Topological sensitivity of energy cost functional for wave-based defect identification

This article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration.     

Sensitivity of a general class of shape functionals to topological changes

The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations. The topological derivative has been successfully applied in the treatment of problems such as topology optimization, inverse analysis and image processing. In this paper, the calculation of the topological derivative for a general class of shape functionals is presented. In particular, we evaluate the topological derivative of a modified energy shape functional associated to the steady-state heat conduction problem, considering the nucleation of a small circular inclusion as the topological perturbation. Several methods were proposed to calculate the topological derivative. In this paper, the so-called topological-shape sensitivity method is extended to deal with a modified adjoint method, leading to an alternative approach to calculate the topological derivative based on shape sensitivity analysis together with a modified Lagrangian method. Since we are dealing with a general class of shape functionals, which are not necessarily associated to the energy, we will show that this new approach simplifies the most delicate step of the topological derivative calculation, namely, the asymptotic analysis of the adjoint state.     

Dynamic void nucleation and growth in solids: A self-consistent statistical theory

We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.     

A Radiation Condition for Layered Elastic Media

This study aims to establish a generalized radiation condition for time-harmonic elastodynamic states in a piecewise-homogeneous, semi-infinite solid wherein the “bottom” homogeneous half-space is overlain by an arbitrary number of bonded parallel layers. To consistently deal with both body and interfacial (e.g. Rayleigh, Love and Stoneley) waves comprising the far-field patterns, the radiation condition is formulated in terms of an integral over a sufficiently large hemisphere involving elastodynamic Green's functions for the featured layered medium. On explicitly proving the reciprocity identity for the latter set of point-load solutions, it is first shown that the layered Green's functions themselves satisfy the generalized radiation condition. By virtue of this result it is further demonstrated that the entire class of layered elastodynamic solutions, admitting a representation in terms of the single-layer, double-layer, and volume potentials (distributed over finite domains), satisfy the generalized radiation condition as well. For a rigorous treatment of the problem, fundamental results such as the uniqueness theorem for radiating elastodynamic states, Graffi's reciprocity theorem for piecewise-homogeneous domains, and the integral representation theorem for semi-infinite layered media are also established.     

The energy-release rate and “self-force” of dynamically expanding spherical and plane inclusion boundaries with dilatational eigenstrain

In the context of the linear theory of elasticity with eigenstrains, the radiated field including inertia effects of a spherical inclusion with dilatational eigenstrain radially expanding is obtained on the basis of the dynamic Green's function, and one of the half-space inclusion boundary (with dilatational eigenstrain) moving from rest in general subsonic motion is obtained by a limiting process from the spherically expanding inclusion as the radius tends to infinity while the eigenstrain remains constrained, and this is the minimum energy solution. The global energy-release rate required to move the plane inclusion boundary and to create an incremental region of eigenstrain is defined analogously to the one for moving cracks and dislocations and represents the mechanical rate of work needed to be provide for the expansion of the inclusion. The calculated value, which is the “self-force” of the expanding inclusion, has a static component plus a dynamic one depending only on the current value of the velocity, while in the case of the spherical boundary, there is an additional contribution accounting for the jump in the strain at the farthest part at the back of the inclusion having the time to reach the front boundary, thus making the dynamic “self-force” history dependent.     

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh-Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.     

Molecular dynamics simulation study of microstructure evolution during cyclic martensitic transformations

Shape memory alloys (SMA) exhibit a number of features which are not easily explained by equilibrium thermodynamics, including hysteresis in the phase transformation and “reverse” shape memory in the high symmetry phase. Processing can change these features: repeated cycling can “train” the reverse shape memory effect, while changing the amount of hysteresis and other functional properties. These effects are likely to be due to formations of localised defects and these can be studied by atomistic methods. Here we present a molecular dynamics simulation study of such behaviour employing a two-dimensional, binary Lennard-Jones model. Our atomistic model exhibits a symmetry breaking, displacive phase transition from a high temperature, entropically stabilised, austenite-like phase to a low temperature martensite-like phase. The simulations show transformations in this model material proceed by non-diffusive nucleation and growth processes and produce distinct microstructures. We observe the generation of persistent lattice defects during forward-and-reverse transformations which serve as nucleation centres in subsequent transformation processes. These defects interfere the temporal and spatial progression of transformations and thereby affect subsequent product morphologies. During cyclic transformations we observe accumulations of lattice defects so as to establish new microstructural elements which represent a memory of the previous morphologies. These new elements are self-organised and they provide a basis of the reversible shape memory effect in the model material.     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

