Lower strain and stress bounds for elastic random composites consisting of two isotropic phases and exhibiting cubic symmetry

This work is concerned with a composite consisting of two linearly elastic isotropic phases which are distributed at the microscopic scale in such a way that the composite exhibits cubic symmetry at the macroscopic scale. Lower bounds on the second-order and higher order moments of the phase stress and strain fields of such a composite undergoing uniform macroscopic loading are provided in terms of the phase volume fractions and moduli. The lower bounds on the phase second-order strain and stress moments are proved to be optimal by showing that the stress and the strain fields inside appropriate finite-rank laminates achieve them.     

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.     

Optimal lower bounds on the local stress inside random thermoelastic composites

A methodology is presented for bounding all higher moments of the local hydrostatic stress field inside random two phase linear thermoelastic media undergoing macroscopic thermomechanical loading. The method also provides a lower bound on the maximum local stress. Explicit formulae for the optimal lower bounds are found that are expressed in terms of the applied macroscopic thermal and mechanical loading, coefficients of thermal expansion, elastic properties, and volume fractions. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to the applied loads and the thermal stresses inside each phase. These bounds are shown to be the best possible in that they are attained by the Hashin–Shtrikman coated sphere assemblage.     

A finite element analysis of plane strain dynamic crack growth at a ductile-brittle interface

In this work, steady, dynamic crack growth under plane strain, small-scale yielding conditions along a ductile-brittle interface is analysed using a finite element procedure. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behaviour. The objectives of this work are to establish the validity of an asymptotic solution for this problem which has been derived recently [12], and to examine the effect of changing the remote (elastic) mode-mixity on the near-tip fields. Also, the influence of crack speed on the stress fields and crack opening profiles near the propagating interface crack tip is assessed for various bi-material combinations. Finally, theoretical predictions are made for the variation of the dynamic fracture toughness with crack speed for crack growth under a predominantly tensile mode along ductile-brittle interfaces. Attention is focused on the effect of mismatch in stiffness and density of the constituent phases on the above aspects.     

Numerical investigation of crack tip strain localization under cyclic loading in FCC single crystals

In this work, the crack tip strain localization in a face centered cubic single crystal subject to both monotonic and cyclic loading was investigated. The effect of constraint was implemented using T-stress and strain accumulation was studied for both isotropic and anisotropic elastic cases with the appropriate application of remote displacement fields in plane strain. Modified boundary layer simulations were performed using the crystal plasticity finite element framework. The consideration of elastic anisotropy amplified the effect of constraint level on stress and plastic strain fields near the crack tip indicating the importance of its use in fracture simulations. In addition, to understand the cyclic stress and strain behavior in the vicinity of the crack tip, combined isotropic and kinematic hardening laws were incorporated, and their effect on the evolution of yield curves and plastic strain accumulation were investigated. With zero-tension cyclic load, the evolution of plastic strain and Kirchhoff stress components showed differences in magnitudes between isotropic and anisotropic elastic cases. Furthermore, under cyclic loading, ratcheting was observed along the localized slip bands, which was shown to be affected by T-stress as well as elastic anisotropy. Negative T-stress increased the accumulation of plastic strain with number of cycles, which was further amplified in the case of elastic anisotropy. Finally, in all the cyclic loading simulations, the plastic strain accumulation was higher near the \(55^0 \) slip band.     

Effect of orthotropic material on finite element modeling of completely dentate mandible

The objective of this investigation is to construct a high quality complete dentate mandible model with detailed biological structures, and assign mandibular bone with inherent orthotropic material characteristics. Three different types of scan data are used to elaborate detailed mandibular structures, including the cortical and cancellous bone, tooth enamel, dentin, periodontal ligament, temporal fossa, TMJ articular disk, temporal cartilage, and condylar cartilage. In addition, an extended orthotropic material assignment methodology based on harmonic fields is used to handle the alveolar ridge region of dentate mandible, to generate compatible orthotropic axes fields. The influence of orthotropic material on the biomechanical behavior of complete dentate mandible is analyzed compared with commonly used isotropic model. The result revealed that the orthotropic model would induce higher stress values and more well-distributed stress pattern than the isotropic model, especially for the cancellous bone. And the orthotropic model would induce lower volumetric strain values than the isotropic model on the cortical bone. It was concluded that elastic orthotropy had a significant effect on the simulated stress value and distribution pattern, as well as the volumetric strain, and demonstrated the mechanical optimality of the mandible.     

Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.     

