T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms

A new interaction integral formulation is developed for evaluating the elastic T-stress for mixed-mode crack problems with arbitrarily oriented straight or curved cracks in orthotropic nonhomogeneous materials. The development includes both the Lekhnitskii and Stroh formalisms. The former is physical and relatively simple, and the latter is mathematically elegant. The gradation of orthotropic material properties is integrated into the element stiffness matrix using a “generalized isoparametric formulation” and (special) graded elements. The specific types of material gradation considered include exponential and hyperbolic-tangent functions, but micromechanics models can also be considered within the scope of the present formulation. This paper investigates several fracture problems to validate the proposed method and also provides numerical solutions, which can be used as benchmark results (e.g. investigation of fracture specimens). The accuracy of results is verified by comparison with analytical solutions.     

T-stress and its implications for crack growth

Some of the “irregular” crack growth behaviour observed in different specimen geometries may not be unrelated. Discrepancies in fatigue crack growth rate have been observed in different specimen geometries of the same material; crack front “tunnelling” and out-of-plane crack growth have been found in mode I tension at elevated temperature. The results presented in this paper seem to indicate the relevance of a crack tip constraint parameter, the elastic T-stress, to the irregular crack growth behaviour that conventional LEFM fails to explain.     

Crack-tip fields in anisotropic shells

Asymptotic crack-tip fields including the effect of transverse shear deformation in an anisotropic shell are presented. The material anisotropy is defined here as a monoclinic material with a plane symmetry at x ₃=0. In general, the shell geometry near the local crack tip region can be considered as a shallow shell. Based on Reissner shallow shell theory, an asymptotic analysis is conducted in this local area. It can be verified that, up to the second order of the crack tip fields in anisotropic shells, the governing equations for bending, transverse shear and membrane deformation are mutually uncoupled. The forms of the solution for the first two terms are identical to those given by respectively the plane stress deformation and the antiplane deformation of anisotropic elasticity. Thus Stroh formalism can be used to characterize the crack tip fields in shells up to the second term and the energy release rate can be expressed in a very compact form in terms of stress intensity factors and Barnett–Lothe tensor L. The first two order terms of the crack-tip stress and displacement fields are derived. Several methods are proposed to determine the stress intensity factors and `T-stresses'. Three numerical examples of two circular cylindrical panels and a circular cylinder under symmetrical loading have demonstrated the validity of the approach.     

Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching

To increase the operational life of defected structures, a repairing method using composite patches has been used to reinforce cracked components. Due to various advantages of composite materials, this method has received much attention from researchers and engineers. Considerable investigations have been performed to highlight the effect of bonded composite patches on the fracture parameters such as stress intensity factors (SIF) and J-integral. However the effect of composite patches on the T-stress, the constant stress term acting parallel to the crack, has not been investigated in the past. In this paper, the finite element method is carried out to analyze the effect of bonded composite patches for repairing cracks in pure mode I and also mixed mode I/II conditions, by computing the stress intensity factors and the T-stress, as functions of the crack length, the crack inclination angle and the type of composite material. In pure mode I condition, the finite element analysis is carried out for three different specimens: centre crack, double edge crack and single edge crack specimens. For mixed mode I/II condition the analysis is conducted on an inclined central crack of various slant angles. For both pure mode I and mixed mode I/II, the numerical results show that composite patching has considerable effect on the T-stress.     

The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading

The purpose of this paper is to revisit the maximum tensile stress (MTS) criterion to predict brittle fracture for mixed mode conditions. Earlier experimental results for brittle fracture of polymethylmethacrylate (PMMA) using angled cracked plates are also re-examined. The role of the T -stress in brittle fracture for linear elastic materials is emphasized. The generalized MTS criterion is described in terms of mode I and II stress intensity factors, K _I and K _II and the T- stress (the stress parallel to the crack), and a fracture process zone, r _c . The generalized MTS criterion is then compared with the earlier experimental results for PMMA subjected to mixed mode conditions. It is shown that brittle fracture can be controlled by a combination of singular stresses (characterized by K ) or non-singular stress ( T -stress). The T -stress is also shown to have an influence on brittle fracture when the singular stress field is a result of mode II loading.     

