软材料粘接结构界面破坏研究综述

软材料已经在软机器人、生物医学及柔性电子等各个领域得到广泛的应用. 实际应用中, 软材料多需要粘附于不同类型的基底上, 与之共同组成工程构件进而实现特定的功能, 粘接界面性能对构件的结构完整性与功能可靠性起着关键性作用. 本文对目前软材料粘接结构界面破坏行为方面的研究进行了系统总结. 首先通过与传统粘接结构的对比, 指出了“软界面”与“软基体”两种软材料粘接结构界面破坏行为的独特性及其物理本质. 接着分别总结了“软界面”与“软基体”两种粘接结构界面破坏行为的实验表征方面的研究成果, 对界面及基体黏弹性耗散对界面破坏机理的影响分别进行了分析. 然后从理论角度, 介绍了针对两种软材料粘接结构界面破坏行为的理论分析方法, 并对已建立的相关理论模型进行了总结. 之后以内聚力模型方法为基础, 介绍了软材料粘接结构界面破坏行为数值模拟方面的相关研究进展. 最后基于已有的研究成果, 提出了目前研究所面临的挑战, 并对可能的软材料粘接结构界面破坏的未来研究方向进行了讨论和展望.      

Three-dimensional analysis of doubly curved functionally graded magneto-electro-elastic shells

Three-dimensional (3D) solutions for the static analysis of doubly curved functionally graded (FG) magneto-electro-elastic shells are presented by an asymptotic approach. In the present formulation, the twenty-nine basic equations are firstly reduced to ten differential equations in terms of ten primary variables of elastic, electric and magnetic fields. After performing through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of two-dimensional (2D) governing equations for various order problems. These 2D governing equations are merely those derived in the classical shell theory (CST) based on the extended Love–Kirchhoffs' assumptions. Hence, the CST-type governing equations are derived as a first-order approximation to the 3D magneto-electro-elasticity. The leading-order solutions and higher-order corrections can be determined by treating the CST-type governing equations in a systematic and consistent way. The 3D solutions for the static analysis of doubly curved multilayered and FG magneto-electro-elastic shells are presented to demonstrate the performance of the present asymptotic formulation. The coupling magneto-electro-elastic effect on the structural behavior of the shells is studied.     

Limit models in the analysis of three different layered elastic strips

The mechanical behavior of three types of laminated strips is investigated. They are made of three layers filled with homogeneous, isotropic and elastic materials; the upper and lower layer are called adherents, the middle layer is called adhesive. The first model studies a strip consisting of three layers made of materials with similar stiffness; the second one concerns with a strip in which the adhesive is soft; in particular, we suppose that the elastic stiffness of the middle layer is two orders of magnitude smaller than that of the upper and lower layers; the third case is a strip in which the core is thinner and stiffer than the two adherents: the elastic modula of the adherents are one order of magnitude bigger that those of the adhesive. After identifying a parameter of smallness ε (which measures the thickness and the stiffness of each layer), the limit of the solution when ε tends to zero has been considered. Afterwards, it has been shown that each solution of the simplified models verifies the so-called limit problems, written using a “weak” and a “strong” formulation. The existence and uniqueness of the solutions of each limit problem have been established. The strong convergence of the exact solutions towards the solution of the limit problem of the first model has been established, too.     

Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales

A three-dimensional (3D) free vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with closed-circuit surface conditions is presented using the method of perturbation. By means of the direct elimination, we firstly reduce the twenty-nine basic equations of 3D magneto-electro-elasticity to ten differential equations in terms of ten primary variables of magnetic, electric and elastic fields. The method of multiple scales is introduced to eliminate the secular terms in various order problems of the present formulation so that the present asymptotic expansion to the primary field variables leads to be uniform and feasible. Through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of governing equations for various order problems. The coupled classical shell theory (CST) is derived as a first-order approximation to the 3D magneto-electro-elasticity. Higher-order modifications can be further determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Some benchmark solutions for the free vibration analysis of FG elastic and piezoelectric plates are used to validate the performance of the present asymptotic formulation. The influence of the material-property gradient index on the natural frequencies and corresponding modal field variables of the FG shells is mainly concerned.     

Verification of deforming polarized structure computation by using a closed-form solution

Various engineering systems exploit the conversion between electromagnetic and mechanical work. It is important to compute this coupling accurately, and we present a method for solving the governing equations simultaneously (at once) without a staggering scheme. We briefly present the theory for coupling the elecgoverning equations as well as the variational formulation that leads to the weak form. This weak form is nonlinear and couples various fields. In order to solve the weak form, we use the finite element method in space and the finite difference method in time for the discretization of the computational domain. Numerical problems are circumvented by selecting the field equations carefully, and the weak form is assembled using standard shape functions. In order to examine the accuracy of the method, for the case of a linear elastic material under small deformations, we present and use an analytic solution. Comparison of the computation to the closed-form solution shows that the computational approach is reliable and models the jump of the electromagnetic fields across the interface between two different materials.     

