Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

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.).     

Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure

The main purpose of this paper is to complete the works presented by Andrieux and Varé (2002) and El Arem et al. (2003) by taking into account the effects of shearing in the constitutive equations of a beam cracked section in bi-axial flexure. The paper describes the derivation of a lumped cracked beam model from the three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. Properties of the potential energy and convex analysis are used to reduce the three-dimensional computations needed for the model identification, and to derive the final form of the elastic energy that determines the nonlinear constitutive equations of the cracked transverse section. We aim to establish a relation of behavior between the applied forces and the resulting displacements field vectors, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis. The approach has been applied to the case of a cracked beam with a single crack covering the half of its circular cross section.     

The stability of crack growth in a beam subject to combined bending and tensile loadings. II: Beam subject to rotations and lateral displacements at the ends

The stability of crack growth in a beam subject to combined bending and tensile loadings is examined for the case where the material is very ductile and where the beam is built-in at one end. Application of a displacement at the free end, and variation of the angle which the displacement makes with the beam, allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The stability analysis is based on the tearing modulus procedure, and the general conclusion is that tensile loadings can promote crack instability.     

Process zone in the Single Cantilever Beam under transverse loading: Part I: Theoretical analysis

Single Cantilever Beam (SCB) specimen loaded with a transverse force parallel to the crack front is proposed for the analysis of crack propagation phenomena under mixed mode conditions. The stress redistribution in the adhesive layer in the vicinity of the crack front so as the beam deformation are estimated using a Timoshenko beam on elastic foundation model. This model emphasizes the Mode II contribution due to flexural beam rotation but also the cleavage due to the beam torsion induced by the eccentricity of the adhesive layer with respect to the beam neutral axis. Finally, one dimensional representation is assessed by comparing analytical solution with finite elements calculations, proving that this analysis is suitable for the analysis of the SCB test under transverse loading.     

CRACK PROPAGATION IN STRUCTURES SUBJECTED TO PERIODIC EXCITATION

In the present paper,a simple mechanical model is developed to predict the dynamic response of a cracked structure subjected to periodic excitation,which has been used to identify the physical mechanisms in leading the growth or arrest of cracking.The structure under consideration consists of a beam with a crack along the axis,and thus,the crack may open in Mode I and in the axial direction propagate when the beam vibrates.In this paper,the system is modeled as a cantilever beam lying on a partial elastic foundation,where the portion of the beam on the foundation represents the intact portion of the beam.Modal analysis is employed to obtain a closed form solution for the structural response.Crack propagation is studied by allowing the elastic foundation to shorten(mimicking crack growth)if a displacement criterion,based on the material toughness,is met.As the crack propagates,the structural model is updated using the new foundation length and the response continues.From this work,two mechanisms for crack arrest are identified.It is also shown that the crack propagation is strongly influenced by the transient response of the structure.     

梁结构中裂纹参数识别方法研究

以等效弹簧模型来模拟裂纹引起的局部软化效应,将该模型同Bernoulli-Euler梁理论、模态分析方法以及断裂力学原理等结合起来,利用传递矩阵法导出含裂纹梁振动的各种边界条件下的特征方程通解。借助于特征方程,提出两种识别裂纹深度和位置参数的数值方法,最后,通过对含裂纹悬臂梁的分析说明文中方法的有效性。     

The stability of crack growth in a beam subject to combined bending and tensile loadings: I: Beam built-in at one end

The stability of crack growth in a beam subject to combined bending and tensile loadings is examined for the case where the material is very ductile and where the beam is built-in at one end. Application of a displacement at the free end, and variation of the angle which the displacement makes with the beam, allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The stability analysis is based on the tearing modulus procedure, and the general conclusion is that tensile loadings can promote crack instability.     

Crack propagation in wedged double cantilevered beam specimens

A closed form analytical solution of crack propagation in double cantilevered beam specimens opened at a constant rate has been found. Hamilton's principle for non-conservative systems was applied to describe the crack motion, under the assumption of a Bernoulli-Euler beam. The criterion of crack propagation is a critical bending moment at the crack tip. The calculations of beam motion take into account wave effects in the Bernoulli-Euler theory of elastic beams. The beam shape during the crack motion is found with a similarity transformation and expressed by Fresnel integrals. The boundary conditions satisfied are the fixed ones of zero bending moment and constant beam opening rate at the load end of the specimen and the moving ones of zero deflection and zero slope of the deflected beam at the tip of the moving crack. The fracture represents a moving critical bending moment. The analytical results show that the specific fracture surface energy is a unique function of the ratio of the crack length squared to the time subsequent to loading and this is computed from the recorded time-dependence of the crack length.     

