Thermoelastic problem of two collinear cracks

A two-dimensional problem of two collinear cracks influenced by concentrated heat sources is studied. The considerations are limited to a problem of normal opening of the crack (Mode I) assuming that cracks of different lengths are thermally insulated and free of loads. The formulas describing displacements and stresses at any point of the body are given. The stress intensity factors at the crack tips and the openings of the cracks are obtained. To avoid possible overlapping of the cracks boundaries, the position of the heat sources and the lengths of the cracks are analyzed to ensure non-negative stress intensities factors (SIF's) at all the crack tips. The solution to the above problem has been derived using the method given by Panasuyk et al. [V.V. Panasuyk, M.P. Savruk, A.P. Datshshyn, Stress Distribution around Cracks in Plates and Shells, Naukova Dumka, Kiev, 1976 (in Russian); V.V. Panasuyk, M.P. Savruk, A.P. Datsyshyn, A general method of solution of two-dimensional problems in the theory of cracks, Eng. Fracture Mech. 9 (1977) 481–497.     

First fundamental problems of anisotropic elastic plane weakened by periodic collinear cracks

Using the method of complex functions, we discuss the first fundamental problems of an anisotropic infinite elastic plane weakened by periodic collinear cracks and with periodic boundary loads on both sides of the cracks. This problem was considered by Cai [Engineering Fracture Mechanics 46（1）, 133-142 （1993）]. However, the previous method is imperfect. Therefore, the results are incorrect. Here, we revise the method and give a correct solution.     

On collinear microcrack–inclusion arrays interaction and related problems

Investigated is a crack problem for an array of collinear microcracks in composite matrix. Inclusions are situated in between the neighbouring microcracks tips and exhibit different elastic properties than matrix. The problem is solved using the technique of distributed dislocations. A developed approximate fundamental solution for a single dislocation lying in a general point between inclusions is employed in the distribution of continuously distributed dislocation to cracks modelling. Stress intensity factor is calculated for various cracks/inclusions geometries and elastic moduli mismatches. Stability and/or instability of the straight microcrack paths is investigated for slowly growing microcracks with inclusions located in between the neighbouring microcracks tips. Applications to periodic microcrack tunnelling and microcracks weakening ahead of the main crack are discussed.     

Interaction of microcracks with a macrocrack yielded in a narrow strip

Plastic zone growth of collinear cracks has had a longstanding interest in ductile fracture. This work further considers yield zone growth in an isotropic, homogeneous elastic–perfectly plastic infinite plate containing a macrocrack with several neighboring microcracks. Normal loading is considered at distances far away from the cracks. The strip yield is adopted where the plastic zone is assumed to be confined to two narrow strips extending from the ends of a finite length crack while the microcracks are assumed to be elastic. The plastic zone length and crack opening displacement are found from asymptotic solution and compared with finite element solution.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

Surface cracks in a plate of finite width under extension or bending

In this paper the problem of a finite plate containing collinear surface cracks is considered. The problem is solved by using the line spring model with plane elasticity and Reissner's plate theory. The main purpose of the study is to investigate the effect of interaction between two cracks or between cracks and stress-free plate boundaries on the stress intensity factors and to provide extensive numerical results which may be useful in applications. First, some sample results are obtained and are compared with the existing finite element results. Then the problem is solved for a single (internal) crack, two collinear cracks and two corner cracks for wide range of relative dimensions. Particularly in corner cracks the agreement with the finite element solution is surprisingly very good. The results are obtained for semielliptic and rectangular crack profiles which may, in practice, correspond to two limiting cases of the actual profile of a subcritically growing surface crack.     

High-velocity penetration of a melting solid by a slender body

The high-velocity penetration of a melting solid by a thermally insulated slender body is considered. Under certain constraints on the dimensionless melting parameters the flow in the molten layer can be described within the framework of lubrication theory. The local angle of inclination of the body and the surfaces of the molten layer with respect to the velocity is assumed to be small and is taken into account in the linear approximation. The heat flow into the solid is found by simulating the body and the molten layer by means of a segment with distributed heat sources. Within the framework of this simple formulation a closed solution of the problem of the fusion zone around a moving slender body is constructed. The dependence of the shape of the molten layer and the structure of the temperature and longitudinal velocity fields in the layer on the shape of the body and the other governing parameters of the problem is investigated. The results obtained also give a solution of the problem of the melting of a solid rubbing at high velocity against a thermally insulated rough substrate, when the characteristic height of the roughness is of the order of the thickness of the layer and the characteristic length of the order of the contact length.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 43–48, November–December, 1992.     

