Displacements, strains and stresses associated with propagating cracks in materials with continuously varying properties

Summary. Crack tip stress, strain and displacement fields for a propagating crack along the direction of property gradation in functionally graded materials (FGMs) are obtained through an asymptotic analysis coupled with a displacement potential approach. The analysis for the opening mode is performed assuming two types of property variation: (a) linearly varying shear modulus with constant density, and (b) exponentially varying shear modulus and density. The first six terms in the series expansion of the stress, strain and displacement fields for the dynamic crack are derived to explicitly bring out the influence of nonhomogeneity on the structure of the displacement, strain and stress fields. The analysis revealed that crack tip stress fields retain the inverse square root singularity and only the higher order terms in the expansion are influenced by material inhomogeneity. Using these stress, strain and displacement fields, contours of constant maximum shear stress, constant first stress invariant and constant in-plane displacements are generated and the effect of the nonhomogeneity parameter on these contours is discussed.     

Analytical expressions for stress and displacement fields in viscoelastic axisymmetric plane problem involving time-dependent boundary regions

An analytical solution is developed in this paper for viscoelastic axisymmetric plane problems under stress or displacement boundary condition involving time-dependent boundary regions using the Laplace transform. The explicit expressions are given for the radial and circumferential stresses under stress boundary condition and the radial displacement under displacement boundary condition. The results indicate that the two in-plane stress components and the displacement under corresponding boundary conditions have no relation with material constants. The general form of solutions for the remaining displacement or stress field is expressed by the inverse Laplace transform concerning two relaxation moduli. As an application to deep excavation of a circular tunnel or finite void growth, explicit solutions for the analysis of a deforming circular hole in both infinite and finite planes are given taking into account the rheological characteristics of the rock mass characterized by a Boltzmann or Maxwell viscoelastic model. Numerical examples are given to illustrate the displacement and stress response. The method proposed in this paper can be used for analysis of earth excavation and finite void growth.     

Evaluation of crack tip stress and strain fields under nonproportional loading in a viscoelastic material

The crack tip strain and stress fields in a viscoelastic material under nonproportional loading conditions are evaluated. In order to evaluate the strain field, the crack tip displacement field is measured by using the digital image correlation (DIC) technique. This displacement field is then approximated by using the theoretically obtained crack tip displacement field in viscoelastic materials. The result shows that the approximation method can smoothly reconstruct the experimentally obtained displacement field. From the approximated displacement field, the crack tip strain field can be precisely obtained by using the differential form of the theoretical displacement. On the other hand, the crack tip stress field is analyzed by using the stress function. This suggests that the strain and stress fields can be independently evaluated. In addition, different time dependencies between stress and strain fields near the crack tip are observed. Based on this experiment, we can discuss the several criteria for the crack propagation directions in viscoelastic materials.     

Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods

A method for evaluating mode I, mode II and mixed-mode stress intensity factors from in-plane displacement fields using the method of nonlinear least-squares is proposed in this paper. Along with stress intensity factors, crack tip location and rigid body displacement components are determined simultaneously from both displacement components obtained using full-field optical methods or numerical methods. The effectiveness is validated by applying the proposed method to mixed-mode displacement fields obtained through digital image correlation, displacement fields obtained by analysis using elasto-plastic finite element method, and displacement fields around a fatigue crack obtained by electronic speckle pattern interferometry. Results show that the proposed method can extract stress intensity factors from the displacement fields both accurately and easily. Furthermore, they can be determined even if the material at a crack tip exhibits small-scale yielding. It is expected that the proposed method is applicable to various fracture problems during experimental and numerical evaluation of structural components.     

Numerical technique for solving truss and plane problems for a new class of elastic bodies

It is customary to use a displacement-based formulation to seek solutions to boundary value problemsfor its computational efficacy. In displacement-based formulations, it is convenient to prescribe the constitutiverelation for stress as an explicit function of displacement gradient. However, from a general theoretical pointof view, the stress and displacement gradient could be related by an implicit function. This study developstechniques to solve boundary value problems when linearized strain and stress are related by an implicitfunction. Here both the stresses and the displacement are taken as unknowns. The stress field is constructed suchthat it satisfies the equilibrium equations identically within the element and the traction continuity requirementsbetween the elements. A continuously differentiable displacement field is constructed, and the linearized strainis computed from this displacement field. Then, the unknown parameters in the stress and displacement fieldare estimated such that the constitutive relation holds in weak integral sense. Though in this procedure thenumber of unknowns has increased in comparison with the displacement formulation, both the strength andserviceability condition can be checked directly without any post-processing. Also, in this procedure, both theequilibrium equations and continuity of displacement are met exactly. The equation that is not satisfied exactlyis the constitutive relation, which is an approximation anyway. The efficacy and accuracy of this method arebenchmarked by studying some standard problems. Planar and three-dimensional truss elements have beendeveloped and benchmarked. Then, a rectangular plane element is implemented and its performance recorded.     

