Elastostatic analysis of edge cracked orthotropic strips

Summary. The elastostatic problem of an edge cracked orthotropic strip is considered. The crack possesses a semi-infinite length. The crack surfaces are subjected to opening mode I fracture, by a concentrated force action, while the strip surfaces are traction free. Fourier transforms and asymptotic analyses are employed to reduce the problem to a first kind singular integral equation. The stress intensity factor is determined in a closed form expression. The effects of geometric and elastic characteristics of the strip on the values of the stress intensity factor are explained.     

The improved D-M model solutions for single edge cracked plates by considering the yielding at the back side

The method of calculating the stress intensity factor (SIF) and the crack opening displacement (COD) for double edge cracks in plates under arbitrary loadings that results in solving a system of Cauchy-type singular integral equations is presented. The improved D-M model is then constructed for edge cracked plates by considering the yielding at the back side. For the cases of tension and bending, the plastic zone sizes and the crack opening displacements are calculated from the improved model solution, and the envelopes for the beginning of backside yielding and ligament yielding are obtained. The numerical results are compared with known solutions which take no account of the yielding at the back side and with experimental results.     

Magnetic stress intensity factor for an edge crack in a soft ferromagnetic elastic half-plane under tension

Summary This paper applies the theory for magnetoelasticity to solve the plane problem of an edge crack in a soft ferromagnetic half-plane subjected to a far-field tension and a uniform magnetic field. Fourier transform techniques are used to formulate the mixed boundary value problem as a singular integral equation. The stress intensity factor is calculated and is shown graphically. Tensile tests are also performed on a cracked ferromagnetic plate with strain gage technique, and the numerical results are compared with the test data.     

Surface and internal cracks in a residually stressed plate

The problem of a surface or an internal crack in a plate which contains residual stresses is examined. The line spring model, which reduces a three-dimensional elasticity problem into a two-dimensional problem in plate theory, is used to model the crack. The Reissner plate theory, which takes into account transverse shear deformations, is used to model the plate. The formulation is based on Fourier Transforms which lead to a pair of singular integral equations that are solved numerically. The line spring method requires the plane strain solution to both the edge and internally cracked strip with crack surface loads representative of tension, bending, and the given residual stress distribution. For general use, plane strain solutions are presented for polynomial loading through the thickness up to the fifth order. Comparisons are made between the results given by the line spring model for the Reissner plate theory and the finite element method.     

Two-dimensional problems of the theory of elasticity for reinforced cracked plates

By using the method of singular integral equations, we develop a general approach to the solution of static problems of the two-dimensional theory of elasticity for thin-walled structural elements with curvilinear cracks reinforced by elastic patches. We analyze two basic types of fastening patches to plates, namely, continuous fastening (via the adhesive layer) and discrete fastening (riveting). It is assumed that both the cracked plate and reinforcing patches are characterized by the generalized two-dimensional stressed state. We constructed integral equations for an infinite plate with curvilinear crack reinforced by linear or two-dimensional elastic patches. We also present a brief survey of the literature devoted to the solution of problems of the indicated type.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 3, pp. 68–82, May–June, 1995.     

Thermal singular stresses of glassfiber reinforced plastics with surface cracks at cryogenic temperatures

Thermal singular stress problem for glassfiber reinforced plastics with surface cracks at cryogenic temperatures is considered. For the case of the crack which is normal to and ends at the interface between orthotropic elastic materials, the order of stress singularity around the tip of the crack is obtained. Fourier transforms are used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss–Jacobi integration formula. Numerical calculations are carried out for the cases of embedded and edge cracks, and the thermal stress intensity factors at different temperatures are shown graphically.     

Modified crack closure integral using six-noded isoparametric quadrilateral singular elements

Six-noded, isoparametric serendipity type quadrilateral regular/singular elements are used for the estimation of stress intensity factors (SIF) in linear elastic fracture mechanics (LEFM) problems involving cracks in two-dimensional structural components. The square root singularity is achieved in the six-noded elements by moving the in-side nodes to the quarter point position. The modified crack closure integral (MCCI) method is adopted which could generate accurate estimates of SIF for a relatively coarse mesh. The equations for strain energy release rate and SIF are derived for mixed mode situations using six-noded quadrilateral elements at the crack tip. The model is validated by numerical studies for a centre crack in a finite plate under uniaxial tension, a single edge notched specimen under uniaxial tension, an inclined crack in a finite rectangular plate and cracks emanating from a pin-loaded lug (or lug attachment). The results compare very well with reference solutions available in the literature.     

Estimation of fracture parameters and stress field for edge cracks in finite elastically graded plates using boundary collocation

Summary The fracture parameters, stress intensity factor and T-stress are obtained for edge cracks aligned along the gradient in finite size elastically graded plates using the technique of boundary collocation. A scheme for extending the recently derived crack tip stress field for elastically graded materials is proposed. Using this extended stress field, the fracture parameters are evaluated for edge cracks subjected to far field tension and three point bending. The results for far field tension agreed well with published theoretical results over a good range of elastic gradients. The maximum shear stress calculated over the entire domain of the cracked plate using boundary collocation agrees very well with that obtained from finite element analysis. The efficacy of the extended stress field in capturing the effects of the elastic gradient on the stresses and fracture parameters is thus established in this study.     

An assumed displacement hybrid finite element model for linear fracture mechanics

This paper deals with a procedure to calculate the elastic stress intensity factors for arbitrary-shaped cracks in plane stress and plane strain problems. An assumed displacement hybrid finite element model is employed wherein the unknowns in the final algebraic system of equations are the nodal displacements and the elastic stress intensity factors. Special elements, which contain proper singular displacement and stress fields, are used in a fixed region near the crack tip; and the interelement displacement compatibility is satisfied through the use of a Lagrangean multiplier technique. Numerical examples presented include: central as well as edge cracks in tension plates and a quarter-circular crack in a tension plate. Excellent correlations were obtained with available solutions in all the cases. A discussion on the convergence of the present solution is also included.     

