On the Mode I stress intensity factor for an external circular crack with fibre bridging

This paper presents a theoretical analysis of an external matrix crack located in a unidirectional fibre-reinforced elastic solid modelled as a transversely isotropic material. The presence of matrix cracking with fibre continuity introduces bridging action that has an influence on the stress intensity factors at the crack tip of the external crack. This paper presents a model for the bridged crack, where the fibre ligaments induce a constant displacement-dependent traction constraint over the external crack. This gives rise to a Fredholm integral equation of the second kind, which can be solved in an approximate fashion. We examine the specific problem where the bridged external circular crack is loaded by a doublet of concentrated forces. Numerical results are presented to illustrate the influence of the fibre–matrix modular ratio and the location of the loading on the bridged-crack opening mode stress intensity factor.     

The mechanics of matrix cracking in fiber reinforced ceramic composites containing a viscous interface

A detailed fracture mechanics analysis of matrix cracking in a fiber reinforced ceramic composite is presented for the case where the fiber—matrix interface exhibits viscous flow as can be the case when ceramic composites containing amorphous interfacial layers are subjected to loads at elevated temperatures. The analysis considers the case where matrix cracks are fully bridged by fibers, and the role of the viscous interface is to introduce a time dependence into the stress-intensity formulations. Such time-dependence arises because the bridging fibers are able to pull out of the matrix by viscous interfacial flow, with the result that the crack opening, as well as the actual (or shielded) matrix crack-tip stress-intensity factor, increase with time under the action of a constant externally applied load to the composite. The differential equation governing the mechanics of the fiber pull-out is derived. This is then applied to obtain expressions for the time-dependence of the crack opening and the effective crack-tip stress-intensity factor in terms of material and microstructural factors. These expressions predict that the matrix crack will exhibit stable crack growth, with the crack growth rate being essentially crack length (and time) independent and a function only of the applied stress and of material and microstructural factors. It is also shown that the composite lifetime is independent of the sizes of pre-existing cracks and is dependent only on a critical microstructure dependent flaw size, applied stress and microstructural factors.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

A solution method for finding dynamic stress intensity factors for arbitrarily oriented cracks in transversely isotropic materials

The transient elastodynamic response of a transversely isotropic material containing a semi-infinite crack under uniform impact loading on the faces is examined. The crack lies in a principle plane of the material, but the crack front does not coincide with a principle direction. Rather, the crack front is at an angle to a principle direction and thus the problem becomes more three-dimensional in nature. Three loading modes are considered, i.e., opening, in-plane shear and anti-plane shear. The solutions for the stress intensity factor history around the crack tip are found. Laplace and Fourier transforms together with the Wiener-Hopf technique are employed to solve the equations of motion directly. The asymptotic expression of stress near the crack tip leads to a closed-form solution for the dynamic stress intensity factor for each loading mode. It is found that the stress intensity factors are proportional to the square root of time as expected. Results given here converge to known solutions in transversely isotropic materials with a crack oriented along a principle direction and isotropic materials as special cases. The results of this analysis are used to find approximate strain energy release rates for dynamically loaded penny shaped cracks.     

Fatigue crack growth with fiber failure in metal-matrix composites

Crack growth during the fatigue of fiber-reinforced metal-matrix composites can be predicted analytically by determining the reduction in the crack tip stress intensity range resulting from fiber bridging. Various canonical functions exist that relate the crack tip stress intensity range to bridged crack geometries and loading for both infinite and finite width specimens; however, comprehensive crack growth predictions incorporating fiber failure require knowledge of the maximum fiber stress in the bridged zone for all notch sizes and crack lengths. Previous modeling efforts have been extended to predict complete growth curves with fiber failure for specimens of finite width. Functions for maximum fiber stresses in the bridged zone are presented here for a center crack in tension and edge cracks in tension and bending. The rapid increase in crack growth when fibers fail emphasizes the importance of determining the loads and notch sizes that mark the beginning of fiber failure. Critical loads for given notch sizes and fiber strengths are easily determined for finite width specimens using the functions presented in this work.     

Matrix fatigue cracking in fiber composites

A model is developed for fatigue growth of matrix cracks in metals reinforced with aligned continuous elastic fibers. The mechanics of elastic cracks bridged by frictionally constrained fibers is used to develop the model, which provides estimates of the tip value of the stress intensity factor amplitude, ΔK_TIP. It is found that when the applied load amplitude is held fixed during fatigue crack growth, ΔK_TIP, and thus the rate of growth approach an asymptotic value independent of crack length. The residual strength after fatigue crack growth is also discussed. In some cases, the residual strength is unaffected by prior fatigue growth. But, in another regime, the matrix crack length allows fibers to begin breaking before the matrix crack grows. The strength is then inversely proportional to the square root of fatigue crack length.     

