Numerical simulations and experimental measurements of the stress intensity factor in perforated plates

A numerical procedure, which combines two hybrid finite element formulations, was developed to analyse the stress intensity factors in cracked perforated plates with a periodic distribution of holes and square representative volume elements. The accuracy of the method in predicting the stress intensity factor was verified by a comparison with experimental measurements, carried out by a photoelasticity method, and by commercial finite element software. Several simulations were executed by varying both the crack length and the hole diameters, and the effects of the holes on the stress intensity factor are illustrated. The method shows high accuracy and efficiency, as small differences were observed when compared with the traditional finite element method, notwithstanding a strong reduction in degrees of freedom and mesh complexity.     

Stress intensity factor calculation based on the work of external forces

The external forces method is a numerical method for K calculation based on the finite element method. It uses the work of the external forces W for the calculation of the energy release rate and is particularly advantageous when that forces are applied far from the crack front. The method was applied to a corner crack geometry with the objective of studying its accuracy. Good results were obtained for a wide range of virtual crack displacements (0.03% < Δa/a < 6%) considering 4 values of W along with a polynomial regression of order 3. For that choice of parameters the inaccuracy of K is mainly due to FEM errors. A great sensitivity of K to FEM errors was observed, however accurate values of K were obtained, with errors lower than 2 percent. So, the use of the external forces method for the calculation of K is recommended, considering its simplicity and accuracy. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical solution of cracked thin plates subjected to bending, twisting and shear loads

A semi-analytical method namely fractal finite element method is presented for the determination of mode I and mode II moment intensity factors for thin plate with crack using Kirchhoff's theory. Using the concept of fractal geometry, infinite many of finite elements is generated virtually around the crack border. Based on the analytical global displacement function, numerous degrees of freedom (DOF) are transformed to a small set of generalised coordinates in an expeditious way. The stress intensity factors can be obtained directly from the generalized coordinates. No post-processing and special finite elements are required to develop for extracting the stress intensity factors. Examples of cracked plate subjected to bending, twisting and shear loads are given to illustrate the accuracy and efficiency of the present method. The influence of finite boundaries on the calculation of the moment intensity factors is studied in details. Very accuracy results when compare with the theoretical and numerical counterparts are found.     

Fatigue fracture in plates in tension and out-of-plane shear

Straight cracks near a stiffening element, or curved cracks, in a pressurized shell can be subjected to out-of-plane tearing stresses in addition to normal tensile stresses due to the membrane stresses in the shell. To predict the rate of fatigue crack growth in such situations a theory and a crack growth rate correlation are needed. Such loadings are modelled as a superposition of plane stress tensile fracture (mode I) and Kirchhoff plate theory shearing fracture (mode 2). Finite element analyses using shell elements are used to compute the energy release rate and stress intensity factors associated with the loading. Three fatigue crack growth rate experiments were carried out on sheets of 2024-T3 aluminium alloy loaded in tension and torsion. The first set of experiments is constant amplitude fatigue crack growth tests. The second consists of experiments where crack closure is artificially eliminated to determine the rate of crack growth in the absence of crack face contact. The third is a set of constant stress intensity factor amplitude tests. The results all show that as the crack grows extensive crack face contact occurs, retarding crack growth. In the absence of crack face contact, however, the addition of out-of-plane shear loading increases the crack growth rate substantially.     

Computations of the stress intensity factors of double-edge and centre V-notched plates under tension and anti-plane shear by the fractal-like finite element method

The fractal-like finite element method (FFEM) is extended to compute the stress intensity factors (SIFs) of double-edge-/centre-notched plates subject to out-of-plane shear or tension loading conditions. In the FFEM, the use of global interpolation functions reduces the large number of unknowns in a singular region to a small set of generalised co-ordinates. Therefore, the computational cost is reduced significantly. Also, neither post-processing techniques to extract the SIFs nor special singular elements are needed. Many numerical examples of double-edge-/centre-notched plates are presented, and results are validated via existing published data. New results of notched plate problems are also introduced.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Non‐linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non‐linear finite element modal reduction method. The problem is first reduced to a set of Duffing‐type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non‐linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied. Copyright © 2005 John Wiley & Sons, Ltd.     

