Precracking and interfacial delamination in a bi-material structure: Static and dynamic loadings

The behavior of a precracked bi-material structure interface under given static and dynamic axial loading is an interest object in the present paper.Firstly,it is shown that the shear-lag model is a proper tool to analyze a delamination process in a precracked bi-material structure undergoing static loading.Secondly,the"shear-lag model"is applied to the structure under dynamic loading.To solve the problem for an interface delamination of the structure and to determine the debond length along the interface,our own 2D boundary element method(BEM)code is proposed in the case of static loading,and the shear-lag model together with the Laplace transforms and half-analytical calculations are used in the case of dynamic loading.The interface layer is assumed as a very thin plate compared with the other two.The parametric(geometric and elastic)analysis of the debond length and interface shear stress is done. The results from the 2D BEM code proved the validity of analytical solutions to the shear-lag model.In the dynamic case,the influence of loading characteristics,i.e.,frequencies and amplitude fluctuations on the shear stress and the value of debond length for an interval of time,is discussed. The analysis of the obtained results is illustrated by an example of the modern ceramic-metal composite,namely cermet, and depicted in figures.     

Buckling of delaminated bi-layer beam-columns

An improved analytical model is presented to analyze the delamination buckling of a bi-layer beam-column with a through-the-width delamination. Both the transverse shear deformation and local delamination tip deformations are taken into consideration, and two delaminated sub-layers as well as two substrates in the intact (un-delaminated) regions are modeled as individual Timoshenko beams. A deformable interface is introduced to establish the continuity condition between the two substrates in the intact regions. Consequently, a flexible joint is formed at the delamination tip, and it is different from the conventional rigid joint given in most of studies in the literature, in which the local delamination tip deformations are completely ignored. In contrast to the local delamination buckling in our previous study (Qiao et al., 2010), the present model accounts for the global deformations of the intact region in the delaminated composite beam-column, thus capable of capturing the buckling mode shape transitions from the global, to global–local coexistent, and to local buckling for asymmetric delamination as the interface delamination increases. Good agreement of the present analytical solutions with the full 2-D elastic finite element analysis demonstrates the local deformation effects around the delamination tip and verifies the accuracy of the present model. Parametric studies are conducted to investigate the effects of loading eccentricity, delaminated sub-layer thickness ratio, and interface compliance on the critical buckling load for the delaminated composite beam-column. Transitions of buckling modes from the global to local delamination buckling are also disclosed as the thickness of one sub-layer reduces from the thick sub-layer to a thin film. The developed delamination buckling solution facilitates the design analysis and optimization of laminated composite structures, and it can be used with confidence in buckling analysis of delaminated composite structures.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

A three-parameter elastic foundation model for interface stresses in curved beams externally strengthened by a thin FRP plate

Externally bonding of fiber reinforced polymer (FRP) plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP–concrete interface are of great importance to the effectiveness of this type of strengthening because high stress concentration along the FRP–concrete interface can lead to the FRP debonding from the concrete beam. In this study, we develop an analytical solution of interface stresses in a curved structural beam bonded with a thin plate. A novel three-parameter elastic foundation model is used to describe the behavior of the adhesive layer. This adhesive layer model is an extension of the two-parameter elastic foundation commonly used in existing studies. It assumes that the shear stress in the adhesive layer is constant through the thickness, and the interface normal stresses along two concrete/adhesive and adhesive/FRP interfaces are different. Closed-form solutions are obtained for these two interfacial normal stresses, shear stress within the adhesive layer, and beam forces. The validation of these solutions is confirmed by finite element analysis.     