Closed-form solution for a mode-III interfacial edge crack between two bonded dissimilar elastic strips

A closed-form solution is obtained for the problem of a mode-III interfacial edge crack between two bonded semi-infinite dissimilar elastic strips. A general out-of-plane displacement potential for the crack interacting with a screw dislocation or a line force is constructed using conformal mapping technique and existing dislocation solutions. Based on this displacement potential, the stress intensity factor (SIF, K_III) and the energy release rate (ERR, G_III) for the interfacial edge crack are obtained explicitly. It is shown that, in the limiting special cases, the obtained results coincide with the results available in the literature. The present solution can be used as the Green’s function to analyze interfacial edge cracks subjected to arbitrary anti-plane loadings. As an example, a formula is derived correcting the beam theory used in evaluation of SIF (K_III) and ERR (G_III) of bimaterials in the double cantilever beam (DCB) test configuration.     

Monitoring structural damage of components using an effective modulus approach

This paper describes a novel nondestructive damage detection method that was developed to study the influence of a crack on the dynamic properties of a cantilever beam subjected to bending. Experimental measurements of transfer functions for the cracked cantilever beam revealed a change in the natural frequency with increasing crack length. A finite element model of a cracked element was created to compute the influence of severity and location of damage on the structural stiffness. The proposed model is based on the response of the cracked beam element under a static load. The change in beam deflection as a result of the crack is used to calculate the reduction in the global component stiffness. The reduction of the beam stiffness is then used to determine its dynamic response employing a modal analysis computational model. Euler–Bernoulli and Timoshenko beam theories are used to quantify the elastic stiffness matrix of a finite element. The transfer functions from both theories compare well with the experimental results. The experimental and computational natural frequencies decreased with increasing crack length. Furthermore the Euler–Bernoulli and Timoshenko beam theories resulted in approximately the same decrease in the natural frequency with increasing crack length as experimentally measured.     

Kinking of a crack emanating from the free surface of a beam under pure bending

A perturbation technique developed by Karihaloo et al. is employed to obtain the stress intensity factors at the tip of a kinking crack that emanates from the free surface of a beam under pure bending. Under the condition that the kink extends in the direction of vanishing K₁₁ the crack path is obtained as well as a path stability condition. From conditions on K₁ a material parameter r^* akin to that of Ramulu and Kobayashi's r_c is obtained. By analysis of the slope of the kinking crack a stability condition is obtained corroborating the stability condition from consideration of vanishing K₁₁. It is shown that for a beam in pure bending the nonsingular remote stress term T must be greater than some positive critical value for kinking to occur confirming the results of Sayir and Schindler.     

Symmetric failure of the halfspace with penny-shaped cracks in compression

The three-dimensional axisymmetric problem is investigated for a halfspace with penny-shaped crack parallel to the free surface. A uniform compression is applied parallel to the crack plane. The Griffith-Irwin theory is not applicable to this configuration of load and crack geometry since all the stress intensity factors are zero. Instead, a stability criterion will be invoked within the framework of the three-dimensional linearized stability theory. Reference can be made to previous works [5,6] involving compressible and incompressible elastic bodies. Use was made of an arbitrary form of the elastic potential for high subcritical deformations and for two variants of the theory of small subcritical deformations that involved equal and unequal roots of the characteristics equation [6]. It was found that consideration of the mutual influence of the subsurface crack and the free surface results in a considerable reduction of the theoretical strength limit. This was previously derived for infinite material with a crack. Examples of potentials with equal roots are discussed: the Bartenev-Khazanovich (incompressible bodies) and harmonic potential (compressible bodies).     