A complex variable approach for asymptotic Mode-III crack-tip fields in an anisotropic functionally graded material

This paper addresses asymptotic full crack-tip fields for an anti-plane (Mode-III) stationary crack in an anisotropic functionally graded material. A monoclinic material that has a material symmetry plane is considered. The complex variable approach and the asymptotic analysis are used to solve a perturbed Laplace equation resulting from material anisotropy and gradation. The out-of-plane displacement and stress solutions are provided for a crack in exponentially and linearly graded anisotropic materials by considering material gradation either parallel or perpendicular to the crack. The characteristics of the asymptotic solutions in an anisotropic functionally graded material are compared with those for anisotropic homogeneous and isotropic graded materials. Finally, engineering significance of the present work is discussed.     

A boundary element method for determining the effect of holes on the stress distribution around a crack

A boundary integral procedure is obtained for examining the effect of a finite number of holes on the crack tip stress intensity factors for a plane crack in a homogeneous anisotropic elastic material. Numerical results for specific examples involving a transversely isotropic material are given.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

PROCESSING AND FRACTURE TOUGHNESS OF SINTERED FIBER REINFORCED CERAMIC MATRIX COMPOSITES

The paper discusses the processing and the resulting mechanical properties of sintered fiber reinforced ceramic matrix composites. In situ observations of the sintering process revealed that stresses which develop due to the differential shrinkage between the fibers and the matrix initiate already during the heating cycle and are of sufficient magnitude and duration that crack like damage forms. Successful methods were employed for reducing and avoiding these stresses during the sintering process. Coarse grained alumina coatings deposited onto the fibers with a coating thickness of up to 10 µm delayed and reduced the stress development. Polymer coated fibers produced fully dense composites on which fracture toughness measurements were performed. Crack propagation and crack/fiber interaction was observed in situ inside a scanning electron microscope. The importance of studying both the crack front and the crack wake phenomena in fiber reinforced composites is illustrated. In specimens where the cracks are already bridged by 10% area fraction of fibers a toughness of 7 MPa √m was obtained. However, in samples where the cracks are not bridged yet by fibers, the crack becomes unstable before reaching the fiber positions and the fibers had no effect in resisting the crack propagation.     

PROCESSING AND FRACTURE TOUGHNESS OF SINTERED FIBER REINFORCED CERAMIC MATRIX COMPOSITES

ABSTRACT

The paper discusses the processing and the resulting mechanical properties of sintered fiber reinforced ceramic matrix composites. In situ observations of the sintering process revealed that stresses which develop due to the differential shrinkage between the fibers and the matrix initiate already during the heating cycle and are of sufficient magnitude and duration that crack like damage forms. Successful methods were employed for reducing and avoiding these stresses during the sintering process. Coarse grained alumina coatings deposited onto the fibers with a coating thickness of up to 10 µm delayed and reduced the stress development. Polymer coated fibers produced fully dense composites on which fracture toughness measurements were performed. Crack propagation and crack/fiber interaction was observed in situ inside a scanning electron microscope. The importance of studying both the crack front and the crack wake phenomena in fiber reinforced composites is illustrated. In specimens where the cracks are already bridged by 10% area fraction of fibers a toughness of 7 MPa √m was obtained. However, in samples where the cracks are not bridged yet by fibers, the crack becomes unstable before reaching the fiber positions and the fibers had no effect in resisting the crack propagation.     

In situ observation of cyclic fatigue crack propagation of SiC-fiber/SiC composite at room temperature

In situ observation of cyclic fatigue crack propagation of SiC-fiber reinforced SiC composite at room temperature has been carried out by laser microscopy. Both smooth (unnotched) and notched specimens are used for tension-tension cyclic fatigue tests. Cracks initiate at the comers of large pores during loading in smooth specimens. In notched specimens cracks are formed at the interfaces between fibers and matrix that are connected to the notch. The balance between the fiber bridging in the wake of propagating crack tip and the breakage of bridged fibers by the degradation of interfaces maintains a steady cyclic crack propagation. Crack propagation rate gradually decreases with time after the maximum load being applied.     

Stress states and growth behavior of bridged cracks at creep regime

In this study growth behavior of bridged cracks, resulting from the growth of pre-nucleated creep cavities with diffusional and dislocation-assisted mechanisms, is investigated numerically. The elements bridging the crack are assumed to be elastic; the bridging behavior ranges from full development of the bridging zones to failure of the bridging elements during the course of crack growth. The results indicate that the bridging traction significantly relaxes even with the overall creep deformation alone. The rate of this relaxation is not influenced by the rate of crack growth. However, the rate of change in the bridging zone length or the density of the bridging elements in the bridging zone strongly affects both the maximum value and the distribution of the traction in the bridging zone. A much weaker stress singularity than the ones described by K or C^* was found ahead of the bridged cracks in the creep regime. In this weak singularity region the cavities, located at increasing distance from the crack tip, grow at similar high rates to each other.     

