Asymptotic Stress Fields Research Articles

Efficient crack growth analyzes by combining fast methods for BEM

AbstractThe combination of fast methods for the boundary element method (BEM) for efficient crack growth analyzes is presented. Due to the nonlinearity of fatigue crack growth an incremental procedure has to be applied. Within each increment a stress analysis is needed. Based on the asymptotic stress field the stress intensity factors (SIFs) are calculated by an extrapolation method. Then, a new crack front is determined by a reliable 3D crack growth criterion. Finally, the numerical model has to be updated for the next increment. The time dominant factor in each increment is the computation of the stress field. Due to the stress concentration problem the BEM is utilized. To speed‐up the calculation several independent fast methods are exploited. An algebraic technique is the adaptive cross approximation (ACA) method which is acting on the system matrix itself. The application of the substructure technique leads to a blockwise band matrix and therefore to reduced memory requirements. Further savings in memory and computation time are reached by modelling cracks with the dual discontinuity method (DDM) and using the ACA method in each substructure. The efficiency of the combined methods is shown by a complex industrial example. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Three-dimensional asymptotic stress field in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., located at the center, edge or corner, and subjected to the far-field torsion (mode III), extension/bending (mode I), and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (1) bimaterial penny-shaped interface anticrack or perfectly bonded thin rigid inclusion, (2) bimaterial penny-shaped interfacial jammed contact, (3) bimaterial penny-shaped interface crack, (4) bimaterial penny-shaped interface crack with partial axisymmetric frictionless slip, and (5) bimaterial penny-shaped interface thin rigid inclusion alongside penny-shaped crack. Solutions to these cases except (3) are hitherto unavailable in the literature. Closed-form expressions for stress intensity factors subjected to various far-field loadings are also presented. Numerical results presented include the effect of the ratio of the shear moduli of the layer materials, and also Poisson’s ratios on the computed lowest real parts of eigenvalues for the case (5). Interesting and physically meaningful conclusions are also presented, especially with regard to cases (1) and (2).

Asymptotic Stress Fields for Thermomechanically Loaded Cracks in FGMs

Abstract The problem of a stationary crack in functionally graded materials (FGM), subjected to a combination of thermal and mechanical loading is considered. An asymptotic analysis coupled with Westergaard's stress function approach is used to characterize the stress field around the crack tip. Thermal and mechanical properties (e.g., elastic modulus, coefficient of thermal expansion, and thermal conductivity) are assumed to vary exponentially. The crack is assumed to be inclined to the direction of the property gradation. The thermal loading is taken to be a uniform heat flow in a direction inclined to the crack. The principal of superposition from linear elasticity is used to solve the problem, whereby the problem is divided into a number of subproblems. The first four terms in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the structure of the stress field. It is observed that the presence of heat flow produced no additional singularity and hence the classical inverse square root singularity still prevails around the crack tip. Using these stress field contours of constant maximum shear stress are generated and the effect of thermal loading on the crack-tip stress field is discussed.

Elastic T-Stress Solutions of Embedded Elliptical Cracks Subjected to Uniaxial and Biaxial Loadings

Abstract The elastic T-stress has been found to be an important parameter in characterizing the near crack tip elastic-plastic stress states of 2-D and 3-D crack problems. In addition to the J-integral, the T-stress provides an effective two-parameter characterization of 2-D and 3-D elastic-plastic crack tip fields in a variety of crack configurations under small scale yielding conditions. Embedded elliptical crack in an infinite solid subjected to uniaxial or biaxial tension conditions represents an excellent model for many engineering components with embedded cracks. Although extensive analyses are available for stress intensity factors (SIFs) for embedded elliptical cracks, to date no analytical T-stress solutions are available. This paper presents the exact close form solutions for elastic T-stress of an embedded elliptical crack in an infinite body under remote uniaxial or biaxial tension. The foundation of the derivation is based on the potential method for three-dimensional elasticity. Solutions of stress components parallel to and perpendicular to the cracked plane are derived first. T-stress is then obtained from the asymptotic stress field near the crack front. Complete solutions of T-stress along the crack front are established. Further, three-dimensional finite element analyses are conducted to verify the derived solutions; excellent agreements are achieved. When combined with the corresponding K or J solutions, these T-stress solutions are suitable for the analysis of constraint effects for embedded elliptical shaped cracks in engineering components.

