Electro-mechanical pre-fracture zones for an electrically permeable interface crack in a piezoelectric bimaterial

A plane strain problem for two piezoelectric half-spaces adhered by a very thin isotropic interlayer with a crack under the action of remote mixed mode mechanical loading and electrical flux is considered. The crack is situated either at an interface or in the interlayer. It is assumed that the substrates are much stiffer than the intermediate layer. Therefore, pre-fracture zones (plastic or damage) arise at the crack continuations. Normal and shear stresses are assumed to be constant in this zones and to satisfy some material equation, which can be taken from theory or derived experimentally. Modeling the pre-fracture zones by the crack continuations with unknown cohesive stresses on their faces reduces the problem to elastic interface crack analysis leading to a Hilbert problem. This problem is solved exactly. The pre-fracture zone lengths and stresses in these zones are found from algebraical and transcendental equations. The latter are derived from the conditions of stress finiteness at the ends of pre-fracture zones and the material equations. The electrical displacement at any point of the pre-fracture zones is found in closed form as well. Particular cases of symmetrical loading and of equivalent properties of the upper and lower bimaterial components are considered. Numerical results corresponding to certain material combinations and interlayer material equations are presented and analysed. In the suggested model, any singularities connected with the crack are eliminated, i.e., all mechanical and electrical characteristics are limited in the near-crack tip region.     

Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading

A plane problem for a tunnel electrically permeable interface crack between two semi-infinite piezoelectric spaces is studied. A remote mechanical and electrical loading is applied. Elastic displacements and potential jumps as well as stresses and electrical displacement along the interface are presented using a sectionally holomorphic vector function. It is assumed that the interface crack includes zones of crack opening and frictionless contact. The problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved analytically. From the obtained solution, simple analytical expressions are derived for all mechanical and electrical characteristics at the interface. A quite simple transcendental equation, which determines the point of separation of open and close sections of the crack, is found. For the analysis of the obtained results, the main attention is devoted to the case of compressive-shear loading. The analytical analysis and numerical results show that, even if the applied normal stress is compressive, a certain crack opening zone exists for all considered loading values provided the shear field is present. It is found that the shear stress intensity factor at the closed crack tip and the energy release rates at the both crack tips depend very slightly on the magnitude of compressive loading.     

Contact Interaction of the Faces of a Rectangular Crack under Normally Incident Tension–Compression Waves

A three-dimensional problem on the contact interaction between the faces of a rectangular crack under a normally incident harmonic tension–compression wave is considered. The problem is solved by using the method of boundary integral equations and an iterative algorithm. The contact forces and the discontinuity in the displacement of the crack faces are studied. The results obtained are compared with those for a finite plane crack.     

Contact zone assessment for a fast growing interface crack in an anisotropic bimaterial

Summary A plane strain problem for a crack with a frictionless contact zone at the leading crack tip expanding stationary along the interface of two anisotropic half-spaces with a subsonic speed under the action of various loadings is considered. The cases of finite and infinite-length interface cracks under the action of a moving concentrated loading at its faces are considered. A finite-length crack for a uniform mixed-mode loading at infinity is considered as well. The associated combined Dirichlet-Riemann boundary value problems are formulated and solved exactly for all above-mentioned cases. The expressions for stresses and the derivatives of the displacement jumps at the interface are presented in a closed analytical form for an arbitrary contact zone length. Transcendental equations are obtained for the determination of the real contact zone length, and the associated closed form asymptotic formulas are found for small values of this parameter. It is found that independently of the types of the crack and loading, an increase of the crack tip speed leads to an increase of the real contact zone length and the correspondent stress intensity factor. The latter increase significantly for an interface crack tip speed approaching the Ragleigh wave speed.     

Effect of friction on the interface crack loaded in shear

The conventional formulation used in the past for problems involving interface cracks leads to a physical contradiction: The two sides of the crack are assumed to be free of tractions in the formulation, but the crack faces are seen to overlap after the solution is constructed. This unsatisfactory feature can be eliminated by introducing contact zones at the tips of an interface crack. The present article investigates the effect of friction in the contact zones for loads that start from zero and are increased monotonically. As an application, shear loading is considered, and the problem is reduced to a singular integral equation with a Cauchy-type kernel which is solved numerically. The results show that one of the contact zones is large and that friction affects the global nature of the stress fields. The results worked out include also the stress intensity factors, crack opening displacement, and the pressure distribution in the larger contact zone.

