the tensile ductility exhibited a near sigmoidal dependence on the porosity content. Compressive ductility, on the other hand, was not affected by the initial porosity content. In addition, direct observations, on an X-ray microscope (XRM), enabled 3-dimensional imaging of the porosity evolution during plastic deformation. Numerical simulations using a homogenized damage model, the Gurson-Tvergaard-Needleman (GTN) approach, was used for predicting the mechanical responses. The continuum model, which accounted for the growth and coalescence of spherical voids, captured the overall experimental patterns reasonably well.

The two-dimensional stress fields induced by a set of infinitely many parallel edge dislocations are difficult to estimate as those of individual dislocations decay slowly. A simple numerical method to compute them is proposed. The method is based on series summation using a convergence factor, (Formula presented.) that decays rapidly with radial distance r from the field point, and letting the positive parameter (Formula presented.) numerically through Richardson extrapolation. The present method is more general than a lattice summation method with explicit spurious stress cancellation that is widely used in the literature. Furthermore, the spurious long-range stresses are cancelled in the present method without explicit evaluation.

The stress state in a shear-lag model of a unidirectional fiber composite with an arbitrary configuration of fiber breaks is obtained by the weighted superposition of the stress state due to a single broken fiber. In a periodic patch comprised of N fibers located at the points of a regular lattice, a method to determine the stress state due to a single break was proposed by Landis et al. (J Mech Phys Solids 48(3):621–648, 2000). This method entails the determination of the eigenspace of an N× N matrix, at a computational cost of O(N3). In the present work, an alternative algorithm is proposed. This algorithm exploits the circulant structure of the matrix describing the inter-fiber interactions. The asymptotic computational complexity of the present algorithm equals that of the discrete Fourier transform: O(Nlog N). Run times of the present method with the eigensolution based method are compared, and shown to be very favorable for the present method, even for small N. Power-law scaling of the overloads due to a single break to much larger distances than previously possible has been verified using the present method.

An isotropic multi-surface model of porous material plasticity is derived and employed to investigate the effects of the third stress invariant in ductile failure. The constitutive relation accounts for both homogeneous and inhomogeneous yielding of a material containing a random distribution of voids. Individual voids are modeled as spheroidal but the aggregate has no net texture. Ensemble averaging is invoked to operate a scale transition from the inherently anisotropic meso-scale process of single-void growth and coalescence to some macroscopic volume that contains many voids. Correspondingly, expressions for effective yield and associated evolution equations are derived from first principles, under the constraint of persistent isotropy. It is found that the well-known vertex on the hydrostatic axis either disappears for sufficiently flat voids or develops into a lower-order singularity for elongated ones. When failure is viewed as the onset of an instability, it invariably occurs after the transition to inhomogeneous yielding with the delay between the two depending strongly upon the Lode parameter. The strain to failure is found to be weakly dependent on the Lode parameter for shear-dominated loadings, but strongly dependent on it near states of so-called generalized tension or compression. Experimentally determined fracture loci for near plane stress states are discussed in light of the new findings.