Elastic moduli of cubic polycrystals

The averaged elastic constants of polycrystals can be found by averaging the stresses (Voigt method [1]) or the strains (Reuss method [2]). Comparison of the elastic moduli, averaged according to Voigt and Reuss, with the experimental values shows that in the first case averaging gives values that are too high, and in the second values that are too low [3]. The reason for this is that direct averaging of the moduli with respect to arbitrary orientations of the crystallites does not take account of correlation effects. There are two ways of allowing for such correlations between polycrystal grains.     

Dynamic bulk and shear moduli due to grain-scale local fluid flow in fluid-saturated cracked poroelastic rocks: Theoretical model

Grain-scale local fluid flow is an important loss mechanism for attenuating waves in cracked fluid-saturated poroelastic rocks. In this study, a dynamic elastic modulus model is developed to quantify local flow effect on wave attenuation and velocity dispersion in porous isotropic rocks. The Eshelby transform technique, inclusion-based effective medium model (the Mori–Tanaka scheme), fluid dynamics and mass conservation principle are combined to analyze pore-fluid pressure relaxation and its influences on overall elastic properties. The derivation gives fully analytic, frequency-dependent effective bulk and shear moduli of a fluid-saturated porous rock. It is shown that the derived bulk and shear moduli rigorously satisfy the Biot-Gassmann relationship of poroelasticity in the low-frequency limit, while they are consistent with isolated-pore effective medium theory in the high-frequency limit. In particular, a simplified model is proposed to quantify the squirt-flow dispersion for frequencies lower than stiff-pore relaxation frequency. The main advantage of the proposed model over previous models is its ability to predict the dispersion due to squirt flow between pores and cracks with distributed aspect ratio instead of flow in a simply conceptual double-porosity structure. Independent input parameters include pore aspect ratio distribution, fluid bulk modulus and viscosity, and bulk and shear moduli of the solid grain. Physical assumptions made in this model include (1) pores are inter-connected and (2) crack thickness is smaller than the viscous skin depth. This study is restricted to linear elastic, well-consolidated granular rocks.     

Perturbation approach to elastic constitutive relations of polycrystals

Herein we obtain a formula for the effective elastic stiffness tensor C^eff of an orthorhombic aggregate of cubic crystallites by the perturbation method. The effective elastic stiffness tensor of the polycrystal gives the relationship between volume average stress and volume average strain. Under Voigt's model, Reuss’ model and Man's theory, the elastic constitutive relation accounts for the effect of the orientation distribution function (ODF) up to terms linear in the texture coefficients. However, the formula derived in this paper delineates the effect of crystallographic texture on elastic response and shows quadratic texture dependence. The formula is very simple. We also consider the influence of grain shape to elastic constitutive relations of polycrystals. Some examples are given to compare computational results of the formula with those given by Voigt's model, Reuss's model, the finite element method, and the self-consistent method. In Section 3, we also present an expression of the perturbation displacement field, in which Green's function for an orthorhombic aggregate of cubic crystallites is included.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries

The Voigt-Reuss-Hill (VRH) average provides a simple way to estimate the elastic constants of a textured polycrystal in terms of its crystallographic texture and the elastic constants of the constituting crystallites. Empirically the VRH estimates were found in most cases to have an accuracy comparable to those obtained by more sophisticated techniques such as self-consistent schemes. In this paper we determine, in the space of fourth-order tensors with major and minor symmetries, a special set of irreducible basis tensors, with which we obtain a simple explicit formula for the VRH average for elastic polycrystals with arbitrary crystal and texture symmetries. Our formula is correct to first order in the texture coefficients.     

Analysis for polycrystal nonproportional cyclic plasticity with a dissipated energy based rule for cross-hardening

Based on a nonclassical hardening law and the Hill’s self-consistent scheme, a new approach is proposed for the analysis of polycrystal nonproportional cyclic plasticity. A novel parameter related to the plastic dissipation on each slip system is proposed and embedded in the Bassani’s definition of cross-hardening. The tangential elastoplastic tensor relating the increments of stress and strain in a single crystal is derived and the corresponding numerical algorithm for polycrystal plasticity is developed. The elastoplastic response of 316 stainless steel subjected to typical biaxial nonproportional strain cycling is analyzed, and the main features are well replicated. The validity of the proposed approach is demonstrated by the satisfactory agreement between the computed results and experimental observation.     

A finite strain elastic-viscoplastic self-consistent model for polycrystalline materials

A large strain elastic-viscoplastic self-consistent (EVPSC) model for polycrystalline materials is developed. At single crystal level, both the rate sensitive slip and twinning are included as the plastic deformation mechanisms, while elastic anisotropy is accounted for in the elastic moduli. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress-strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tomé (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response due to lack of elastic deformation. It is also demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model.     