A micromechanical model based on hypersingular integro-differential equations for analyzing micro-crazed interfaces between dissimilar elastic materials

The current work models a weak(soft) interface between two elastic materials as containing a periodic array of micro-crazes. The boundary conditions on the interfacial micro-crazes are formulated in terms of a system of hypersingular integro-differential equations with unknown functions given by the displacement jumps across opposite faces of the micro-crazes. Once the displacement jumps are obtained by approximately solving the integro-differential equations, the effective stiffness of the micro-crazed interface can be readily computed. The effective stiffness is an important quantity needed for expressing the interfacial conditions in the spring-like macro-model of soft interfaces. Specific case studies are conducted to gain physical insights into how the effective stiffness of the interface may be influenced by the details of the interfacial micro-crazes.     

基于哈密顿原理的两种材料界面裂纹奇性研究

研究了两种材料组成的弹性体在交界面上含裂纹时的裂纹尖端奇异场。通过变量代换及变分原理，将平面弹性扇形域的方程导向哈密体系，从而可通过分离变量及共轭辛本征函数展开法解析法求解扇形域方程，得到求解双材料界面裂纹尖点奇性的一般表达式，由此为该类问题的求解开辟了一条新途径。     

On Saint Venant - Kirchhoff imperfect interfaces

Using matched asymptotic expansions with fractional exponents, we obtain original transmission conditions describing the limit behavior for soft, hard and rigid thin interphases obeying the Saint Venant-Kirchhoff material model. The novel transmission conditions, generalizing the classical linear imperfect interface model, are discussed and compared with existing models proposed in the literature for thin films undergoing finite strain. As an example of implementation of the proposed interface laws, the uniaxial tension and compression responses of butt joints with soft and hard interphases are given in closed form.     

插值矩阵法分析与复合材料界面相交的平面裂纹奇异应力场

提出了用插值矩阵法分析与各向异性材料界面相交的平面裂纹应力奇异性。基于V形切口尖端附近区域位移场渐近展开,将位移场的渐近展开式的典型项代入线弹性力学基本方程,得到关于平面内与复合材料界面相交的裂纹应力奇异性指数的一组非线性常微分方程的特征值问题,运用插值矩阵法求解,获得了平面内各向异性结合材料中与界面以任意角相交的裂纹尖端的应力奇异性指数随裂纹角的变化规律,数值计算结果与已有结果比较表明,本文方法具有很高的精度和效率。     

Comparative study of an interface crack for different wedge-interface models

Summary  An interface crack problem is investigated under various assumptions on an interface between two elastic materials. The interface is modeled by an additional third structure (thin elastic wedge of differing elastic properties) matching the bonded materials, or by introducing special boundary conditions on the crack line ahead. The main emphasis of the paper is placed on a comparison of the asymptotic expansion of the elastic solutions near the crack tip obtained for the different models. In particular, the behaviour of the stress singularity exponent and the generalized SIF are discussed. Numerical examples are presented. Received 16 August 2000; accepted for publication 26 May 2001     

Nonlinear axisymmetric deformations of an elastic tube under external pressure

The problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the C++ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the “corner bulging” of short tubes, and multiple modes of deformation for longer tubes.     

Vibration of two bonded composites: effects of the interface and distinct periodic structures

Small elastic vibrations of two particulate composites that are caused by a non-plane time-harmonic wave are investigated. Effects of the adhesive interface and distinct periodic structures on the transmission and reflection of acoustic waves are rigorously analyzed. A two-scale asymptotic expansion with interfacial correctors is introduced to account for the macro- and micromechanical effects on wave propagation. An efficient algorithm is developed for computing first and second order corrections for the coefficients that depend on the composites microstructure and the interfacial constraint.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

Wave-modulated orbits in rate-and-state friction

A frictional spring-block system has been widely used historically as a model to display some of the features of two slabs in sliding frictional contact. Putelat et al. (2008) [7] demonstrated that equations governing the sliding of two slabs could be approximated by spring-block equations, and studied relaxation oscillations for two slabs driven by uniform relative motion at their outer surfaces, employing this approximation. The present work revisits this problem. The equations of motion are first formulated exactly, with full allowance for wave reflections. Since the sliding is restricted to be independent of position on the interface, this leads to a set of differential-difference equations in the time domain. Formal but systematic asymptotic expansions reduce the equations to differential equations. Truncation of the differential system at the lowest non-trivial order reproduces a classical spring-block system, but with a slightly different “equivalent mass” than was obtained in the earlier work. Retention of the next term gives a new system, of higher order, that contains also some explicit effects of wave reflections. The smooth periodic orbits that result from the spring-block system in the regime of instability of steady sliding are “decorated” by an oscillation whose period is related to the travel time of the waves across the slabs. The approximating differential system reproduces this effect with reasonable accuracy when the mean sliding velocity is not too far from the critical velocity for the steady state. The differential system also displays a period-doubling bifurcation as the mean sliding velocity is increased, corresponding to similar behaviour of the exact differential-difference system.     