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.     

Thermal stress analysis of a three-dimensional anticrack in a transversely isotropic solid

A three-dimensional analysis is performed for an infinite transversely isotropic elastic body containing an insulated rigid sheet-like inclusion (an anticrack) in the isotropy plane under a remote perpendicularly uniform heat flow. A general solution scheme is presented for the resulting boundary-value problems. Accurate results are obtained by constructing suitable potential solutions and reducing the thermal problem to a mechanical analog for the corresponding isotropic problem. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and analyzed. This solution is compared with that corresponding to a penny-shaped crack problem.     

Dynamic stress intensity factors of two collinear cracks in orthotropic medium subjected to time-harmonic disturbance

Dynamic stresses are obtained for an infinite orthotropic medium weakened by two collinear cracks. Time-harmonic elastic waves are interrupted at normal incidence by the line cracks. Fourier transform is applied reducing the problem to solving a pair of dual integral equations. Solution method involves expanding the crack surface displacement in a series of functions that vanish along a collinear line outside the cracks. The unknown coefficients in the series are evaluated by using the Schmidt method. Dynamic stress intensity factors are computed and displayed graphically for an orthotropic medium that corresponds to the elastic properties of boron-epoxy composite.     

Circulation flow at the front surface of a sphere in a supersonic wake

Several theoretical and experimental studies of supersonic flow past a blunt body located in the wake behind another body have been made [1–7]. It has been shown that a reverse-circulation flow can occur in the shock layer at the front surface. The possibility of such a flow forming depends on the nonuniformity of the freestream flow and the Reynolds number. This paper presents new results of the theoretical study of the structure of the shock wave at the front surface of such a sphere, obtained on the basis of numerical solution of Navier-Stokes equations. It is shown that for a fixed nonuniformity of the freestream flow, an increase in the Reynolds number and cooling of the surface of the body lead to the formation of a secondary vortex in the region where the contour of the body intersects the axis of symmetry. A study is made of the variations of the drag and heat transfer parameters over the front surface of a cooled and thermally insulated sphere. The possibility of numerical simulation of the flow on the basis of the Euler equations is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–148, May–June, 1985.     

Flow Around a Crack in a Porous Matrix and Related Problems

The equations governing plane steady-state flow in heterogeneous porous body containing cracks are presented first. Then, a general transformation lemma is presented which allows extending a particular solution obtained for a given flow problem to another configuration with different geometry, behaviour and boundary conditions. An existing potential solution in terms of discharges along the cracks, established by Liolios and Exadaktylos (J Solids Struct 43:3960–3982, 2006) for non-intersecting cracks in isotropic matrix, is extended to intersecting cracks in anisotropic matrix. The basic problem of a single straight crack in an infinite body submitted to a pressure gradient at infinity is then investigated and a closed-form solution is presented for the case of void cracks (infinite conductivity), as well as a semi-analytical solution for the case of cracks with Poiseuille type conductivity. These solutions, derived first for an isotropic matrix, are then extended to anisotropic matrices using the general transformation lemma. Finally, using the solution obtained for a single crack, a closed-form estimation of the effective permeability of micro-cracked porous materials with weak crack density is derived from a self-consistent upscaling scheme.     

On the stress-strain state of a transversally isotropic body with a paraboloidal inclusion

A closed solution is presented for the three-dimensional problem of the stress-strain state of an unbounded elastic body with a soldered-in transversally isotropic inclusion in the form of a paraboloid of revolution. Here, it is assumed that the body is under axisymmetric tension (compression). A solution of the corresponding problem for a paraboloidal recess is obtained as a special case. Podil’chuk [2, 3] has investigated similar problems for isotropic bodies with an inclusion assuming the form of a paraboloid of revolution or an elliptical paraboloid. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 16–22, November, 1998.     

Contact Zone Approach for a Moving Interface Crack in a Piezoelectric Bimaterial under Thermoelectromechanical Loading

An interface crack of a finite length moving with a constant subsonic speed v along an interface of two semi-infinite piezoelectric spaces is considered. It is assumed that the bimaterial compound is loaded by a remote mixed mode mechanical loading and a thermoelectrical field and that a frictionless contact zone arises at the leading crack tip. Electrically permeable and electrically insulated cases of the open part of the crack are involved into the consideration. By introducing a moving coordinate system at the crack tip the problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved exactly. For both cases of the electrical conditions the transcendental equations are obtained for the determination of the real contact zone length, and moreover, the associated closed form asymptotic formulas are found for small values of this parameter. Variations of the contact zone length and the stress intensity factor with respect to the crack speed and the loading have been investigated both for electrically permeable and electrically insulated cases.     