Fracture of a surface layer bonded to a half space

The plane problem of a cracked elastic surface layer bonded to an elastic half space is considered. The surface layer is assumed to contain a transverse crack whose surface is subjected to uniform compression. The problem is formulated in terms of a singular integral equation, the derivative of the crack surface displacement being the density function. By using appropriate quadrature formulas, the integral equation reduces to a system of linear algebraic equations. This system is solved; the stress intensity factors and the crack surface displacement for various crack geometries, namely for internal crack, edge crack, crack touching the interface, and completely broken layer cases, are obtained.     

Analytical stress around mode II crack parallel to principle axis in orthotropic composite plate

Stress analyses for orthotropic composite materials containing a through crack under remote shear loads (Mode II) are conducted. By employing the complex theory, a harmonic differential equation was established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry. An analytical complex function was introduced by following the Westergaard approach. Stress around a mode II crack in the orthotropic composite plate is deduced to have an analytical form. In addition, the analytical solution for a mode II crack was examined in the case of isotropic materials. It demonstrated that the analytical solution obtained is correct for the mode II cracked orthotropic composite plates.     

A mode II crack problem with sliding contact in an orthotropic strip

The problem of a centrally cracked, linear elastic orthotropic strip loaded in bending by three point forces is analyzed and discussed. Coulomb friction is assumed between the crack faces to study the influence of the friction coefficient on the strain energy release rate. Under certain simplifying assumptions the problem is reduced to the solution of a singular integral equation which is evaluated numerically. The results are compared with the solution of the same problem obtained using the beam theory; limits of the application of beam theory for the reduction of experimental data are discussed.     

Elastic problem of an adhesion crack in T-junction of two orthotropic thin plates

In this study, the general solution is derived for stresses in a T-junction of two thin plates with an adhesion crack. The plates are orthotropic, and shear force is applied to the crack surface. The analysis is based on the supposition that the stresses in each plate can be approximated by the condition of plane stress. The results obtained are verified through numerical calculation using the finite element method. A singular stress field is obtained from the solution in the vicinity of a crack tip.     

Analysis of multiple interfacial cracks in an orthotropic bi-material subjected to anti-plane shear loading

The present work concerns with the elasto-static problem of double interfacial cracks located between two dissimilar orthotropic plates. The dimensions of the bi-material composite, are assumed to be finite. The crack faces are subjected to anti-plane shear traction. Finite Fourier transforms are applied to reduce the problem to a triple series equations, and then to a system of singular integral equations with Cauchy type singularity. That are solved numerically using Gauss-Chebyshev integration formulae. The stress intensity factors, are determined in a closed form expressions. The obtained results agreed with the previous analytical ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the composite on the values of the stress intensity factors.     

USE OF THE DISTRIBUTED DISLOCATIONS METHOD TO DETERMINE THE T-STRESS

Abstract— This paper demonstrates a method to determine the elastic T -stress for a semi-infinite half-plane containing a surface-breaking crack which is loaded by an arbitrarily distributed far-field tension. The method consists of representing the crack by a continuous distribution of edge dislocations and forming singular integral equations to determine the equilibrium dislocation distributions. By numerically solving the integral equations, stress intensity factors and T -stresses are obtained for the example case of a crack which is normal and inclined to the free-surface of a half-plane and loaded by a uniform far-field tension.     

含裂纹损伤有限加筋板弹塑性承载力分析

基于含直裂纹问题的复应力函数解法,提出了用Dugdale模型分析和求解弹塑性条件下含中心裂纹的有限加筋板承载力问题的方法。通过将加筋板离散为筋、板的结构,将含裂纹有限加筋板的问题转化为边界受切向力作用的含裂纹有限板的问题进行求解。计算了筋、板相对刚度不同的情况下,含中心裂纹有限加筋板裂纹尖端开口位移CTOD(Crack Tip Opening Displacement)值随裂纹长度及承载力情况变化的系列值。     

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.     

A thin cracked layer bonded to an elastic half-space under an antiplane concentrated load

The antiplane elasticity problem for a thin cracked layer bonded to an elastic half-space under an antiplane concentrated load is considered. The fundamental solution is obtained as a rapidly convergent series in terms of the complex potentials via iterations of Möbius transformation. The singular integral equation with a logarithmic singular kernel is derived to model a crack problem that can be solved numerically in a straightforward manner. The dimensionless mode-III stress intensity factors obtained for various crack inclinations and crack lengths are discussed in detail and provided in graphic form. A strip problem with an arbitrarily oriented crack is also considered.     

The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

考虑裂纹效应的弹性板振动分析模型

针对含裂纹板的动力学问题,提出了一种耦合裂纹效应的弹性板动力学建模方法。该方法依据变形等效原则用虚拟外部载荷代替裂纹作用,并通过力学平衡原理建立了耦合裂纹项的弹性板运动方程,且基于Rice和Levy应力关系式推导出裂纹项表达式;在此基础上,结合Galerkin法和Berger经验,把含裂纹弹性板振动系统简化成一单自由度非线性振动模型进行动力学特性分析。通过算例探讨了裂纹尺度、阻尼以及激励力位置对弹性板振动特性的影响。结论表明,裂纹尺度和板尺寸对振动非线性作用明显,动应力幅值受阻尼与激励力位置的控制。