PERSPECTIVES IN MECHANICS OF HETEROGENEOUS SOLIDS

The Microand Nano-mechanics Working Group of the Chinese Society of Theoretical and Applied Mechanics organized a forum to discuss the perspectives,trends,and directions in mechanics of heterogeneous materials in January 2010.The international journal,Acta Mechanica Solida Sinica,is devoted to all fields of solid mechanics and relevant disciplines in science,technology,and engineering,with a balanced coverage on analytical,experimental,numerical and applied investigations.On the occasion of the 30 th anniversary of Acta Mechanica Solida Sinica,its editor-in-chief,Professor Q.S.Zheng invited some of the forum participants to review the state-of-the-art of mechanics of heterogeneous solids,with a particular emphasis on the recent research development results of Chinese scientists.Their reviews are organized into five research areas as reported in different sections of this paper.§I firstly brings in focus on microand nano-mechanics,with regards to several selective topics,including multiscale coupled models and computational methods,nanocrystal superlattices,surface effects,micromechanical damage mechanics,and microstructural evolution of metals and shape memory alloys.§II shows discussions on multifield coupled mechanical phenomena,e.g.,multi-fields actuations of liquid crystal polymer networks,mechanical behavior of materials under radiations,and micromechanics of heterogeneous materials.In §III,we mainly address the multiscale mechanics of biological nanocomposites,biological adhesive surface mechanics,wetting and dewetting phenomena on microstructured solid surfaces.The phononic crystals and manipulation of elastic waves were elaborated in §IV.Finally,we conclude with a series of perspectives on solid mechanics.This review will set a primary goal of future science research and engineering application on solid mechanics with the effort of social and economic development.     

Limiting velocity of crack propagation in elastic materials

A crack is represented as a continuous set of linear dislocations. Simple analytical expressions are obtained for the potential and kinetic energies of the environment of moving cracks and the attached mass of cracks for an arbitrary form of the stress applied to the crack P(x). It is shown that the indicated analytical expressions are bilinear integrals of the functions P(x) and ∂P(x)/∂x. These integrals are calculated in explicit form for a constant stress over the entire crack length and the stress due to the action of molecular adhesion forces in a narrow region near the crack openings. It is shown that the calculation results does not depend on the form of molecular adhesion forces. The proposed approach to describing cracks and calculations based on it has made it possible for the first time to obtain a complete analytical expression for the limiting crack propagation velocity in elastic materials as a function of the main mechanical characteristics of such materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 158–166, July–August, 2009.     

粉末热压扩散与应力场耦合的力学模型

以弹性接触应力场为初始条件,建立了热压条件下球形颗粒表面扩散与应力场耦合的力学模型. 引入包含表面能项级数形式的应力函数,以描述随时间演化的表面扩散过程及扩散对应力场演化的影响. 而应力场通过改变化学势梯度,又会促进(或阻止)表面扩散结合的进程.利用该模型分析了压力、温度和界面区应力场演化对致密化参数的影响. 比较了满足粘着或非粘着对结合宽度和应力分布的影响,将考虑粘着的弹性接触应力场作为初始条件,分析了弹性变形和表面扩散共同驱动的粉末冶金热压烧结致密化规律.      

Perturbation analysis of Mode III interfacial cracks advancing in a dilute heterogeneous material

The paper addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as microcracks, rigid line inclusions and voids. It is assumed that the bonding between dissimilar elastic materials is weak so that the interface is a preferential path for the crack. The perturbation analysis is made possible by means of the fundamental solutions (symmetric and skew-symmetric weight functions) derived in Piccolroaz et al. (2009). We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate “effective” tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects. For a stable propagation of the crack, the perturbation of the stress intensity factor induced by the defects is then balanced by the elongation of the crack along the interface, thus giving an explicit asymptotic formula for the calculation of the crack advance. The method is general and applicable to interfacial cracks with general distributed loading on the crack faces, taking into account possible asymmetry in the boundary conditions.The analytical results are used to analyse the shielding and amplification effects of various types of defects in different configurations. Numerical computations based on the explicit analytical formulae allows for the analysis of crack propagation and arrest.     

正交异性材料平面裂纹尖端应力场

根据正交各向异性材料力学性能确定出了用应力函数表示的弹性力学基本方程,利用坐标变换和复变函数方法求解了正交异性材料平面裂纹体的应力边值问题。借鉴一般断裂力学解法构造了I型和II型裂纹问题的应力函数,推导出了正交各向异性板裂纹尖端区的奇异应力场。通过数值计算说明了裂纹尖端应力表达式的正确性,验证了裂尖前沿应力变化规律,即σx与材料特征参数h2成正比,而σy和τxy不随材料特性变化。     

Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials

Using the hypersingular integral equation method based on body force method, a planar crack in a three-dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed. This crack problem is reduced to solve a set of hypersingular integral equations. Compare with the crack problems in elastic isotropic materials, it is shown that for the impermeable crack, the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials. Based on the exact analytical solution of the singular stresses and electrical displacements near the crack front, the numerical method of the hypersingular integral equation is proposed by the finite-part integral method and boundary element method, which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied. Finally, the numerical solutions of the stress and electric field intensity factors of some examples are given.     

