Crack opening displacement in plate with bonded repair patch

Mathematical techniques are extended to compute crack opening displacements in a cracked plate with an adhesively bonded composite patch. The plate and the patch are considered as orthotropic materials. The problem is reduced to the solution of integral equations. A software program is written to compute shear stresses in adhesive, stress intensity factors in the plate and the crack openings at the centreline of the crack. The effects of adhesive thickness, adhesive modulus, patch thickness and plate thickness on crack openings are investigated. A test program is carried out to obtain crack opening displacements in plate with bonded patch. A good agreement with analytical predictions is obtained. The effects of patches bonded on one or both sides of a plate on stress intensity factors are evaluated.     

裂纹板贴补应力分析

本文采用含裂纹无限大板特殊基本解和合力边界条件,用体积力法对含裂纹金属薄板的胶贴补强问题进行应力分析。使用一满足胶贴层位移连续条件的剪切单元,把问题转化为对裂纹板和贴片的分析。由于使用的特殊基本解精确满足裂纹面自由力边界条件,避免了对裂纹尖端附近的奇异场进行离散处理,因而可以比较精确地求出裂纹尖端附近的应力分布,同时由于单位集中力引起的裂纹尖端应力强度因子可以解析得到,因而可以较准确地反映出用应力强度因子的降低来表征的贴补效果。作为贴补计算的例子,文中计算了受拉力和剪力作用时,含中心裂纹的金属裂纹板在贴补前后裂纹尖端应力强度因子的降低,给出了贴片的厚度、弹性模量和尺寸及肢贴层厚度等对贴补效果的影响。     

Fatigue crack growth analysis of stiffened cracked panel repaired with bonded composite patch

Fatigue crack growth behavior in a stiffened thin 2024-T3 aluminum panel repaired with one-sided adhesively bonded composite patch was investigated through experiments and analyses. The patch had three plies of unidirectional boron/epoxy composite. 2024-T3 aluminum stiffeners were riveted as well as bonded on the panel. Stiffeners were oriented in the loading direction and were spaced at either 102 mm or 152 mm with a crack centered between them. Also, un-repaired cracked panel with and without stiffeners were studied. Experiment involved tension-tension fatigue at constant amplitude with maximum stress of 120 MPa and stress ratio of 0.05. Bonded composite patch repair increased fatigue life about five-fold in the case of stiffened panels while it increased about ten fold in the case of un-stiffened panels. Fatigue life also increased with decrease of the distance between the stiffeners for both repaired and un-repaired panels. A three-dimensional finite element method was used to analyze the experiments. Residual thermal stresses, developed during patch bonding, requires the knowledge of temperature at which adhesive becomes effective in creating a bond between the structure and patch in the analysis. A simple method to estimate the effective curing temperature range is suggested in this study. The computed stress intensity factor versus measured crack growth relationships for all panel configurations were consistent and in agreement with the counterpart from the test material. Thus, the present approach provides a means to analyze the fatigue crack growth behavior of stiffened structures repaired with adhesively bonded composite patch.     

Stress analysis of a bonded repair over a corrosion grind-out using an inclusion model with a second ordered eigenstrain theory

The problem of an isotropic sheet with an internal elliptical grind-out repaired with an octagonal bonded patch is analyzed, following Rose's two-stage analysis procedure [Theoretical analysis of crack patching, in: A.A. Baker, R. Jones (Eds.), Bonded Repair of Aircraft Structures, Kluwer Academic Publishers, Dordrecht, 1988] for a crack patching problem. For simplicity, the bending effect is ignored in the present analysis. In the first stage, an infinite sheet reinforced by an octagonal patch under a prescribed far-field stress and a uniform thermal field is analyzed using an inclusion model with a second ordered eigenstrain theory, without considering a grind-out cavity. The stresses (σ_ij^*) inside the patched area at a center of the prospective grind-out location are calculated and later used as far field boundary conditions for a second (stage II) problem. In the second stage, the patch is assumed to be infinite and an integral part of a sheet. Stage II analysis then involves solving a problem of an infinite patched sheet containing an embedded elliptical grind-out cavity under far field stresses and initial strain field. The latter problem is also solved using an inclusion model with a second ordered eigenstrain. This approach had been used previously by the present authors very successfully in solving for a similar repair problem but with an elliptical patch and for a grind-out with a uniform depth [Analytical approach to bonded repair of elliptical dent, corrosion grind-out and cut-out, Theoretical and Applied Fracture Mechanics, 2000, submitted] in a relatively thin sheet. However, our recent 3-D finite element study showed that the use of such inclusion analogy in a stage II stage analysis would not yield sufficiently accurate results for a repair problem of a very deep grind-out cavity in a thick sheet. The original approach given in (Duong et al., loc. cit.) therefore has been modified in this paper. The present paper also deals in particular with additional complexities such as octagonal shaped patch, a grind-out cavity with a spherical depth, and effects of thermal stresses due to curing and/or cruising.     