Crack in a 2D beam lattice: Analytical solutions for two bending modes

We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass-spring lattices. Considering the static problems we use the discrete Fourier transform and the Wiener-Hopf technique. Monotonically distributed bending moments ahead of the crack are determined for the symmetric mode, and a self-equilibrated transverse force distribution is found for the antisymmetric mode. It is shown that in the latter case only the crack-front beam resists to the fracture development, whereas the forces in the other beams facilitate the fracture. In this way, the macrolevel fracture energy is determined in terms of the material strength. The macrolevel energy release is found to be much greater than the critical strain energy of the beam, especially in the hyper-singular mode. In both problems, it is found that among the beams surrounding the crack the crack-front beam is maximally stressed, and hence its strength defines the strength of the structure.     

裂纹在冲击载荷作用下起裂的临界载荷面

采用有限元方法研究裂纹在I型短脉冲载荷作用下应力强度因子随时间的变化 ,用应力强度因子的初始上升时间Tr 对时间坐标无量纲化 ,对应力强度因子初始上升段进行曲线拟合 ,得到了上升段的曲线表达式。运用简单弹性梁理论和Lagrangian运动方程 ,获得载荷与时间对裂纹作用的关系式 ,结合有限元的结果 ,得到了上升时间Tr 的计算表达式 ,并进一步推出了裂纹在冲击载荷作用下起裂的临界载荷面。     

Stability of an infinite solid with a circular cylindrical crack under compression using the Treloar potential

The fracture stability of a circular cylindrical crack in an infinite incompressible solid subjected to an axial compression is considered. A state of subcritical initial strain is assumed. The failure criterion is based on the local stability loss. The investigation is carried out in a single form for the hyper-elastic bodies with an arbitrary type of an elastic potential. Critical loads are determined for axisymmetric forms of a stability loss in the region local to the crack. The linearized problem reduced to the eigenvalue problem is solved numerically. Numerical results are obtained for solids with Treloar potential.     

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 67–77, November 2005.     

Nonaxisymmetric compressive failure of a circular crack parallel to a surface of halfspace

Considered in this work is the fracture stability of a circular crack parallel to the surface of a halfspace subjected to around applied compression. A state of subcritical initial strain is assumed. The failure criterion is based on the loss of local stability and quantified in terms of a critical eigen value. Analysis involves reducing the problem to a system of Fredholm equations of the second kind where the solutions are identified with harmonic potential functions. Critical loads are determined for nonaxisymmetric forms of loss in stability in the region local to the crack; their values would depend on the material properties and geometric parameters. Numerical results are displayed graphically for a crack in an elastic solid and a composite made of aluminum/boron/silicate glass with epoxy-maleimic resin. They depend on the number of angularly dependent harmonies that introduce the nonaxisymmetry in the problem.     

Process zone in the Single Cantilever Beam under transverse loading - Part II: Experimental

This paper describes an experimental arrangement to evaluate stress/strain fields in the process zone of asymmetric adhesively bonded joints. A transparent polycarbonate flexible beam was bonded to an aluminium alloy rigid block with an epoxy adhesive in a Single Cantilever Beam (SCB) configuration. The flexible adherend was loaded in the direction parallel to the initial crack front at constant rate. To monitor strains induced by bending and shear along the beam, electric strain gauges were attached to the upper surface of the flexible adherend. Thus strain distribution was measured above the bonded surface, which could be used to monitor crack propagation and investigate stress redistribution in the process zone. A Timoshenko beam lying on a Pasternak elastic foundation model was used for the analysis of experimental findings. Subsequently, the Digital Image Correlation technique was used to measure the flexible substrate in-plane displacement field in the vicinity of the crack front and to assess the specimen kinematics. We found that strain gauge instrumentation of the fracture mechanics specimen was a very sensitive technique for experimental analysis of crack propagation under complex loading, offering fine investigation of stress distribution in the cohesive zone.     

Stress-strain field near crack tip and calculation of critical stress of crack propagation in a pure bending beam of rectangular section with one-sided mode i crack

This paper studies the stress-strain field near crack tip in a pure bending beam of rectangular section with one-sided mode I crack by the analytic method of Ref. [1], then it gives the stress and strain components at the crack tip when the crack propagates and further it obtains the formulas of calculating the elastic deformed area width, the deformed intensity area width and the equation groups of calculating the critical stress of crack propagation, last the equation group of calculating critical stress of crack propagation is verified by calculating instance. The maximum error is 0.18%. First Received May 7, 1994.