Asymptotic modelling of interfaces taking contact conditions into account: Asymptotic expansions and numerical implementation

In the present theoretical and numerical study on thin layers, a method is presented for modelling bonding processes and interfaces. When the thickness and the rigidity of the layer tend to zero, the asymptotic problem leads to a limit problem with a mechanical constraint on the surface, to which the layer shrinks. The formulation of the limit problem includes the mechanical and geometrical properties of the layer. The contact and pseudo-friction conditions between the adhesive and the adherents are also taken into account. Theoretical results are obtained using asymptotic expansions and numerical data. The asymptotic laws are implemented in a finite element software. Numerical algorithms are detailed and numerical examples are presented.     

壁厚不连续不对称圆柱壳和开口壳的自由振动分析

视壁厚不连续不对称的圆柱壳或开口壳为层合壳。以弹性力学中的Hamilton正则方程为基础,分别应用Hamilton正则方程的半解析法构建层合壳每一层的线性方程。考虑到每两层连接界面上应力和位移的连续性,联立各层的方程得到系统的控制方程和特征方程。主要的优点是系统的控制方程不限制圆柱壳的厚度,数学模型和数值方法也适应分析壁厚不连续且壁厚不对称的复合材料层合壳问题。     

Analysis of fracture and delamination in laminates using 3D numerical modelling

The static failure behaviour of the fibre-metal laminate GLARE is examined using 3D finite element simulations. The configuration analysed is a centre-cracked tensile specimen composed of two aluminium layers sandwiching a cross-plied, fibre-epoxy layer. The crack and delamination growths are simulated by means of interface elements equipped with a mixed-mode damage model. The mode-mixity is derived from an energy criterion typically used in linear elastic fracture mechanics studies. The damage kinetic law is rate-dependent, in order to simulate rate effects during interfacial delamination and to avoid numerical convergence problems due to crack bifurcations. The numerical implementation of the interface damage model is based on a backward Euler approach. In the boundary value problem studied, the failure responses of GLARE specimens containing elastic aluminium layers and elasto-plastic aluminium layers are compared. The development of plastic deformations in the aluminium layers stabilizes the effective failure response, and increases the residual strength of the laminate. For a ‘quasi-brittle’ GLARE specimen with elastic aluminium layers, the residual strength is governed by the toughness for interfacial delamination, and is in close correspondence with the residual strength obtained from a closed-form expression derived from energy considerations. Conversely, for a ‘ductile’ GLARE specimen with elasto-plastic aluminium layers, the residual strength is also determined by the relation between the fracture strength and the yield strength of the aluminium. The amount of constraint by the horizontal displacements at the vertical specimen edges has a moderate to small influence on the residual strength. Furthermore, the ultimate laminate strength is lower for a larger initial crack length, and shows to be in good correspondence with experimental values.     

Light propagation in two-layer tissues with an irregular interface

We study light propagation in a half-space composed of two homogeneous layers each having different optical properties from the other. This problem is a model for light propagation in tissues composed of a thin epithelial layer supported from below by a thick stromal layer. The interface between the two layers is irregular. Assuming that this interface is a small perturbation of a plane that is parallel to the boundary surface, we obtain an asymptotic approximation to the solution. We give a numerical method to compute this asymptotic approximation. Finally, we show how to recover this irregular interface surface from boundary measurements when the optical properties of the two layers are known.     

