Dynamic stress intensity factors for homogeneous and smoothly heterogeneous materials using the interaction integral method

Dynamic stress intensity factors (DSIFs) are important fracture parameters in understanding and predicting dynamic fracture behavior of a cracked body. To evaluate DSIFs for both homogeneous and non-homogeneous materials, the interaction integral (conservation integral) originally proposed to evaluate SIFs for a static homogeneous medium is extended to incorporate dynamic effects and material non-homogeneity, and is implemented in conjunction with the finite element method (FEM). The technique is implemented and verified using benchmark problems. Then, various homogeneous and non-homogeneous cracked bodies under dynamic loading are employed to investigate dynamic fracture behavior such as the variation of DSIFs for different material property profiles, the relation between initiation time and the domain size (for integral evaluation), and the contribution of each distinct term in the interaction integral.     

Thermal fracture analysis of nonhomogeneous piezoelectric materials using an interaction energy integral method

This paper presents a modified interaction energy integral method to analyze the thermal stress intensity factors (TSIFs) and electric displacement intensity factor (EDIF) in nonhomogeneous piezoelectric materials under thermal loading. This modified method is demonstrated to be domain-independent, even when the nonhomogeneous piezoelectric materials contain interfaces with thermo-electro-mechanical properties. As a result, the method is shown to be convenient for determining the TSIFs and EDIF in nonhomogeneous piezoelectric materials with interfaces. Several examples are shown, and they successfully verify the domain-independence of the present interaction energy integral. The study results also show that the mismatch of material properties can significantly influence the TSIFs and EDIF, particularly when the crack tip is close to the interface. Crack angles and temperature boundary conditions are also shown to significantly influence the TSIFs and EDIF.     

Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements

Three-dimensional enriched finite elements are used to compute mixed-mode stress intensity factors (SIFs) for three-dimensional cracks in elastic functionally graded materials (FGMs) that are subject to general mixed-mode loading and constraint conditions. The method, which advantageously does not require special mesh configuration/modifications and post-processing of finite element results, is an enhancement of previous developments applied so far on isotropic homogeneous and isotropic interface cracks. The spatial variation of FGM material properties is taken into account at the level of element integration points. To validate the developed method, two- and three-dimensional mixed-mode fracture problems are selected from the literature for comparison. Two-dimensional cases include: inclined central crack in a large FGM medium under uniform tensile strain and stress loadings, a slanted crack in a finite-size FGM plate under exponentially varying tensile stress loading and an edge crack in a finite-size plate under shear traction load. The three-dimensional example models a deflected surface crack in a finite-size FGM plate under uniform tensile stress loading. Comparisons between current results and those from analytical and other numerical methods yield good agreement. Thus, it is concluded that the developed three-dimensional enriched finite elements are capable of accurately computing mixed-mode fracture parameters for cracks in FGMs.     

An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading

A plane crack problem of nonhomogeneous materials with interfaces subjected to static thermal loading is investigated. A modified interaction energy integral method (IEIM) is developed to obtain the mixed-mode thermal stress intensity factors (TSIFs). Compared with the previous IEIM, the original point of this paper is: the domain-independence of the modified IEIM still stands in nonhomogeneous materials with interfaces under thermal loading. Therefore, the modified IEIM can still be applied to obtain the TSIFs of nonhomogeneous material even if the integral domain includes interfaces. The modified IEIM is combined with the extended finite element method (XFEM) to solve several thermal fracture problems of nonhomogeneous materials. Good agreement can be obtained compared with the analytic solutions and the domain-independence of the IEIM is verified. Therefore, the present method is effective to study the TSIFs of nonhomogeneous materials even when the materials contain interfaces. The influence of the discontinuity of the material properties (thermal expansion coefficient, thermal conductivity and Young’s modulus) on the TSIFs is investigated. The results show that the discontinuity of both thermal expansion coefficient and Young’s modulus affects the TSIFs greatly, while the discontinuity of thermal conductivity does not arouse obvious change of the TSIFs.     

The interaction integral for fracture of orthotropic functionally graded materials: evaluation of stress intensity factors

The interaction integral is an accurate and robust scheme for evaluating mixed-mode stress intensity factors. This paper extends the concept to orthotropic functionally graded materials and addresses fracture mechanics problems with arbitrarily oriented straight and/or curved cracks. The gradation of orthotropic material properties are smooth functions of spatial coordinates, which are integrated into the element stiffness matrix using the so-called “generalized isoparametric formulation”. The types of orthotropic material gradation considered include exponential, radial, and hyperbolic-tangent functions. Stress intensity factors for mode I and mixed-mode two-dimensional problems are evaluated by means of the interaction integral and the finite element method. Extensive computational experiments have been performed to validate the proposed formulation. The accuracy of numerical results is discussed by comparison with available analytical, semi-analytical, or numerical solutions.     