An efficient approximate numerical method for modeling contact of materials with distributed inhomogeneities

This research explores the influence of distributed non-interpenetrating inhomogeneities on the contact of inhomogeneous materials via a new efficient numerical model based on Eshelby’s Equivalent Inclusion Method. The half-space contact of a sphere with an inhomogeneous material is considered, and the solutions take into account interactions between all inhomogeneities. The efficiency and solution accuracy of the proposed method are demonstrated through comparative studies with those of an existing numerical method and the finite element method. The influence of spatial inhomogeneity orientations on the contact elastic field is investigated and parametric studies are conducted for the effect of arbitrarily distributed inhomogeneities on the stress field of the materials. The significance of the influences of inhomogeneity distribution parameters on the inverse volumetric stress integral is quantified and the corresponding data are fitted into selected several formulas as a step towards understanding the rolling contact fatigue life of the materials.     

Dynamic stress intensity factor KⅢ and dynamic crack propagation characteristics of anisotropic materials

Based on the mechanics of anisotropic materials,the dynamic propagation problem of a mode Ⅲ crack in an infinite anisotropic body is investigated.Stress,strain and displacement around the crack tip are expressed as an analytical complex function,which can be represented in power series.Constant coefficients of series are determined by boundary conditions.Expressions of dynamic stress intensity factors for a mode Ⅲ crack are obtained.Components of dynamic stress,dynamic strain and dynamic displacement around the crack tip are derived.Crack propagation characteristics are represented by the mechanical properties of the anisotropic materials,i.e.,crack propagation velocity M and the parameter α.The faster the crack velocity is,the greater the maximums of stress components and dynamic displacement components around the crack tip are.In particular,the parameter α affects stress and dynamic displacement around the crack tip.     

Contact analysis in the presence of an ellipsoidal inhomogeneity within a half space

Many materials contain inhomogeneities or inclusions that may greatly affect their mechanical properties. Such inhomogeneities are for example encountered in the case of composite materials or materials containing precipitates. This paper presents an analysis of contact pressure and subsurface stress field for contact problems in the presence of anisotropic elastic inhomogeneities of ellipsoidal shape. Accounting for any orientation and material properties of the inhomogeneities are the major novelties of this work. The semi-analytical method proposed to solve the contact problem is based on Eshelby’s formalism and uses 2D and 3D Fast Fourier Transforms to speed up the computation. The time and memory necessary are greatly reduced in comparison with the classical finite element method. The model can be seen as an enrichment technique where the enrichment fields from the heterogeneous solution are superimposed to the homogeneous problem. The definition of complex geometries made by combination of inclusions can easily be achieved. A parametric analysis on the effect of elastic properties and geometrical features of the inhomogeneity (size, depth and orientation) is proposed. The model allows to obtain the contact pressure distribution – disturbed by the presence of inhomogeneities – as well as subsurface and matrix/inhomogeneity interface stresses. It is shown that the presence of an inclusion below the contact surface affects significantly the contact pressure and subsurfaces stress distributions when located at a depth lower than 0.7 times the contact radius. The anisotropy directions and material data are also key elements that strongly affect the elastic contact solution. In the case of normal contact between a spherical indenter and an elastic half space containing a single inhomogeneity whose center is located straight below the contact center, the normal stress at the inhomogeneity/matrix interface is mostly compressive. Finally when the axes of the ellipsoidal inclusion do not coincide with the contact problem axes, the pressure distribution is not symmetrical.     

Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress

The fundamental framework of micromechanical procedure is generalized to take into account the surface/interface stress effect at the nano-scale. This framework is applied to the derivation of the effective moduli of solids containing nano-inhomogeneities in conjunction with the composite spheres assemblage model, the Mori-Tanaka method and the generalized self-consistent method. Closed-form expressions are given for the bulk and shear moduli, which are shown to be functions of the interface properties and the size of the inhomogeneities. The dependence of the elastic moduli on the size of the inhomogeneities highlights the importance of the surface/interface in analysing the deformation of nano-scale structures. The present results are applicable to analysis of the properties of nano-composites and foam structures.     

横观各向同性压电材料中裂纹问题的边界元法

采用Somigiliana公式给出了三维横观各向同性压电材料中的非渗漏裂纹问题的一般解和超奇异积分方程，其中未知函数为裂纹面上的位移间断和电势间断．在此基础上，使用有限部积分和边界元结合的方法，建立了超奇异积分方程的数值求解方法，并给出了一些典型数值算例的应力强度因子和电位移强度因子的数值结果，结果令人满意．     

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.     

