Differentiation of the energy functional in the equilibrium problem for a Timoshenko plate containing a crack

The variational formulation of the equilibrium problem for a Timoshenko plate containing a vertical plane crack is considered. Nonpenetration conditions in the form of inequalities (Signorini type conditions) are specified on the crack faces. The behavior of the solution and the corresponding energy functional of the plate with variation in the crack length is analyzed. A formula for the derivative of the energy functional along the crack length is obtained. The solutions are found to continuously depend on the parameter characterizing the crack length.     

Differentiation of energy functionals in the problem of a curvilinear crack in a plate with a possible contact of the crack faces

A problem of equilibrium of a cracked plate is considered within the framework of the Kirchhoff-Love model. Non-penetration conditions in the form of inequalities (Signorini-type conditions) are set on the crack faces. The behavior of the energy functional is studied for the case of a rather smooth perturbation of the domain of the general form. Sufficient conditions for the existence of the energy functional derivative with respect to the parameter of domain perturbation are derived. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 153–168, September–October, 2008.     

Regular Perturbation Methods for a Region with a Crack

The paper considers a model problem for Poisson's equation for a region containing a crack or a set of cracks under arbitrary linear perturbation. Variational formulation of the problem using smooth mapping of regions yields a complete asymptotic expansion of the solution in the perturbation parameter, which is a generalized shape derivative. This global asymptotic expansion of the solution was used to derive representations of arbitraryorder derivatives for the potential energy function, stress intensity factors, and invariant energy integrals in general form and for basis perturbations of the region (shear, tension, and rotation). The problem of the local growth of a branching crack for the Griffith fracture criterion and the linearized problem of optimal location of a rectilinear crack in a body with the energy function as a cost functional were formulated.     

On safe crack shapes in elastic bodies

According to the Griffith criterion, a crack propagation occurs, provided that the derivative of the energy functional with respect to the crack length reaches some critical value. We consider a generalization of this criterion to the case of nonlinear cracks satisfying a nonpenetration condition and investigate the dependence of the shape derivative of the energy functional on the crack shape. In the paper, we find the crack shape which gives the maximal deviation of the energy functional derivative from a given critical value and, in particular, prove that this optimality problem admits a solution.     

Stable growth of a crack interacting with a circular inclusion

The problem of failure of a plate containing a circular inclusion and a crack is studied. The crack is oriented along a diameter of the inclusion and the plate is subjected to a remote uniaxial stress perpendicular to the crack axis. The process of slow stable crack growth from initiation to termination is studied by the strain energy density theory. The crack growth is simulated by predicting finite increments of crack extension when material elements near the crack tip absorb a critical amount of strain energy density level, . Unstable crack growth occurs when the strain energy density factor S reaches a critical value where r_c is the critical size of the final crack increment prior to instability. The stress at crack initiation and the critical stress and crack length at failure are determined. The influence of the mechanical properties of the plate and the inclusion, the relative position of the inclusion and the crack and the crack length on the characteristic quantities of stable crack growth is analyzed. The dependence of the stable crack growth process on the loading rate is also investigated. Results are displayed in graphical form.     

Dependence of the Critical Load on the Geometric Characteristics of a Hinged Plate with a Crack

The plane problem of three-dimensional stability of a hinged plate with a central crack under uniaxial loading along the crack is considered. The net approach is used to solve the problem. The variational difference and gradient methods are used, respectively, to construct a difference scheme and to solve difference problems. The dependence of the critical load on two parameters — the crack length and the thickness ratio — is derived. Formulas for calculation of the critical load are given     

Asymptotic behavior of the energy functional for a three-dimensional body with a rigid inclusion and a crack

A model of a three-dimensional elastic body containing a rigid inclusion and a crack located on the interface between the inclusion and the body is considered. Natural boundary conditions are imposed on the crack. A derivative of the energy functional with respect to the perturbation parameter is derived for an arbitrary, rather smooth perturbation of the domain, in particular, the Griffith formula is obtained.     

Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks

This paper considers the equations of two-dimensional elasticity theory in nonsmooth domains. The domains contain curvilinear cracks of variable length. On the crack faces, conditions are specified in the form of inequalities describing mutual nonpenetration of the crack faces. It is proved that the solutions of equilibrium problems with a perturbed crack converge to the solution of the equilibrium problem with an unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the length of a curvilinear crack is obtained.     

Nonlinear vibration analysis of isotropic plate with inclined part-through surface crack

The four modes of vibration of an isotropic rectangular plate with an inclined crack are investigated. It is assumed that the crack remains continuous and its center is located at the center of the plate. The governing nonlinear equation of the transverse vibration of the plate with the plate boundary conditions being simply-supported on all edges is developed. The multiple scale perturbation method is utilized as the solution procedure to find the steady-state frequency response equations for all the four modes of vibration. The equations for the free and forced vibrations are derived and their frequency responses are presented. A special case of large-scale excitation force has also been considered. The parameter sensitivity analysis for the angle of crack, length of crack and the position of the external applied excitation force is performed. It has been shown that according to the aspect ratio of the plate, the vibration modes can have either nonlinear hardening effect or nonlinear softening behavior.     