A contour integral method for dynamic stress intensity factors

The contour integral method previously used to determine static stress intensity factors is applied to dynamic crack problems. The required derivatives of the traction in the reference problem are obtained numerically by the displacement discontinuity method. Stress intensity factors are determined by an integral around a contour which contains a crack tip. If the contour is chosen as the outer boundary of the body, the stress intensity factor is obtained from the boundary values of traction and displacement. The advantage of this path-independent integral is that it yields directly both the opening-mode and sliding-mode stress intensity factors for a straight crack. For dynamic problems, Laplace transforms are used and the dynamic stress intensity factors in the time domain are determined by Durbin's inversion method. An indirect boundary element method, incorporating both displacement discontinuity and fictitious load techniques, is used to determine the boundary or contour values of traction and displacement numerically.     

A domain-independent interaction integral for fracture analysis of nonhomogeneous piezoelectric materials

Piezoelectric materials and structures contain more or less electromechanical interfaces in engineering applications. It is difficult to obtain the fracture parameters efficiently of the piezoelectric materials with complex interfaces. This paper presents a domain-independent interaction integral for material nonhomogeneity and discontinuity which can be used for solving the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) of piezoelectric materials with complex interfaces efficiently. The interaction integral is based on the J-integral by superimposition of two admissible states and the present formulation does not involve any derivatives of mechanical and electric properties. Moreover, it is proved that the interface in the integral domain does not affect the value of the interaction integral and thus, the present method does not require electromechanical parameters of piezoelectric materials to be continuous. The interaction integral method combined with the extended finite element method (XFEM) is used to investigate the influences of material continuity on the SIF and the EDIF and the results show that the material parameters and their first-order derivatives affect both the SIF and the EDIF greatly, while the higher-order derivatives affect both of them slightly.     

An interaction integral method for 3D curved cracks in nonhomogeneous materials with complex interfaces

This work derives an interaction integral for the computation of mixed-mode stress intensity factors (SIFs) in three-dimensional (3D) nonhomogeneous materials with continuous or discontinuous properties. The present method is based on a two-state integral by the superposition of actual and auxiliary fields. In 3D domain formulation of the interaction integral derived here, the integrand does not involve any derivatives of material properties. Furthermore, the formulation can be proved to be still valid even when the integral domain contains material interfaces. Therefore, it is not necessary to limit the material properties to be continuous for the present formulation. On account of these advantages, the application range of the interaction integral can be greatly enlarged. This method in conjunction with the finite element method (FEM) is employed to solve several representative fracture problems. According to the comparison between the results and those from the published lectures, good agreement demonstrates the validation of the interaction integral. The results show that the present interaction integral is domain-independent for nonhomogeneous materials with interfaces.     

求解混合型裂纹应力强度因子的围线积分法

本文用复变函数理论推导出裂纹的辅助场，并用Betti功互等定理给出求解混合型裂纹应力强度因子的远场围绕积分法．此方法与积分路径的选择无关，用有限元法计算出远离裂纹尖端的位移场和应力场，就可通过计算绕裂端的围线积分，精确地给出混合型裂纹的应力强度因子KⅠ和KⅡ的数值解．     

Using moiré interferometry and boundary integral hybrid method to determine mixed-mode stress intensity factor

The paper describes a hybrid experimental and numerical method of Moiré Interferometry and the boundary-integral-element method. The interference patterns used for the evaluation of the displacement vector are obtained by Moiré Interferometry. The boundary displacements obtained experimentally are conveniently used for the calculation of the stress intensity factor in the body by the boundary-integral-method. Some examples bear witness to the effectiveness and accuracy of the hybrid technique. Project is supported by the Science Fundation of the State Education Commission of China.     

Computing bounds to mixed-mode stress intensity factors in elasticity

A bounding procedure combined with an effective error bound method for linear functionals of the displacements and a simple two points displacement extrapolation method is presented to compute the lower and upper bounds to the stress intensity factors in elastic fracture problems. First, the displacements of two nodes (or node pairs) on the crack edges are used to construct the linear extrapolation to obtain the stress intensity factors at the crack tip, so that stress intensity factors are explicitly expressed as linear functionals of the displacements. Then, a posteriori bounding method is utilized to compute the bounds to the stress intensity factors. Finally, the bounding procedure is verified by a mixed-mode homogenous elastic fracture problem and a bimaterial interface crack problem.     