Thickness effects in microlayer composites of polycarbonate and poly(styrene-acrylonitrile)

Coextruded microlayer sheet consisting of alternating layers of polycarbonate (PC) and styrene-acrylonitrile copolymer (SAN) exhibits improved properties such as toughness and ductility as the number of layers is increased. In this study, the composition was kept essentially constant, as was the sheet thickness at 1.2 mm, and the layer thickness was changed by varying the total number of layers from 49 to 776. All the compositions exhibited macroscopic yielding in uniaxial tension but the fracture strain, which represents neck propagation, increased with the number of layers. The increased ductility was attributed to a transition in the microdeformation behaviour observed when microspecimens were stretched in the optical microscope. When the layers were thicker, individual layers exhibited behaviour characteristic of the bulk, that is SAN crazed or cracked while shear bands initiated in PC from the craze tips. As the layer thickness decreased, crazing or cracking of the SAN was suppressed and shear bands that extended through several layers produced shear yielding of both PC and SAN. Calculations showed that when the layer thickness is sufficiently small, impingement of a PC shear band on the interface creates a local shear stress concentration. As a result the shear band continues to grow through the SAN layer and subsequently, at the point of instability, shear yielding can occur in both PC and SAN layers.     

A multivalued boundary integral equation for adhesive contact problems

The present paper proposes a boundary integral equation (BIE) formulation for adhesive contact interface problems, i.e. problems involving interfaces glued by adhesives where unilateral contact conditions also hold. A non-monotonic, multi-valued law is assumed to describe the behaviour of the adhesive tangential to the interface direction, which leads to a hemivariational inequality problem. For the numerical treatment of the non-convex-non-smooth optimization problem, a new method is proposed which reduces the initial problem to a sequence of simple quadratic programming problems.     

Fracture toughness of layered structures: Embrittlement due to confinement of plasticity

The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)-cohesive zone (CZ) model. The fracture mechanism of an elastic-plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jump-like increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance.     

Mixed‐mode delamination in 2D layered beam finite elements

In this work, a 2D finite element (FE) formulation for a multi‐layer beam with arbitrary number of layers with interconnection that allows for mixed‐mode delamination is presented. The layers are modelled as linear beams, while interface elements with embedded cohesive‐zone model are used for the interconnection. Because the interface elements are sandwiched between beam FEs and attached to their nodes, the only basic unknown functions of the system are two components of the displacement vector and a cross‐sectional rotation per layer. Damage in the interface is modelled via a bi‐linear constitutive law for a single delamination mode and a mixed‐mode damage evolution law. Because in a numerical integration procedure, the damage occurs only in discrete integration points (i.e. not continuously), the solution procedure experiences sharp snap backs in the force‐displacements diagram. A modified arc‐length method is used to solve this problem. The present model is verified against commonly used models, which use 2D plane‐strain FEs for the bulk material. Various numerical examples show that the multi‐layer beam model presented gives accurate results using significantly less degrees of freedom in comparison with standard models from the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

陶瓷复合装甲粘结层效应和抗多发打击性能的数值模拟研究

粘结层性能对陶瓷复合装甲抗多发打击性能有重要影响。建立了研究粘结层和多发打击的数值模拟方法,解决了传统方法不能模拟“脱粘”和多发打击的问题。基于文献弹道试验,研究了环氧树脂和聚氨酯两种粘结层材料及其厚度对陶瓷/铝合金复合装甲抗7.62mm 穿甲弹单发和两发打击性能的影响。结果表明:单发打击的数值模拟可不建粘结层,而多发打击应采用建粘结层的方法;抗单发打击时,粘结层越薄,极限速度越大;抗多发打击时,陶瓷复合装甲应采用聚氨酯粘结层,且其抗两发打击的较优厚度约为0.40mm。     

The dynamic behaviour of surface-bonded piezoelectric actuators with debonded adhesive layers

The performance of smart structures depends on the electromechanical behaviour of piezoelectric actuators and the bonding condition along the interface, which connects the actuators and the host structures. This paper provides a theoretical study of the effect of partially debonded adhesive layers on the coupled electromechanical behaviour of piezoelectric actuators subjected to high-frequency electric loads. An actuator model with an imperfect adhesive bonding layer, which undergoes a shear deformation, is proposed to simulate the two-dimensional electromechanical behaviour of the integrated system. An analytical solution of the problem is provided by solving the resulting integral equations in terms of the interfacial stress. Numerical simulation is conducted to study the effect of the bonding layer upon the actuation process. The effect of interfacial debonding on the dynamic response of the layered structure and on the interlaminar strain and stress transfer mechanisms is discussed.     