An analytical delamination model for laminated plates including bridging effects

A penalised interface model, whose strain energy is the penalty functional related to interface adhesion constraint, is introduced in conjunction with a damageable interface whose local constitutive law, in turn, represents bridging stress effects, in order to analyse delamination and bridging phenomena in laminated plates. The laminate is modelled by means of first-order shear deformable layer-wise kinematics and the governing equations are formulated in the form of a non-linear differential system with moving intermediate boundary conditions related to opportune delamination and bridging growth conditions. The problem is solved through an analytical approach. The model leads to an accurate and self-consistent evaluation of the energy release rate and its mode components due to the inclusion of significant contributions arising from coupling between in-plane and transverse shear stresses, and to an asymptotic estimate of interlaminar stresses. The salient features of the proposed model are investigated in the context of an energy balance approach and of a J-integral formulation, thus providing simple results useful to model delamination growth and bridging behaviour when mixed mode loading is involved. The accuracy of the proposed model is substantiated through comparisons with results from continuum analysis obtained by a finite element (FE) procedure. The effectiveness of the proposed model is highlighted by showing the solution of a two-layered plate scheme subjected to pure and mixed mode loading conditions and to fibre bridging stresses. The results point out that the present model, despite its low computational cost in comparison with more complex FE analyses, is an efficient tool to predict delamination and bridging evolution.     

Modelling of stresses and strains in bonded concrete overlays subjected to differential volume changes

The differential volume changes between repair material and substrate concrete induces stresses in concrete overlays. These stresses could cause the repair material cracking or the interface delamination, which facilitate the penetration of harmful substances into concrete and hence accelerate the deterioration of concrete overlays. In order to investigate the characteristics of these stresses, an analytical model has been developed based on the plate theory and the assumption of the linear relation between shear stress and slip at the interface. This model is able to calculate the tensile stress in the repair material, shear and normal stresses at the interface and the corresponding strains. With this model, the influence of the interface shear stiffness, the dimension of the concrete overlay, and the elastic moduli of two materials on stresses and strains was studied.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

Finite element analysis of ink-tack delamination of paperboard

Finite element analysis of ink-tack delamination of paperboard were presented. The paperboard was modeled as a multilayered structure with a softening interface model connecting the paperboard plies. The paperboard plies were modeled as orthotropic linear elastic. The ink-tack loading was applied to the board in the form of a moving displacement boundary condition. The purpose of the analysis was to assess the influence from the elastic moduli of the individual layers on the ink-tack delamination event. The results indicated that in most cases of practical interest the board delaminated between the outer plies of the board, although the interface strength was lower in the middle of the board. This observation helped to explain why traditional tests for out-of-plane testing of paper by standardized methods could not uniquely predict the propensity for ink-tack delamination.     

Analysis of crack extension in anisotropic materials based on local normal stress

The normal stress ratio theory is applied to predict crack extension behavior in center-notched unidirectional graphite-epoxy of arbitrary fiber axis orientation, subjected to arbitrary far-field planar loading. The theory is applied within analytical solutions for two infinite plate geometries: a plate with a sharp center crack, and a plate with an elliptical center flaw. A critical analytical case is identified suggesting that application of the theory within a stress solution modelling crack tip shape may increase the accuracy of crack growth direction predictions. Crack extension direction, location of crack extension, and critical stress predictions of the theory are compared to those obtained from experiments on specimens subjected to tensile, shear, and mixed-mode far-field loading. The comparison shows that, applied within each analytical solution, the normal stress ratio theory provides verifiable predictions of crack growth behavior. By modelling actual notch tip shape, the elliptical notch solution is able to provide accurate qualitative predictions of the origin of crack extension along the periphery of a cut notch tip in a way that the sharp crack analysis cannot. The sharp notch solution appears to provide slightly more accurate crack growth direction predictions, however. Also, in predicting critical applied far-field stresses, the sharp crack solution appears to exhibit a stronger ability to model subtle experimental trends.     

Energy release rate and phase angle of delamination in sandwich beams and symmetric adhesively bonded joints

Delamination in sandwich structures along the interface between the face sheet and the core, or along the adherend/adhesive interface in adhesively bonded joints, is one of the most common failure modes of this type of tri-layer structure. This delamination is usually modeled as an interface crack problem, for which the energy release rate and phase angle can be calculated using interface fracture mechanics solutions. Existing interface fracture mechanics solutions, however, ignore the effect of transverse shear deformation, which can be significant for short crack. In an effort to overcome this shortcoming, this study presents new analytical solutions for the energy release rate and for the phase angle of the interface crack in sandwich structures or adhesively bonded joints. Since the new solutions incorporate relative rotation at the tip of the delamination, transverse shear effects are taken into account in this study. Typical delaminated sandwich and adhesively bonded joint specimens are analyzed by using the new solutions, as well as by the existing solutions. The energy release rate predicted by the present model agrees very well with that predicted by FEA, and furthermore it is considerably more accurate relative to existing models. As the existing model neglects the transverse shear force, it underestimates the total energy release rate. A stress field analysis is also conducted in this study in order to clarify some misunderstandings in the literature on the determination of the phase angle of adhesively bonded joints using an interface stress-based method.     

