Westergaard Stress Function Research Articles

A strain gage technique for measuring the mode II stress intensity factors

In this paper, we propose a strain gage technique that allows an accurate measurement of the mode II stress intensity factors (SIFs). In order to ensure a good KII measurement, the size of the correct radial location of the strain gages is identified by using a methodology based on finite element analysis (FEA) and consolidated by solid theoretical foundations. The extent r1,max and r2,max of the validity of the five- and six-term representation of generalized Westergaard stress functions, which requires the use of two and three strain gages, respectively, is determined using the proposed method, and its dependence on the crack length to width ratio is also investigated. Numerical results obtained are in good agreement with the theoretical predictions, and confirm that the present method makes it possible to determine the KII with high precision when two (or three) strain gages are well placed within r1,max (or r2,max). Also, the numerical experiments show that the use of three strain gages is necessary when the size of the linear behavior region r1,max is very small. In addition, the effect of friction between the crack lips was checked and shown to have no effect on KII in pure mode II.

Analytical solutions for crack opening displacements of eccentric cracks in thin-walled metallic plates

In the context of the prevalence of thin-walled metallic aerospace structures, the added resistance to crack propagation offered by a built-up structure is desirable from a damage tolerance standpoint. The analysis of failure in such structures, however, is limited by the lack of crack opening solutions. This paper develops analytical models that calculate crack opening displacements (CODs) for a more general cracking scenario, i.e. non-symmetric cracks. The proposed models are based on the Westergaard stress functions. It is then found that the COD solution of one model is particularly accurate. The potential significance of the obtained solutions lies in analysing failure in built-up structures containing non-symmetric cracks. The crack opening solution is particularly useful in estimating the load transfer between cracked body and intact bridging structures in built-up structures using the principle of displacement compatibility.

Dynamic analysis of crack problems in functionally graded materials using a new graded singular finite element

In this research a recently developed graded singular element is used for studying dynamic crack problems in linear elastic isotropic functionally graded materials (FGMs) with spatially varying elastic parameters. The formulation for the graded singular element is obtained by analyzing the crack tip stress field using the Westergaard stress function method. Shape functions are extracted by presenting expressions for displacements near the crack tip in terms of the complex functions with unknown coefficients. The stiffness and consistent mass matrices for the singular element are then determined based on the principle of minimum potential energy and kinetic energy of the element, respectively. The global stiffness and mass matrices are then generated by assembling matrices for the singular and adjacent regular elements by considering the displacement continuity at their common boundary. The results from the current method are compared with existing solutions to evaluate the accuracy of the technique. A set of simulations are performed using this element for analyzing dynamic crack problems in FGMs whose elastic properties vary normal to the crack line. Compared with traditional elements, the proposed element includes higher order terms in addition to the singular term, which leads to accurate description of the stress field near the crack tip. It also significantly reduces the calculation time needed in comparison with using very fine mesh of traditional elements. With the use of the singular element, no post processing technique is required for calculating fracture parameters such as stress intensity factors, energy release rate, and mode mixity.

Study of Cyclic Fracturing in Vertical CBM Wells

Background:Coal-bed methane productivity of single well is very low, and has been the bottleneck of the coal-bed methane industry in China.Objective:Although hydraulic fracturing is the only stimulation measure to develop CBM, it cannot increase production effectively, conventional fracturing method to create opening fractures should be improved. How to make good use of natural fractures, which are plentiful in CBM reservoirs, is also an important subject for hydraulic fracturing.Method:In this paper, the plastic deformation of coal rock is analyzed by harnessing a pseudo-Maxwell creep phenomenon, which is normally present in rock. The Kelvin-Voigt model is utilized to describe pseudo-plastic behavior of coal rock to determine pressurization and decay cyclic time for cyclic fracturing design. The mechanical requirement for shearing natural fractures is also analyzed, and shearing distance between the faces of natural fracture can be calculated by Westergaard stress function. Ultimately, the cyclic fracturing method is proposed according to theories about stress alteration and shearing of natural fractures. This method includes such periods as fracturing, pumping shut-down and so on.Conclusion:A complex fracture system can be created, which consists of opened and sheared fractures, then, large SRV(stimulated reservoir volume)and flowing drainage area can be acquired. In comparison with conventional fracturing method, this new way can make full use of the characteristics of CBM reservoirs and is more suitable to CBM. This method will lead to a significant increase of CBM production, and will achieve huge economic benefits.