The continuum elastic and atomistic viewpoints on the formation volume and strain energy of a point defect

We discuss the roles of continuum linear elasticity and atomistic calculations in determining the formation volume and the strain energy of formation of a point defect in a crystal. Our considerations bear special relevance to defect formation under stress. The elasticity treatment is based on the Green's function solution for a center of contraction or expansion in an anisotropic solid. It makes possible the precise definition of a formation volume tensor and leads to an extension of Eshelby's [Proc. R. Soc. London Ser. A 241 (1226), 376] result for the work done by an external stress during the transformation of a continuum inclusion. Parameters necessary for a complete continuum calculation of elastic fields around a point defect are obtained by comparing with an atomistic solution in the far field. However, an elasticity result makes it possible to test the validity of the formation volume that is obtained via atomistic calculations under various boundary conditions. It also yields the correction term for formation volume calculated under these boundary conditions. Using two types of boundary conditions commonly employed in atomistic calculations, a comparison is also made of the strain energies of formation predicted by continuum elasticity and atomistic calculations. The limitations of the continuum linear elastic treatment are revealed by comparing with atomistic calculations of the formation volume and strain energies of small crystals enclosing point defects.     

Topological derivative for an anisotropic and heterogeneous heat diffusion problem

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of physical phenomenon modeled by partial differential equations, considering homogeneous and isotropic constitutive behavior. In fact, only a few works dealing with heterogeneous and anisotropic material behavior can be found in the literature, and, in general, the derived formulas are given in an abstract form. In this work, we derive the topological derivative in its closed form for the total potential energy associated to an anisotropic and heterogeneous heat diffusion problem, when a small circular inclusion of the same nature of the bulk phase is introduced at an arbitrary point of the domain. In addition, we provide a full mathematical justification for the derived formula and develop precise estimates for the remainders of the topological asymptotic expansion. Finally, the influence of the heterogeneity and anisotropy are shown through some numerical examples of heat conductor topology optimization.     

SH波对浅埋弹性圆柱及裂纹的散射与地震动

采用Green函数、复变函数和多极坐标等方法研究含圆柱形弹性夹杂的弹性半空间中任意位置、任意方位有限长度裂纹对SH波的散射与地震动. 构造了含圆柱形弹性夹杂的半空间对SH波的散射波,并求解了适合本问题Green函数,即含有圆柱形弹性夹杂的半空间内(表面)任意一点承受时间谐和的出平面线源载荷作用时位移函数的基本解答. 利用裂纹``切割'方法在任意位置构造任意方位的裂纹,可以得到基体中圆柱形弹性夹杂和裂纹同时存在条件下的位移场与应力场. 通过数值算例,讨论各种参数对夹杂上方地表位移的影响.      

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In this paper we study the antiplane problem of concentrated point force moving with constant velocity and oscillating with constant frequency in unbounded homogeneous anisotropic elastic medium.The explicit representation of the elastodynamic Green's function is obtained by using Fourier integral transform techniques for all rates of source motion as a sum of the integrals over the finite interval. The dynamic and quasistatic components of the Green's function are extracted. The stationary phase method is applied to derive an asymptotic approximation at the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too.It is shown that the motion brings some differences in the far field properties, such as, for example, fast and slow waves appearance under superseismic motion and modification of the wave propagation zones and their numbers.The case of isotropic medium is considered separately. For isotropic material all main formulae are obtained in explicit forms.     

Unsteady motions of degrading or aging linearized elastic solids

Many structures are made of concrete and steel. The response characteristics of such structures that are exposed to the environment change due to its impact such as oxidation due to moisture. Usually the structural element “degrades” in that the load bearing capacity decreases; it also decreases due to “aging”. On the other hand, chemicals are infused into biological systems in order to enhance their strength. Thus far, not much attention has been paid to studying such problems. In this investigation, we study the response of a linearized elastic solid whose material properties change due to the diffusion of a chemical.     