Analysis of cracked plates with a bonded patch

The problem of a cracked plate repaired by an adhesively bonded patch is studied. A shear spring model is adopted to reduce the problem to the analysis of a cracked plate and a patch subjected to external loads and interacting adhesive shear. While the patch is treated by the finite element method, the cracked plate is analyzed by the boundary element method, in which a special fundamental solution satisfying the boundary condition on the crack surface is introduced. The present formulation provides comparable results on the stress intensity factor of the patched crack with less computational effort.     

Nonlinear adhesive behavior effects in a cracked orthotropic sheet stiffened by a semi-infinite orthotropic sheet

The stress-intensity factors are determined for a cracked orthotropic sheet adhesively bonded to an orthotropic stringer where the adhesive layer is modeled with a nonlinear stress-strain curve. Since the stringer is modeled as a semi-infinite sheet, the solution is most appropriate for a crack tip located near a stringer edge. By the use of Green's functions and the complex variable theory of orthotropic elasticity developed by Lekhnitskii, a set of integral equations is obtained. The integral equations are replaced by an equivalent set of algebraic equations, which are solved to obtain the shear stress distribution in the adhesive layer. With these adhesive stresses, the crack-tip stress-intensity factors are found.When the adhesive was modeled with a nonlinear stress-strain curve, the peak shear stresses in the adhesive were considerably reduced in comparison to the solution for the linear elastic adhesive. This resulted in increases in the stress-intensity factors for the nonlinear adhesive solution compared to the linear adhesive solution. The nonlinear adhesive did not have a significant effect on the stress-intensity factor unless the near crack tip was beneath the stringer. The present investigation assumes that the adhesive bond remains intact. Onset of adhesive failure is predicted to occur at decreasing levels of applied stress as the crack propagates beneath the stringer.     

Disbond effect on the stress intensity factor for repairing cracks with bonded composite patch

It is well known that the stress intensity factor is considerably reduced by the bonded composite repair. The finite element method is used to compute the stress intensity factor for repairing cracks with bonded composite patch taking account of the disbond. In the case of a disbond, the increase of patch thickness reduce the negative effects of disbond. The curves plotted show the concordance with the model [Thermal residual stresses in composite repairs on cracked metal structures, Ph.D. Thesis, University of British Columbia, 1998].     

Progressive edge cracked aluminium plate repaired with adhesively bonded composite patch under full width disbond

In this study, the crack growth behaviour of an aluminium plate cracked at the tip and repaired with a bonded boron/epoxy composite patch in the case of full-width disbond was investigated. This effect is the imperfection which could result during the bonded patch of the repaired structure. Disbonds of various sizes and situated at different positions with respect to the crack tip as well as the effect of adhesive and patch thickness on repair performance were examined. An analysis procedure involving the efficient finite element modelling applied to cracked plate, adhesive and composite patch was used to compute the stress intensity factors. The crack growth rate is dominated by the stress intensity factor near the location and size of the pre-existing disbonds. The cracked plate and disbond propagation result in an increase in the patch deformation. The patch does not have an influence on the crack growth when the ratio 2a/d_R exceeds 0.8.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

