Influence of fiber’s shape and size on overall elastoplastic property for micropolar composites

A general micromechanical method is developed for a micropolar composite with ellipsoidal fibers, where the matrix material is idealized as a micropolar material model. The method is based on a special micro–macro transition method, and the classical effective moduli for micropolar composites can be determined in an analytical way. The influence of both fiber’s shape and size can be analyzed by the proposed method. The effective moduli, initial yield surface and effective nonlinear stress and strain relation for a micropolar composite reinforced by ellipsoidal fibers are examined, it is found that the prediction on the effective moduli and effective nonlinear stress and strain curves are always higher than those based on classical Cauchy material model, especially for the case where the size of fiber approaches to the characteristic length of matrix material. As expected, when the size of fiber is sufficiently large, the classical results (size-independence) can be recovered.     

State dependent linear moduli in ferroelectrics

A novel experimental method is used to measure the evolution of the linear elastic, dielectric and piezoelectric moduli of a soft ferroelectric ceramic during loading. The applied loading states are combinations of uniaxial compressive stress and electric field. Short pulses of electric field and stress are used to increment the remanent strain and polarization state of the material, while the rates of change of electric displacement and strain during unloading are used to assess the moduli. The remanent quantities are treated as state variables, with a view to expressing the moduli as functions of the material state. The piezoelectric moduli are found to vary approximately linearly with polarization, regardless of the remanent strain state, whilst the dielectric moduli and elastic compliances show more complex behaviour. A simple model of the state dependence of the moduli, based on varying the volume fractions of six crystal variants in the tetragonal system, is used to interpret the results.     

On estimation methods for effective moduli of microcracked solids

Summary  Most of the conventional methods for estimating the overall elastic moduli of microcracked solids are defined based on the concept of effective medium or effective field. The formal similarity of these methods is examined in this paper. A one-to-one correspondence relation exists between the effective medium methods and the effective field methods in the sense that they yield identical results. In addition to the conventional estimation techniques, any other number of such approaches may be constructed by appropriately specifying the effective matrix compliance (or stiffness) tensor and the effective stress (or strain) field which a microcrack is assumed to be subjected to. To generate continuous spectra of new methods for estimating the effective elastic moduli, two simple and straightforward approaches are proposed, which contain one or two adjustable parameters in order to yield results of good accuracy. The discussion in this paper can be extended to other kinds of heterogeneous materials. Received 4 October 2000; accepted for publication 30 January 2001     

On the experimental determination of yield surfaces and some results of annealed 304 stainless steel

Factors affecting the experimental determination of yield surfaces are discussed. They include the elastic moduli and the zero offset strain, the strain domain used to determine the yield stress, the probing path, and the strain rate of probing. To obtain yield surfaces consistently, it is necessary to account for these factors. The initial and subsequent yield surfaces of annealed AISI type 304 stainless steel have been experimentally determined in the axial-torsional stress space. Three loading paths have been studied. They are a pure axial path, a pure torsional path, and a proportional axial-torsional path. Each path includes loading, unloading, reloading, and the cyclically steady state.     

Elastic interaction of partially debonded circular inclusions. II. Application to fibrous composite

A complete analytical solution has been obtained of the elasticity problem for a plane containing periodically distributed, partially debonded circular inclusions, regarded as the representative unit cell model of fibrous composite with interface damage. The displacement solution is written in terms of periodic complex potentials and extends the approach recently developed by Kushch et al. (2010) to the cell type models. By analytical averaging the local strain and stress fields, the exact formulas for the effective transverse elastic moduli have been derived. A series of the test problems have been solved to check an accuracy and numerical efficiency of the method. An effect of interface crack density on the effective elastic moduli of periodic and random structure FRC with interface damage has been evaluated. The developed approach provides a detailed analysis of the progressive debonding phenomenon including the interface cracks cluster formation, overall stiffness reduction and damage-induced anisotropy of the effective elastic moduli of composite.     

The total creep of viscoelastic composites under hydrostatic or antiplane loading

The problem of bounding the total creep (or total stress relaxation) of a composite made of two linear viscoelastic materials and subjected to a constant hydrostatic or antiplane loading is considered. It is done by coupling the immediate and the relaxed responses of the composite, which are pure elastic. The coupled bounds provide the possible range of the total deformation at infinite time as a function of the initial deformation of the composite. For antiplane shear existing bounds for coupled two-dimensional conductivity yield the required coupled bounds, and these are attained by doubly coated cylinder assemblages. The translation method is used to couple the effective bulk moduli of a viscoelastic composite at zero and infinite time. A number of microgeometries are found to attain the bulk modulus bounds. It is shown that the Hashin's composite sphere assemblage does not necessarily correspond to the maximum or minimum overall creep, although it necessarily attains the bounds for effective bulk moduli. For instance, there are cases when the doubly coated sphere microstructure or some special polycrystal arrangements attain the bounds on the total creep.     

