Fracture analysis of materials with periodic microstructure by the representative cell method

The periodic structure of some natural and especially man-made materials can be manifested not only on an atomic but also on a larger scale. Investigation of mechanical properties of these materials usually hinges on well-developed homogenization methods. On the other hand, these methods are not suitable for fracture analysis where the knowledge of the local stress-strain fields near a flaw (a crack) is required. The result is obtained by the use of the representative cell method based on the discrete Fourier transform. This method enables one to determine the exact stress distribution in a periodic structure subjected to arbitrary loading. Direct application of the method is impossible since the crack violates the translational symmetry defined by the material microstructure. This obstacle is overcome by application of the fictitious loading to the uncracked body at the line where the crack is to be located. The amplitude of the loading is adjusted in order to fulfill the boundary conditions imposed on the crack faces. The compatibility equation for deriving this amplitude is obtained by the use of the corresponding Green function, which is found in a closed form. Fracture problems for the two types of materials with a periodic microstructure are considered. The first one is a composite material consisting of dissimilar isotropic elastic layers arranged periodically. The second periodic microstructure is a 2D infinite beam lattice modeling a cellular material. The analysis of the failure process in the latter case shows that in contrast to the case of homogeneous material, the crack propagation path is not defined by the condition of zero Mode II stress intensity factor.     

Crack morphology models for fracture toughness and fatigue strength analysis

Planar cracks represent an approximation, largely adopted in fracture mechanics and fatigue problems, of the physical reality, where cracks feature complex geometric morphologies related to material microstructure, residual stresses, material properties dispersions and so on. In the present paper, firstly a model to describe the influence of roughness and friction of the crack surfaces is reviewed in relation to the resulting near‐tip stress field and the fracture resistance under monotonic loading. Such a model is based on the Distributed Dislocation Technique, and considers a periodic profile of the crack. Then, some approximate theoretical models describing periodically kinked cracks are reviewed in their application to the estimation of fatigue strength of materials. In particular, the influence of crack path meandering on fatigue propagation is analysed by modelling the crack profile as a piecewise linear periodic curve in two dimensions. The same type of model is discussed within the framework of self‐similar fractal geometries. In the paper, emphasis is given to the effect of crack size on the fracture resistance and fatigue strength, where such an effect depends on the ratio between the characteristic length of the crack morphology and the nominal length of the crack itself.     

High-cycle fatigue mechanisms in 1045 steel under non-proportional axial-torsional loading

Detailed microscopic analyses have been made on the high-cycle mechanisms in 1045 steel under various stress-controlled axial-torsional loadings. A special attention has been paid to a critical example of non-proportional loading, i.e., 90° out-of-phase loading with different stress ratios. The replica technique has been used to monitor crack initiation and propagation from the microstructure scale. The orientations of persistent slip bands and Stage I cracks are in good agreement with the critical plane concept. The evolutions of crack length with cycle life as well as the crack aspect ratios depend on the loading condition. However at a given life, the data are consolidated in terms of crack depth versus cycle life. The McDiarmid parameter correlates stress-life data under proportional loadings. However, it underestimates fatigue lives under out-of-phase loading at high stress ratio and it overestimates them in the case where all planes experience the same shear stress amplitude (stress ratio = 0.5). More damaging mechanisms are involved in crack initiation and crack propagation. It is recommended to test the fatigue performance of materials in this last condition that involves the worst damage mechanisms.     

Anrißerzeugung in WC-Co-Hartmetall-Proben für die Messung des kritischen Spannungsintensitätsfaktors KIC

Precracking of WC-Co-Hardmetal-Specimens for Fracture-Toughness Testing The determination of a valid critical stress intensity factor K_IC requires an extremely sharp, well defined initial crack. Methods producing such a crack are well known for metallic materials, but they often can not be used with brittle materials, like cemented carbides or ceramics. Their low fracture toughness makes a controlled crack growth under pure tensile stress nearly impossible. More useful are precracking methods, utilizing a stress gradient to stop the crack at defined depth. A very simple methods uses the indentation of a hardness tester to produce a semi-elliptical surface crack, interfered with residual stresses. For different areas of application and specimen geometries, bridge indentation, wedge indentation and composite bending method produce cracks with a straight front. Also under cyclic loading, under tensile as well as under compressive stress, the creation of a sharp precrack, applicable in K_IC measurement, is possible.     

