J. W. Ju

On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects

Abstract Novel energy-based coupled elastoplastic damage theories are presented in this paper. The proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials. At variance with Lemaitres work on damage-elastoplasticity, the present formulation renders rational thermodynamic potential and damage energy release rate. In contrast to previous work by Simo and Ju (featuring an additive split of the stress tensor), current formulation assumes an additive split of the strain tensor. It is shown that the “strain split” damage-elastoplasticity formulation leads to more robust tangent moduli than the “stress split” formulation. The plastic flow rule and hardening law are characterized in terms of the effective quantities; viz. the effective stress space plasticity . This mechanism is both physically well-motivated and computationally efficient. Further, a fourth-order anisotropic damage mechanism is proposed for brittle materials. Rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain-rate dependency of microcrack growth. Efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the “operator splitting” methodology. In particular, new three-step operator split algorithms are presented Application is made to a class of inviseid and rate-dependent cap-damage models for concrete and mortar. Experimental validations are also given to illustrate the applicability of the proposed damage models.

Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities

SummaryA micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations, and elastic properties of constituent phases of a linear elastic particulate composite. A renormalization procedure is employed to render absolutely convergent integrals. Therefore, the micromechanical equations and effective elastic properties of a statistically homogeneous composite are independent of the shape of the RVE. Various micromechanical models can be developed based on the proposed ensemble-volume averaged constitutive equations. As a special class of models, inter-particle interactions are completely ignored. It is shown that the classical Hashin-Shtrikman bounds, Walpoles bounds, and Willis bounds for isotropic or anisotropic elastic multiphase composites are related to the “noninteracting” solutions. Further, it is demonstrated that the Mori-Tanaka methodcoincides with the Hashin-Shtrikman bounds and the “noninteracting” micromechanical model in some cases. Specialization to unidirectionally aligned penny-shaped microcracks is also presented. An accurate, higher order (in particle concentration), probabilistic pairwise particle interaction formulation coupled with the proposed ensemble-volume averaged equations will be presented in a companion paper.

Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities

SummaryBased on the general micromechanical framework proposed in a companion paper, effective elastic moduli of two-phase composites containing randomly dispersedspherical inhomogeneities are investigated in this paper. At variance with existing micromechanical pairwise interaction models (accurate up to the second-order in particle volume fraction ϕ), the proposed approximate, probabilistic pairwise particle interaction formulationcoupled with the general ensemble-volume averaged field equations leads to a novel, higher-order (in ϕ), and accurate method for the prediction of effective elastic moduli of two-phase composites containing randomly located spherical particles. The relevant ensemble integrals in the proposed formulation are absolutely convergent due to a “renormalization” procedure employed in a companion paper. In accordance with the analogy between the effective shear modulus of an incompressible elastic composite with randomly dispersed rigid spheres and the effective shear viscosity of a colloidal dispersion with randomly dispersed rigid spheres (at high shear rates), the proposed ensemble-micromechanical approach is extended to predict effective shear viscosities of colloidal dispersions at the high-shear limit. Comparisons with experimental data, classical variational bounds, improved three-point bounds, the second-order particle interaction model, and other micromechanical models are also presented. It is observed that significant improvement in predictive capability for two-phase composites with randomly dispersed spheres can be achieved by using the proposed method.

Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal inhomogeneities. Part I: micromechanics-based formulation

Abstract Based on the framework of Ju and Chen (Ju, J.W., Chen, T.M., 1994. J. Engng. Mater. Tech. ASME 116, 310–318) and Ju and Tseng (Ju, J.W., Tseng, K.H., 1996. Int. J. Solids Struct. 33, 4267–4291; Ju, J.W., Tseng, K.H., 1997. J. Engng. ASCE 123, 260–266), we study the effective elastoplastic behavior of two-phase metal matrix composites (MMCs) containing randomly located yet unidirectionally aligned spheroidal inhomogeneities. Specifically, the particle phase is assumed to be linearly elastic and the matrix phase is elastoplastic. The ensemble-volume averaging procedure is employed to micromechanically derive the effective yield function of MMCs based on the probabilistic spatial distribution of aligned spheroidal particles and the particle-matrix influences. The transversely isotropic effective elasticity tensor is explicitly derived. Further, the associative plastic flow rule and the isotropic hardening law are postulated according to the continuum plasticity. As a result, we can characterize the overall elastoplastic stress–strain responses of aligned spheroid-reinforced MMCs under three-dimensional loading and unloading histories. The overall elastoplastic continuum tangent tensor of MMCs is also explicitly presented.

Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal Inhomogeneities. Part II: applications

Abstract Based on the proposed formulation in Part I of this sequel (Ju, J.W., Sun, L.Z., Int. J. Solids Struct. 38, 183–201), effective elastoplastic constitutive relations are implemented in this article for metal matrix composites (MMCs) with randomly located and unidirectionally aligned spheroidal particles. First, we investigate the uniaxial elastoplastic stress–strain behavior of MMCs. In particular, we perform comparisons among the theoretical uniaxial stress–strain predictions, existing finite element results and experimental data for MMCs to illustrate the capability of the proposed method. Furthermore, the effect of stress triaxiality is discussed under either the purely hydrostatic or axisymmetric loading on the overall elastoplastic behavior of composites. The proposed initial effective yield surfaces for composites are demonstrated and compared with those of the experimental data. As a special case of the incompressible ductile material containing aligned spheroidal voids, the initial effective yield criterion is studied and compared with that of mathematical upper bound. Finally, viscoplastic extension is briefly presented.