Orthotropically textured elastic aggregates of cubic crystals

The linear orthotropic relations between stress and infinitesimal strain require only seven, instead of the usual nine, independent elastic moduli, and one of them can be identified as a bulk modulus coincident with that common to all the grains. Each of the remaining six overall moduli is placed between upper and lower, “Voigt-Reuss-Hill”, and also “Hashin- Shtrikman”, bounds, in terms of the grain moduli and of three measurable parameters that take account of the particular mix of lattice orientations. One or more of them can be determined at once in exceptional cases where the grains all have a particular fixed or somewhat variable lattice orientation: the upper and lower bounds come to the appropriate coincidence then. Generally the vagaries of the configuration have an influence in keeping each pair of bounds apart, but effective estimates of the overall elastic moduli can be offered, except perhaps when the grains have a very pronounced cubic anisotropy. We shall refer in particular to the more symmetrical, tetragonal and transversely isotropic, textures for which correspondingly fewer overall moduli and orientation parameters are required.     

Tailored heterogeneity increases overall stability regime of composites having a negative-stiffness inclusion

Recent theoretical and experimental results have shown the possibility of enormous increases in composite material overall elastic stiffness, damping, thermal expansion, piezoelectricity, etc., when the composite contains a tuned non-positive-definite (i.e., negative stiffness) constituent. For such composite materials to have practical utility, they must be stable. Recent research has shown they can be, for a limited range of constituent negative stiffness. This research has treated linear elastic composite materials with homogeneous phases, via the energy method and full dynamic stability analyses.In the present work, we first show how to analyze the composites previously treated by the comprehensive but simpler static stability approach, obtaining closed-form results. We then employ this approach to show that permitting heterogeneity of the positive-definite phase can substantially increase the range of constituent negative stiffness while maintaining overall composite stability. We first treat the positive-definite phase heterogeneity as piecewise homogeneous, and then treat it as continuously-varying. In the continuously-varying heterogeneity case, we seek the radially optimal distribution of the elastic moduli in the coatings, under constant coating average moduli constraint, to permit the most negative possible inclusion stiffness while maintaining overall composite stability. This is accomplished for three coating cases: constant bulk modulus but arbitrarily radially-varying shear modulus; constant shear modulus but arbitrarily radially-varying bulk modulus; and both moduli arbitrarily radially varying. We find the optimal coatings to be: a heterogeneous one with shear modulus being a specific continuously decreasing function of radius for the first case; a homogeneous one for the second case; and a heterogeneous one with both moduli being either Dirac-delta or Heaviside-step decreasing functions of radius for the last case (if the coating moduli are unrestricted in magnitude or have upper limits, respectively). The results show a substantial increase in the permissible inclusion negative stiffness range is provided by coating heterogeneity, while maintaining overall composite stability. Such an increased range of constituent negative stiffness provides an enlarged tuning parameter range for the development of novel, high-performance composite materials.     

A plastic flow rule at a yield vertex

A yield vertex having three distinct facets at the uniaxial stress point is considered. An associated plastic flow rule is constructed using a new 3 × 3 coupled hardening matrix. This has the property that the incipient shear modulus is less than the elastic value. The approach is novel and distinct from previous work having that conclusion. In particular, all the relevant incipient moduli governing fully active in-plane loading can be fitted, if desired, to those values which J₂-deformation theory would require. To that extent the proposed incremental theory therefore legitimizes the use of the latter moduli, for example in certain ‘paradoxical’ buckling problems. When the moduli of the incremental theory are so chosen, the domain of stress-rate vectors enforcing loading is a calculable pyramid which contracts from the Mises half-space as the stress increases beyond yield. This domain becomes wider, at a given stress, as the initial smooth curvature of the stress-strain curve is imagined to become sharper.     

A continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Dilatational plastic constitutive equations and their application to ductile fracture analysis of axisymmetric specimens

In many ductile materials voids nucleate and grow under large strain and triaxial stress, which yield volumetric plastic expansion. A constitutive equation is presented, which accounts for this plastic dilatancy. The plastic moduli involved in this equation can be calibrated by using necking tests of axisymmetric bars, void model analysis and computer simulation. To verify the rationality of such a constitutive equation and adjust the values of plastic moduli, the constitutive equation with its moduli to be determined is applied to analyse the ductile fracture behaviour of axisymmetric bars.     

Hybrid theoretical,experimental and numerical study of vibration and buckling of composite shells with scatter in elastic moduli

Hybrid theoretical, experimental and numerical method is proposed for free vibration and buckling of composite shell with unavoidable scatter in elastic moduli. Based on the Goggin’s measurement techniques, the elastic moduli for material T300-QY8911 are measured, and a set of experimental points are obtained. The measurements of elastic moduli are quantified by either (1) the smallest ellipsoid and (2) the smallest four-dimensional uncertainty hyper-rectangle. Then uncertainty propagation in vibration and buckling problems of composite shell by ellipsoidal analysis and interval analysis are, respectively, studied from the theoretical standpoint. Comparison between these analyses is performed numerically.     

Matched asymptotic expansions in nonlinear fracture mechanics—I. longitudinal shear of an elastic perfectly-plastic specimen

A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the Jintegral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.     