A coupled interface-multilayer approach for mixed mode delamination and contact analysis in laminated composites

An accurate laminate model developed by using multilayer shear deformable plate modeling and interface elements, based on fracture mechanics and contact mechanics, is proposed to analyze mixed mode delamination in composite laminates. Perfect adhesion along the undelaminated portion of the delamination plane is simulated by treating interface stiffnesses as penalty parameters, whereas to enforce interface displacement continuity between plate elements constituting each sub-laminate above or below the delamination plane, the Lagrange multiplier method is used. The governing differential equations are derived through a variational procedure by using a modified total potential energy functional. Results are obtained by numerical integration of the non-linear three-point boundary value problem modeling mixed-mode delamination of the laminate plate subjected to end loading, which accounts also for the frictionless contact condition.The coupling of a penalty procedure with the Lagrange multiplier method, results in an accurate and direct energy release rate evaluation. Comparisons with results available from the literature obtained with a local continuum approach, show that mode partition may be performed to the desired accuracy by refining multilayer plate models for each sub-laminate. In addition, original analytical formulas for mode partition are obtained by coupling the interface approach and fracture mechanics concepts, evidencing the effectiveness of the proposed approach and gaining a better insight into the influence of shear effects on mode decomposition. Numerical computations for practical problems, evidence both the relative simplicity and the efficiency of the proposed model to represent mixed mode interlaminar fracturing as well as crack–face interaction.     

含孔边非均匀材料圆环的无限大薄板的应力集中系数

对于含圆孔及孔边非均匀材料圆环的无限大薄板，假设非均匀材料的弹性模量沿径向按照指数函数变化，而泊松比为常数，分别导出了双轴拉伸和纯剪切作用时孔边及界面处的应力集中系数的解析解．通过数值算例详细分析了非均匀材料圆环的弹性模量的变化对无限大薄板的孔边及界面处的应力集中系数的影响．研究结果表明，合理选择孔边非均匀材料圆环的材料性能变化参数可有效地缓解薄板的孔边应力集中程度．本文的研究结果可为含圆孔的薄板的设计提供一定的参考．     

PLANE ELASTIC PROBLEM ON RIGID LINES ALONG CIRCULAR INCLUSION

The plane elastic problem of circular-arc rigid line inclusions is considered. The model is subjected to remote general loads and concentrated force which is applied at an arbitrary point inside either the matrix or the circular inclusion. Based on complex variable method, the general solutions of the problem were derived. The closed form expressions of the sectionally holomorphic complex potentials and the stress fields were derived for the case of the interface with a single rigid line. The exact expressions of the singular stress fields at the rigid line tips were calculated which show that they possess a pronounced oscillatory character similar to that for the corresponding crack problem under plane loads. The influence of the rigid line geometry, loading conditions and material mismatch on the stress singularity coefficients is evaluated and discussed for the case of remote uniform load.     

Delamination mechanism maps for a strong elastic coating on an elastic–plastic substrate subjected to contact loading

Hard wear resistant coatings that are subjected to contact loading sometimes fail because the coating delaminates from the substrate. In this report, systematic finite element computations are used to model coating delamination under contact loading. The coating and substrate are idealized as elastic and elastic–plastic solids, respectively. The interface between coating and substrate is represented using a cohesive zone law, which can be characterized by its strength and fracture toughness. The system is loaded by an axisymmetric, frictionless spherical indenter. We observe two failure modes: shear cracks may nucleate just outside the contact area if the indentation depth or load exceeds a critical value; in addition, tensile cracks may nucleate at the center of the contact when the indenter is subsequently removed from the surface. Delamination mechanism maps are constructed which show the critical indentation depth and force required to initiate both shear and tensile cracks, as functions of relevant material properties. The fictitious viscosity technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces allows us to explore a wider parametric space that a conventional cohesive model cannot handle. Numerical results have also been compared to analytical analyses of asymptotic limits using plate bending and membrane stretching theories, thus providing guidelines for interpreting the simulation results.     