A micromechanical damage model for effective elastoplastic behavior of partially debonded ductile matrix composites

Abstract A micromechanical damage model considering progressive partial debonding is presented to investigate the effective elastoplastic-damage behavior of partially debonded particle reinforced ductile matrix composites (PRDMCs). The effective, evolutionary elastoplastic-damage responses of three-phase composites, consisting of perfectly bonded spherical particles, partially debonded particles and a ductile matrix, are micromechanically derived on the basis of the ensemble-volume averaging procedure and the first-order effects of eigenstrains. The effects of random dispersion of particles are accommodated. Further, the evolutionary partial debonding mechanism is governed by the internal stresses of spherical particles and the statistical behavior of the interfacial strength. Specifically, following Zhao and Weng (1996) , a partially debonded elastic spherical isotropic inclusion is replaced by an equivalent, transversely isotropic yet perfectly bonded elastic spherical inclusion. The Weibulls probabilistic function is employed to describe the varying probability of progressive partial particle debonding. The proposed effective yield criterion, together with the assumed overall associative plastic flow rule and the hardening law, forms the analytical framework for the estimation of the effective elastoplastic-damage behavior of ductile matrix composites. Finally, the present predictions are compared with the predictions based on Ju and Lees (2000) complete particle debonding model, other existing numerical predictions, and available experimental data. It is observed that the effects of partially debonded particles on the stress–strain responses are significant when the damage evolution becomes rapid.

On two-dimensinal self-consistent micromechanical damage models for brittle solids

Abstract Two-dimensional self-consistent micromechanical damage models are presented for microcrack-weakedned brittle solids under “cleavage l” deformation processes. The proposed frame-work basically follows the previous work of Horii and Nemat-Nasser (1983. J. Mech. Phys. Solids 31 (2). 155–171) and Sumarac and Krajcinovic (1987, Mech. Mater . 6 . 39–52). Thermodynamics basis, microcrack opening displacements and damage-induced inelastic compliances are derived. Microcrack evolutions (growth) are characterized through the use of fracture mechanics stability criteria and microstructural microcrack geometry. Mode I, mode ll and mixed mode microcrack growth are considered. Simple and efficient computational algorithms as well as three detailed numerical simulations are also presented to illustrate the potential capability of the proposed micromechanical damage models. In particular, no fitted “material parameters” are needed. More-over loading unloading stress paths and microcks status changes in opening closing are trivially accommodated in this work.

A micromechanical damage model for effective elastoplastic behavior of ductile matrix composites considering evolutionary complete particle debonding

A micromechanical damage model is presented to predict the overall elastoplastic behavior and damage evolution in ductile matrix composites. The effective elastic moduli of three-phase composites are predicted by a micromechanical formulation. To estimate the overall elastoplastic-damage responses, an effective yield criterion is derived based on the ensemble-volume averaging process and the first-order effects of eigenstrains due to the existence of spherical inclusions. The effects of random dispersion of inclusions are accommodated. The proposed effective yield criterion, together with the assumed overall associative plastic flow rule and the hardening law, constitutes the analytical foundation for the estimation of effective elastoplastic behavior of ductile matrix composites. An evolutionary interfacial particle debonding model is subsequently considered in accordance with the Weibulls statistical function to describe the varying probability of complete particle debonding. The interfacial debonding process is controlled by internal stresses of particles and a Weibull interfacial strength parameter. The completely debonded particles are regarded as voids for simplicity. The proposed elastoplastic-damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Efficient step-by-step iterative computational algorithms are also presented to implement the proposed damage model. Furthermore, the present predictions under various loading conditions are compared with other theoretical predictions and some available experimental data with modest particle concentrations.

Elastoplastic modeling of metal matrix composites with evolutionary particle debonding

Abstract A microstructural-overall two-level elastoplastic and damage model is proposed to predict the overall mechanical behavior of particle-reinforced metal matrix composites. Unidirectionally aligned spheroidal elastic particles, some of which are partially debonded from the matrix, are randomly distributed in the ductile medium. These imperfect particles are modeled by fictitious orthotropic inclusions without debonding. An ensemble-volume averaged homogenization procedure is employed to estimate the effective yield function of the said composites. The associative plastic flow rule and the hardening law are postulated based on the continuum plasticity theory. The evolution of volume fraction of debonding particles is considered in accordance with Weibull’s statistical function to characterize the varying probability of reinforcement debonding. The uniaxial elastoplastic stress–strain behavior of particle composites is investigated as the first application. In particular, comparisons between the proposed uniaxial stress–strain predictions and experimental data are performed to illustrate the capability of the proposed method. Furthermore, the effect of stress triaxiality is discussed under either the purely hydrostatic or axisymmetric loading on the overall elastoplastic behavior of composites.

Effective elastoplastic behavior of two-phase ductile matrix composites: A micromechanical framework

Abstract A micromechanics-based framework is presented to predict effective elastoplastic behavior of two-phase particle-reinforced ductile matrix composites (PRDMCs) containing many randomly dispersed elastic spherical inhomogeneities. Specifically, the inclusion phase (particle) is assumed to be elastic and the matrix phase is elastoplastic. A complete second-order formulation is presented based on the probabilistic spatial distribution of spherical particles, pairwise particle interactions and the ensemble-volume averaging procedure. Two non-equivalent formulations are considered in detail to derive the effective yield functions. In addition, the plastic flow rule and hardening law are postulated according to continuum plasticity and, together with the micromechanically derived effective yield function, are employed to characterize the plastic behavior of PRDMCs under three-dimensional arbitrary loading/unloading histories. Initial effective yield criteria for incompressible ductile matrix containing many randomly dispersed spherical voids are also studied. Furthermore, uniaxial elastoplastic stress-strain behavior of PRDMCs is investigated. Comparison between our theoretical uniaxial stress-strain predictions and experimental data for PRDMCs is also performed to illustrate the capability of the proposed framework.