识别薄板中兰姆波和体波的阶梯板方法

斜探头在某些频率下激励出的兰姆波,其群速度与体波的传播速度相近,所以通过判断传播速度不易区分出兰姆波和体波。通过数值模拟和实验,分别研究了激励频率为2 MHz的纵波和S0模态兰姆波在阶梯板上的反射特性,发现:在阶梯板上入射S0模态兰姆波时,有反射回波;而入射纵波时,无反射回波。基于这种反射特性的差别,提出了一种利用阶梯板区别薄板中兰姆波和体波的方法,该方法可用于确认探头的激励特性。     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

Analysis of fracture behaviour of the cement mantle of reconstructed acetabulum

In this study, the finite element method was used to analyse the crack behaviour in the cement of a reconstructed acetabulum by computing the stress intensity factors at the crack tip. Three loading cases were examined (Fig. 6). These cases present the different human body postures. Both positions and orientations of crack effect on the SIF variation were analysed. When valuating the crack position effect, one notices no risk of crack propagation under the load type 1; however, under the load type 2 and the load type 3 this risk is more important. Load type 3 is the most dangerous loading condition. When computing crack orientation, one noted that the risk of crack propagation is higher when the crack inclination is 20° and 100°.     

Curved crack propagation in homogeneous and graded materials

Deflection and deviation of cracks commonly occurs because of asymmetry in crack‐tip stresses in both homogeneous materials and functionally graded materials (FGMs); yet the analysis of curved cracks has been limited to simple crack shapes, otherwise the analysis would involve extensive levels of computation. The present study investigates the approximation of curved cracks with simplified shapes. A simple analytical model justifying the use of crack‐shape approximations, developed in an earlier study on stationary curved cracks in homogeneous materials, is outlined. Then, the approach is applied to propagating cracks in both homogeneous and graded material structures. Results are presented from finite element (FE) simulations of crack propagation using exact and simplified crack shapes. The use of an approximated crack shape can provide basic estimates for crack propagation path and critical load. However, systematic divergence can occur between predictions for exact and approximated crack shapes, particularly in inhomogeneous material configurations, and so the development of solutions for non‐straight cracks in FGMs would be expedient.     

Characterization of a fracture specimen for crack growth in epoxy due to thermal fatigue

Finite element simulations are carried out to characterize a new fracture specimen, consisting of an outer circular epoxy ring bonded to an inner circular invar plate for accelerated thermal fatigue testing. Radial cracks are introduced in the epoxy ring. The growth of these radial cracks is correlated to the applied energy release rate G. We studied the dependence of G on the crack length, the specimen geometry and the elastic modulus. For short cracks, G is obtained in closed form. Analysis is carried out to determine the critical thermal buckling load the specimen can withstand. Experimental results show that the fatigue crack growth rate per thermal cycle da/dN is given by da/dN = 0.51(ΔG)^0.38 for cycling between 4 and 100 °C but by da/dN = 0.25(ΔG)^0.24 for cycling between 20 and 85 °C, where ΔG is the difference of the energy release rate between the highest and lowest temperatures during a thermal cycle. More severe thermal cycles produce considerably larger fatigue crack growth rates than less severe ones at the same ΔG. This result also implies that isothermal fatigue tests will probably be inadequate to predict thermal fatigue crack growth in epoxies.     

Application of the substructured finite element/extended finite element method (S-FE/XFE) to the analysis of cracks in aircraft thin walled structures

The substructured finite element/extended finite element (S-FE/XFE) approach is used to compute stress intensity factors in large aircraft thin walled structures containing cracks. The structure is decomposed into a ‘safe’ domain modeled with classical shell elements and a ‘cracked’ domain modeled using three-dimensional extended finite elements. Two applications are presented and discussed, supported by validation test cases. First a section of stiffened panel containing a through-thickness crack is investigated. Second, small surface cracks are simulated in the case of a generic ‘pressure membrane’ with realistic crack configurations. These two semi-industrial benchmarks demonstrate the accuracy, robustness and computational efficiency of the substructured finite element/extended finite element approach to address complex three-dimensional crack problems within thin walled structures.     

The least‐squares finite element method in elasticity. Part II: Bending of thin plates

A least‐squares finite element method (LSFEM) for bending problems of thin plates is developed. This LSFEM is based on the first‐order deflection‐slope‐moment‐shear force formulation. Four compatibility conditions are added into the first‐order system; thus, the method can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples show that the method achieves an optimal rate of convergence for all eight variables. Copyright © 2002 John Wiley & Sons, Ltd.     