---

Résumé La formulation habituelle utilisée dans le passé pour des problèmes de fissures interfaciales conduit à une contradiction physique: Les deux lèvres de la fissure doivent être libres de contraintes, mais la solution peut impliquer une interpénétration des bords. Il est possible d'éliminer cette contradiction physique en introduisant des zones de contact aux pointes de la fissure. Le présent article étudie l'effet du frottement dans les zones de contact pour des charges croissantes à partir de zero. Comme application, on considère une charge de cisaillement et le problème se réduit à une équation intégrale singulière avec un noyau de type Cauchy. Cette équation est résolue numériquement. Les résultats montrent que l'une des zones de contact est grande et que le frottement modifie l'ensemble du champ de contraintes. On a aussi obtenu les facteurs d'intensité de contraintes, l'ouverture de la fissure et la distribution du pression dans la plus grande zone de contact.

     

Modeling of pre-fracture zones for limited permeable crack in piezoelectric materials

A plane-strain problem for a limited permeable crack in an adhesive thin interlayer between two semi-infinite piezoelectric spaces is considered. The tensile mechanical stress and the electric displacement are applied at infinity. The interlayer is assumed to be softer than the connected materials; therefore, the zones of mechanical yielding and electric saturations can arise at the crack tips on the continuations of the crack. These zones are considered in this work. It was assumed that the length of electric saturation zones is larger than the length of mechanical yield zones. The zones of mechanical yielding are modeled by the crack continuations with normal compressive stresses applied at its faces. The electric saturation zones are modeled by segments at the crack continuations with prescribed saturated electric displacements. These electric displacements can linearly vary along the mechanical yielding zones. The problem is reduced to the Hilbert–Riemann problem of linear relationship, which is solved exactly. The equation for the determination of the yielding zones length, the expressions for the crack-opening displacement jump, electric potential jump, and J-integral is obtained in an analytical form. In case of finite size body, the finite elements method is used and the variation in the fracture mechanical parameters with respect to this size is demonstrated.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

Periodic set of limited electrically permeable interface cracks with contact zones

Plane problem for an infinite space composed of two different piezoelectric or piezoelectric/dielectric semi-infinite spaces with a periodic set of limited electrically permeable interface cracks is considered. Uniformly distributed electromechanical loading is applied at infinity. The frictionless contact zones at the crack tips are taken into account. The problem is reduced to the combined Dirichlet–Riemann boundary value problem by means of the electromechanical factors presentation via sectionally analytic functions, assuming that the electric flux is uniformly distributed inside the cracks. An exact solution of the problem is proposed. It permits to find in a closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux value. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region.Formulae for stresses, electric displacement vector, elastic displacements and electric potential jump at the interface as well as the intensity factors at the crack tips are given. Equation for the contact zone length determination is presented. Calculations for certain material combinations are carried out. The influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters is analyzed.     

Study of Iterative Algorithms for Solution of Dynamic Contact Problems for Elastic Cracked Bodies

A plane problem is solved for the contact interaction between the faces of a rectilinear crack under the action of a normally incident harmonic tension–compression wave. Iterative algorithms are presented to solve the problem for both given initial distribution of contact forces and given initial discontinuity in the displacements of the crack faces. The convergence rates of the algorithms, the maximum contact forces, and displacement discontinuities are compared.     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.     

Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding

A plane problem for two identical piezoelectric semi-infinite spaces adhered by means of a thin isotropic interlayer is considered. It is assumed that a crack of a limited electric permeability occurs in the interlayer parallel to its faces. Combined electromechanical loading is prescribed at infinity. It is assumed that the interlayer is softer than the adherent materials. To avoid the singularities, which are typical for the Griffith crack model, two distinct zones – a zone of mechanical yielding and a zone of electrical saturation – of unknown lengths are introduced as crack continuations. These lengths can be essentially different, with the zone of mechanical yielding significantly longer or shorter than the zone of electrical saturation. Assuming that the interlayer thickness tends to zero, a constant normal stress is prescribed in the zone of mechanical yielding and a saturated electrical displacement is prescribed in the zone of electrical saturation. Outside of these zones, the semi-infinite spaces are assumed to be perfectly bonded. This formulation results in a linear fracture mechanics problem with unknown pre-fracture zone lengths. The problem, formulated mathematically by a system of two equations of linear relationship, is solved exactly. The unknown yield and saturated zones lengths are found from the conditions of finiteness of stress and electrical displacement at the ends of these zones for the both cases when the electrical saturated zone is longer and shorter than the zone of mechanical yielding. It is shown that the same equation as for the Griffith crack model can be used for the determination of the electrical displacement in the crack region. The main results of the paper are obtained in the form of simple analytical equations which are convenient for engineering applications. Some numerical illustrations in graphical and tabular form show dependencies of the pre-fracture zone lengths, the energy release rate, the mechanical displacement and electrical potential jumps on the electromechanical loading and the electrical permeability of the crack medium.     