Application of variational principles to the axial extension of a circular cylindical nonlinearly elastic membrane

In this paper stationary potential-energy and complementary-energy principles are formulated for boundary-value problems for compressible or incompressible nonlinearly elastic membranes, and full justification for adoption of the complementary principle is provided. The stationary principles are then extended to extremum principles, which provide upper and lower bounds on the energy functional associated with the solution of a given problem. The principles are then illustrated by their application to the nonlinear problem of the axially symmetric static deformation of an isotropic elastic membrane. In its undeformed natural configuration the membrane has the form of a circular cylindrical surface. The cylinder is subject to a prescribed (tensile) axial force with the ends of the cylinder constrained so that their radii remain constant. The alternative boundary condition in which the axial displacement of the ends is prescribed instead of the axial force is also considered. The extremum principles are applied first without restriction on the form of strain-energy function in order to obtain primitive bounds on the energy of Voigt and Reuss type commonly used in composite-material mechanics. Then, for particular forms of strain-energy function, specific bounds are obtained by selecting suitable trial deformation and stress fields and the bounds are optimized using a numerical procedure (which is readily adapted for other forms of strain-energy function). It is found that these bounds are very close and hence give a good estimate of the actual energy. The associated deformed geometry of the membrane is described together with the resulting principal stresses.     

Estimations for the overall properties of some locally-ordered composites

Summary Simple estimations for the overall conductivity and elastic properties of some isotropic locally-ordered composites are deduced from two different variational approaches. The estimations based on limited information about microgeometry of the composites lie inside the Hashin-Shtrikman bounds over the whole range of parameters.     

Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity

The aim of this paper is to study the elastic stress and strain fields of dislocations and disclinations in the framework of Mindlin’s gradient elasticity. We consider simple but rigorous versions of Mindlin’s first gradient elasticity with one material length (gradient coefficient). Using the stress function method, we find modified stress functions for all six types of Volterra defects (dislocations and disclinations) situated in an isotropic and infinitely extended medium. By means of these stress functions, we obtain exact analytical solutions for the stress and strain fields of dislocations and disclinations. An advantage of these solutions for the elastic strain and stress is that they have no singularities at the defect line. They are finite and have maxima or minima in the defect core region. The stresses and strains are either zero or have a finite maximum value at the defect line. The maximum value of stresses may serve as a measure of the critical stress level when fracture and failure may occur. Thus, both the stress and elastic strain singularities are removed in such a simple gradient theory. In addition, we give the relation to the nonlocal stresses in Eringen’s nonlocal elasticity for the nonsingular stresses.     

Some elastic properties of reinforced solids,with special reference to isotropic ones containing spherical inclusions

Based on Mori and Tanaka's concept of “average stress” in the matrix and Eshelby's solutions of an ellipsoidal inclusion, an approximate theory is established to derive the stress and strain state of constituent phases, stress concentrations at the interface, and the elastic energy and overall moduli of the composite. Both “stress-free” strain (polarization strain) and “strain-free” stress (polarization stress) are employed in these derivations under the traction- and displacement-prescribed conditions. The theory was developed first for a general multiphase, anisotropic composite with arbitrarily oriented anisotropic inclusions; explicit results are then given for a suspension of uniformly distributed, multiphase isotropic spheres in an isotropic matrix. Numerical results for stress concentrations in the spherical inclusions and at the interface are given for a 2-phase composite. Further, it is shown that the derived moduli are related to the Hashin-Shtrikman bounds and that, when the shear moduli are equal, the overall bulk modulus of a 2-phase composite reduces to Hill's exact solution. As compared with experimental data, the theory also provides reasonably accurate estimates for the Young's modulus of some 2- and 3-phase composites.     

复合材料的等效弹性性能

用细观力学的分析方法研究了复合材料的宏观等效弹性性能。在严格满足组分相间界面的连续性条件下,正确反映了组分相间的相互作用, 考虑了弹性张量各分量之间的相互关系, 分析了层合介质的宏观等效弹性性能。进一步用统计平均的思想, 得到了总体横观各向同性及总体各向同性复合材料的等效弹性性能的解析表达式。与有关的理论及实验结果比较, 得到了非常满意的结果。      

Method of complex potentials in linear microstretch elasticity

The paper is concerned with the plane strain of homogeneous and isotropic microstretch elastic bodies. We give a new representation of the solution in terms of complex potentials. The method is useful for the treatment of the constitutive equations established by Eringen in [A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999], where the introduction of stress functions leads to difficulties. The complex variable technique is used to study Kirsch problem in the context of the theory presented in [A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999].     

Stress and couple-stress fields produced by Frank disclinations in an isotropic elastic micropolar continuum

The stress and couple-stress fields due to Frank disclinations in an infinitely extended isotropic, elastic and micropolar continuum are estimated. The author starts with a brief review on the expressions for the elastic fields due to distributed dislocations and disclinations. Those expressions are converted into line-integral expressions for the elastic fields due to Frank disclinations. They are specialized to the case of infinite straight disclinations. The stress and couple-stress fields are calculated for both twist and wedge disclinations.     

