New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.     

J-integral estimate for biaxially stressed Mode I cracks based on COD strain

This paper proposes a simple equation for evaluating J-integral values for biaxially stressed Mode I cracks. Finite element analyses were carried out to obtain the relationship between J-integral and crack opening displacement by varying the inelastic constitutive relationship, crack length and stress/strain biaxiality. A quantitative relationship was derived between the J-integral and crack opening displacement for various kinds of stress–strain relationships. A new equation for estimating the J-integral was proposed based on the crack opening displacement. The equation evaluated J-integral values for biaxially stressed Mode I cracks within a 25% error if the yield stress, strain hardening coefficient and exponent were known.     

Micromechanical analysis of fatigue cracks in cord–rubber composites

Analyses of two different types of cracks due to fatigue of cord–rubber composites is carried out using micromechanical two-dimensional (2D) and three-dimensional (3D) finite element analysis. The fracture parameter, tearing energy (TE)/J-integral that characterizes the severity of crack tip stresses in rubber composites, is computed from the finite element results of stresses and strains. The results obtained are validated with existing analytical methods in the literature. Numerical results of J-integral values are presented for two crack types, and crack sizes under transverse strain and shear strain loading conditions. The results presented illustrate that crack type, loading, and crack size have a strong effect on the values of J-integral. The results of the J-integral should help our understanding in estimating the severity of local failures in cord–rubber composites.     

Mechanics of mixed mode small fatigue crack growth

An elastic-plastic analysis of a mixed mode crack was conducted under general yielding conditions using a finite element method. Based on the finite element analysis, a formula of the J-integral was developed, and the behavior of the crack opening or sliding displacement was investigated. Crack growth mechanics was discussed to explain mixed mode growth. A model expressed by the J-integral was proposed assuming that the crack growth is determined by the linear summation of relations of pure Mode I and II crack-tip deformation. The crack growth rate obtained using Inconel 718 thin-wall tubular specimens was correlated with the range of ΔJ evaluated from the proposed formula. The crack growth equations in terms of ΔJ for three different biaxial strain ratios were compared with the relations expected from the proposed crack growth model. The difference between the experimental results and the estimation was discussed from the viewpoint of crack closure and the geometry of the surface crack.     

On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics

In this paper, goal-oriented a posteriori error estimators of the averaging type are presented for the error obtained while approximately evaluating theJ-integral in nonlinear elastic fracture mechanics. Since the value of the J-integral is one component of the material force acting on the crack tip of a pre-cracked elastic body, the appropriate mechanical framework to be chosen is the one named after Eshelby rather than classical Newtonian mechanics. However, in a finite element setting, the discretized Eshelby problem is generally not solved explicitly. Rather, its solution is approximated by the finite element solution of the corresponding discretized dual Newton problem. As a consequence, discrete material forces arise not only at the crack tip but also at other nodes of the current finite element mesh. It is the objective of this paper to establish goal-oriented a posteriori error estimators in both the framework of Eshelbian and Newtonian mechanics and to elaborate their dual relations. This allows to control the error of the J-integral while, at the same time, no further discrete material forces arise during the adaptive mesh refinement process which could lead to misleading mechanical interpretations of the results obtained by the finite element method. The paper is concluded by numerical examples that illustrate our theoretical results. Dedicated to the memory of the esteemed colleague Professor Karl Popp, University of Hannover, who unexpectedly passed away on April 24, 2005.     

大坝有限元分析应力取值的研究

提出了基于误差控制下的自适应网格的有限元应力取值标准:即给定一全局误差限作为自适应有限元网格剖分的准则,以此网格计算所得应力即为有限元应力取值。应用适用于工程计算的Z2后验误差估计方法以及h-型自适应策略,对一个典型的重力坝剖面进行了线弹性自适应有限元计算。计算结果表明:给定一个全局误差限,网格剖分调整若干次后即可满足误差要求,不会出现因角缘应力集中出现剖分不收敛的情况;存在一个全局误差限,使得当继续降低误差限时,坝踵和坝趾的角缘应力趋于稳定值。     

A new method for evaluation of stress intensities for interface cracks

A new method is presented for calculating the values of K_I and K_II in the elasticity solution at the tip of an interface crack. The method is based on an evaluation of the J-integral by the virtual crack extension method. Expressions for calculating K_I and K_II by using the displacements and the stiffness derivative of the finite element solution and asymptotic crack tip displacements are derived. The method is shown to produce very accurate solutions even with coarse element mesh.     