A Self-Similar Crack Expansion method for three-dimensional cracks in an infinite or semi-infinite medium

The Self-Similar Crack Expansion (SSCE) method is used to calculate stress intensity factors for three-dimensional cracks in an infinite medium or semi-infinite medium by the boundary integral element technique, whereby, the stress intensity factors at crack tips are determined by calculating the crack-opening displacements over the crack surface. For elements on the crack surface, regular integrals and singular integrals are precisely evaluated based on closed form expressions, which improves the accuracy. Examples show that this method yields very accurate results for stress intensity factors of penny-shaped cracks and elliptical cracks in the full space, with errors of less than 1% as compared with analytical solutions. The stress intensity factors of subsurface cracks are in good agreement with other analytical solutions.     

Mechanics of crack propagation in materials with initial (residual) stresses (review)

Major results on the mechanics of crack propagation in materials with initial (residual) stresses are analyzed. The case of straight cracks of constant width that propagate at a constant speed in a material with initial (residual) stresses acting along the cracks is examined. The results were obtained, based on linearized solid mechanics, in a universal form for isotropic and orthotropic, compressible and incompressible elastic materials with an arbitrary elastic potential in the cases of finite (large) and small initial strains. The stresses and displacements in the linearized theory are expressed in terms of analytical functions of complex variables when solving dynamic plane and antiplane problems. These complex variables depend on the crack propagation rate and the material properties. The exact solutions analyzed were obtained for growing (mode I, II, III) cracks and the case of wedging by using methods of complex variable theory, such as Riemann–Hilbert problem methods and the Keldysh–Sedov formula. As the initial (residual) stresses tend to zero, these exact solutions of linearized solid mechanics transform into the respective exact solutions of classical linear solid mechanics based on the Muskhelishvili, Lekhnitskii, and Galin complex representations. New mechanical effects in the dynamic problems under consideration are analyzed. The influence of initial (residual) stresses and crack propagation rate is established. In addition, the following two related problems are briefly analyzed within the framework of linearized solid mechanics: growing cracks at the interface of two materials with initial (residual) stresses and brittle fracture under compression along cracks     

Effect of surface/interface stress on the plastic deformation of nanoporous materials and nanocomposites

Surface and interface play an important role on the overall mechanical behaviors of nanostructured materials. We investigate the effect of surface/interface stress on the macroscopic plastic behaviors of nanoporous materials and nanocomposites, where both the surface/interface residual stress and surface/interface elasticity are taken into account. A new second-order moment nonlinear micromechanics theory is developed and then reduced to macroscopically isotropic materials. It is found that the effect of surface/interface residual stress is much more prominent than that of the surface/interface elasticity, causing strong size effect as well as asymmetric plastic deformation for tension and compression. The variation of yield strength is more prominent with smaller pore/inclusion size or higher pore/inclusion volume fraction. For a representative nanoporous aluminum, the surface effect becomes significant when the pore radius is smaller than about 50 nm. When hard inclusions are embedded in a ductile metal matrix, the interface effect and resulting size effect are much smaller than that of nanoporous materials. The results may be useful for evaluating the mechanical integrity of nanostructured materials.     

The line spring model with arbitrary loads on crack surfaces and its applications in thermal shock analysis

In this paper the line spring model taking account of arbitrary loads on crack surfaces, and the corresponding constitutive relations, are proposed. The general expressions of the additional outfield loads, which are equivalent to the distributed loads on crack surfaces, are derived. The model is used to compute stress intensity factors in a hollow cylinder with an axial surface crack subjected to thermal shock. Several results of calculations are presented and discussed.     

An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic elastic plane is studied. An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, resulting in fiber rupture and hence the crack propagation would occur in a self-similarity manner. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two moving loads given is obtained, respectively. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be obtained.     

Fracture of a body with a periodic set of coaxial cracks under forces directed along them: an axisymmetric problem

Linearized solid mechanics is used to solve an axisymmetric problem for an infinite body with a periodic set of coaxial cracks. Two nonclassical fracture mechanisms are considered: fracture of a body with initial stresses acting in parallel to crack planes and fracture of materials compressed along cracks. Numerical results are obtained for highly elastic materials described by the Bartenev–Khazanovich, Treloar, and harmonic elastic potentials. The dependence of the fracture parameters on the loading conditions, the physical and mechanical characteristics of the material, and the geometrical parameters is analyzed Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 3–18, February 2009.     

Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads

Mixed-mode fracture problems of orthotropic functionally graded materials (FGMs) are examined under mechanical and thermal loading conditions. In the case of mechanical loading, an embedded crack in an orthotropic FGM layer is considered. The crack is assumed to be loaded by arbitrary normal and shear tractions that are applied to its surfaces. An analytical solution based on the singular integral equations and a numerical approach based on the enriched finite elements are developed to evaluate the mixed-mode stress intensity factors and the energy release rate under the given mechanical loading conditions. The use of this dual approach methodology allowed the verifications of both methods leading to a highly accurate numerical predictive capability to assess the effects of material orthotropy and nonhomogeneity constants on the crack tip parameters. In the case of thermal loading, the response of periodic cracks in an orthotropic FGM layer subjected to transient thermal stresses is examined by means of the developed enriched finite element method. The results presented for the thermally loaded layer illustrate the influences of the material property gradation profiles and crack periodicity on the transient fracture mechanics parameters.