Mixed-mode stress intensity factors for graded materials

In this paper, we present a general method for the calculation of the various stress intensity factors in a material whose constitutive law is elastic, linear and varies continuously in space. The approach used to predict the stress intensity factors is an extension of the interaction integral method. For this type of material, we also develop a systematic method to derive the asymptotic displacement fields and use it to achieve better-quality results. A new analytical asymptotic field is given for two special cases of graded materials. Numerical examples focus on materials with space-dependent Young modulus.     

基于p型有限元法和围线积分法计算复合型应力强度因子

传统有限元法由于采用低阶插值计算应力强度因子时,需要划分的网格数较多,收敛速度较慢,得到的应力强度因子精度不足。p型有限元法在网格确定时通过增加插值多项式的阶数来提高计算精度,具有网格划分少、收敛速度快、精度高、自适应能力强等特点。本文采用基于p型有限元法的有限元计算软件StressCheck计算得到应力场和位移场,并由围线积分法导出混合型应力强度因子(SIFs)。通过几个经典算例,分析了围线的选择对计算精度的影响,计算了不同裂纹长度、不同裂纹角度和裂纹在应力集中区域不同位置时的应力强度因子。并将数值结果、理论解与文献中其他数值计算方法所得的部分结果进行了对比分析,结果表明自由度数不大于7000时,导出的应力强度因子相对误差最大不超过1.2%,数值解表现出较高的精度及数值稳定性。     

Semi-weight function method on computation of stress intensity factors in dissimilar materials

Semi-weight function method is developed to solve the plane problem of two bonded dissimilar materials containing a crack along the bond. From equilibrium equation, stress and strain relationship, conditions of continuity across interface and free crack surface, the stress and displacement fields were obtained. The eigenvalue of these fields is lambda. Semi-weight functions were obtained as virtual displacement and stress fields with eigenvalue-lambda. Integral expression of fracture parameters, KⅠ and KⅡ, were obtained from reciprocal work theorem with semi-weight functions and approximate displacement and stress values on any integral path around crack tip. The calculation results of applications show that the semi-weight function method is a simple, convenient and high precision calculation method.     

Determination of stress intensity factors by the dynamic method of caustics for optically isotropic materials

Summary The method of caustics is an optical method for experimental determination of stress intensity factors at crack tips. The paper generalizes the method of caustics to dynamic situations and the dynamic stress intensity factor at the tip of a running crack in an optically isotropic material is determined. Higher order terms of the Westergaard type stress functions are included and their effect on the shape and extension of the highly constrained zone surrounding a crack tip is discussed. Analytical equations for the caustic are presented. For the singular solution it is found that dynamic K-values associated with larger shadow spots are lower than their static counterparts. Higher order terms induce a generalized evaluation formula for the stress intensity factor where powers of the order n + 5/2 (n = 0, ...) of the caustic diameter are present. The effect of superposition of dynamic and higher order term corrections on the K-value is discussed. The dynamic correction implies that the K(c)-characteristic (c ... crack velocity) is to be modified towards lower values of K. This correction is negligible for small and moderate crack velocities justifying the use of static equations for practical purposes. The K-values for crack branching, however, turn out to be smaller than assumed hitherto, a fact which is of particular interest in connection with SEN-type fracture test specimens.

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Übersicht Die schattenoptische Methode — auch Methode der Kaustik genannt — ist ein wichtiges Verfahren zur experimentellen Bestimmung des Spannungsintensitätsfaktors K an einer Rißspitze. Die vorliegende Arbeit verallgemeinert die schattenoptische Methode auf dynamische Rißausbreitungsvorgänge in optisch isotropen Materialien. Die Bestimmung des dynamischen Spannungsintensitätsfaktors an der Rißspitze schnellaufender Risse aus der Geometrie des Schattenflecks wird aufgezeigt und eine dynamische Korrekturfunktion angegeben.Glieder höherer Ordnung in den Westergaardschen Spannungsfunktionen werden beibehalten und deren Einfluß auf die Gestalt und Ausdehnung der Zone großer Verzerrungen um die Rißspitze sowie die Form und Größe der Kaustik werden untersucht. Bei Berücksichtigung des alleinigen singulären Anteiles des Spannungsfeldes um die Rißspitze zeigen die Rechnungen, daß die mit laufenden Rissen verbundenen K-Werte kleiner als die entsprechenden statischen sind, der Unterschied für praktische Zwecke jedoch erst bei Rißgeschwindigkeiten in der Umgebung der Verzweigungsgeschwindigkeit berücksichtigt werden muß.Die Glieder höherer Ordnung beeinflussen die Struktur der Beziehung zwischen Spannungsintensitätsfaktor und Kaustikdurchmesser. Die Auswerteformel, die im (statischen und dynamischen) singulären Fall eine D ^5/2- Abhängigkeit des Spannungsintensitätsfaktors vom Kaustikdurchmesser zeigt, weist bei Berücksichtigung höherer Terme Abhängigkeiten der höheren Ordnungen n + 5/2(n = 1, ...). auf. Die Korrekturfaktoren für die K-Werte, die im singulären Fall nur von der Rißgeschwindigkeit abhängig sind, werden explizite Funktionen des Kaustikdurchmessers. Das Zusammenwirken von dynamischen Effekten und Einflüssen zufolge des nichtsingulären Spannungsfeldanteiles kann bei bestimmten Arten der Probenbelastung und Probengeometrie zu einer erheblichen Korrektur des K-Wertes führen.Im experimentell gewonnenen Schaubild, das die Rißgeschwindigkeit in Abhängigkeit des Spannungsintensitätsfaktors darstellt, verursacht die dynamische Korrektur eine Verschiebung der Kurve zu etwas niedrigeren K-Werten. Dies bedeutet, daß der K-Wert bei der Rißverzweigung kleiner ist als bisher angenommen; eine Tatsache, die besonders in Verbindung mit der SEN-Bruchprobe von Interesse ist.