拉压不同弹性模量薄板上圆孔的应力分析

采用弹性理论研究了拉压不同弹性模量薄板上圆孔的孔边应力集中问题．采用广义虎克定律推导出了拉压不同弹性模量薄板上圆孔边的应力平衡方程,并联合利用应力函数及边界条件得到了拉压不同弹性模量薄板上圆孔边的应力表达式．算例分析表明,当薄板材料的拉压弹性模量相差较大时,采用经典弹性理论研究薄板上圆孔的孔边应力是不合适的,当经典弹性理论与拉压不同弹性模量弹性理论的计算结果间的差别超过工程允许误差5%时,应该采用拉压不同弹性模量弹性理论进行计算．     

A simple method for calculating interaction of numerous microcracks and its applications

The effects of microcrack interaction on the failure behavior of materials present one problem of considerable interest in micromechanics, which has been extensively argued but has not been resolved as yet. In the present paper, a simple and effective method is presented based on the concept of the effective field to analyze the interaction of microcracks of a large number or of a high density. To determine the stress intensity factors of a microcrack embedded in a solid containing numerous or even countless microcracks, the solid is divided into two regions. The interaction of microcracks in a circular or elliptical region around the considered microcrack is calculated directly by using Kachanov’s micromechanics method, while the influence of all other microcracks is reflected by modifying the stress applied in the far field. Both the cases of tensile and compressive loading are considered. This simplified scheme may yield an estimate for stress intensity factors of satisfactory accuracy, and therefore provide a potential tool for elucidating some phenomena of material failure associated with microcracking. As two of its various promising applications, the above scheme is employed to investigate the size effects of material strength due to stochastic distribution of interacting microcracks and to calculate the effective elastic moduli of elastic solids containing distributed microcracks. Some conventional micromechanics methods for estimating the effective moduli of microcracked materials are evaluated by comparing with the numerical results. Only two-dimensional problems have been considered here, though the three-dimensional extension of the present method is of greater interest.     

On the compressive strength of open-cell metal foams with Kelvin and random cell structures

Two families of finite element models of anisotropic, aluminum alloy, open-cell foams are developed and their predictions of elastic properties and compressive strength are evaluated by direct comparison to experimental results. In the first family of models, the foams are idealized as anisotropic Kelvin cells loaded in the <100> direction and in the second family more realistic models, based on Surface Evolver simulations of random soap froth with N³ cells are constructed. In both cases the ligaments are straight but have nonuniform cross sectional area distributions that resemble those of the foams tested. The ligaments are modeled as shear deformable beams with elasto-plastic material behavior. The calculated compressive response starts with a linearly elastic regime. At higher stress levels, inelastic action causes a gradual reduction of the stiffness that eventually leads to a stress maximum, which represents the strength of the material. The periodicity of the Kelvin cell enables calculation of the compressive response up to the limit stress with just a single fully periodic characteristic cell. Beyond the limit stress, deformation localizes along the principal diagonals of the microstructure. Consequently beyond the limit stress the response is evaluated using finite size 3-D domains that allow the localization to develop. The random models consist of 3-D domains of 216, 512 or 1000 cells with periodicity conditions on the compressed ends but free on the sides. The compressive response is also characterized by a limit load instability but now the localization is disorganized resembling that observed in experiments. The foam elastic moduli and strengths obtained from both families of models are generally in very good agreement with the corresponding measurements. The random foam models yield 5–10% stiffer elastic moduli and slightly higher strengths than the Kelvin cell models. Necessary requirements for this high performance of the models are accurate representation of the material distribution in the ligaments and correct modeling of the nonlinear stress–strain response of the aluminum base material.     

在役设备材料弹塑性力学参数测定方法综述--小冲杆实验力学研究进展之一

20多年以来,采用小型试件的小冲杆试验技术来测量在役设备材料的各种力学参数已经取得了很大进展,这个方法已经用来确定材料的弹性模量、屈服强度、塑性性能、抗拉强度、韧一脆转变温度、断裂韧度、蠕变性能和黏塑性性能等各种力学性能。由于从小冲杆试验的测量结果来确定材料的力学性能是一个反问题,因此,与此有关的反问题分析方法也得到了相应的发展。本文系统综述小冲杆试验的测量技术及从测量数据来确定材料弹塑性参数的各种经验方法和计算方法,例如有限元分析和参数法、反向有限元法、有限元和反方法、反向识别和人工神经网络、有限元优化和试验变形形状以及杂交反方法等。     

Overall moduli of solids with microcracks: Load-induced anisotropy

For a linearly elastic brittle solid containing microcracks that may be closed or may undergo frictional sliding, a general method is developed for estimating the overall instantaneous moduli which depend on the loading conditions. When the cracks are all open and when they are randomly distributed, then the overall response is isotropic. The moduli for this case have been obtained by Budiansky and O'C onnell (1976). On the other hand, when some cracks close, and when some closed cracks undergo frictional sliding, then the overall response becomes anisotropic and dependent on the loading conditions, as well as on the loading path. The self-consistent method is used to estimate the overall moduli. The effects of crack closure and loadinduced anisotropy are included. Several illustrative examples are worked out, showing the important influence of the load path on the overall response when crack closure and frictional sliding are involved.