T -stress in piezoelectric solid

The non-singular and bounded terms for stresses near the crack tip were investigated. The crack problem in a transversely isotropic piezoelectric solid for the plane problem was dealt with. The principle of superpsition and the Plemelj formulation were introduced. The non-singular terms are given by solving Rieman-Hilbert problem. It is shown that the non-singular terms are influenced by the elastic and electric constants. Contributed by WANG Biao Foundation items: the Nature Science Foundation of Shandong of China (Q99F15); the Post Doctoral Science Foundation of Heilongjiang Provice of China Biography: MA Hao (1967≈), Professor, Doctor     

Non-singular terms for multiple cracks in anisotropic elastic solids

The non-singular and bounded terms for the stresses near the crack tip are investigated. This paper deals with the multiple crack problem for an infinite plate. The original problem is decomposed into three elementary subproblems: (1) the problem for remote uniform stress filed without cracks; (2) the single crack problem with traction applied along the crack face and (3) the problem for the influence of the other cracks. Several examples for collinear and non-collinear cracks are discussed and the results are shown.     

Biaxial Load Effect on Crack Initiation for Orthotropic Materials

This paper is concerned with the study of the elastostatic fracture response of an orthotropic plate with an inclined crack and subjected at infinity to a biaxial uniform load. To this end an unconventional approach to the derivation of the complex variable expressions of the elastic fields is proposed. The above formulation has been used to solve the boundary value problem as superposition of Mode-I and Mode-II crack problems and it is shown that the near tip asymptotic expressions of stress and displacement fields are affected by non-singular terms originated by load biaxiality. The maximum circumferential tensile stress criterion is applied in order to investigate the effects of non-singular terms on the angle of crack extension.     

Limitations to the small scale yielding approximation for crack tip plasticity

Recent finite-element results by S.G. Larsson and A.J. Carlsson suggest a limited range of validity to the ‘small scale yielding approximation’, whereby small crack tip plastic zones are correlated in terms of the elastic stress intensity factor. It is shown with the help of a model for plane strain yielding that their results may be explained by considering the non-singular stress, acting parallel to the crack at its tip, which accompanies the inverse square-root elastic singularity. Further implications of the non-singular stress term for crack tip deformations and fracturing are examined. It is suggested that its effect on crack tip parameters, such as the opening displacement and J-integral, is less pronounced than its effect on the yield zone size.     

Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic-plastic materials

The plane strain elastic-plastic state at a crack tip is determined for compact tension, bend, double edge-cracked and centre-cracked specimens using a finite element method with triangular constant-strain elements. The solutions are found to differ by 10 to 30 per cent at the ASTM-limit as regards fracture surface displacement, normal stress and plastic zone size. In order to bring the boundary layer solution for the crack problem into agreement with the solution for a specific specimen one has to modify this solution. The modification consists of an addition to the boundary tractions for the boundary layer problem of tractions corresponding to the non-singular, constant second term in a series expansion of the normal stress parallel to the crack plane.     

Crack energy density of a piezoelectric material under general electromechanical loading

Crack energy density is considered and used as a possible fracture parameter in piezoelectricity under arbitrary electromechanical remote loads. The closed-form solution of a crack in a piezoelectric infinite plate subjected to general static electromechanical loading is obtained through a method alternative to the more common Stroh’s formalism. This analytical method, which is based on the spectral theorem of linear algebra, involves a transformation of similarity induced by the fundamental matrix in order to express the equations governing the problem in terms of complex potentials. The application of the mechanical boundary condition of stress-free crack and of one of the three considered electric boundary conditions (impermeable, permeable or semipermeable) leads then to the formulation of a Hilbert problem whose solution yields the stress and displacement fields. The crack energy density factors for mixed mode are then calculated under different mechanical and electrical loadings, as well as under different electric boundary conditions. The non-singular terms of the stress expressions are retained as well. The definition of the minimum energy density fracture criterion, as proposed by Sih, is given, and the influence of load biaxiality and positive or negative applied electric field on the criterion results is analyzed. The prediction of the incipient branching angle as from the energy density approach is also compared to that arising from the maximum circumferential stress theory for a mixed mode loading condition. Numerical results and graphs are presented and discussed for a PZT-4 piezoelectric ceramic.     

Calculation of near-tip non-singular stresses for pressurized cracks

When the crack surfaces are traction-free, there is only one constant term T in the near-tip stress field, which contributes uniformly to the stress component acting in the direction parallel to the crack flank. As to pressurized cracks, the non-singular part of the asymptotic stresses appears to be more complicated and is no longer characterized only by the constant T. In this work, an effective numerical approach is developed for calculation of the non-singular parts of the asymptotic near-tip stresses under the action of nonuniform crack surface pressures. With this approach, the near-tip non-singular stress field can be accurately evaluated by direct use of regular numerical methods such as finite elements.     