Multiple cracks on the arc-shaped interface in a semi-cylindrical magneto-electro-elastic composite with an orthotropic substrate

The present work investigates the problem of multiple cracks on the arc-shaped interface of a semi-cylindrical magneto-electro-elastic layer bonded onto an orthotropic substrate. Continuously distributed dislocation is used to simulate the anti-plane interfacial cracks. The problem is formulated as a Cauchy singular integral equation by integrating the Green’s function of an interfacial point dislocation. Both the theoretical derivation and numerical computation are verified in special cases. The effects of the interface end, crack space, layer thickness, stiffness ratio and material orthotropy are surveyed, among which the fracture behavior of the interface end deserves special attention in design.     

A new efficient numerical method and the dynamic analysis of composite laminates with piezoelectric layers

Firstly, a numerical method for the inversion of Laplace transform is developed and its accuracy is shown through examples. Then, a state-vector equation for the dynamic problems of piezoelectric plates is deduced directly from a modified mixed variational principle for piezoelectric bodies and its exact solution for the dynamic problems of simply supported rectangle piezoelectric plate is simply given. For multilayered hybrid plates, we derive the solution in terms of the propagator matrices. The techniques accounts for the compatibility of generalized displacements and generalized stresses on the interface both the elastic layers and piezoelectric layers, and the transverse shear deformation and the rotary inertia of laminate are also considered in the global algebraic equation of structure. Meanwhile, there is no restriction on the thickness and the number of layers. As an application of the numerical inversion of Laplace transform presented in this paper, typical numerical examples of the harmonic vibration and transient response are proposed and discussed. Since the highly accurate numerical results, they can serve as benchmarks to test various thick plate theories and various numerical methods, such as the finite and boundary element methods for transient response problems.     

Numerical and experimental investigation on lap shear fracture of Al/CFRP laminates

This paper presents a new approach to numerically investigate the lap shear fracture of a hybrid laminate made of Carbon Fibre Reinforced Plastic (CFRP) and metal foil plies (e.g. aluminium), validated by corresponding experiments. The numerical Finite Element (FE) model of the hybrid laminate, subjected to lap shear fracture, is composed of five laminas with alternating metal/CFRP layers with cohesive elements lying within Al/CFRP interface. In the FE model, individual CFRP laminas are assumed as an orthotropic homogenized continuum under plane stress, and aluminium facesheets are modelled as an elastic–plastic continuum. The Al/CFRP interface is represented via quadratic cohesive elements, the constitutive law of which is an exponentially decaying law representing the degrading behaviour of the interface (implemented as user element in ABAQUS). The numerical model captures the experimentally obtained results with minimal error, and predicts the failure modes successfully. The influence of specimen geometry (e.g. overlap length, total length, and total width) on lap shear fracture response is analyzed in detail in this study, too, in order to confirm the specimen design for the test, as there is still no corresponding test standard for hybrid laminates.     

Asymptotic Solution of Thermocapillary Convection in a Differentially Heated Thin Annular Two-Layer Pool

The steady laminar two-dimensional thermocapillary convection of two immiscible liquid layers in a thin annular pool with one free surface, one liquid/liquid interface subjected to a radial temperature gradient was investigated using asymptotical analysis. The pool is heated from the inner cylindrical wall and cooled at the outer wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the lower layer thickness to the gap width, trends to zero. The numerical experiments are also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. The results indicate that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.     

Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation

The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws takes into account the randomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, both the fractal law and the fractal interface are considered to be graphs of two different fractal interpolation functions which are the “fixed points” of two contractive operators. Using this method, the fractal friction law is approximated by a sequence of nonmonotone possibly multivalued classical C ⁰-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone problem by a sequence of monotone subproblems. Numerical applications from the static analysis of cracked structures with a prescribed fractal geometry and fractal interface laws are included in order to illustrate the theory.     