变幅载荷下纤维金属层板的疲劳与寿命预测

文章建立了纤维金属层板等幅疲劳载荷下的疲劳裂纹扩展速率与寿命预测模型。在此基础上对玻璃纤维-铝合金层板（GLARE）的疲劳裂纹扩展与分层扩展行为进行了试验研究,探讨了层板过载疲劳行为的机理,提出了纤维金属层板变幅载荷下疲劳寿命预测的等效裂纹闭合模型,并在GLARE层板上得到了验证。     

Crack resistance in two-dimensional periodic materials of medium and low porosity

The plane problem of crack resistance in elastic materials with doubly periodic system of voids is investigated. The relatively high fractional density (medium and low porosity) of above 0.6 which is typical for the partially-sintered materials is addressed. The stress field in the voided plane with embedded Mode I crack of arbitrary length bridging the voids is evaluated. In the framework of stress criterion of fracture this field allows the determination of the fracture toughness of the considered material with periodic microstructure in terms of the solid (parent) material tensile strength.The mathematical modeling of this problem is complicated due to the fact that the “ligaments” between the voids are so thick that they cannot be treated as beams. As a result, theoretical methods used to evaluate the fracture toughness of low-density cellular materials are not applicable. Accordingly, the present problem is solved by means of a novel analysis which is based on the combined use of the representative cell method, based on the discrete Fourier transform, in conjunction with a higher-order theory and a micromechanical model. These combined three approaches allow the performance of an accurate continuum mechanics analysis of the highly non-uniform stress field in the voided material with a crack.The analysis of this field for different crack lengths has shown which length must be employed in the fracture toughness evaluation. The dependence of the fracture toughness upon the relative density is examined and found to vary linearly in the majority of the considered density range. This result agrees with the known experimental observations and other theoretical predictions.     

An eigenfunction expansion-variational method based on a unit cell in analysis of a generally doubly periodic array of cracks

A new variational functional for a unit cell of a heterogeneous solid with periodic microstructures is constructed by incorporating the quasi-periodicity of the displacement field and the periodicity of the stress and strain fields into the strain energy functional. The functional can accommodate a broad class of periodic structures including the case where symmetry or antisymmetry properties of the unit cell may not exist. Then the functional is applied to deal with a doubly periodic array of cracks under plane and anti-plane loading. By combining with the eigenfunction expansions of the complex potentials satisfying the traction-free condition on the crack surfaces, an eigenfunction expansion-variational method based on a unit cell is developed. Numerical examples are presented and compared with existing results to demonstrate the high accuracy and efficiency, and wide application scope of the present method. Some interesting phenomena of multi-crack interaction, which do not occur in the case of symmetrical arrays of cracks, are revealed and discussed.     

The numerical solution of crack problems in plane elasticity in the case of loading discontinuities

The direct quadrature method of numerical solution of Cauchy type singular integral equations encountered in plane elasticity crack problems is applied to the case where the loading distribution along the crack edges presents jump discontinuities. This is made by using a well-known modification of the quadrature method which is free of undesirable errors due to the loading discontinuities. Hence, the method is ideal to treat the aforementioned class of crack problems and, particularly, crack problems where the Dugdale-Barenblatt elastic-perfectly plastic model is adopted. Finally, a numerical application of the method to the problem of a periodic array of cracks with a loading distribution presenting a jump discontinuity is made. The numerical results obtained in this problem compare favorably with the corresponding theoretical results available in this special problem.     

Elastic–plastic fatigue crack growth analysis under variable amplitude loading spectra

Most fatigue loaded components or structures experience a variety of stress histories under typical operating loading conditions. In the case of constant amplitude loading the fatigue crack growth depends only on the component geometry, applied loading and material properties. In the case of variable amplitude loading the fatigue crack growth depends also on the preceding cyclic loading history. Various load sequences may induce different load-interaction effects which can cause either acceleration or deceleration of fatigue crack growth. The recently modified two-parameter fatigue crack growth model based on the local stress–strain material behaviour at the crack tip [1,2] was used to account for the variable amplitude loading effects. The experimental verification of the proposed model was performed using 7075-T6 aluminum alloy, Ti-17 titanium alloy, and 350WT steel. The good agreement between theoretical and experimental data shows the ability of the model to predict the fatigue life under different types of variable amplitude loading spectra.     