On the asymptotics of the stress and strain fields near a crack tip

The problem is solved under the plane strain conditions for a crack of general form, which in general is neither a mode I nor a mode II crack. We assume that the strains are small and the material is nonlinearly elastic. The mathematical statement of the problem is reduced to the eigenvalue problem for a system of ordinary nonlinear differential equations. Its solution is obtained numerically. We show that, for an incompressible material with power-law relations between the stress and strain deviators, the solution (the well-known HRR-asymptotics [1, 2]) exists only for mode I and II cracks. In the general case, we can only speak of approximate solutions. A similar conclusion can be made for different-modulus materials. We analyze the results of the preceding papers [1–7], where specific cases of the problem were considered.     

Elastic interaction of partially debonded circular inclusions. II. Application to fibrous composite

A complete analytical solution has been obtained of the elasticity problem for a plane containing periodically distributed, partially debonded circular inclusions, regarded as the representative unit cell model of fibrous composite with interface damage. The displacement solution is written in terms of periodic complex potentials and extends the approach recently developed by Kushch et al. (2010) to the cell type models. By analytical averaging the local strain and stress fields, the exact formulas for the effective transverse elastic moduli have been derived. A series of the test problems have been solved to check an accuracy and numerical efficiency of the method. An effect of interface crack density on the effective elastic moduli of periodic and random structure FRC with interface damage has been evaluated. The developed approach provides a detailed analysis of the progressive debonding phenomenon including the interface cracks cluster formation, overall stiffness reduction and damage-induced anisotropy of the effective elastic moduli of composite.     

Hypersonic flow past a flat body in a regime of intense radiative heat transfer

The paper is devoted to the investigation of hypersonic flow regimes in which radiative transfer plays a significant part. A numerical solution is obtained to the two-dimensional steady problem of hypersonic flow past a flat thermally insulated body of an inviscid radiating gas with allowance for radiative transfer of energy in the approximation of radiative thermal conductivity. It is noted that a heated region is formed around the body, its dimensions exceeding by an order of magnitude those of the body itself; the temperature is effectively equalized, and the gas velocity is close to the velocity of the oncoming flow. Heated gas flows past the body at a moderate Mach number (M ～ 3–6). A thin region of strongly compressed gas is formed directly in front of the body.     

Stress intensity factors for two offset parallel circular cracks: Part II — Semi-infinite solid

With the aid of the formulation in [1] (R. Muki, Progress in Solid Mechanics (North-Holland, 1961)) for general three-dimensional asymmetric problems and the superposition principle, Part II of this work makes use of the method in Part I (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 207–225 (1994) [2]) to examine the interaction of arbitrarily located penny-shaped cracks in an infinite elastic solid to the case of a semi-infinite solid. As in Part I for the infinite body, the problem of a semi-infinite solid containing two penny-shaped cracks is reduced to a system of Fredholm integral equations of the second kind. These integral equations are then solved for some special cases when cracks are far apart and far away from the boundary. Some asymptotic solutions are presented and comparisons are made with the results for the special case where there is only one crack under axisymmetric loading.     

Electrothermal stress in conductive body with collinear cracks

The temperature and stress field in a thin plate with collinear cracks interrupting an electric current field are determined. This is accomplished by using a complex function method that allows a direct means of finding the distribution of the electric current, the temperature and stress field. Temperature dependency for the heat-transfer coefficient, coefficient of linear expansion and the elastic modulus are considered. As an example, temperature distribution is calculated for an alloy (No. GH2132) plate with two collinear cracks under high temperature. Relationships between the stress, temperature, electric density and crack length are obtained. Crack trajectories emanating from existing crack are predicted by application of the strain energy density criterion which can also be used for finding the load carrying capacity of the cracked plate.     

Prediction of thermal crack spacing

When the thermally induced stress in a shrinking pavement layer reaches the tensile strength of asphalt, regularly spaced thermal cracks form across the width of the pavement. A one-dimensional analytical solution for the stress distribution in a thermally shrinking elastic pavement layer placed on an elastoplastic, cohesive–frictional base is developed and validated by comparison with a 2D numerical solution. From the analytical model, a prediction of a length parameter that provides bounds on the thermal crack spacing is obtained. Predicted bounds on crack spacing are validated by comparison with field observations. It is demonstrated that the proposed formulation can also be applied to estimate the average crack density observed in thin ceramic films subjected to the application of an axial strain; in the latter system, the crack spacing is six decades smaller than that observed in pavement systems.