Fracture investigation of U‐notch made of tungsten–copper functionally graded materials by means of strain energy density

The averaged strain energy density over a well‐defined control volume was employed to assess the fracture of U‐notched specimens made of tungsten–copper functionally graded materials under prevalent mode II loading. The boundary of control volume was evaluated by using a numerical method. Power law function was employed to describe the mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness and ultimate tensile stress) through the specimen width. The effect of notch tip radius and notch depth on notch stress intensity factors and mode mixity parameter χ were assessed. In addition, a comparison based on fracture load between functionally graded and homogeneous W–Cu was made. Furthermore, in this research, it was shown that the mean value of the strain energy density over the control volume can be accurately determined using coarse meshes for functionally graded materials.     

The effect of notch depth on J-integral and critical fracture load in plates made of functionally graded aluminum–silicone carbide composite with U-notches under bending

The present paper deals with the effect of notch depth on J-integral and critical fracture load in a plate made of functionally graded aluminum–silicone carbide composite (Al–SiC) with U-notch under bending. The weight fraction of SiC particles varies from 0% to 20% through the specimen width. Using three criteria namely mean stress (MS), point stress (PS), and averaged strain-energy density (ASED), the critical fracture load has been predicted and its variation with respect to the notch depth has been studied. A comparison of the J-integral between functionally graded and homogeneous Al–SiC composite were made, where the notch tip in the functionally graded material is situated in a layer with same mechanical properties as the homogeneous composite. The results indicated that in the case where the notch scene is toward brittleness increment the critical J-integral in functionally graded material (FGM) is larger than that of in homogeneous material with the same mechanical properties at the notch tip. Therefore, FGM is more convenient than homogeneous material against fracture.     

K‐dominance of static crack tip in functionally gradient materials

K‐dominance of static crack tip in functionally gradient materials (FGMs) with a crack oriented along the direction of the elastic gradient is studied through coherent gradient sensing (CGS), digital speckle correlation method (DSCM) and finite element method (FEM). In the direction of crack propagation, the shear modulus has a linear variation with constant mass density and Poisson's ratio. First, the CGS and DSCM governing equations related to the measurements and the elastic solutions at mode I crack in FGMs are obtained in terms of the stress intensity factor, material constants and graded index. Secondly, two kinds of FGMs specimens and one homogenous specimen are prepared to observe the influences of the property variation on the K‐dominance. Then, CGS and DSCM experiments using three‐point‐bending of FGMs and homogenous beams are performed. Thirdly, based on the results of the experiments, the stress intensity factors of three kinds of specimens are calculated by CGS and DSCM. Meanwhile, the stress intensity factors are obtained by FEM. Finally, comparing the results from CGS, DSCM and FEM, the K‐dominance of mode‐I static crack tip in FGMs is discussed in detail.     

Transient thermo-mechanical analysis of dynamic curving cracks in functionally graded materials

Mixed-mode dynamic crack growth behavior along an arbitrarily smoothly varying path in functionally graded materials (FGMs) under transient thermo-mechanical loading is studied. An asymptotic analysis in conjunction with displacement potentials is used to develop transient thermo-mechanical stress fields around the propagating crack-tip. Asymptotic temperature field equations are derived for exponentially varying thermal properties, and later, these equations are used to derive transient thermo-mechanical stress fields for a curving crack in FGMs. The effect of the transient parameters (loading rate, crack-tip acceleration, and temperature change) and temperature gradient on the maximum principal stress and circumferential stress associated with the propagating crack-tip is discussed. Finally, using the minimum strain energy density criterion, the effect of temperature gradient, crack-tip speeds, and T-stress on crack growth directions is determined and discussed.     