压电材料中渗透性周期共线反平面裂纹问题

本文利用复变函数方法,借助于Riemann-Schwarz延拓技术和保形映照方法,研究了渗透性边界条件下周期共线反平面裂纹问题,获得了解的表达式,得到了力学和电学强度因子。结果表明在裂纹尖端应力和电位移的奇异性都与远场作用的应力载荷和裂纹长度有关,其中应力的奇异性与材料无关,电位移的奇异性则与材料有关,电载荷对裂尖的奇异性没有影响。最后,运用数值算例,给出周期裂纹间的干涉效应和裂纹的尺度效应。     

Threshold stress intensity factor and crack growth rate prediction under mixed-mode loading

A new mixed-mode threshold stress intensity factor is developed using a critical plane-based multiaxial fatigue theory and the Kitagawa diagram. The proposed method is a nominal approach since the fatigue damage is evaluated using remote stresses acting on a cracked component rather than stresses near the crack tip. An equivalent stress intensity factor defined on the critical plane is proposed to predict the fatigue crack growth rate under mixed-mode loading. A major advantage is the applicability of the proposed model to many different materials, which experience either shear or tensile dominated crack growth. The proposed model is also capable to nonproportional fatigue loading since the critical plane explicitly considers the influence of the load path. The predictions of the proposed fatigue crack growth model under constant amplitude loading are compared with a wide range of fatigue results in the literature. Excellent agreements between experimental data and model predictions are observed.     

A phenomenological model of fatigue macrocrack initiation near stress concentrators

Fatigue macrocrack initiation is considered to be a two-parameter process. It is governed by the local or strain amplitude, and a certain linear parameter of the material. Corresponding parameters have been proposed, i.e. the local stress range Δσ*y and a characteristic distance d *, the prefracture zone size. The formation of this zone is conditioned by a decrease in yield strength within the material’s surface layers, microstructure, loading amplitude, cyclic strain hardening and environment. The value of d * is estimated experimentally by several methods and is assumed to be a certain material constant, independent of both notch and specimen geometry. At the prefracture zone boundary, a major barrier exists that retards the growth of a physically small fatigue crack. The moment when the physically small crack overcomes the prefracture zone boundary is assumed to be a quantitative criterion, a_{i = d *}, for the micro- to macrocrack transition. The proposed relationships, Δσ*y versus N_i , and d * versus N_i , can be used as a basis for the establishment of the materials resistance to macrocrack initiation.     

Fatigue strength of small-notched specimens under variable amplitude loading within the fatigue limit diagram

Although the fatigue limit diagram is defined in principle for constant stress amplitude, it is often considered that fatigue failure would not occur, even in varying loading, if applied stresses were kept within the fatigue limit diagram. However, it was shown in the case of small‐notched specimens that fatigue failure occurred in some special cases of variable amplitude loading, even when all stress amplitudes were kept within the fatigue limit diagram. The cause of this phenomenon was examined using two‐step stress and repeated two‐step stress patterns in which the first step stress was chosen to be equal to the fatigue limit with zero mean stress and a mean stress was superposed on the second step stress. A non‐propagating crack was formed by the first step stress. This crack functioned as a pre‐crack for the second step stress with high mean stress. Consequently, fatigue failure occurred even when all stress amplitudes were kept within the fatigue limit diagram. It was an unexpected fracture caused by the interference effect of a non‐propagating crack and a mean stress change.     

INTERACTION OF OVERLOADS ON FATIGUE CRACK GROWTH FOR 40CrNi STEEL

This work was designed to improve the general understanding of loading sequence effects on fatigue crack growth and lead to the development of improved methods for predicting crack propagation behaviour. Two loading histories were selected (1) a baseline amplitude with periodic overloads or underloads, and (2) several overloads without interactive effects. The specimen used was of a Wedge Opening Loading type and the material was a low alloy high strength steel, i.e. 4OCrNi. It was found that the Linear Summation of Damage (LSD) assumption could be applied in predicting fatigue crack growth rate (FCGR) under periodic overloads or underloads within the scatterband of the constant amplitude data obtained using an Alternating Current Potential Drop technique for measuring crack length for multiple specimens with several load amplitudes. A discrepancy existed between the FCGR predicted from LSD and the one actually measured during several hundreds of loading cycles immediately following every non-interactive overload of the latter loading history although the overload ratio was the same as that of periodic overloading. The causes of this phenomenon are discussed.     