Stress concentration around cracks in long bones

By modeling a long bone (e.g. femur) as a cylindrical shell of transversely isotropic material, the deformation and the concentration of stresses around penny-shaped cracks are studied in the paper. Expressions for the stress intensity factor, the shape of the crack, and the crack energy are obtained. A quantitative analysis is made in respect of the normal stress and displacement in the neighbourhood of the crack. The magnitudes of these quantities have also been calculated for another non-isotropic material (magnesium) and an isotropic material (aluminium) and compared with those for bone material.     

Micromechanical modelling of strain hardening and tension softening in cementitious composites

This paper will describe a procedure for modelling the complete macroscopic response (including strain hardening and tension softening) of two short fibre reinforced cementitious composites and show how their microstructural parameters influence this response. From a mathematical point of view it is necessary to examine how bridging forces imposed by the fibres alter the opening of multiple cracks in elastic solids under unidirectional tensile loading. The strain hardening is essentially due to elastic bridging forces which are proportional to crack opening displacements. After a certain critical crack opening displacement is reached, some fibres progressively debond from the elastic matrix and thereafter provide a residual bridging force by frictional pull-out, while others continue to provide full bridging. This results in a kind of elasto-plastic bridging law which governs the initial tension softening response of the composite. Besides the usual square-root singularity at crack tips, the elasto-plastic bridging law introduces a logarithmic singularity at the point of discontinuity in the bridging force. These singularities have been analytically isolated, so that only regular functions are subjected to numerical integration. Unbridged multiple crack problems have in the past been solved using double infinite series which have been found to be divergent. In this paper a superposition procedure will be described that eliminates the use of double infinite series and thus the problem of divergence. It is applicable to both unbridged and bridged multiple cracks. The paper will end by showing how the model of multiple bridged cracks can accurately predict the prolonged nonlinear strain hardening and the initial tension softening response of two cementitious composites.     

超高分子量聚乙烯架桥纤维与裂缝的交互微观力学

使用拉曼光谱研究了架桥纤维与裂缝微观力学,以超高分子量聚乙烯(UHMWPE)纤维为例,将纤维搭桥试样进行微拉伸试验,着重分析架桥纤维的止裂作用和架桥纤维/环氧树脂界面的应力分布,并对不同位置架桥试样的裂缝扩展速度和应力分布进行分析,并进一步运用剪切滞后模型,对架桥纤维在不同拉伸载荷下的应力分布进行了拟合分析,结果表明:架桥纤维能够分散部分外载应力,对于裂纹扩展具有显著的止裂作用。在低于UHMWPE纤维最大应变拉伸时,发现在裂缝中心位置处架桥纤维所承受的应力最大,其应力不超过2GPa,而基体树脂的应力可达到12GPa,架桥纤维/基体界面的应力传递达不到100%。以UHMWPE为架桥的应力传递模型呈"正抛物线"型,应力分布存在于粘结区、脱粘区和架桥区。     

Screw dislocations interacting with a partially debonded interface in cylindrically anisotropic composites

Interaction between screw dislocations and a partially debonded interface in cylindrically anisotropic composites subjected to uniform stress at infinity is investigated in this paper. Using Muskhelishvili’s complex variable method, the closed forms of complex potentials are obtained for a screw dislocation and a screw dislocation dipole located inside either matrix or inhomogeneity. Explicit expressions of stress intensity factors at the crack tips, image forces and image torques acting on dislocation or the center of dipole are provided. The results show that the crack and dislocation geometry combination plays an important role in the interaction between screw dislocations and interface crack. Furthermore, it is found that the anisotropy of solids may change the shielding and anti-shielding effects arising from screw dislocations and the equilibrium position of screw dislocations. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some novel or previously known results.     

Analysis of cracked transversely isotropic and inhomogeneous solids by a special BIE formulation

In this paper, a special boundary integral equation (BIE) formation is proposed to analyze the fracture problem in transversely isotropic and inhomogeneous solids. In this formulation, the single-domain boundary element method (BEM) is utilized to discretize the cracked matrix and the displacement BEM to the surface of the embedded inhomogeneity. The two regions are then connected through the continuity conditions along their joint interface. The conventional and three special nine-node quadrilateral elements are utilized to discretize the inhomogeneity–matrix interface and the crack surface. From the crack-opening displacements on the crack surface, the mixed-mode stress intensity factors (SIFs) are calculated, using the well-known asymptotic expression in terms of the Barnett–Lothe tensor. In the numerical analysis, the distance between the inhomogeneity and the crack as well as the orientation of the isotropic plane of the transversely isotropic media is varied to show their influences on the mixed-mode SIFs along the crack fronts.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.