Solution of multiple crack problem in a finite plate using coupled integral equations

This paper studies a numerical solution of multiple crack problem in a finite plate using coupled integral equations. After using the principle of superposition, the multiple crack problem in a finite plate can be converted into two problems: (a) the multiple crack problem in an infinite plate and (b) a usual boundary value problem for the finite plate. For the former problem, the Fredholm integral equation is used. For the latter problem, a BIE based on complex variable is suggested in which a Cauchy singular kernel exists. For the proposed BIE, after using the inverse matrix technique, the dependence of the traction at a domain point from the boundary tractions is formulated indirectly. This is a particular advantage of the present study. Several numerical examples are provided and the computed results for stress intensity factor and T-stress at crack tips are given.     

Torsion of a cylinder with a shallow external crack

The torsional problem of a finite elastic cylinder with a circumferential edge crack is studied in this paper. An efficient solution to the problem is achieved by using a new form of regularization applied to dual Dini series equations. Unlike the Srivastav approach, this regularization transforms dual equations into a Fredholm integral equation of the second kind given on the crack surface. Hence, exact asymptotic expansions of the Fredholm equation solution, the stress intensity factor and the torque are derived for the case of a shallow crack. The asymptotic expansions are certain power-logarithmic series of the normalized crack depth. Coefficients of these series are found from recurrent relations. Calculations for a shallow crack manifest that the stress intensity factor exhibits the rather weak dependence upon the cylinder length when the torque is fixed and the triple length is larger than the diameter.     

Estimating the error of the beam approximation in the plane stability problem for a rectangular plate with a central crack

The plane stability problem for a rectangular, linearly elastic, isotropic plate with a central crack is solved. The dependence of the critical load on the crack length is studied using exact (the three-dimensional linearized theory of stability of elastic bodies) and approximate (beam approximation) approaches. The results produced by the beam approach are evaluated.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 117–126, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

The motion of a brittle crack

The acceleration phase and the subsequent motion of a brittle crack of finite length extending in an infinite sheet loaded in the extensional mode are determined. The energy release is assumed to depend on crack length and on crack-tip velocity but not on higher time-derivatives of crack length. This energy release should equal the fracture energy, which, for a given material is supposed to be dependent only on the crack-tip velocity. For polymethylmethacrylate this velocity dependence has been determined from recent experiments on the long strip configuration. The results obtained show good agreement with earlier experimental investigations of crack motion in large sheets of polymethyl-methacrylate. Since the validity of existing exact analytical solutions of the accelerating crack problem seems to be severely restricted from a practical point of view, the approximate method in the present paper could offer a promising alternative approach.     

Growth of a mode II crack in an orthotropic plate made of a viscoelastic composite material

The growth of a straight mode II crack in a viscoelastic orthotropic plate is examined. The plate material is modeled by a viscoelastic anisotropic medium. The shear displacement in the fracture process zone is determined as a function of time using the corresponding elastic solution, the Volterra principle, and the method of operator continued functions. The time dependence of the crack length is constructed as integral equations of three phases of stable growth. The solution of these equations gives kinetic curves __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 89–97, September 2006.     

Investigation of a griffith crack subject to uniform tension using the non-local theory by a new method

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Equilibrium problem for a Timoshenko plate with an oblique crack

The condition of mutual nonpenetration of the crack faces is proposed for a Timoshenko plate with an oblique crack, whose initial state is defined by a surface the normal to which makes a small angle with the middle plane. Unique solvability of the variational problem of plate equilibrium with the nonpenetration conditions for the crack faces specified on the curve describing the crack is proved. A differential formulation of the problem equivalent to the original formulation for sufficiently smooth solutions is proposed. For the one-dimensional case (beam with a cut), an analytical solution is obtained, and the cases of longitudinal tension and compression are examined.     

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.     

利用焦耳效应提高含裂纹金属构件抗裂性能问题的研究

设一无限大金属薄板中含有一个线裂纹,对金属板施加恒定的电流场,在两个裂尖处产生的热量远远大于其余地方产生的热量,可简化成两个点热源．经求解得到了问题的解析解,包括裂纹尖端附近区域温度、应力、应变、应变能密度因子的解析表达式．计算结果表明,裂纹尖端处的材料发生熔化而形成一个焊点,裂纹尖端明显纯化,可抑制裂纹的进一步扩展,提高含裂纹金属构件的抗裂性能．     

Fracture toughness from the standpoint of softening hyperelasticity

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack/notch is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture where the crack sharpness is of minor importance.¹To explain the experimental observations qualitatively we simulate tension of a thin plate with a small crack of a finite and varying sharpness. In simulations, we introduce the average bond energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find, in agreement with experiments, that the magnitude of the fracture toughness cannot be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack, the lower is the toughness.Based on the obtained results, we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance, etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of continuum fails.     

Impact response of a functionally graded piezoelectric plate with a vertical crack

The dynamic fracture problem for a functionally graded piezoelectric plate containing a crack perpendicular to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction. Integral transform techniques and dislocation density function are employed to reduce the problem to the solution of a singular integral equation. Mode I dynamic energy density factors are presented for an internal crack as well as an edge crack for various values of dimensionless parameters representing the size and location of the crack and the material nonhomogeneity.