An Automated Procedure for Determining Asymptotic Elastic Stress Fields at Singular Points

The paper describes an analytical tool in MATLAB for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity. The user is prompted to input the local geometry of the system, the material properties, and the boundary conditions (and interface conditions in the case of composite bodies or problems involving contact between two or more bodies). The tool then computes the dominant eigenvalue and provides as output the equations defining the singular stress and displacement fields and contour plots of these fields. No knowledge of the asymptotic analysis procedure is required of the user.The tool is tested against previously published results where available and proves to be robust and accurate. It is potentially useful for the development of special finite elements for singular points or for characterizing failure at such points. It can be downloaded from the website http://www-personal.umich.edu/∼jbarber/asymptotics/intro.html

Analytical solution for an interfacial crack subjected to dynamic anti-plane shear loading

A closed form solution for an interfacial crack problem is obtained. The crack lies on the interface of two bonded dissimilar orthotropic strips. Its surfaces are subjected to a dynamic anti-plane shear traction. Separation of variables technique is employed, to reduce the problem to a singular system of triple series equations, then to a singular integral equation. That is solved exactly, such that the asymptotic stress field distribution and the stress intensity factor are obtained in closed form expressions. The validity of the solution is proved. Further, a parametric study is introduced to investigate the effects of elastic and geometric characteristics of the composition on the values of the dynamic stress intensity factor (DSIF).

Calculation of stress intensities at the interface corner of anisotropic/isotropic bonded joints with a wedge

The behavior of bonded joints is of interest in many engineering fields. In order to analyze the failure at the interface of a bi-material structure, it is important to investigate stress intensities at the interface corner. In this paper, the expanded Stroh formalism is used to determine asymptotic stress and displacement fields near the interface corner between anisotropic and isotropic adherends. This simple procedure makes it possible to obtain stress singularities more easily than before. Three examples are presented to help understand the method. To obtain stress intensities at the interface corner, a path-independent conservative line integral, derived from Betti's reciprocal principle, is used. Finally the method to calculate stress intensities at the interface corner of bonded joints is presented in detail.

A boundary layer framework considering material gradation effects

This paper describes the development and application of a novel modified boundary layer (MBL) model for graded nonhomogeneous materials, e.g. functionally graded materials (FGMs). The proposed model is based on a middle-crack tension, M(T), specimen with traction boundary conditions applied to the top and lateral edges of the model. Finite element analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates elements with graded elastic and plastic properties. Elastic crack-tip fields obtained from the proposed MBL model show excellent agreement with those obtained from full models of the cracked component for homogeneous and graded nonhomogeneous materials. The K– T dominance of FGMs is investigated by comparing the actual stress fields with the asymptotic stress fields (the Williams’ solution). The examples investigated in the present study consider a crack parallel to the material gradient. Results of the present study provide insight into the K– T dominance of FGMs and also show the range of applicability of the proposed MBL model. The MBL model is applied to analyze the elastic–plastic crack-tip response of a Ti/TiB FGM SE(T) specimen. The numerical results demonstrate that the proposed MBL model captures the effect of T-stress on plastic zone size and shape, constraint effects, etc. for such configurations.

Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial wedge associated with matrix cracking or fiber break

A combined approach for prediction of the singular stress fields at the interfacial fronts of fiber breaks and matrix cracks is presented. A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of a point located at the front of a bimaterial wedge of general (unsymmetric) geometrical configuration (with respect to bimaterial interface), and subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far field loading and free–free wedge-side boundary condition. Each material is isotropic and elastic, but with different material properties. The material 2 (fiber in the case of matrix cracking or matrix in the case of fiber break) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying matrix cracking or fiber break incidence angles at the interfaces. Numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) for various wedge aperture angles of the material 1, subjected to the afore-mentioned wedge-side boundary condition, are also presented. Variation of the same with the shear moduli ratio of the component material phases is also an important part of the present investigation. The conclusion drawn from the present asymptotic analysis of the matrix cracking problem is in agreement with that of an earlier investigation based on the energy based criterion. Similar conclusion drawn from the present asymptotic analysis of fiber break problem is in agreement with that of an energy based criterion derived here in analogy to the corresponding matrix crack problem by another investigator.