基于应力超收敛恢复技术的广义特征值问题后验误差估计

根据有限元解的超收敛特性提出了一种基于应力超收敛恢复技术的广义特征值问题后验误差估计。通过对单元内的应力超收敛点以及相邻单元的应力超收敛点进行插值或外推处理,得到单元内其它点处更高精度的应力解。通过高精度的应力值可以得到结构处理后改进的势能。将改进的势能代入瑞利商,最终得到比原始有限元解更高精度的特征解。将后处理特征解作为“准精确解”代替误差估计因子中未知的精确解,实现后验误差估计过程。数值计算结果表明,所提出的后验误差估计是渐进精确的,因此可作为结构广义特征值问题自适应有限元方法的误差估计因子。     

Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh. The Zienkiewicz-Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors. The optimization of crack-tip singular finite element size is achieved through the adaptive mesh strategy. Finally, several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of computational algorithm in calculation of fracture parameters and prediction of crack path pattern.     

A posteriori error estimates in finite element solution of structure vibration problems with applications to acoustical fluid-structure analysis

We present an a posteriori error estimator for piecewise linear finite element approximations of structure vibration problems. We prove that this estimator is equivalent to the energy norm of the error. Also, we introduce an adaptive mesh-refinement procedure based on the proposed estimator to analize fluid-structure interaction problems. Finally, numerical results for some test examples are presented which show the efficiency of the error estimator and the mesh-refinement techniques.     

Stress intensity factors for corner cracks emanating from fastener holes under tension

In this paper, previous work associated with the stress intensity factor for corner cracks at fastener holes in finite thickness plates is briefly reviewed. The stress intensity factors for two symmetric quarter-elliptical corner cracks subjected to remote tension are evaluated by using both the quarter-point displacement and J-integral methods based on three-dimensional finite element analyses. The geometry ratios analyzed cover a wide range, i.e. depth ratio a/t: 0.2–0.95, aspect ratio a/c: 0.2–5, and hole radius ratio r/t: 0.5–3. Analysis of the J-integral path independence and mutual comparison of the stress intensity factor results between the two methods demonstrate that the present results are of good numerical accuracy. Deviation of the present results from some other solutions found in the literature is also revealed, particularly from Newman and Raju's equations. It is shown that the difference among these results obtained by the different methods is generally within a reasonable bound of error, but Newman and Raju's equations systematically underestimate (up to 15%) the stress intensity factor for cracks of depth ratio larger than 0.8.     

Cyclic AE count rate and crack growth rate under low cycle fatigue fracture loading

In the low cycle fatigue fracture testing with KS (or JIS) SS41 steel, crack growth rate, AE count rate and J-integral range are measured to get empirical relationships between crack growth rate and J-integral range, AE count rate and J-integral range as well as AE count rate and crack growth rate. All the relationships are shown to be linear on the log—log graphs. It is also shown that the linear relationships can be formulated by using Dunegan's assumption and elastic-plastic fracture mechanics along with the well-known relationships of crack growth rate and J-integral range. It is concluded that the differences between experimental and theoretical values are due to the problem of Dunegan's assumption.     

The nonlinear and biaxial effects on energy release rate, J-integral and stress intensity factor

A nonlinear finite element analysis is performed for a finite center-cracked specimen subjected to biaxial loading. A Ramberg-Osgood type stress-strain relation is used to characterize the material property. It is found that the energy release rate, J-integral, stress intensity factor, strain intensity factor depend not only on applied stress perpendicular to the crack but also on applied stress parallel to the crack. Biaxial effects on fracture toughness parameters increase as applied stress increases. The coupling between biaxial effects and material nonlinearity has been indicated.     