Stress intensity factors for an edge interface crack in a bonded semi-infinite plate for arbitrary material combination

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary crack lengths and material combinations. In this paper the stress intensity factors of an edge interface crack in a bonded strip are considered under tension with varying the crack length and material combinations systematically. Then, the limiting solutions are provided for an edge interface crack in a bonded semi-infinite plate under arbitrary material combinations. In order to calculate the stress intensity factors accurately, exact solutions in an infinite bonded plate are also considered to produce proportional singular stress fields in the analysis of FEM by superposing specific tensile and shear stresses at infinity. The details of this new numerical solution are described with clarifying the effect of the element size on the stress intensity factor. It is found that for the edge interface crack the normalized stress intensity factors are not always finite depending upon Dunders’ parameters. This behavior can be explained from the condition of the singular stress at the end of bonded strip. Convenient formulas are also given by fitting the computed results.     

Finite Fracture Mechanics at elastic interfaces

In the present paper we provide a method to determine the load causing delamination along an interface in a composite structure. The method is based on the elastic interface model, according to which the interface is equivalent to a bed of linear elastic springs, and on Finite Fracture Mechanics, a crack propagation criterion recently proposed for homogeneous structures. The procedure outlined is general. Details are given for the pull–push shear test. For such geometry, the failure load is obtained and compared with the estimates provided by stress concentration analysis and Linear Elastic Fracture Mechanics. It is seen that Finite Fracture Mechanics provides intermediate values. Furthermore, it is shown that the predictions provided by Finite Fracture Mechanics are almost coincident with the ones provided by the Cohesive Crack Model. As far as we are concerned with the determination of the failure load, the advantage of using Finite Fracture Mechanics with respect to the Cohesive Crack Model is evident, since a troublesome analysis of the softening taking place in the fracture process zone is not necessary. A final comparison with classical fracture criteria based on critical distances, such as the average stress criterion, concludes the paper.     

On the load transfer from a rigid cylindrical inclusion into an elastic half space

The present paper examines the problems related to the axial, lateral, and rotational loading of a rigid cylindrical inclusion which is embedded in bonded contact at the boundary of an isotropic elastic half space. The rigid inclusion is modeled as a field of distributed forces which represent the normal and shear tractions that act on the inclusion-elastic-medium interface. The intensities of these distributed tractions are determined by enforcing displacement compatibility conditions at discrete locations of the interface. These compatibility conditions are derived from rigid-body displacement modes appropriate for each loading. The results derived from this numerical scheme are compared with equivalent results derived via analytical techniques which focus on the solution of the governing integral-equation schemes and other approximate-solution schemes. The numerical results presented in the paper illustrate the manner in which the generalized stiffnesses of the embedded inclusion are influenced by its geometry and Poisson's ratio of the half-space region.     

双材料界面断裂力学模型与实验方法

纤维增强聚合物(FRP)质轻、高强, 可提高结构的刚度、强度、抗震性能和耐久性, 近年来在结构加固及工程改造中得到广泛应用. FRP与传统复合材料之间形成双材料黏结界面, 界面断裂特性是决定双材料结构性能的关键因素. 对双材料界面裂纹尖端应力场理论、界面裂纹模型、黏结界面I型、II型及混合型断裂试验及理论研究现状进行综合评述和分析. 界面模型主要有经典梁/板理论和刚性节点模型、考虑剪切变形的双亚层理论和半刚性节点模型、基于双亚层理论的柔性节点模型、考虑剪切变形的多层亚层理论和多亚层柔性节点模型、弹性地基梁模型以及黏聚模型. 还介绍了双材料界面断裂力学在FRP-混凝土研究中的应用.