Effect of material gradation on K-dominance of fracture specimens

This work describes the effect of material gradation (parallel to the crack plane) on stress intensity factors and K-dominance, i.e. the dominance of the singular region, of fracture specimens; SE(T), SE(B) and C(T). The extent of K-dominance is investigated by comparing the actual stress field with the Williams’ asymptotic stress field. Linear-elastic finite element analyses are performed using graded elements which incorporate graded material properties at the element level. Material gradation and crack geometry are systematically varied to perform the parametric study. Results reveal that the effect of material gradation on K I is most pronounced when a short crack is located on the stiffer side of the fracture specimen. For a given specimen and crack geometry, the extent of K-dominance yields a curve with a peak point at a certain material gradation. Results of the present study provide valuable insight into the K-dominance of FGMs.

A refined CLNA model in fretting fatigue using asymptotic characterization of the contact stress fields

ABSTRACTUsing the Atzori–Lazzarin criterion, the first author has recently proposed a unified model for fretting fatigue (FF), called the crack‐like notch analogue (CLNA) model. Two possible types of behaviour were suggested: either ‘crack‐like’ or ‘large blunt notch,’ and these are immediately studied in the typical condition of constant normal load and in phase oscillating tangential and bulk loads. The former condition (‘crack‐like’) was treated by approximating the geometry to the perfectly flat one of the crack analogue (CA), improved in some details, reducing all possible geometries to a single contact problem. The latter (‘large blunt notch’), with a simple peak stress condition i.e. a simple notch analogue model. In the present paper, the calculation of the ‘crack‐like’ behaviour is improved using the recent asymptotic characterisation developed by Dini, Hills and Sackfield, which extracts the asymptotic singular stress field of the fretting contact. A significant difference is found in the ‘equivalent’ geometric factor obtained for the Hertzian geometry, particularly near full sliding, where the new criterion is more conservative, but still not large enough to permit to find, for example in Nowell's FF experimental data, if the refinement is an improvement of predictive capabilities. In flatter geometries, the difference is expected to be even smaller than in the case of the Hertzian geometry, and in this case, the original CLNA model, for its simplicity, remains a very convenient simple closed form criterion.

On the Asymptotic Stress Field in Angularly Nonhomogeneous Materials

The problem of multi-material junctions composed of angularly nonhomogeneous elastic wedges in plane elasticity is addressed. For this new type of grading the governing equation for the Airy stress function is derived and, by applying the eigenfunction expansion method, a fourth-order ODE with nonconstant coefficients for the eigenequation is obtained. The solution to this ODE permits the formulation of an eigenvalue problem similar to that valid for material junctions between homogenous different materials. It is mathematically demonstrated that the angular grading influences the order of the stress-singularity. The potentials of the use of this new class of materials in joining technology are carefully investigated and some illustrative examples are deeply discussed. Comparisons with the corresponding results obtained from homogeneous materials are made.

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh( β)/ π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.

Numerical evaluation of generalized stress-intensity factors in multi-layered composites

The problem of the evaluation of the generalized stress-intensity factors for re-entrant corners in multi-layered structural components is addressed. An approximate analytical model based on the theory of multi-layered beams is presented. This approach provides a simple closed-form solution for the direct computation of the Mode I stress-intensity factor for the general problem of a re-entrant corner symmetrically meeting a bi-material interface. For the self-consistency of the theory, re-entrant corners in homogeneous materials and cracks perpendicular to bi-material interfaces can also be gained as limit cases of this formulation. According to this approach, the effects of the elastic mismatch parameters, the value of the notch angle and the thicknesses of the layers on the stress-intensity factor are carefully quantified and the results are compared with FE solutions. FE results are obtained by applying a combination of analytical and numerical techniques based on the knowledge a priori of the asymptotic stress field for re-entrant corners perpendicular to a bi-material interface and on the use of generalized isoparametric singular finite elements at the notch tip. A good agreement between approximate and analytical/numerical predictions is achieved, showing the effectiveness of this approach.

A through interface crack between a ±45° transversely isotropic pair of materials

In this study, the first term of the asymptotic displacement and stress fields is determined analytically for a straight through crack along the interface between a ±45° transversely isotropic pair of materials. Since with this configuration, there is full coupling between the modes, this problem requires a three-dimensional treatment. To calculate stress intensity factors, a three-dimensional M-integral is derived using the asymptotic fields as auxiliary solutions. The displacement extrapolation method is derived as well, and used to check the results obtained by the M-integral. Two numerical test cases are employed to examine the accuracy of both methods. Results obtained for other mechanical problems are presented as well.