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. A more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. This approach is very similar in spirit but differs in its details from earlier work of Willis, showing how Hashin-Shtrikman bounds and certain classes of self-consistent estimates may be related. These self-consistent estimates always lie within the bounds for physical choices of the crystal elastic constants and for all the choices of crystal symmetry considered. For cubic symmetry, the present method reproduces the self-consistent estimates obtained earlier by various authors, but the formulas for both bounds and estimates are generated in a more symmetric form. Numerical values of the estimates obtained this way are also very comparable to those found by the Gubernatis and Krumhansl coherent potential approximation (or CPA), but do not require computations of scattering coefficients.     

立方晶粒任意集合下的格林函数和有效弹性刚度张量

为了推导多晶体材料的有效弹性刚度张量,给出立方晶粒任意集合的格林函数封闭但近似的表达式,该格林函数表达式包含三个单晶弹性常数和多晶体材料五个织构系数,它考虑取向分布函数的影响直至织构系数的线性项,它适用于弱织构多晶体材料或具有弱各向异性晶粒的多晶体材料(如金属铝),它与Nishioka格林函数近似式的比较通过三个算例给出;Synge的格林函数积分式则直接通过数值计算完成,它可作为问题的精确解供参考．该文还简单介绍了多晶体材料有效弹性刚度张量的推导过程,并把所得结果和有限元计算结果进行比较。     

确定裂纹体等效弹性模量的边界元方法

采用边界元方法计算含有序分布裂纹的裂纹体在压缩载荷作用下的等效弹性模量，利用一种能适当考虑裂纹有间相作用的自洽理论，建立了相应的迭代格式，通过算例研究了裂纹方向，裂纹面间摩擦系数对裂纹体等效弹性模量的影响。     

Homogenization methods to approximate the effective response of random fibre-reinforced Composites

In this article a fibre-reinforced composite material is modelled via an approach employing a representative volume element with periodic boundary conditions. The effective elastic moduli of the material are thus derived. In particular, the method of asymptotic homogenization is used where a finite number of fibres are randomly distributed within the representative periodic cell. The study focuses on the efficacy of such an approach in representing a macroscopically random (hence transversely isotropic) material. Of particular importance is the sensitivity of the method to cell shape, and how this choice affects the resulting (configurationally averaged) elastic moduli. The averaging method is shown to yield results that lie within the Hashin–Shtrikman variational bounds for fibre-reinforced media and compares well with the multiple scattering and (classical) self-consistent approximations with a deviation from the latter in the larger volume fraction cases. Results also compare favourably with well-known experimental data from the literature.     

Revised Bounds on the Elastic Moduli of Two-Dimensional Random Polycrystals

Our earlier derived bounds on the elastic moduli of two-dimensional random polycrystals [1, 2] involve a geometric restriction through an assumption on the form of an isotropic eight-rank tensor. The general form of the tensor is used in this study to reconstruct the bounds, which are expected to approach the scatter range for the moduli of the irregular aggregate.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations

Reservoir porous rocks usually consist of more than two types of matrix materials,forming a randomly heterogeneous material.The determination of the bulk modulus of such a medium is critical to the elastic wave dispersion and attenuation.The elastic moduli for a simple matrix-inclusion model are theoretically analyzed.Most of the efforts assume a uniform inclusion concentration throughout the whole single-material matrix.However,the assumption is too strict in real-world rocks.A model is developed to estimate the moduli of a heterogeneous bimaterial skeleton,i.e.,the host matrix and the patchy matrix.The elastic moduli,density,and permeability of the patchy matrix differ from those of the surrounding host matrix material.Both the matrices contain dispersed particle inclusions with different concentrations.By setting the elastic constant and density of the particles to be zero,a double-porosity medium is obtained.The bulk moduli for the whole system are derived with a multi-level effective modulus method based on Hashin's work.The proposed model improves the elastic modulus calculation of reservoir rocks,and is used to predict the kerogen content based on the wave velocity measured in laboratory.The results show pretty good consistency between the inversed total organic carbon and the measured total organic carbon for two sets of rock samples.     

Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

In the paper the theoretical analysis of bounds and self-consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented. In the study two invariant decompositions of Hooke’s tensors are used. The applied method enables derivation of novel expressions for estimates of the bulk and shear moduli, which depend on invariants of local stiffness tensor. With use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained.     

Estimates for the elastic moduli of random trigonal polycrystals and their macroscopic uncertainty

Explicit expressions of the upper and lower estimates on the macroscopic elastic moduli of random trigonal polycrystals are derived and calculated for a number of aggregates, which correct our earlier results given in Pham [Pham, D.C., 2003. Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals. Int. J. Solids Struct. 40, 4911–4924]. The estimates are expected to predict the scatter ranges for the elastic moduli of the polycrystalline materials. The concept of effective moduli is reconsidered regarding the macroscopic uncertainty of the moduli of randomly inhomogeneous materials.     

含微裂纹弹性体的应力应变关系

本义建立了考虑裂纹闭合和裂纹表面摩擦影响的含微裂纹弹性体的应力应变关系，给出了柔度张量增量的显式表达式。对于二维平面应力和平面应变状态，给出了等效工程弹性系数。数值计算结果表明，裂纹闭合和裂纹面摩擦对裂纹体的应力应变关系和等效工程弹性系数有重要影响。