Transient elastic wave propagation in a spherically symmetric bimaterial medium modeling the thorax

The transient response resulting from an impact wave on an elastic bimaterial, made out of a “hard” medium and a “soft” medium, welded at a spherical interface, have been investigated by using an integral transform technique. This technique permits isolation of the pressure and shear waves contributions to the wave field. The method of solution makes use of the generalized ray/Cagniard-de Hoop (GR/CdH) method associated with a “flattening approximation” (FA) technique, similar to the Earth flattening transformation used in geophysics. The GR/CdH method and the FA technique are briefly presented, together with their numerical implementations. The FA has proved to be useful in geophysical application, however, as far as the authors know, it has never been investigated for other applications. For the purpose of this paper, numerous tests of the method have been performed in order to check that the FA is appropriate to compute transient responses in the special case presented here. We could determine appropriate values for some parameters involved in the FA. This paper follows Grimal et al. [Int. J. Solid Struct. 39 (2002) 5345] in which we investigated the same bimaterial with a plane––instead of spherical––interface. Numerical examples are concerned with the propagation of an impact wave in the thorax modeled as a bimaterial (thoracic wall-lung). In addition to the effects of the weak coupling of the two media already observed in our previous study, we found that, for interface curvatures characteristic of those measured in the thorax, focalization of energy is manifest.     

Axisymmetric membrane in adhesive contact with rigid substrates: Analytical solutions under large deformation

The large deformation of an elastic axisymmetric membrane in adhesive contact with a rigid flat punch is studied. Detachment of membrane is analyzed using a critical energy release rate criterion. Two types of incompressible hyperelastic material models are considered: neo-Hookean and a class of materials whose elastic energy density functions are independent of the trace of the Cauchy–Green tensor (I₂-based material). We also include pre-stretch in our formulation and study the stability of detachment process. Closed form analytical solutions for the membrane stresses, deformed profiles and energy release rate are obtained in the regime of large longitudinal stretch. For the I₂-based material, we discover an interesting “pinching” instability where the contact angle suddenly increases in a displacement controlled test. The region of validity of our analytical solutions is determined by comparing them with numerical solutions of the governing equations. We found that the accuracy of our solution improves with pre-stretch; for pre-stretch ratios greater than 1.3, our analytical solution also works well in the small deformation regime.     

Mechanics of interface failure in the trilayer elastic composite

An exact analysis of the mechanics of interface failure is presented for a trilayer composite system consisting of geometrically and materially distinct linear elastic layers separated by straight nonlinear, uniform and nonuniform decohesive interfaces. The technical significance of this system stems from its utility in representing two slabs joined together by a third adhesive layer whose thickness cannot be neglected. The formulation, based on exact infinitesimal strain elasticity solutions for rectangular domains, employs a methodology recently developed by the authors to investigate both solitary defect as well as multiple defect interaction problems in layered systems under arbitrary loading. Interfacial integral equations, governing the normal and tangential displacement jump components at the interfaces, are solved for the uniformly loaded trilayer system. Interfacial defects, taken in the form of interface perturbations and nonbonded portions of interface, are modeled by coordinate dependent interface strengths. They are examined in a variety of configurations chosen so as to shed light on the various interfacial failure mechanisms active in layered systems.     

Elastic layers bonded to flexible reinforcements

Elastic layers bonded to reinforcing sheets are widely used in many engineering applications. While in most of the earlier applications, these layers are reinforced using steel plates, recent studies propose to replace “rigid” steel reinforcement with “flexible” fiber reinforcement to reduce both the cost and weight of the units/systems. In this study, a new formulation is presented for the analysis of elastic layers bonded to flexible reinforcements under (i) uniform compression, (ii) pure bending and (iii) pure warping. This new formulation has some distinct advantages over the others in literature. Since the displacement boundary conditions are included in the formulation, there is no need to start the formulation with some assumptions (other than those imposed by the order of the theory) on stress and/or displacement distributions in the layer or with some limitations on geometrical and material properties. Thus, the solutions derived from this formulation are valid not only for “thin” layers of strictly or nearly incompressible materials but also for “thick” layers and/or compressible materials. After presenting the formulation in its most general form, with regard to the order of the theory and shape of the layer, its applications are demonstrated by solving the governing equations for bonded layers of infinite-strip shape using zeroth and/or first order theory. For each deformation mode, closed-form expressions are obtained for displacement/stress distributions and effective layer modulus. The effects of three key parameters: (i) shape factor of the layer, (ii) Poisson’s ratio of the layer material and (iii) extensibility of the reinforcing sheets, on the layer behavior are also studied.     

尖端具有线性粘着力作用的加层空间的界面裂纹对稳态弹性波的散射

本文研究一类粘着型界面裂纹的弹性波散射问题．文中利用积分变换和积分方程方法推导了确定这类问题的奇异积分方程组．采用围道积分技术和切比雪夫多项式展开技术，得到了待定系数的非线性代数方程组．最后本文给出裂纹尖端粘着区的大小和界面应力的数值结果．     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.