Three-point bounds and other estimates for strongly nonlinear composites

A variational procedure due to Ponte Castañeda et al. [Phys. Rev. B 46, 4387 (1992)] is used to determine three-point bounds and other types of estimates for the effective response of strongly nonlinear composites with random microstructures. The variational procedure makes use of estimates for the effective properties of linear comparison composites to generate corresponding estimates for nonlinear composites. Several equivalent forms of the variational procedure are derived. In particular, it is shown that the mean-field theory of Wan et al. [Phys. Rev. B 54, 3946 (1996)], which also makes use of a linear comparison composite, together with a certain decoupling approximation, leads to results that are precisely identical to those that can be obtained from the earlier variational procedure. Finally, three-point bounds and other estimates are computed for power-law composites with cell-type microstructures, and the results are compared with random resistor network simulations available from the literature

Second-order theory for nonlinear dielectric composites incorporating field fluctuations

This paper deals with the development of an improved second-order theory for estimating the effective behavior of nonlinear composite dielectrics. The theory makes use of the field fluctuations in the phases of the relevant linear comparison composite to generate improved Maxwell-Garnett (MGA) and effective-medium (EMA) types of approximations for nonlinear media. Similar to the earlier version of the theory, the resulting MGA and EMA predictions are exact to second-order in the contrast, but—unlike the earlier version—the estimates satisfy all known bounds. In particular, the EMA estimates exhibit a nonlinearity-independent percolation threshold, and critical exponents that are consistent with recently developed bounds on these exponents. In addition, the MGA and EMA estimates are shown to yield reasonable predictions for strongly nonlinear composites with threshold-type nonlinearities, which are extreme cases where earlier methods have been known to sometimes fail

Influence of the Lode parameter and the stress triaxiality on the failure of elasto-plastic porous materials

International audienceThis work makes use of a recently developed ''second-order'' homogenization model to investigate failure in porous elasto-plastic solids under general triaxial loading conditions. The model incorporates dependence on the porosity and average pore shape, whose evolution is sensitive to the stress triaxiality and Lode parameter L. For positive triaxiality (with overall tensile hydrostatic stress), two different macroscopic failure mechanisms are possible, depending on the level of the triaxiality. At high triaxiality, void growth induces softening of the material, which overtakes the intrinsic strain hardening of the matrix phase, leading to a maximum in the effective stress-strain relation for the porous material, followed by loss of ellipticity by means of dilatant shear localization bands. In this regime, the ductility decreases with increasing triaxiality and is weakly dependent on the Lode parameter, in agreement with earlier theoretical analyses and experimental observations. At low triaxiality, however, a new mechanism comes into play consisting in the abrupt collapse of the voids along a compressive direction (with small, but finite porosity), which can dramatically soften the response of the porous material, leading to a sudden drop in its load-carrying capacity, and to loss of ellipticity of its incremental constitutive relation through localization of deformation. This low-triaxiality failure mechanism leads to a reduction in the ductility of the material as the triaxiality decreases to zero, and is highly dependent on the value of the Lode parameter. Thus, while no void collapse is observed at low triaxiality for axisymmetric tension (L=-1), the ductility of the material drops sharply with decreasing values of the Lode parameter, and is smallest for biaxial tension with axisymmetric compression (L=+1). In addition, the model predicts a sharp transition from the low-triaxiality regime, with increasing ductility, to the high-triaxiality regime, with decreasing ductility, as the failure mechanism switches from void collapse to void growth, and is in qualitative agreement with recent experimental work

Effective properties of nonlinear inhomogeneous dielectrics

We develop a general procedure for estimating the effective constitutive behavior of nonlinear dielectrics. The procedure is based on a variational principle expressing the effective energy function of a given nonlinear composite in terms of the effective energy functions of the class of linear comparison composites. This provides an automatic procedure for converting well-known information for linear composites, in the form of estimates and bounds for their effective dielectric constants, into corresponding estimates and bounds for the effective behavior of nonlinear composites. Further, the procedure is easily implemented, and leads in some cases to exact results. This, exact estimates are given herein for isotropic weakly nonlinear composites with general nonlinearity, and bounds of the Hashin-Shtrikman type are given for the class of two-phase, isotropic dielectric composites with strongly and perfectly non-linear constitutive behavior. The optimality of the bounds is addressed briefly

Homogenization-Based Predictions for Texture Evolution in Halite

International audienceThe “variational” homogenization method developed by deBotton and Ponte Castaneda [2] is used here to predict texture development in halite polycrystals at room and high temperatures accounting for hardening and grain shape changes. The new predictions are compared with those of the Taylor and “tangent” model of Molinari et al. [5] for uniaxial tension and compression. The predictions of the “variational” model are found to be intermediate between the Taylor and “tangent” predictions, although not too different from either, as a consequence of the relatively high isotropy of the halite single crystal grains