     

一种确定结合材料界面端应力强度因子的数值方法

基于弹性力平面问题的基本方程,给出了结合材料界面端的应力奇异性特征方程以及位移场和奇异应力场。提出了一种确定结合材料界面端应力强度因子的数值外插方法。对界面端区域进行了有限元网格单元划分。经过具体实例检验进一步确定了求解应力强度因子的最佳方向,该数值外插法的计算结果精度符合工程应用的要求,为工程材料强度的评价提供了有效的计算途径。     

Stress intensity factors for a moving cylindrical crack in a nonhomogeneous cylindrical layer in composite materials

Stresses are determined for a finite cylindrical crack that is propagating with a constant velocity in a nonhomogeneous cylindrical elastic layer, sandwiched between an infinite elastic medium and a circular elastic cylinder made from another material. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. An internal gas pressure is then applied to the crack surfaces. The solution is derived by dividing the nonhomogeneous interfacial layer into several homogeneous cylindrical layers with different material properties. The boundary conditions are reduced to two pairs of dual integral equations. These equations are solved by expanding the differences in the crack surface displacements into a series of functions that are equal to zero outside the crack. The Schmidt method is then used to solve for the unknown coefficients in the series. Numerical calculations for the stress intensity factors were performed for speeds and composite material combinations.     

The computation of stress intensity factors in dissimilar materials

A reciprocal work contour integral method for calculating stress intensity factors is extended to treat the problem of two bonded dissimilar materials containing a crack along the bond. The method is based on Betti's Reciprocal work theorem from which the singular stress intensities at the crack tip may be evaluated in terms of an integral involving tractions and displacements on a contour remote from the crack tip.     

Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials

The antiplane shear deformation problem of two edge-bonded dissimilar isotropic wedges is considered. In the case when the sum of the two apex angles is equal to 2π, the problem reduces to that of two edge-bonded dissimilar materials with an interfacial crack subjected to concentrated antiplane shear tractions on the crack faces. An explicit expression is extracted for the stress intensity factor at the crack tip. In the special cases of different combinations of the apex angles, the obtained expression for the stress intensity factor may be simplified and relations of a simpler form are given for the stress intensity factor. It is shown that the stress intensity factor is dependent on the material properties as well as the geometry and loading. However, in special cases of equal apex angles as well as the case of similar materials the dependency of the stress intensity factor on the material properties disappears.     

A domain-independent interaction integral for magneto-electro-elastic materials

Magneto-electro-elastic (MEE) materials usually consist of piezoelectric (PE) and piezomagnetic (PM) phases. Between different constituent phases, there exist lots of interfaces with discontinuous MEE properties. Complex interface distribution brings a great difficulty to the fracture analysis of MEE materials since the present fracture mechanics methods can hardly solve the fracture parameters efficiently of a crack surrounded by complex interfaces. This paper develops a new domain formulation of the interaction integral for the computation of the fracture parameters including stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF) for linear MEE materials. The formulation derived here does not involve any derivatives of material properties and moreover, it can be proved that an arbitrary interface in the integral domain does not affect the validity and the value of the interaction integral. Namely, the interaction integral is domain-independent for material interfaces and thus, its application does not require material parameters to be continuous. Due to this advantage, the interaction integral becomes an effective approach for extracting the fracture parameters of MEE materials with complex interfaces. Combined with the extended finite element method (XFEM), the interaction integral is employed to solve several representative problems to verify its accuracy and domain-independence. Good results show the effectiveness of the present method in the fracture analysis of MEE materials with continuous and discontinuous properties. Finally, the particulate MEE composites composed of PE and PM phases are considered and four schemes of different property-homogenization level are proposed for comparing their effectiveness.