Stress and strain partition in elastic and plastic deformation of two phase alloys

A stress and strain partition theory for two phase alloys was developed on the basis of the modified rules of mixtures. The extreme value condition of macroscopic strain energy density was found through Lagrangian multiplier method. Expressions for macroscopic elastic constants of two phase alloys were derived from the extreme value condition by assuming the strain linearity between constituent phases. Governing equation for stress and strain partition in plastic deformation was also obtained from the extreme value condition. The calculated elastic constants of WC-Co alloys fell invariably within the Hashin and Shtrikman's bounds. According to the governing equation the stress ratio between constituent phases was plotted as a function of strain increment ratio. By applying the governing equation to spheroidized carbon steel and duplex stainless steel, it was shown that the stress ratios, strain ratios, macroscopic stress-strain curves, and internal stresses could be evaluated from thein situ stress-strain curves of constituent phases.     

Elastic moduli for a class of porous materials

Summary The effective elastic moduli for a class of porous materials with various distributions of spheroidal voids are given explicitly. The distributions considered include the unidirectionally aligned voids, three-dimensionally and two-dimensionally, randomly oriented voids, and voids with two types of biased orientations. While the 3-d random orientation results in a macroscopically isotropic solid, the porous media associated with the other arrangements are transversely isotropic. The five independent elastic constants for each arrangement, as well as the two for the isotropic case, are derived by means of Mori-Tanaka's mean field theory in conjunction with Eshelby's solution. Specific results for long, cylindrical pores and for thin cracks with the above orientations are also obtained, the latter being expressed in terms of the crack-density parameter. Before we set out the analysis, it is further proven that, in the case of long, circular inclusions, the five effective moduli of a fiber composite derived from the Mori-Tanaka method coincide with Hill's and Hashin's lower bounds if the matrix is the softer phase, and coincide with their upper bounds if the matrix is the harder.With 5 Figures     

Universal theorems for total energy of the dynamics of linearly elastic heterogeneous solids

In this paper we consider a sample of a linearly elastic heterogeneous composite in elastodynamic equilibrium and present universal theorems which provide lower bounds for the total elastic strain energy plus the kinetic energy, and the total complementary elastic energy plus the kinetic energy. For a general heterogeneous sample which undergoes harmonic motion at a single frequency, we show that, among all consistent boundary data which produce the same average strain, the uniform-stress boundary data render the total elastic strain energy plus the kinetic energy an absolute minimum. We also show that, among all consistent boundary data which produce the same average momentum in the sample, the uniform velocity boundary data render the total complementary elastic energy plus the kinetic energy an absolute minimum. We do not assume statistical homogeneity or material isotropy in our treatment, although they are not excluded. These universal theorems are the dynamic equivalent of the universal theorems already known for the static case [Nemat-Nasser and Hori, 1993] and [Nemat-Nasser and Hori, 1995]. It is envisaged that the bounds on the total energy presented in this paper will be used to formulate computable bounds on the overall dynamic properties of linearly elastic heterogeneous composites with arbitrary microstructures.     

CONDITIONS FOR ADAPTATION OF DAMAGED ELASTIC–PLASTIC BODIES TO CYCLIC LOADING

Conditions for adaptation of isotropically damaged elastic–plastic bodies with isotropic strain hardening are investigated in the framework of the energy-based coupled elastic–plastic damage model by Ju. The yield function is assumed to be a homogeneous function of the first order in the stress tensor components. Due to this assumption, the notion of effective yield stress can be introduced. The loading program is supposed to be prescribed. Features of the stress path at the post-adaptation stage are considered, which lead to new necessary shakedown conditions expressed by a set of inequalities, and, in turn, to a problem of mathematical programming whose solution yields lower estimates for the damage and strain-hardening parameters. In the event, if the calculated value of the damage parameter is greater than its critical value, an adaptation to a given loading program is impossible. This condition is also necessary for adaptation in the case if only bounds for applied loads are prescribed. A correction of the constitutive material model is proposed which possibly could be good for ductile damage. The derived shakedown condition is not only necessary, but also sufficient for the plastic adaptation. The developed method is expounded in an example.     

Upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks in three-dimensional elasticity

An elementary method for obtaining upper bounds for the stress intensity factors along the boundaries of interacting coplanar cracks inside an infinite isotropic elastic medium is presented. This method is based on the singular integral equation of the aforementioned elasticity problem and on the solutions of this equation for each particular crack problem, assumed known. The method is applied to the simple problem of interaction of two circular cracks, as well as to the similar problem of two cracks having the shape of a straight strip. The present results constitute a generalization of the corresponding method for crack problems in two-dimensional elasticity and can easily be further generalized to apply to more complicated crack problems in three-dimensional elasticity.     

Non-symmetric extension of a crack

Summary The dynamic in-plane problem of the non-symmetric extension of a crack in an infinite, isotropic elastic medium under normal stress is analyzed. Following Cherepanov [8], Cherepanov and Afanas'ev [9] the general solution of the problem is derived in terms of an analytic function of complex variable. The results include the expressions for the stress intensity factors at the crack tips and the rate of energy flux into the cxtending crack edges. For a particular case, numerical calculations for the stress intensity factor and the energy flux rate are carried out.