Contact problem for a plane crack under a normally incident antiplane shear wave

The contact-interaction problem for a stationary plane crack with friction between its edges under the action of a normal (to the crack plane) harmonic shear wave is addressed. Antiplane deformation conditions are considered. The distribution of contact forces and displacement discontinuity of crack edges are studied Published in Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 138–142, May 2007.     

Modeling of Cracks with Nonlinear Effects at the Tip Zones and the Generalized Energy Criterion

Starting with a plane anisotropic homogeneous elasticity problem in a domain with an interior crack, we develop a mathematical frame where nonlinear effects in the tip zones like crack kinking or plastic zones can be modeled in an enlarged state space with the help of additional conditions at the crack tips. Using generalized Green’s formulae, we show that the solutions to these problems turn out to minimize energy functionals which contain terms additional to the classical elastic energy and work of external forces. They can be interpreted as performed work and energy stored in the crack tips. Within the theory of matched asymptotic expansions, the general type of these energy functionals can be characterized in a form applicable to mechanical problems.     

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam.The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.     

Elastic-plastic mechanics of steady crack growth under anti-plane shear

For a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.     

Interface crack with a contact zone in an isotropic bimaterial under thermomechanical loading

A plane problem for a thermally insulated interface crack with a contact zone in an isotropic bimaterial under tension–shear mechanical loading and a temperature flux is considered. The expressions for the stresses and the electrical flux as well as for the derivatives of the displacement and the temperature jumps at the material interfaces via sectionally holomorphic mechanical and thermal potential functions are given. After the solution of the thermal problem the inhomogeneous combined Dirichlet–Riemann boundary value problem is formulated and solved exactly. The stresses at the interface and the stress intensity factors at the singular points are presented in a clear analytical form. Special attention is devoted to the case of a small contact zone when the stress intensity factors can be presented in form similar to the associated presentation for an “open” crack model. A transcendental equation and an asymptotic analytic formula for the determination of the real contact zone length are derived. It is shown that for a certain bimaterial this length as well as the correspondent stress intensity factor are defined by a single parameter which depends on the normal-shear loading and the heat flux.     

On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

On dynamical fracture mechanics in the case of polyharmonic loading by SH-waves

We solve a fracture-mechanics problem for a cracked material under polyharmonic loading (waves of various lengths propagating through an elastic material). The frictional contact interaction of the finite crack edges in a plane is analyzed for the case of normal incidence of two harmonic shear waves with different frequencies. The forces of contact interaction and displacement discontinuity are determined. The effect of the wave frequency on the stress intensity factor for different normalized wave numbers is examined     

Crack emanating from an open notch

This paper is concerned with the plane elastostatic problem of a crack emanating from an open notch where it is assumed that the crack is directed parallel to one side of the notch. The problem is reduced to two simultaneous Fredholm integral equations which are then solved numerically for three loading conditions of the crack faces. In particular, it is shown that the crack tip intensity factors are almost independent of crack length if the singular stress field associated with the uncracked notch is used as the loading of the crack faces.     

Analysis of the Singularity for the Vertical Contact Problem of Line Crack-Inclusion

ＩｎｔｒｏｄｕｃｔｉｏｎＵｐｔｏｎｏｗ ,ｔｈｅｔｅｃｈｎｉｃａｌｌｉｔｅｒａｔｕｒｅｏｎｓｅｐａｒａｔｅｃｒａｃｋｓ,ｖｏｉｄｓ,ｉｎｃｌｕｓｉｏｎｓａｎｄｔｈｅｉｎｔｅｒａｃｔｉｏｎｓｂｅｔｗｅｅｎｃｒａｃｋｓａｎｄｉｎｃｌｕｓｉｏｎｓｈａｖｅｂｅｅｎｑｕｉｔｅｅｘｔｅｎｓｉｖｅ.Ｈｏｗｅｖｅｒ,ｔｈｅｃｏｎｔａｃｔｐｒｏｂｌｅｍｓｏｆｃｒａｃｋ＿ｉｎｃｌｕｓｉｏｎｄｏｎｏｔｓｅｅｍｔｏｂｅａｓｗｉｄｅｌｙｓｔｕｄｉｅｄ .Ｔｈｉｓｐａｐｅｒｃａｎｂｅｒｅｇａｒｄｅｄａｓｔｈｅｆｕｒｔｈｅｒｒｅｓｅ…