Multiaxial fatigue of V‐notched steel specimens: a non‐conventional application of the local energy method

The paper deals with the multi‐axial fatigue strength of notched specimens made of 39NiCrMo3 hardened and tempered steel. Circumferentially V‐notched specimens were subjected to combined tension and torsion loading, both in‐phase and out‐of‐phase, under two nominal load ratios, R=?1 and R= 0, also taking into account the influence of the biaxiality ratio, λ=τ_a/σ_a. The notch geometry of all axi‐symmetric specimens was a notch tip radius of 0.1 mm, a notch depth of 4 mm, an included V‐notch angle of 90° and a net section diameter of 12 mm. The results from multi‐axial tests are discussed together with those obtained under pure tension and pure torsion loading on plain and notched specimens. Furthermore the fracture surfaces are examined and the size of non‐propagating cracks measured from some run‐out specimens at 5 million cycles. Finally, all results are presented in terms of the local strain energy density averaged in a given control volume close to the V‐notch tip. The control volume is found to be dependent on the loading mode.     

Investigation of fracture characterizations of functionally graded materials

In this paper, the mode I crack problem of functionally gradient materials (FGMs) with the gradient direction parallel to the crack is discussed, and the differences of stress distribution between the gradient materials and the homogeneous materials are analyzed. It is shown that a mode I crack problem of FGMs with the gradient direction parallel to the crack direction can become a mixed‐mode crack problem. In FGMs, the crack initiation angles are determined by the fracture toughness gradient, elastic modulus and crack mode. If the gradient coefficients are small, the crack initiation angles in FGMs are the same as those in homogeneous materials. If the elastic modulus gradient is large, the principal stress terms without the gradient coefficients can be ignored in obtaining the crack initiation angle. In this study, all the above results are generalized to the mixed‐mode crack problems with arbitrary angle between the gradient direction and the crack direction.     

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.     

Numerical and experimental investigation of mixed‐mode fracture parameters on silicon nitride using the Brazilian disc test

Engineering applications of ceramics can often involve mixed‐mode conditions involving both tensile and shear loading. Mixed‐mode fracture toughness parameters are evaluated for applicability to ceramics using the Brazilian disc test on silicon nitride. Semi‐elliptical centrally located surface flaws are induced on the disc specimens using Vickers indentation and compression loaded to fracture with varying levels of mode mixity. The disc specimens are modelled via 3D finite element analysis and all three modes of stress intensity factors computed along the crack front, at failure load. We present a numerical and experimental investigation of four widely used mixed‐mode fracture criteria and conclude that the critical strain energy release rate criterion is simple to implement and effective for silicon nitride under mixed‐mode conditions.     

聚合物梯度材料黏弹性断裂的双控制参数

针对组分材料体积分数任意分布的聚合物功能梯度材料,研究其在蠕变加载条件下Ⅰ型裂纹应力强度因子（SIFs）和应变能释放率的时间相依特征。由Mori-Tanaka方法预测等效松弛模量,在Laplace变换域中采用梯度有限元法和虚拟裂纹闭合方法计算断裂参数,由数值逆变换得到物理空间的对应量。分析边裂纹平行于梯度方向的聚合物功能梯度板条,分别考虑均匀拉伸和三点弯曲蠕变加载。结果表明,聚合物梯度材料应变能释放率随时间增加,其增大的程度与黏弹性组分材料体积分数正相关;材料的非均匀黏弹性性质产生应力重新分布,导致裂纹尖端应力场强度随时间变化,当裂纹位于黏弹性材料含量较低的一边时,应力强度因子随时间增加,反之,随时间减小。而且,材料的应力强度因子与时间相依的变化范围和体积分数分布以及加载方式有关,当体积分数接近线性分布时,变化最明显,三点弯曲比均匀拉伸的变化大。SIFs随时间的延长增加或减小、加剧或减轻裂纹尖端部位的“衰坏”,表明黏弹性功能梯度裂纹体的延迟失稳需要联合采用应力强度因子与应变能释放率作为双控制参数。     

Acoustic emission imaging of shear failure in large reinforced concrete structures

The averaged value of the strain energy density over a well-defined volume is used to predict the static strength of U-notched specimens under mixed-mode conditions due to combined bending and shear loads. The volume is centered in relation to the maximum principal stress present on the notch edge, by rigidly rotating the crescent-shaped volume already used in the literature to analyse U- and V-shaped notches subject to mode I loading. The volume size depends on the ultimate tensile strength σ _u and the fracture toughness K _IC of the material. In parallel, an experimental programme was performed. All specimens are made of polymethyl-metacrylate (PMMA), a material which exhibits quasi-brittle behaviour at -60°C. Good agreement is found between experimental data for the critical loads to failure and theoretical predictions based on the constancy of the mean strain energy density over the control volume.     