非均匀复合材料中反平面裂纹的动态断裂力学研究

对于非均匀复合材料中多个裂纹的动态断裂力学问题, 提出了一种分析方法, 假设复合材料为正交各向异性并含有多个垂直于厚度方向的裂纹, 材料参数沿厚度方向为变化的, 沿该方向将复合材料划分为许多单层, 假设单层材料参数为常数, 应用柔度矩阵/刚度矩阵方法及Fourier变换法, 在L aplace 域内推导出了控制问题的奇异积分方程组, 并用虚位移原理求解, 给出了应力强度因子及能量释放率的表达式, 然后利用Laplace 数值反演, 得出了裂纹尖端的动态应力强度因子和能量释放率。作为算例, 研究了带有两个裂纹的功能梯度结构, 分析了材料参数的优化对降低应力强度因子的意义。      

The elastic contact problem for dissimilar orthotropic semi-infinite and infinite strips

This article examines the stress state which develops when the end of a symmetrically loaded orthotropic semi-infinite strip is supported on an orthotropic infinite strip. Both smooth and bonded contact between the strips are considered. By using Fourier transforms the problem is reduced to a system of singular integral equations. The dependence of the order of the stress singularity on the orthotropic properties of the materials is investigated. Numerical solutions are presented for the contact stresses corresponding to different loadings and material properties.     

Stress Intensity Factor of an Edge Crack in Bonded Orthotropic Materials

The article presents the problem of an edge crack under normal point loading terminating perpendicular to the surface of an orthotropic strip of finite thickness which is bonded to another orthotropic half plane. Expressing the displacements and stresses in plane strain condition in terms of harmonic functions, the problem is reduced to a pair of simultaneous integral equations with Cauchy type singularities, which are finally been solved by the Hilbert transform technique. The analytical expression of stress intensity factor (SIF) at the crack tip for large thickness of the strip is calculated, which corresponds to the weight function of a crack under normal loading. The influences of elastic constants of two different orthotropic materials, distinct arbitrary locations of normal point loading on the crack surface and length of the crack on the dynamic SIF are depicted through graphs.     

Symmetric Edge Cracks in an Orthotropic Strip Under Normal Loading

The problem of two edge cracks of finite length, situated symmetrically in an orthotropic infinite strip of finite thickness 2 h, under normal point loading has been discussed. The displacements and stresses in plane strain conditions are expressed in terms of two harmonic functions. The problem is addressed by seeking the solution of a pair of simultaneous integral equations with Cauchy type singularities solved by finite Hilbert Transform technique. For large h, analytical expression for the stress intensity factor at the crack tip is obtained.     

The effects of disbonds on the pure mode II stress intensity factor of aluminium plate reinforced with bonded composite materials

Adhesively bonded composite patch repair has been widely used to restore or extend the service life of cracked structural components due to its effectiveness to mechanical repair technique. In this work, the finite element method is applied to analyse the performance of the bonded composite patch for repairing cracks emanating from semicircular notch root in pure mode II. The stress intensity factor was computed at the crack tip repaired using a boron/epoxy patch for different orientation of fibers, taking into account the disbond. In this case, the increase of a patch thickness reduces the negative effects of disbond. When this effect is significant between the patch and the plate, it reduces the repair effectiveness. The maximum reduction obtained by using a boron/epoxy of fibers in the x-direction is of the order of 20% more important compared to a patch having its fibers in the y-direction. The stress intensity factor exhibits an asymptotic behaviour as the disbond size increases.     

Mixed-Mode Crack Extension in Orthotropic Materials Under Biaxial Load

The problem of an infinite orthotropic material with a central crack inclined with respect to the principal axes of material symmetry is analyzed. The material is subjected to uniform biaxial load along its boundary. By considering the effect of the horizontal load, the distribution of stresses at the crack tip is analyzed. The direction of crack extension in orthotropic solids is examined. The results based on the normal stress ratio theory show significant effects of biaxial load, crack inclination angle, fiber angle and orthotropic material properties on the crack extension.     