A special crack tip displacement discontinuity element

Based on the analytical solution to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid and the note of the crack tip element by Crouch, in the present paper, the special crack tip displacement discontinuity element is developed. Further the analytical formulas for the stress intensity factors of crack problems in general plane elasticity are given. In the boundary element implementation the special crack tip displacement discontinuity element is placed locally at each crack tip on top of the non-singular constant displacement discontinuity elements that cover the entire crack surface. Numerical results show that the displacement discontinuity modeling technique of a crack presented in this paper is very effective.     

Crack kinking in a piezoelectric solid

The intrinsic coupling between the mechanical and the electric fields assigns a uniquefeature for the fracture in a piezoelectric solid. We model the kink of a crack by continuousdistribution of edge dislocations and electric dipoles. The problem admits an approach based onthe Stroh formalism. A set of coupled singular integral equations are derived for the dislocationand electric dipole density functions associated with a kinked crack. Numerical results indicatethat the crack tends to propagate in a straight line under a tensile stress and a positive electricfield. For a crack subjected to the mixed mode mechanical loading, a superimposed positiveelectric field tends to reduce the kink angle. The influence of the non-singular T-stress-chargeparallel to a crack is also investigated. It is shown that a transverse tensile stress or a positivetransverse electric field will lead to further deviation of the kinked crack from the crackextension line.     

A numerical analysis of perpendicular cracks under general in-plane loading with a hybrid displacement discontinuity method

In this paper, a numerical analysis of perpendicular cracks under general in-plane loading is performed by using a hybrid displacement discontinuity method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results show that the numerical approach is simple, yet very accurate for calculating numerically stress intensity factors for perpendicular cracks under general in-plane loading.     

On partial sliding along a planar crack: the case of a circular sliding zone

Summary A problem of partial sliding along a planar crack with a local drop in frictional resistance is investigated. A sliding zone initiates in the area of reduced friction, and then propagates as the applied shear load is monotonously increased. The problem is formulated in general terms, and then solved for the case when sliding spreads as a penny-shaped zone. Conditions under which the front of the zone stays circular during sliding are analyzed. It is observed that the axisymmetry of the profile of frictional resistance does not necessarily guarantee uniform propagation of sliding in the radial direction. The circular shape becomes the most favorable growth condition only if the shear modes are related in a certain way. The problem is studied based on the criterion of propagation that stress intensity factors(SIFs) for II and III modes vanish on the boundary of the sliding zone. The singular integrals in expressions for the SIFs are reduced to non-singular ones. Analytical solutions are derived for a number of special cases where the radius of the sliding zone is related to the applied shear load.This work is supported by the National Science Foundation through grant DGE-0209543 to the University of New Mexico.     

A boundary element modeling of fatigue crack growth in a plane elastic plate

This paper presents an extension of a boundary element method to fatigue growth analysis of mixed-mode cracked plane elastic bodies. The method consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity element due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the modified maximum strain energy density criterion. In numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the boundary element method. Crack growth is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. Some numerical results of fatigue growth in a plane elastic plate with a center-inclined crack under uniaxial cyclic loading are given.     

Limitation on crack speed for a strip-craze model applied to PMMA

A strip-craze model is proposed to study crack propagation in polymers. A nonlinear differential equation is derived to govern the dynamic process of crack propagation. The viscous feature of the material in the craze zone is taken into account by means of an experimentally determined relationship between the craze stress and crack speed. By fitting experimental data of PMMA into the model, some parameters including the strip-craze length are deduced. A non-singular stress is introduced to control the crack propagation with a strip craze at its tip. Variations of the crack length and the crack speed with time are computed and their dependence on the non-singular stress is investigated. For PMMA, three stages of crack propagation are identified in terms of initial non-singular stress σ_ns0. When σ_ns0<60 MPa, the crack speed mm/s and the crack is basically stationary; when 60 <σ_ns0<95 MPa, then mm/s the crack is in slow propagation; when σ_ns0>95 MPa, then mm/s and the crack is in rapid propagation. The proposed model is applicable only in slow crack propagation.     

FATIGUE GROWTH MODELING OF MIXED-MODE CRACK IN PLANE ELASTIC MEDIA

This paper presents an extension of a displacement discontinuity method with cracktip elements （a boundary element method） proposed by the author for fatigue crack growth analysis in plane elastic media under mixed-mode conditions. The boundary element method consists of the non-singular displacement discontinuity elements presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or right crack-tip element is placed locally at the corresponding left or right crack tip on top of the non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the maximum circumferential stress criterion. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the numerical approach. Crack growth is modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characteristics of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the fatigue growth process of cracks emanating from a circular hole in a plane elastic plate is simulated using the numerical simulation approach.     