基于ABAQUS平台考虑T应力的I型裂纹扩展模拟开发

工程结构在制造工艺过程中或使用期间会产生裂纹,对结构断裂路径的预测和研究是防治工程安全问题发生的重要手段。在考虑裂纹尖端应力场常数项T应力的基础上对传统的最大周向应力准则(Maximum tangential stress criterion, MTS)和最小应变能密度因子准则(Minimum strain energy density criterion, SED)进行修正,采用Python语言对ABAQUS的前、后处理和有限元计算模块进行二次开发,通过计算最优解的粒子群算法(Particle swarm optimization, PSO)将修正后的准则编入裂纹自动扩展程序脚本中。利用上述二次开发程序对初始纯I型裂纹的扩展路径进行模拟,结果表明：采用ABAQUS脚本程序模拟结果与相关文献实验结果吻合,表明了程序的有效性,进而实现考虑T应力的多种断裂准则对裂纹扩展路径的预测;当T应力值处于一定范围内时,修正的MTS准则无法预测裂纹发生的偏转现象,扩展路径呈直线,此时可采用修正的SED准则进行预测。     

爆生气体驱动双共线Ⅰ型裂纹的扩展行为

在光面预裂爆破中相邻炮孔间的共线裂纹相向扩展时会先互相排斥后相互吸引,裂纹扩展路径呈“勾连”状现象。为解释这一现象,基于T应力修正后的最大拉应力准则,联合Abaqus软件与Hypermesh软件研究了爆生气体驱动相向扩展的双共线Ⅰ型裂纹的起裂和扩展行为。结果表明：模型尺寸(模型宽度W与高度H的比值)、初始裂纹尺寸(初始裂纹长度a和圆孔半径r的比值)和裂纹面上爆生气体压力对裂纹起裂行为的影响显著,Ⅰ型裂纹扩展中裂纹扩展路径出现“勾连”状现象的先决条件是“Ⅰ型裂纹起裂时或扩展中发生偏转”;裂纹面上的爆生气体减弱了Ⅰ型裂纹扩展中发生偏转的程度,使裂纹扩展路径更难出现呈“勾连”状的现象。通过裂纹增量扩展法模拟了裂纹扩展路径,Ⅰ型裂纹扩展会产生“勾连”状的裂纹扩展路径;裂纹扩展路径呈“勾连”状的现象与裂纹间的相互作用有关,该相互作用可通过裂纹扩展角θ的正负来表示。     

An analysis of fracture under biaxial loading using the non-singular T-stress

Finite element analysis using a two-dimensional modified-boundary-layer approach was used to model the effects of biaxial loading on crack tip stress fields. Loadings were applied corresponding to an elastic KI field, non-singular T-stress and a biaxial stress. For through-thickness cracks the T-stress inherent in the specimen geometry is augmented by the external biaxial stress. For surface-notched specimens the biaxial stress acts out of the crack plane. This effect was modelled with generalized plane strain elements. Results were analysed using the Anderson-Dodds approach for cleavage and the Beremin model in the ductile regime. Biaxial loading is predicted to have a large effect on the toughness of a through-thickness crack but little effect on a surface crack. Experimental results from a previous series of large-scale biaxial fracture tests are generally consistent with these predictions.     

Determination of higher order coefficients and zones of dominance using a singular integral equation approach

A method to determine higher order coefficients from the solution of a singular integral equation is presented. The coefficients are defined by , which gives the radial stress at a distance, r, in front of the crack tip. In this asymptotic series the stress intensity factor, k₀, is the first coefficient, and the T-stress, T₀, is the second coefficient. For the example of an edge crack in a half space, converged values of the first 12 mode I coefficients (k_n and T_n, n = 0, … , 5) have been determined, and for an edge crack in a finite width strip, the first six coefficients are presented. Coefficients for an internal crack in a half space are also presented. Results for an edge crack in a finite width strip are used to quantify the size of the k-dominant zone, the kT-dominant zone and the zones associated with three and four terms, taking into account the entire region around the crack tip.     

Determination of weight functions for elastic T-stress from reference T-stress solutions

ABSTRACT This paper presents the application of the weight function method for the calculation of elastic T -stress. First, the background of the weight function method for the calculation of T -stress is summarized. Then an analysis of known weight functions for T -stress revealed that it is possible to approximate them with one universal mathematical form with three unknown parameters with high accuracy. The existence of this weight function form significantly simplified the determination of weight functions for T -stress. For any particular crack geometry, the unknown parameters can be determined from reference T -stress solutions. The general weight function expression, with suitable reference T -stress solutions, was used to derive the weight functions for single edge cracked plate, double edge cracked plate and center cracked plate specimens. These weight functions were then further used to calculate the T -stress solutions for cracked specimens under several nonlinear stress fields and were compared to available numerical data.