Mixed-mode dynamic crack propagation in graded materials under thermo-mechanical loading

Mixed-mode dynamic crack growth behavior in functionally graded materials (FGMs) under thermo-mechanical loading is studied. Asymptotic analysis in conjunction with displacement potentials has been used to develop thermo-mechanical stress fields for a mixed mode propagating crack in a FGM. The shear modulus, mass density, thermal conductivity and coefficient of thermal expansion of the FGM are assumed to vary exponentially along the gradation direction. First, asymptotic temperature fields are derived for an exponential variation of thermal conductivity and later these temperature fields are used in deriving stress fields. Using asymptotic thermo-mechanical stress fields the variation of maximum shear stress, circumferential stress and strain-energy density as a function of temperature around the crack tip are generated. Finally, utilizing the minimum strain-energy density criterion and the maximum circumferential stress criterion, the crack growth direction for various crack-tip speeds, non-homogeneity coefficients and temperature fields are determined.     

Fracture assessment of U-notches under mixed mode loading: two procedures based on the ‘equivalent local mode I’ concept

Two fracture criteria are proposed and applied to blunt-notched components made of brittle materials loaded under mixed mode; the former is based on the averaged strain energy density over a given control volume, the latter on the cohesive crack zone model. In both instances use of the equivalent local mode I hypothesis is made. Only two material properties are needed: the ultimate tensile strength and the fracture toughness. Numerical predictions of rupture loads from the two criteria are compared with experimental measurements from more than 160 static tests with notched beams. The samples are made of PMMA and tested at − 60°C to assure a bulk behaviour almost linear elastic up to rupture. Notch root radii range from 0.2 to 4.0 mm and load mixicity varies from pure mode I to a prevailing mode II. The good agreement between theory and experimental results adds further confidence to the proposed fracture criteria.     

Calculation of stress intensity factors for functionally graded materials by using the weight functions derived by the virtual crack extension technique

The weight function method provides a powerful approach for calculating the stress intensity factors for a homogeneous cracked body subjected to mechanical loadings. In this paper, the basic equations of weight function method for mode I and mixed mode crack problems in a two-dimensional functionally graded crack system are derived based on the Betti’s reciprocal theorem. The weight functions derived by the virtual crack extension technique are further used to calculate the stress intensity factors of functionally graded materials (FGMs). The practicability and accuracy of this proposed method has been confirmed by the comparison with theoretical or numerical solutions available in the literatures. On account that the repeated extractions of the distributions of normal stress and shear stress in the uncracked component along the prospective crack line under different loadings can be avoided using the method presented in this paper, this method can be potentially used for optimal design for FGMs under multiple-load cases.     

Fracture Mechanics Analysis Model for Functionally Graded Materials with Arbitrarily Distributed Properties

Functionally Graded Materials (FGMs) have been developed as super-resistant materials for propulsion systems and airframe of space-planes in order to decrease thermal stresses and to increase the effect of protection from heat. It has been experimentally observed that surface cracking in FGMs is the most common failure mode of a metal-ceramic FGM when it is subjected to a thermal shock. Therefore, it is very important to consider the thermally induced fracture behaviors of FGMs. In this paper, a functionally graded material strip containing an embedded or a surface crack perpendicular to its boundaries is considered. The graded region is treated as a large number of single layers, with each layer being homogeneous material. The problem is reduced to an integral equation and is solved numerically. Unlike most of the existing researches, which considered only certain assumed material distributions, the method developed in this paper can be used to investigate functionally graded materials with arbitrarily varied material properties.     