Homogenization estimates for texture evolution in halite

In this work, the recently developed “second-order” self-consistent method [Liu, Y., Ponte Castaneda, P., 2004a. Second-order estimates for the effective behavior and field fluctuations in viscoplastic polycrystals. J. Mech. Phys. Solids 52 467–495] is used to simulate texture evolution in halite polycrystals. This method makes use of a suitably optimized linear comparison polycrystal and has the distinguishing property of being exact to second order in the heterogeneity contrast. The second-order model takes into consideration the effects of hardening and of the evolution of both crystallographic and morphological texture to yield reliable predictions for the macroscopic behavior of the polycrystal. Comparisons of these predictions with full-field numerical simulations [Lebensohn, R.A., Dawson, P.R., Kern, H.M., Wenk, H.R., 2003. Heterogeneous deformation and texture development in halite polycrystals: comparison of different modeling approaches and experimental data. Tectonophysics 370 287–311], as well as with predictions resulting from the earlier “variational” and “tangent” self-consistent models, included here for comparison purposes, provide insight into how the underlying assumptions of the various models affect slip in the grains, and therefore the texture predictions in highly anisotropic and nonlinear polycrystalline materials. The “second-order” self-consistent method, while giving a softer stress-strain response than the corresponding full-field results, predicts a pattern of texture evolution that is not captured by the other homogenization models and that agrees reasonably well with the full-field predictions and with the experimental measures

Exact results for weakly nonlinear composites and implications for homogenization methods

Weakly nonlinear composite conductors are characterized by position-dependent dissipation potentials expressible as an additive composition of a quadratic potential and a nonquadratic potential weighted by a small parameter. This additive form carries over to the effective dissipation potential of the composite when expanded to first order in the small parameter. However, the first-order correction of this asymptotic expansion depends only on the zeroth-order values of the local fields, namely, the local fields within the perfectly linear composite conductor. This asymptotic expansion is exploited to derive the exact effective conductivity of a composite cylinder assemblage exhibiting weak nonlinearity of the power-law type (i.e., power law with exponent m = 1 + δ, such that |δ| ¿ 1), and found to be identical (to first order in δ) to the corresponding asymptotic result for sequentially laminated composites of infinite rank. These exact results are used to assess the capabilities of more general nonlinear homogenization methods making use of the properties of optimally selected linear comparison composites.Fil: Furer, Joshua. University of Pennsylvania; Estados UnidosFil: Idiart, Martín Ignacio. Universidad Nacional de La Plata. Facultad de Ingeniería. Departamento de Aeronáutica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Ponte Castañeda, Pedro. University of Pennsylvania; Estados Unido

Rheology of a Suspension of Elastic Particles in a Viscous Shear Flow

In this paper we consider a suspension of elastic solid particles in a viscous liquid. The particles are assumed to be neo-Hookean and can undergo finite elastic deformations. A polarization technique, originally developed for analogous problems in linear elasticity, is used to establish a theory for describing the finite-strain, time-dependent response of an ellipsoidal elastic particle in a viscous fluid flow under Stokes flow conditions. A set of coupled, nonlinear, first-order ODEs is obtained for the evolution of the uniform stress fields in the particle, as well as for the shape and orientation of the particle, which can in turn be used to characterize the rheology of a dilute suspension of elastic particles in a shear flow. When applied to a suspension of cylindrical particles with initially circular cross-section, the theory confirms the existence of steady-state solutions, which can be given simple analytical expressions. The two-dimensional, steady-state solutions for the particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspension, are in excellent agreement with direct numerical simulations of multiple-particle dispersions in a shear flow obtained by using an arbitrary Lagrangian–Eulerian (ALE) finite element method (FEM) solver. The corresponding solutions for the evolution of the microstructure and the rheological properties of suspensions of initially spherical (three-dimensional) particles in a simple shear flow are also obtained, and compared with the results of Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) in the steady-state regime. Interestingly, the results show that sufficiently soft elastic particles can be used to reduce the effective viscosity of the suspension (relative to that of the pure fluid)

Shape Dynamics and Rheology of Soft Elastic Particles in a Shear Flow

The shape dynamics of soft, elastic particles in an unbounded simple shear flow is investigated theoretically under Stokes flow conditions. Three types of motion—- steady-state, trembling, and tumbling—- are predicted, depending on the shear rate, elastic shear modulus, and initial particle shape. The steady-state motion is found to be always stable. In addition, the existence of a trembling regime is documented for the first time in nonvesicle systems, and a complete phase diagram is developed. The rheological properties of dilute suspensions of such soft particles generally exhibit shear-thinning behavior and can even display negative intrinsic viscosity for sufficiently soft particles