Improved test method and analytical modelling for fatigue crack growth in coarse‐grain titanium alloy with rough fatigue surfaces

Fatigue crack growth rate properties are typically determined by experimental methods in accordance with ASTM Standard E647. These traditional methods use standard notched specimens that are precracked under cyclic tensile loads before the main test. The data that are produced using this approach have been demonstrated elsewhere to be potentially adversely affected by the test method, particularly in the threshold region where load reduction (LR) methods are also required. Coarse‐grained materials that exhibit rough and tortuous fatigue surfaces have been observed to be strongly affected by the tensile precracking and LR, in part because the anomalies caused by crack closure and roughness‐induced closure become more important. The focus of the work reported in this paper was to further develop methods to determine more accurate fatigue crack growth rate properties from threshold through to fracture for coarse‐grained, β‐annealed, titanium alloy Ti‐6Al‐4V extra low interstitial thick plate material. A particular emphasis was put upon the threshold and near threshold region, which is of strong importance in the overall fatigue life of components. New approaches that differ from the ASTM Standard included compression precracking, LR starting from a lower load level and continuing the test beyond rates where crack growth would otherwise be considered below threshold. For the threshold regime, two LR methods were also investigated: the ASTM method and a method where the load is reduced with crack growth such that the crack mouth opening displacement is held constant, in an attempt to avoid remote closure. Constant amplitude fatigue crack growth rate data were produced from threshold to fracture for the titanium alloy at a variety of stress ratios. Spike overload tests were also conducted These data were then used to develop an improved analytical model to predict crack growth under spectrum loading and the predictions were found to correlate well with test results.     

Prediction of crack initiation plane direction in high‐cycle multiaxial fatigue with in‐phase and out‐of‐phase loading

A new method for predicting crack plane direction in high‐cycle multiaxial fatigue is proposed. This method considers material properties and loading conditions. Two situations are considered: (i) in‐phase loading, where the crack plane direction only depends on the loading condition and material properties have little influence on it, and (ii) out‐of‐phase loading, where the crack plane direction is affected by both loading conditions and material properties. The prediction accuracy is assessed by comparison with several experimental results, including different loading conditions and materials. The results show that the proposed method provides a good prediction capability for these experiments.     

Effect of microstructural stability on fatigue crack growth behaviour of nanostructured Cu

Constant amplitude fatigue crack growth tests were carried out on commercial and high purity nanostructured copper processed by High Pressure Torsion (HPT). Due to strong grain refinement the HPT processed materials show higher tensile strength but also faster crack growth rates when compared to coarse grained material. Crack growth curves of nanostructured copper determined at different stress levels, however, showed that the occurrence of grain coarsening at low stress amplitudes leads to a retardation of crack growth in commercial and high purity HPT Cu. This effect was not observed for high purity HPT Cu with a bimodal microstructure. Crack propagation rates depend significantly on the coarsening phenomenon which on the other hand depends on the applied stress amplitude. A comparison of these results with cyclic deformation tests in the high cycle fatigue regime suggests that grain coarsening during crack growth depends more on the stored energy of the materials while a similar coarsening during cyclic deformation depends more on the activation enthalpy for annealing of defects.     

Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.     

Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.     

Fracture toughness of short carbon fibre reinforced thermoplastic polyimide

The fracture toughness testing of short fibre reinforced thermoplastic materials were performed. Materials tested were the polyimide resin and also that reinforced with 20 wt% or 30 wt% short carbon fibre. For introducing the initial crack, the tapping method, the sliding method and the bridge indentation method were examined. Among them, the sliding method was found to be effective for every case. The fracture tests were conducted by the three-point bending test with several loading rates. Stable crack growth was observed for the neat material while unstable fracture occurred for the reinforced materials. The critical values of the stress intensity factor at crack initiation were greater for the reinforced materials than for the neat resin. The fracture toughness of the 30 wt% reinforced material was independent of loading rate while that of 20 wt% reinforced material increased with loading rate. In order to investigate the fracture mechanisms, fractographic observations were also performed.