A new graded singular finite element for crack problems in functionally graded materials

A new graded singular finite element is proposed for analyzing crack problems in linear elastic isotropic functionally graded materials (FGMs) with spatially varying elastic parameters. The general formulation of the suggested singular element is obtained by analyzing the crack tip stress field using the Westergaard stress function method. The general shape function is generated by integrating the strains obtained from the stress function. The stiffness matrix for the singular element is then determined using the principle of minimum potential energy. Using the displacement continuity between the singular and the adjacent regular elements, stiffness matrices are assembled. This new element is characterized by containing both singular and higher order terms, which provides more precise description of the crack tip fields. The validity of the new element is demonstrated by comparing with existing solutions. This element is implemented for simulating crack problems in FGMs whose elastic properties vary normal to the crack line. Stress intensity factors, energy release rates, levels of non-singular stresses, and stress distributions near the crack tip are investigated. Numerical results reveal that the introduced graded singular element is more efficient than conventional finite elements as it provides more accurate description of the crack tip field with less efforts.

A novel approach to simulate the stress and displacement fields induced by hydraulic fractures under arbitrarily distributed inner pressure

Stress and displacement fields induced by hydraulic fractures have been studied by a lot of researchers since they seriously affect the hydraulic fracture geometry. In previous studies, the distribution of inner pressure was assumed to be uniform or diminishing. Therefore, taking the complexly distributed inner pressures, such as geo-stresses with random fluctuations, into consideration in the classical models was unrealizable. A convenient and stable superposition model is developed to simulate the induced stress and displacement fields around artificial or natural fractures under arbitrarily distributed inner pressures in conjunction with complex variable method in theory of elasticity, which includes Westergaard Stress Function Method. The new model is validated by the analytical solution of induced displacement field around a fracture under linear load. It could be inferred that the superposition model is always convergent with a large enough number of discrete segments since the structure of the superposition model is similar to that of the numerical integrations. Some conclusions are drawn from the simulation results of the stress and displacement fields around fractures, when taking the inner pressure drop, asymmetrical propagation and fluctuating geo-stress into consideration. In consideration of the inner pressure drop in artificial fracture, narrower fracture width will form and the risk of early screen-out will increase significantly. Hence, in this case a greater pad volume or a higher pumping rate is needed. The symmetry of the induced stress and displacement fields would be broken by the asymmetrical propagation of artificial fracture. However, the difference of the results computed by classical model and superposition model is negligible in the far field (where y > a). The influence of fluctuating geo-stress on displacement distribution is non-ignorable since the most important displacement computing data is fracture width, which is the double of the displacement at y = 0.

Antiplane Problem of Periodically Stacked Parallel Cracks in an Infinite Orthotropic Plate

The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. The antiplane problem is turned into the boundary value problem of partial differential equation. By constructing proper Westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks degenerates into an algebra problem. Using the complex variable function method and the undetermined coefficients method, as well as with the help of boundary conditions, the boundary value problem of partial differential equation can be solved, and the analytic expressions for stress intensity factor, stress, and displacement near the periodical parallel cracks tip are obtained. When the cracks spacing tends to infinity, the antiplane problem of the periodic parallel cracks degenerates into the case of the antiplane problem of a single central crack.

A new hybrid element analysis for exponentially varying FGM based on an asymptotic stress field

A hybrid finite element for exponentially varying Functionally Graded Material (FGM) is established based on a derived stress field for an intact plate. Elements with both single layer and two layers of gradient are presented. The general stress field is obtained through an asymptotic analysis coupled with Westergaard's stress function approach. With the aid of boundary collocation technique, convergence of the derived stress field is verified. Effectiveness and efficiency of the element is investigated through calculation and comparison. Based on that element, stress field of FGM plate under different loading cases is studied briefly.

Elasto-Plastic Fracture Analysis of Finite-Width Cracked Stiffened Plate

Thispaper adopts the crack line-field method to analyze finite-width stiffenedplates with central through I-type crack under uniaxial tensile loading. Themethod completely abandons the small scale yield hypothesis. The plastic stressand strain fields at crack tips and the plastic-zone length can be accurately determinedby combining with equivalent shear stress of Westergaard stress function in theposition of stiffener. It can be seen from the illustrative example that theresults of the paper agree well with those by finite element method.

Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic Plate

The mechanical behavior near crack tip for periodic parallel cracks in an orthotropic composite plate subjected to the uniformly distributed load within the cracks surface is studied. The mechanical problem is turned into the boundary value problem of partial differential equation. By using the periodicity of the hyperbolic function in the complex domain and constructing proper Westergaard stress function, the periodicity of parallel cracks can be removed. Using the complex variable function method and the undetermined coefficients method, the boundary value problem of partial differential equation can be solved with the help of boundary conditions. The analytic expressions for stress intensity factor, stress, and displacement near the crack tip of periodical parallel cracks are obtained. When the vertical distance of cracks tends to infinity, the stress intensity factor degenerates into a single central crack situation. The stress intensity factor around the crack tip of periodic parallel cracks in an orthotropic composite plate depends on the shape factor. The interaction happens between the cracks. Finally, a numerical analysis of the stress and displacement changed with the polar angle is done.

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

Influence of material ductility and crack surface roughness on fracture instability

This paper presents a stability analysis for fractal cracks. First, the Westergaard stress functions are proposed for semi-infinite and finite smooth cracks embedded in the stress fields associated with the corresponding self-affine fractal cracks. These new stress functions satisfy all the required boundary conditions and according to Wnuk and Yavari's (2003 Eng. Fract. Mech. 70 1659–74) embedded crack model they are used to derive the stress and displacement fields generated around a fractal crack. These results are then used in conjunction with the final stretch criterion to study the quasi-static stable crack extension, which in ductile materials precedes the global failure. The material resistance curves are determined by solving certain nonlinear differential equations and then employed in predicting the stress levels at the onset of stable crack growth and at the critical point, where a transition to the catastrophic failure occurs. It is shown that the incorporation of the fractal geometry into the crack model, i.e. accounting for the roughness of the crack surfaces, results in (1) higher threshold levels of the material resistance to crack propagation and (2) higher levels of the critical stresses associated with the onset of catastrophic fracture. While the process of quasi-static stable crack growth (SCG) is viewed as a sequence of local instability states, the terminal instability attained at the end of this process is identified with the global instability. The phenomenon of SCG can be used as an early warning sign in fracture detection and prevention.

Generalized Westergaard Stress Functions as Fundamental Solutions

Generalized Westergaard Stress Functions as Fundamental Solutions

Stress Field Analysis of Mode II Periodic Cracks-Tip

The cracks-tip field on ModeⅡperiodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to the concentrated force was studied. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. Owing to the distribution of periodic cracks, stress intensity factor(SIF) depends on the shape factor, which is greatly influenced by the crack spacing and the crack length. The results show that interaction happens between the periodic cracks, and that scale effect of the stress intensity factor in cracks-tip is obvious.

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.

Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E11, E22, G12, ν12 are replaced by an effective stiffness \({E=\sqrt {E_{11} E_{22}}}\), a stiffness ratio \({\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-\nulldelimiterspace} {E_{22}}} \right)}\), an effective Poisson’s ratio \({\nu =\sqrt {\nu_{12}\nu _{21}} }\) and a shear parameter \({k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-\nulldelimiterspace} {2G_{12}}}\right)-\nu }\). An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.

Mode-I crack-tip stress fields for inhomogeneous orthotropic medium

Asymptotic analysis coupled with Westergaard stress function approach is used to develop quasi-static stress fields for a crack oriented along one of the principal axes of inhomogeneous orthotropic medium. In the formulation, four independent engineering constants, E 11, E 22, G 12, ν 12 are replaced by an effective stiffness E = E 11 E 22 , a stiffness ratio δ = ( E 11/ E 22), an effective Poisson’s ratio ν = ν 12 ν 21 , and a shear parameter k = ( E/2 G 12) − ν. It is assumed that the effective stiffness varies exponentially along one of the principal axes of orthotropy. The first two terms in the expansion of stress field are derived to explicitly bring out the influence of non-homogeneity on the structure of the stress field. Using the derived stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack-tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.

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.

T-Stresses Across Static Crack Kinking

This paper is concerned with the T-stress change before and after crack kinking in two-dimensional elastic solids. By using asymptotic analysis and the Westergaard stress function method, approximate analytical formulas for calculating the T-stress as well as stress intensify factors of an infinitesimal kink are given. Contributions from the T-stress before crack kinking, to the T-stress and the stress intensity factors of the kinked crack, are clearly described. It is noted that since the sign of the T-stress of a kinked open crack might be different from that of a main crack, simply using the sign of the T-stress before crack kinking is not sufficient to determine crack growth stability as observed in recent experiments.