New ‘ηpl'' and ‘γ'' functions to evaluate J–R curve from cracked pipes and elbows. Part II: experimental and numerical validation

Experimental evaluation of J–R curve in a crack growth situation requires ‘η_pl' and ‘γ' functions that are specific to a cracked geometry and loading condition. In Part I [Engng. Fract. Mech., in press] of this paper, new η_pl and γ functions, which are not available in the literature, for pipe and elbow geometry with various crack configurations under different loading conditions have been derived. In this paper, some of these newly proposed η_pl and γ functions have been validated experimentally through comparison of crack initiation load and J–R curve. In few cases, numerical validation has also been provided by comparing the J-integral values calculated through η factor approach and finite element method.     

The values of J-integral within the plastic zone

A study has been conducted to characterize the J-integral within the plastic zone for different strain hardening materials. The relation between the calculated J-integral and plastic energy enclosed by the selected integration contour is explored for different strain hardening materials. The equations between the J-integral and plastic energy around the crack tip for 7075-T651 aluminum alloy and HY-130 steel have been derived. It is shown that the J-integral is path dependent if the selected integration contour goes across the plastic zone. Results also indicated the plastic energy enclosed by the selected integration contour provides the dominant contribution to the J-integral.     

The energy release rate and the J-integral in nonlocal micropolar field theory

The recent theory of nonlocal micropolar continuum is used to derive explicit expressions for both the energy release rate and J-integral. It was shown that this J-integral has a physical meaning of the energy release rate and, for homogeneous body with the straight crack, is path independent. It was also shown that the influence of nonlocality is contained in J-integral which allows one to discuss several special cases.     

Some problems in the J-integral

For a blunt crack the j-integral is path dependent on contours which are very close to the crack tip even for elastic material. Using the incremental J-integral theory we introduce a new parameter J_t, characterizing the behavior of a crack tip and prove that the J-integral is almost path independent on contours whose radii are greater than several COD if σ_ij,1Δε_ij — ε_ijΔσ_ij = 0 in plastic regions for elasto-plastic material.     

The J-integral applied to cyclic loading

The influence of notch geometry on the life to fatigue crack initiation is determined using a modified two-dimensional J-integral analysis. A notched plate that is subjected to fully reversed cyclic nominal strain control is analyzed in order to determine the strain history at the notch root. The expression for strain concentration has the same form as that obtained from the Neuber approach. After the local stress-strain history has been determined, a life prediction for fatigue crack initiation could be made. The modified J-integral and Neuber approaches are compared. Justification for the use of the modified J-integral may be found in the appendices.     

Fatigue of sandwich structures with inserts

The main purpose of the study was to evaluate fatigue in sandwich structures with inserts. Cross-linked PVC foam with closed cells as the core material was used in this investigation because this foam type has gained widespread use for maritime applications. Fatigue tests for a four-point bending test on a sandwich-beam containing an insert were investigated using numerical fatigue calculations in addition to the finite element method. A small crack initiated in the numerical model was propagated by reducing the stiffness of the finite elements at the crack tip. The J-integral was used when estimating the energy release rate. The stiffness of the insert material was used as a design parameter. It was seen from the investigation that the stiffness of the insert material had almost no influence on the fatigue crack propagation.     

Goal-oriented explicit residual-type error estimates in XFEM

A goal-oriented a posteriori error estimator is derived to control the error obtained while approximately evaluating a quantity of engineering interest, represented in terms of a given linear or nonlinear functional, using extended finite elements of $Q1$ type. The same approximation method is used to solve the dual problem as required for the a posteriori error analysis. It is shown that for both problems to be solved numerically the same singular enrichment functions can be used. The goal-oriented error estimator presented can be classified as explicit residual type, i.e. the residuals of the approximations are used directly to compute upper bounds on the error of the quantity of interest. This approach therefore extends the explicit residual-type error estimator for classical energy norm error control as recently presented in Gerasimov et al. (Int J Numer Meth Eng 90:1118–1155, 2012a). Without loss of generality, the a posteriori error estimator is applied to the model problem of linear elastic fracture mechanics. Thus, emphasis is placed on the fracture criterion, here the $J$ -integral, as the chosen quantity of interest. Finally, various illustrative numerical examples are presented where, on the one hand, the error estimator is compared to its finite element counterpart and, on the other hand, improved enrichment functions, as introduced in Gerasimov et al. (2012b), are discussed.