Elastic T-stress solutions for penny-shaped cracks under tension and bending

This paper presents the exact close form solutions for elastic T -stress of a penny-shaped crack in an infinite body under remote tension and bending. Penny-shaped crack problems have been analysed extensively in the literature, but all the focus has been on the determination of stress intensity factors. In the current work, using the potential method and Hankel transformation technique, the complete solutions of stress components parallel to and perpendicular to the cracked plane are derived first. Elastic T -stress is then obtained from the asymptotic stress field near the crack front. Further, three-dimensional finite element analysis is conducted to verify the derived exact solutions. It is found that under tension loading, the elastic T -stress is uniformly compressive along the crack front; while under bending, the elastic T -stress is varying from compressive to tensile along the crack front. In both tension and bending cases, T -stress strongly depends on the Poisson's ratio of the material.

Asymptotic Analysis of a Mode III Stationary Crack in a Ductile Functionally Graded Material

The dominant and higher-order asymptotic stress and displacement fields surrounding a stationary crack embedded in a ductile functionally graded material subjected to antiplane shear loading are derived. The plastic material gradient is assumed to be in the radial direction only and elastic effects are neglected. As in the elastic case, the leading (most singular) term in the asymptotic expansion is the same in the graded material as in the homogeneous one with the properties evaluated at the crack tip location. Assuming a power law for the plastic strains and another power law for the material spatial gradient, we derive the next term in the asymptotic expansion for the near-tip fields. The second term in the series may or may not differ from that of the homogeneous case depending on the particular material property variation. This result is a consequence of the interaction between the plasticity effects associated with a loading dependent length scale (the plastic zone size) and the inhomogeneity effects, which are also characterized by a separate length scale (the property gradient variation).

Three-dimensional asymptotic stress field in the vicinity of the circumferential tip of a fiber–matrix interfacial debond

A heretofore unavailable asymptotic solution pertaining to the stress field in the neighborhood of the circumferential tip of an interfacial debond, between the fiber or inclusion and the unreinforced plate made of the matrix material, and subjected to far-field extension-bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, is presented. A local orthogonal curvilinear coordinate system ( ρ, φ, θ), is selected to describe the local deformation behavior of the afore-mentioned plate in the vicinity of the afore-mentioned circumferential line of interfacial debond. One of the components of the Euclidean metric tensor, namely g 33, is approximated ( ρ/ a≪1) in the derivation of the kinematic relations and the ensuing governing system of three partial differential equations. A recently developed eigenfunction approach coupled with a novel local curvilinear coordinate system, is utilized to compute the asymptotic displacement and stress fields. The oscillatory behavior at the debond tip may be considered to be a first-order approximation of the interdiffusion of the component phases followed by molecular entanglement and other similar microscopic phenomena studied by materials scientists.

Evaluation of asymptotic stress field around a crack tip for Neo–Hookean hyperelastic materials

Abstract The near-tip asymptotic mechanical field in plane Neo–Hookean hyperelastic media is characterized by the appearance of two singular terms and, as a consequence, needs to be described by four stress parameters. Direct computation of the asymptotic stress field therefore appears to be difficult. In this paper, we propose two pairs of contour integrals. The integrals are shown to be path-independent in a modified sense and so they can be evaluated by using finite element approximations with sufficient accuracy. Also, the relationship between these integrals and the stress parameters is analytically derived. Once the integrals are accurately computed, the stress parameters and, consequently, the asymptotic stress field can then be properly determined. The formulation is considered to be suited for problems under large deformation and subjected to general far-field mixed-mode loads. No particular singular elements are used in the calculations.

Interfacial crack tip field in anisotropic/isotropic bimaterials

The co-cured joint is more efficient than the adhesively bonded joint for composite structures because of its several advantages. However, failure analysis of the co-cured joint has just a little been reported since the co-cured joining method was introduced. Observing based on the experimental results, failure starts at the interface corner of the co-cured joint. Therefore, it is important to consider stress intensities at the interface corner of the co-cured joint. In this paper, the eigenvalue problem was used to determine asymptotic stress and displacement fields near the interface corner between composite and steel adherends. For obtaining stress intensities at the interface corner a path independent conservative line integral was used.