The rigid movable inclusion in elastostatics and elastodynamics

The problem of a single rigid movable inclusion is solved both for elastostatics and elastodynamic, using the null field approach. It is also shown how the solution can be obtained as the limit of the solution for an elastic inclusion. Numerical results for scattering by a superspheroidal inclusion are given.     

Second order topological sensitivity analysis

The topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal holes. However, for practical applications, we need to insert holes of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we apply the topological-shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the Poisson’s equations, taking the total potential energy as cost function and the state equation as constraint. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). Finally, we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion, which has two main features: it allows us to deal with hole of finite size and provides a better descent direction in optimization process.     

Identification of an ellipsoidal defect in an elastic solid using boundary measurements

Elastostatic problem of identification of an ellipsoidal cavity or inclusion (rigid or linear elastic) in an isotropic, linear elastic solid is considered. The reciprocity gap functional method is used for solving the problem. It is shown that the parameters of the ellipsoidal defect (coordinates of its center, the directions and magnitudes of the semiaxes and elastic moduli in the case of isotropic, linear elastic inclusion), located in an infinite elastic solid are expressed by means of the values of the reciprocity gap functional. The values of the reciprocity gap functional can be calculated if the loads and displacements corresponding to uniaxial tension (compression) of an infinite solid are known on the closed surface containing the defect inside. Applications of the results to the problem of ellipsoidal defect identification in a bounded body are discussed. A number of numerical examples showing the efficiency of the developed identification method are considered.     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

Detachment of a rigid solid from an elastic wavy surface: Experiments

The mechanics of detachment of a rigid solid from an elastic wavy surface has been analyzed in a recent article, in which the axisymmetric case of a sphere and the plane strain case of a cylinder were considered. Due to the qualitative similarities, the discussion was limited to the axisymmetric case only. It was shown that the surface waviness makes the detachment process proceed in alternating stable and unstable segments and each unstable jump dissipates mechanical energy. As a result, the external work and the peak force required to separate a wavy interface are higher than the corresponding values for a flat interface, i.e., waviness causes interface toughening as well as strengthening. In this paper, a systematic experimental investigation is presented which examines the above theoretical analysis, by measuring adhesion between a “rigid” wavy punch and a soft “elastic” material, which is a block of gelatin here. The observed increase in adhesion due to waviness closely agrees with the theoretical predictions within the experimental and material uncertainties. The experiments not only validate the theory, but also demonstrate that adhesion of a soft material can be substantially enhanced by topographic optimization alone, without modifying the surface chemistry.     

M积分与夹杂/缺陷弹性模量的显式关系

M积分在材料构型力学中表征着缺陷自相似扩展的能量释放率,而有效弹性模量下降量在传统损伤力学中是一个具有内变量属性的损伤参数. 探讨了两者之间的特定关系,以此为材料构型力学与损伤力学搭建桥梁.借助穆斯海里什维利（Muskhelishvili）复势函数方法获取无限大弹性平面含圆形夹杂的弹性场解,根据M 积分的复势函数解析表达式得到M 积分与夹杂弹性模量的显式表达式. 随后通过有限元分析,对含复杂缺陷群的弹塑性材料进行数值模拟,结果表明内部缺陷区域的有效弹性模量下降与M 积分存在着特定关系. 基于此,提出利用材料构型力学中的外变量参数（M 积分）来替代损伤力学中的内变量参数（弹性模量下降量）描述材料的缺陷演化.      

BOUNDARY VALUE PROBLEMS OF TWO-DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

Based upon Stroh formalism we derive a novel and convenient scheme for determiningthe elastic fields of a two-dimensional anisotropic body with a parabolic boundary subject to two kindsof boundary conditions, which are free surface and rigid surface, respectively. The correspondingGreen's functions are found by using the conformal mapping method. When the parabolic curve de-generates into a half-infinite crack or rigid inclusion, the singular stress fields near the tip of defectsare obtained. In particular, those Green's functions for a concentrated moment M_0 applied at a pointon the parabolic curve are also studied. It is easily found that arbitrary parabolic boundary value prob-lems can be solved by using these Green's functions and associate integrals.