An elastic strip with periodic surface cracks under thermal shock

The problem of two periodic edge cracks in an elastic infinite strip located symmetrically along the free boundaries under thermal shock is investigated. It is assumed that the infinite strip is initially at constant temperature. Suddenly the surfaces containing the edge cracks are quenched by a ramp function temperature change. Very high tensile transient thermal stresses arise near the cooled surface resulting in severe damage. The degree of the severity for a subcritical crack growth mode is measured by determining the stresses intensity factors. The thermoelastic problem is treated as uncoupled quasi-static. The superposition technique is used to solve the problem. The thermal stresses obtained from the uncracked strip with opposite sign are utilized as the only external loads to formulate the perturbation problem. By expressing the displacement components in terms of finite and infinite Fourier transforms, a hypersingular integral equation is derived with the crack surface displacement as the unknown function. Numerical results for stress intensity factors are carried out and presented as a function of time, cooling rate, crack length, and periodic crack spacing.     

Elastodynamic Response of Cracked Orthotropic Strip Under Impact Loading

The paper presents the elastodynamic response of an infinite orthotropic strip of finite width containing a central crack opened by suddenly applied stresses. Integral transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane solved by iterations in the low frequency case. Analytic expressions for the dynamic stress intensity factors are obtained. Numerical results for two particular materials are given. The influence of different orthotropic constants on the magnitude of the overshoots in the stress intensity factors K ₁(t) and K ₂(t) are found.     

Fatigue crack closure analysis of bridged cracks representing composite repairs

This article presents an analytical and numerical study of the fatigue crack‐closure behaviour of a bridged crack representing a crack that has been repaired by a composite patch. It is shown that, provided that the plate stress beneath the patch is less than 40% of the material’s yield stress, the crack‐closure stress of a patched crack is approximately equal to that of an unbridged crack under small‐scale yielding, depending only on the stress ratio. Furthermore, it is shown that the transient crack‐closure behaviour of a patched crack subjected to variable amplitude loading can be determined by analysing an unpatched crack subjected to the same stress intensity factor history. Based on these findings, it is proposed that the fatigue crack closure of a patched crack can be determined by analysing an unpatched centre crack subjected to an adjusted stress, for which an explicit expression is given. Predictions based on the proposed method are shown to correlate very well with experimental results obtained under two aircraft loading spectra.     

Plastic zones in an orthotropic plate of finite width containing a Griffith crack

The problem of an orthotropic infinite plate of finite width containing a centrally located stressed Griffith crack is considered. The crack is located perpendicular to the edges of the orthotropic plate. The crack tips are fully yielded and in the inelastic zones the material carries only constant normal stresses equal to the yield stress. Dugdale's model is employed to find the effects of the material anisotropy on the size of the plastic zones around the crack tips. Graphical results showing the effects of anisotropy on the length of the plastic zone are also presented.     

Transverse Stress Decay in a Specially Orthotropic Strip under Localized Normal Edge Loading

Solutions are presented for the stresses in a specially orthotropic infinite strip which is subjected to localized uniform normal loading on one edge while the other edge is either restrained against normal displacement only, or completely fixed. The solutions are used to investigate the diffusion of load into the strip and in particular the decay of normal stress across the width of the strip. For orthotropic strips representative of a broad range of balanced and symmetric angle-ply composite laminates, minimum strip widths are found that ensure at least 90% decay of the normal stress across the strip. In addition, in a few cases where, on the fixed edge the peak shear stress exceeds the normal stress in magnitude, minimum strip widths that ensure 90% decay of both stresses are found. To help in putting these results into perspective, and to illustrate the influence of material properties on load diffusion in orthotropic materials, closed-form solutions for the stresses in similarly loaded orthotropic half-planes are obtained. These solutions are used to generate illustrative stress contour plots for several representative laminates. Among the laminates, those composed of intermediate-angle plies, i.e., from about 30 degrees to 60 degrees, exhibit marked changes in normal stress contour shape with stress level. The stress contours are also used to find 90% decay distances in the half-planes. In all cases, the minimum strip widths for 90% decay of the normal stress exceed the 90% decay distances in the corresponding half-planes, in amounts ranging from only a few percent to about 50% of the half-plane decay distances. The 90% decay distances depend on both material properties and the boundary conditions on the supported edge.