An interface crack in a functionally graded piezoelectric bi-layer under anti-plane shear impact

The transient response of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear impact loading is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness, and the two layers are connected weak-discontinuously. Laplace transform and Fourier transform are used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate are presented for the FGPM to show the effects on the electric loading, variation and gradient of material properties, and thickness of layers. Following things are helpful to increase the resistance of transient fracture of interface crack in FGPMs: (a) increase of the material properties from the interface to the upper or lower free surface; (b) decrease of weak discontinuity at the interface; (c) increase of the gradient of material properties; (d) certain direction and magnitude of the electric loading; and (e) increase of the thickness of the FGPM layer.     

The analytical solution of the contact problem of spherical indenters on layered materials: application for the investigation of TiN films on silicon

Utilising an analytical solution of the contact problem of spherical indentation into layered materials we investigated TiN layers of various thicknesses on single-crystal silicon. We used Heaviside functions to describe the discontinuity of the mechanical parameters at the interface of the TiN-silicon compound. A spherical 5 μm diamond indenter was used to obtain the penetration depth-force curves. In spite of this small indenter we only obtained good utilizable results for film thicknesses of more than 0.8 μm. For thinner films the influence of failure effects such as roughness of the film, not an exactly known film thickness and substrate parameters, deviation of the shape of the diamond indenter from the ideal spherical one and the failure caused by the indentation apparatus themselves becomes too influential. Thus, here we show an investigation of a 1.4 and an 0.8 μm TiN layer on silicon including calculation of the films Young's modulus and the stresses, and for an 0.4 and an 0.2 μm TiN layer on silicon a stress calculation only using estimated Young's moduli from the thicker films. We paid special attention to the investigation of the non-elastic behaviour of the compounds utilising the von-Mises criterion.     

Diffusion kinetics of explosively treated and plasma nitrided Ti-6Al-4V alloy

Ti-6Al-4V alloys, which were exposed to an explosive shock process, were nitrided in nitrogen plasma in the temperature range of 700-900°C for 3-12 h. During the plasma nitriding, the surface layer consisted of TiN (δ), Ti₂N (ε) and nitrogen solid solution layers (α-Ti). The growth rate of nitride and solid solution layers were found to be controlled by the diffusion of nitrogen. An effective nitriding was achieved due to high dislocation density and vacancy concentration. Based on the present layer growth data, an analytical model for multiphase diffusion was used to estimate the effective nitrogen atom diffusion coefficient in the nitride layers. The interface velocity equations were derived from Fick's law and a numerical method has been used to compute the diffusion coefficients of nitrogen in a binary multiphase Ti-TiN system. Depending on temperature and layer thickness, the activation energies of nitrogen in TiN and Ti₂N phases were found to be 18,950 (±2116) and 27,925 (±1105) cal/mole, respectively.     

Influence of the thickness and area of NiCr/Ag electrodes on the characteristics of BaTiO3- ceramic based positive-temperature-coefficient thermistors

The electrical behaviour of commercial BaTiO3-based positive-temperature-coefficient (PTC) thermistors with NiCr/Ag electrodes was investigated by impedance analysis in the frequency domain. The contact resistance and thus the total resistance of the PTC increases with decreasing thickness as well as with decreasing area of the NiCr layer at constant Ag top layer. The increase in the total resistance in both cases is explained by the increase in the contact area of the Ag layer to the ceramic which is represented by a model. The Ag forms a blocking contact to the ceramic in contrast to the NiCr which shows an ohmic behaviour. Using our model, the electrode impedance contribution of very thin NiCr layers can be interpreted in terms of incomplete wetting of the ceramic surface by the NiCr metal during the deposition. Supplementary scanning electron microscopy and high-resolution transmission electron microscopy analyses have been used to study the structure of the electrode interface and the structure of the layers. © 1998 Kluwer Academic Publishers