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.     

Analytical solutions of the displacement and stress fields of the nanocomposite structure of biological materials

Biological materials, such as bone and nacre, are nanocomposites of protein and mineral with superior mechanical properties. The basic building blocks of these materials feature a generic nanocomposite structure with staggered alignment of mineral platelets in protein matrix. Because of the structural complexity of the generic structure, its displacement and stress fields are currently still unknown. In this study, a perturbation method was applied for analytically solving the displacement and stress fields of the nanocomposite structure under uniaxial tension. The effect of the elastic modulus, aspect ratio and volume fraction of mineral and protein on the displacement and stress fields in the nanocomposite structure was studied. A non-dimensional parameter γ was then suggested for characterizing the stress and strain fields in this nanostructure. We showed that the assumption of uniform shear stress distribution at the mineral-protein interface in the TSC model is valid when γ is less than 4 which is broadly applicable to most biological materials. The analytical solutions of displacement and stress fields obtained in this study provide a solid basis for further analyses of mechanical properties, such as the buckling and the fracture behaviors of biological materials.     

Micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in piezoelectric solids

A dislocation emission mechanism for micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in a piezoelectric solid is proposed in the present paper. For a rigid line inhomogeneity embedded in a piezoelectric matrix, dislocations of one sign are driven away from the tip due to high stress level, while the stationary dislocations of the opposite sign are left behind near the tip of the inhomogeneity. As a result, a micro-Zener–Stroh crack is initiated ahead of the line inhomogeneity. In the current study, a dislocation pileup mechanism for micro-crack initiation at the inhomogeneity tip is proposed. An interesting result is that the critical stress intensity factors for a line inhomogeneity perpendicular to the poling direction can be related to the fracture toughness of a conventional crack in the same material. Analytical solutions show that the critical plane shear stress intensity factor depends on the plane shear mechanical and displacement loadings, and the critical opening stress and electric displacement intensity factors depend on not only the mechanical and displacement loadings, but the electric field and displacement loadings as well.     

Higher-order zig-zag model for analysis of thick composite beams with inclusion of transverse normal stress and sublaminates approximations

A modified zig-zag technical theory, suitable for the analysis of thick composite beams with rectangular cross section, general lay-up and in cylindrical bending is developed and tested. An equivalent single-layer model and a multiple-layer model are implemented. The displacement field of both these models is postulated as to allow for appropriate jumps in the strains, so that the transverse shear and the transverse normal stress and stress gradient continuity at the interfaces are met. A third-order piecewise approximation for the in-plane displacement and a fourth-order piecewise approximation for the transverse displacement are assumed in the two models. Their predictive capability is investigated in sample cases wherein the exact three-dimensional elasticity and other approximate solutions are available. On the basis of this numerical investigation, they appear to predict accurately and efficiently the displacement and stress fields of composite beams with layers of different materials.     

基于Newmark法敏度计算的刚架结构动力优化

提出一种有效的求解结构最小质量设计,同时满足动位移和动应力约束的二阶优化设计方法。在有限元法和纽马克法基础上导出一种高效的动应力、动位移对设计变量一阶导数和二阶导数的算法。建立含时间参数,以结构质量最小化为目标,同时满足动位移、动应力和设计变量约束的优化数学模型,通过积分型内点罚函数将含时间参数的不等式约束优化问题转变为一系列不含时间参数的无约束优化问题。利用动位移、动应力对设计变量一阶导数和二阶导数的信息计算内点罚函数的梯度和海森矩阵,利用梯度和海森矩阵构造求解优化设计问题高效有效的二阶优化算法。算例结果表明该文的优化设计方法能获得刚架结构的局部最优设计,优化的效率高于增广拉格朗日乘子法。     

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.     

铆接界面端三维奇异性应力场的研究

针对铆接结构的特点,应用特征函数扩展技术分析柱坐标下接触界面端的应力奇异性问题。建立了柱坐标下圆柱体端面接触边缘附近的三维渐近位移场和应力场渐近表达式,并根据铆钉/被铆接件接触界面端的位移和应力边界条件,建立一个非线性特征方程组。据此方程组可求解界面端邻域的应力奇异性指数、位移和应力角分布函数的数值解。通过与有限元方法计算结果相对比,验证了该方法的有效性。分析了平头、沉头以及半圆头铆钉构成的铆接结构的应力奇异性问题,考察了铆钉材料、几何形式和摩擦系数对接触界面端应力奇异性指数和应力场角分布的影响。     