Probabilistic fracture mechanics by using Monte Carlo simulation and the scaled boundary finite element method

A numerical technique to model the effect of uncertainties in the crack geometry on the reliability of cracked structures is presented. The shape sensitivity analysis of stress intensity factors to the crack size and orientation is performed by using the scaled boundary finite element method (SBFEM). Only a single boundary mesh is required. The varying crack size and orientation are represented by simply moving the scaling center and without the need for remeshing. The reliability assessment is performed by Monte Carlo simulations. Numerical examples are analyzed to verify the accuracy and demonstrate the efficiency and simplicity of the proposed technique.     

Stress intensity factors of a central interface crack in a bonded finite plate and periodic interface cracks under arbitrary material combinations

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary material combinations. This paper deals with a central interface crack in a bonded finite plate and periodic interface cracks. Then, the effects of material combination and relative crack length on the stress intensity factors are discussed. A useful method to calculate the stress intensity factor of interface crack is presented with focusing on the stress at the crack tip calculated by the finite element method.     

Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique

This paper describes an interface element to calculate the strain energy release rates based on the virtual crack closure technique (VCCT) in conjunction with finite element analysis (FEA). A very stiff spring is placed between the node pair at the crack tip to calculate the nodal forces. Dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. This interface element leads to a direct calculation of the strain energy release rate (both components G_I and G_II) within a finite element analysis without extra post-processing. Several examples of stationary cracks under impact loading were examined. Dynamic stress intensity factors were converted from the calculated transient strain energy release rate for comparison with the available solutions by the others from numerical and experimental methods. The accuracy of the element is validated by the excellent agreement with these solutions. No convergence difficulty has been encountered for all the cases studied. Neither special singular elements nor the collapsed element technique is used at the crack tip. Therefore, the fracture interface element for VCCT is shown to be simple, efficient and robust in analyzing crack response to the dynamic loading. This element has been implemented into commercial FEA software ABAQUS^® with the user defined element (UEL) and should be very useful in performing fracture analysis at a structural level by engineers using ABAQUS^®.     

Three dimensional virtual crack closure-integral method (VCCM) with skewed and non-symmetric mesh arrangement at the crack front

In this paper, an extension of virtual crack closure-integral method (VCCM) for three dimensional linear fracture mechanics analysis using hexahedron finite elements is presented. In conventional three dimensional VCCM, there are some inherent requirements on the finite element model. They are (i) the faces of finite elements across the crack front have the same areas and (ii) they must be arranged symmetrically across the crack front. In present study, we developed a three dimensional VCCM without such requirements by considering work required to open one element face area whose shape is arbitrary. Though we assume the use of an ordinary 20 node serendipity element, present approach can be applied to other types of hexahedron elements.     

SiC纤维增强复合材料基体裂纹偏移机理的有限元分析

运用基于能量的裂纹偏移准则, 分别建立了两相和三相复合材料基体裂纹偏移/ 穿透的轴对称有限元模型, 考察了纤维体积分数、描述材料特性弹性失配的Dundurs 参数α和相对裂纹扩展长度a_d / a_p 对相对能量释放率Gd / Gp 的影响。将两相复合材料的有限元结果与He 等人的结果进行对比, 进一步考察了三相复合材料界面层厚度和Dundurs 参数α₁ 和α₂ 对G_d / G_p 的影响。分别将碳涂层SiC 纤维增强复合材料SiC/ C/ Ti-6A1-4V 和碳涂层陶瓷基增强复合材料SiC/ C/ SiC 运用于有限元分析中, 结果表明, 所建立的模型能够准确地预测和比较基体裂纹偏移的机理。      

Regularized BE formulations for the analysis of fracture in thin plates

A boundary integral formulation for the analysis of cracks in thin Kirchhoff plates is presented. The numerical solution of the relevant equations is addressed following three different approaches: two single integration methodologies initially introduced for 2D elastic solids are here reformulated, compared with a third (Galerkin) double integration approach and extended to the analysis of cracks in thin plates. exploiting an analogy with 2D elastic fracture mechanics. Comparative numerical testing, in terms of stress intensity factors, is performed with reference to straight and curved cracks in unbounded domains. This revised version was published online in August 2006 with corrections to the Cover Date.