Fracture analysis of V-notched components – Effects of first non-singular stress term

The effect of the first non-singular stress term on the fracture behavior of notched structures was investigated under symmetric geometry and loading conditions. According to the Williams series expansion, for a large domain of notch angles the non-singular stress terms of sharp notches are functions of complex eigenvalues and their corresponding complex coefficients. Hence, a new representation of stress field near the notch tip was developed in which the higher order terms are expressed as several explicit functions of real and imaginary parts of both the complex eigenvalues and their complex coefficients. A critical stress criterion was then applied to the new stress formulations to assess the influence of the first non-singular stress term on the apparent fracture toughness. Several finite element analyses were also performed on two laboratory specimens in order to show the effects of first non-singular term on the near field stress distribution of notched specimens. The results demonstrated that neglecting the first non-singular stress term could lead to significant errors in predicting the apparent fracture toughness of notched components.     

Interface crack extension at any constant speed in orthotropic or transversely isotropic bimaterials––I. General exact solutions

A semi-infinite crack along the interface of two dissimilar half-spaces extends under in-plane loading. Each half-space belongs to a class of orthotropic or transversely isotropic elastic materials, the crack can extend at any constant speed, and all six possible relations between the four body wave speeds are considered. A steady dynamic situation is treated, and exact full displacement fields derived. A key step is a factorization that produces, despite anisotropy, simple solution forms and compact crack speed-dependent functions that exhibit the Rayleigh and Stoneley speeds as roots. These roots are calculated for various representative bimaterials.Closed-form crack opening displacement gradient and interface stress fields are also derived from a general set of coupled singular integral equations. The equation eigenvalues can, depending on crack speed, be complex/imaginary conjugates, purely real, or zero. This suggests possibilities observed in other studies: oscillations and square-root singular behavior at the crack edge, non-singular behavior, singular behavior not of square-root order, and the radiation of displacement gradient discontinuities at crack speeds beyond the purely sub-sonic range.These possibilities are explored further in terms of two important special cases in Part II of this study [Int. J. Solids Struct., 39, 1183–1198].     

Boundary element weight function analysis of a strip yield crack in a rotating disk

The boundary element method combined with subtration of Bueckner singular fields are used to obtain weight functions for an internal edge crack in a rotating annular disk. A previously developed, general representation of the weight function is used which leads to integrals that can be evaluated analytically to obtain the stress intensity factor and surface displacements of the crack. The determination of crack tip opening displacements for the strip yield crack is reduced to a non-singular integral which can be evaluated in closed form. The strip yield zone length and crack tip opening displacement are determined for an internal radial crack in a rotating annular disk for a range of crack lengths and rotational speeds.     

A crack along a circular interface between dissimilar media

Summary The plane problem dealing with an arc crack along the interface of a circular elastic inclusion and an unbounded different elastic medium is theoretically studied. The closed form solutions of the stresses and displacements along the contour are derived under general biaxial stresses at infinity. The stress field near the crack tip is evaluated and the presence of non-singular terms is pointed out to infer their effects on the quantities predicted by local fracture criteria. An overall stress intensity factor connected with the rate of strain energy release is also introduced to obtain crack growth information on the interface.

---

Sommario Viene studiato teoricamente il problema piano di una fessura all'interfaccia tra un inclusione circolare elastica ed un mezzo con proprietà elastiche diverse. Vengono ricavate le espressioni degli sforzi e degli spostamenti sul contorno in regine di carico biassiale e rilevata l'importanza dei termini non singolari, che compaiono nelle espressioni dei campi locali degli sforzi, in vista dell'applicazione dei criteri locali di frat tura. Viene inoltre definito un fattore globale di intensità degli sforzi, connesso con il rilascio di energia di deformazione, mediante il quale si possono ottenere informazioni sulla propagazione della fessura all'interfaccia.

---

---

Financial support of the National Research Council (C.N.R.) through research contribution n. 80.02203.07.     

Spread of plasticity from stacked stress concentrations

A large elastic solid containing an infinite sequence of slitlike relaxed cracks with a constant distance of vertical separation is considered. The solid is deforming under plane strain shear conditions (mode II). The plastic relaxation around each of the cracks is represented by a suitable distribution of edge dislocations coplanar with the crack itself, the distribution being determined from a singular integral equation. This equation is solved numerically using an expansion of the non-singular part of the kernel in a series of Chebyshev polynomials. Solutions are obtained for the extent of spread of plasticity from each crack and for the associated dislocation distribution as a function of the crack spacing and the applied load. The results are applied to a brief discussion of the fracture process at stress concentrations using the crack opening displacement criterion.     

Crack front curvature effect on stress intensity factor of through and part-through thickness cracks

The character of the local stresses and displacements are determined for a through crack with finite radius of curvature in a finite thickness plate. Numerical results obtained from the boundary element method show that the solutions are sufficiently accurate for /a ≤ 0.03 and 0.03 ≤ /a ≤ 0.1, where and a represent, respectively, the crack front radius of curvature and crack dimension such that a is the width of a through thickness crack and the depth of a part-through crack. For /a ≤ 0.1, the asymptotic singular stress field dominates such that the Mode I stress intensity factor K₁ can be evaluated. As the crack border radius of curvature is increased for /a ≥ 0.1, the non-singular terms become significant such that K_I would no longer dominate. Other failure criteria would have to be invoked to address fracture initiation.