正交各向异性材料结构热变形和热应力分析的无网格法计算模型及应用正交各向异性材料结构热变形和热应力分析的无网格法计算模型及应用

利用无网格伽辽金法（Element-free Galerkin,EFG）建立了正交各向异性材料结构热变形和热应力分析的计算模型,并推导了正交各向异性材料结构热弹性问题的EFG法离散控制方程。选择复合材料冷却栅管算例验证了计算模型和程序的正确性,利用该计算模型分析了具有不同材料方向角及热导率因子、热膨胀系数因子和主次泊松比因子的汽轮机叶轮,得到了其热变形总位移和Mises应力,讨论了材料方向角和上述正交各向异性材料因子对其热变形总位移和Mises应力的影响规律,给出了这些参数的合理取值范围,并选取一组参数与各向同性材料结构进行了热变形和热应力对比分析。结果表明,基于EFG法的热变形总位移和Mises应力的计算精度比有限元法高,材料方向角同时影响热变形总位移和Mises应力的大小和方向,而正交各向异性材料因子只影响热变形总位移和Mises应力的大小,不影响其方向。在复合材料结构设计过程中,合理选取材料方向角和正交各向异性材料因子可有效减小结构热变形和热应力。      

压电材料中渗透性周期共线反平面裂纹问题

本文利用复变函数方法,借助于Riemann-Schwarz延拓技术和保形映照方法,研究了渗透性边界条件下周期共线反平面裂纹问题,获得了解的表达式,得到了力学和电学强度因子。结果表明在裂纹尖端应力和电位移的奇异性都与远场作用的应力载荷和裂纹长度有关,其中应力的奇异性与材料无关,电位移的奇异性则与材料有关,电载荷对裂尖的奇异性没有影响。最后,运用数值算例,给出周期裂纹间的干涉效应和裂纹的尺度效应。     

Plane Elastic Problem of a Crack Normal to and Terminating at Bimaterial Interface of Isotropic and Orthotropic Half Plates

In this paper, the displacement and stress fields for a crack normal to and terminating at a bimaterial interface of isotropic and orthotropic half planes are studied as a plane problem. The eigenequation, by which the order of stress singularity is determined, is given in an explicit form. A discriminant function is presented to judge whether the stress singularity at the crack tip is greater than -1/2 or not. An explicit closed form expression is derived for the displacement and stress distribution near the crack tip.     

A numerical analysis of cracks emanating from a square hole in a rectangular plate under biaxial loads

This note concerns with stress intensity factors of cracks emanating from a square hole in rectangular plate under biaxial loads by means of the boundary element method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results illustrate that the present approach is very effective and accurate for calculating stress intensity factors of complicated cracks in a finite plate and can reveal the effect of the biaxial load and the cracked body geometry on stress intensity factors.     

Stress intensity factors by enriched mixed finite elements

An enriched finite element model for linear elastic fracture mechanics is developed for a mixed variational statement. The independent approximations for the displacement and stress components are enriched by adding the near-field analytic expressions for a cracked body to the polynomial approximations of a conventional element. This allows for an accurate representation of the stress and displacement fields near the crack tip and also results in the direct calculation of the appropriate stress intensity factors. The accuracy of this formulation is demonstrated through several numerical examples.     

Automatic Evaluation of Mixed-mode Stress Intensity Factors Utilizing Digital Image Correlation

Abstract:  A method for determining mixed-mode stress intensity factors from displacement fields obtained by digital image correlation is studied. To perform automatic evaluation, not only stress intensity factors but crack-tip location, higher-order terms in the series expansion of displacement fields and rigid-body displacement components are also determined simultaneously using nonlinear least squares based on the Newton–Raphson method. Experimental results show that the mixed-mode stress intensity factors are evaluated accurately from either radial or circumferential displacement components. As mixed-mode stress intensity factors can be evaluated easily, simply and automatically by the technique with digital image correlation and nonlinear least squares, it is expected that the proposed method can be applied to solve various fracture problems.     

Doubly periodic arrays of cracks and thin inhomogeneities in an infinite magnetoelectroelastic medium

A two-dimensional boundary element method for the analysis of a magnetoelectroelastic medium containing doubly periodic sets of cracks or thin inclusions is developed in this paper. The integral equations and closed-form expressions for corresponding kernels are obtained. Based on the quasi-periodicity of extended displacement and stress function, the integral representations for average stress, strain, electric displacement, magnetic induction etc. are developed. The algorithm of effective properties determination is given. The numerical examples prove the efficiency and high accuracy of the proposed approach in determination of stress, electric displacement and magnetic induction intensity factors and effective properties of the material containing doubly periodic arrays of cracks or thin inclusions.     

Time transient validation of residual stress prediction models for aluminium alloy quenching

Finite element models that predict residual stress states in relatively large quenched aluminium alloy products tend to give reliable results. However, even though there is confidence that the predicted stress state is correct, there is no validation indicating that the stress/displacement development during the quench is comparable to the experimental case. Combined with this, finite element predictions for small samples tend to underestimate surface stress. This paper uses a ‘Navy C-Ring’ benchmark design to monitor displacement during quenching. The heat transfer coefficient is found to be the most dominant boundary condition and is critical to ensuring displacement and residual stress predictions that match the experimental case.

This paper is part of a Themed Issue on Measurement, modelling and mitigation of residual stress.