倾斜应力作用下垂直于功能梯度材料夹层的币型裂纹扩展问题

分析了功能梯度材料中币型裂纹扩展问题。该裂纹体受有与裂纹面成任意角度的张应力或压应力,裂纹垂直于无限域中功能梯度材料夹层。假定非均匀介质的功能梯度材料夹层与两个半无限域完全结合,其弹性模量沿厚度方向变化。利用已发表的裂纹应力强度因子数据和线弹性断裂力学的叠加原理,将应力强度因子耦合于最小应变能密度因子断裂判据,讨论了裂纹扩展的临界荷载;并讨论了荷载方向和材料性质对临界荷载的影响。     

A numerical method for evaluation of J‐integral in plates made of functionally graded materials with sharp and blunt V‐notches

The main purpose of the paper is to propose a numerical method for evaluation of J‐integral in plates made of functionally graded materials (FGM) with sharp and blunt V‐notches under Mode I loading. The material properties have been assumed to be varied exponentially along the specimen width (notch direction). Using the proposed method, the effect of material gradient on the J‐integral for two cases of sharp and blunt V‐notches has been studied. The results have shown that in FGMs with sharp V‐notches, the J‐integral is not proportional to . So, the parameter J_L is path dependent. It has been observed that the material gradient has larger effect on the J‐integral in sharp V‐notch compared with that in blunt V‐notch.     

Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy

The main purpose of this research is to re-analyse experimental results of fracture loads from blunt V-notched samples under mixed mode (I + II) loading considering different combinations of mode mixity ranging from pure modes I to II. The specimens are made of polymethyl-metacrylate (PMMA) and tested at room temperature. The suitability of fracture criterion based on the strain energy density (SED) when applied to these data is checked in the paper. Dealing with notched samples, characterized by different notch angles and notch root radii, the SED criterion used in combination with the concept of local mode I, valid in the proximity of the zone of crack nucleation, permits to provide a simple approximate but accurate equation for the SED in the control volume. This proposal unifies predictions for the experimental results obtained under modes I, II and mixed mode loading.     

Fracture assessment of U-notches under three point bending by means of local energy density

The main purpose of the paper is twofold. First, to provide a new set of experimental results on fracture of U-notched samples, made of two different materials; second, to apply a fracture criterion based on the strain energy density (SED) averaged over a control volume to assess the fracture load of blunt-notched components under three point bending. Two different materials are considered in the tests: a composite material (Al–15%SiC) tested at room temperature and a steel with a ferritic–pearlitic structure tested at −40 °C. All samples are weakened by U-notches characterized by different values of notch root radius and notch depth. The theoretical loads to failure as determined according to the SED criterion are compared with the experimental data from more than 40 static tests and with a SED-based scatter band recently reported in the literature for a number of materials exhibiting a brittle behaviour under static loads.     

Verification of brittle fracture criteria for elements with V-shaped notches

In this paper an analysis of crack initiation in plane elements with V-shaped notches under biaxial loading (mode I and II) was presented. The following fracture criteria were used to evaluate the critical loads and directions of crack initiation: strain energy release rate criterion; strain energy density criterion; modified McClintock's stress criterion; non-local stress criterion.Results of numerical analysis obtained using the boundary element method and path independent H and J integrals were compared with experimental data.     

Evaluation of R-curve behavior of ceramic-metal functionally graded materials by stable crack growth

This paper deals with the fracture toughness and R-curve behavior of ceramic-metal functionally graded materials (FGMs). A possibility of stable crack growth in a three-point-bending specimen is examined based on the driving force and resistance for crack growth in FGMs, and the distribution of fracture toughness or R-curve behavior is evaluated on FGMs fabricated by powder metallurgy using partially stabilized zirconia (PSZ) and stainless steel (SUS 304). The materials have a functionally graded surface layer (FGM layer) with a thickness of 1 mm or 2 mm on a SUS 304 substrate. Three-point-bending tests are carried out on a rectangular specimen with a very short crack in the ceramics surface. On the three-point-bending test, a crack is initiated from a short pre-crack in unstable manner, and then it propagates in stable manner through the FGM layer with an increase in the applied load. From the relationship between applied load and crack length during the stable crack growth in the FGM layer, the fracture toughness is evaluated. The fracture toughness increases with an increase in a volume fraction of SUS 304 phase.