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.     

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.     

Energy functional derivative with respect to the length of a curvilinear oblique cut in the equilibrium problem for a Timoshenko plate

This paper describes the dependence of the solution of the equilibrium problem for a Timoshenko plate and the total energy functional of the plate on the perturbation of an oblique crack. The nonlinearity of the problem is caused by the boundary conditions in the form of inequalities (conditions such as the Signorini conditions), which describe mutual nonpenetration of the opposite crack faces. The continuous dependence of the solution of the problem on the perturbation of the crack length is established. A formula for the energy functional derivative of the perturbation of the crack length is obtained.     

Differentiation of energy functionals in the problem on a curvilinear crack with possible contact between the shores

We consider the N-dimensional (N = 2 or 3) model of a one-dimensional anisotropic elastic body containing a curvilinear or surface crack. On the crack shores, the nonpenetration conditions in the form of inequalities (Signorini type conditions) are posed. For the general form of a sufficiently smooth perturbation of the domain, we obtain the derivative of the energy functional with respect to the perturbation parameter. We derive sufficient conditions for the existence of invariant integrals over an arbitrary closed contour. In particular, we obtain an invariant Cherepanov-Rice integral for curvilinear cracks.     

On the existence and uniqueness of solutions for a class of variational inequalities with applications to the signorini problem in mechanics

In this paper,we introduce a new unified and general class of variational inequalities,and show some existence and uniqueness results of solutions for this kind of variationalinequalities.As an application,we utilize the results presented in this paper to study theSignorini problem in mechanics.     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.     

Formation of the plastic zone in an anisotropic body with a crack

The influence of the length of a mode I crack on the plastic zone in an anisotropic body under hard loading is studied. The case of a generalized plane stress state is examined. A boundary-value problem is solved numerically to study the behavior of the main plastic zone at the crack tip, the additional plastic zone on the lateral face of the body, and the merged plastic zone Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 36–52, September 2008.     

Study of the plastic zone near a crack in an anisotropic body

The plastic zone at a crack tip in a finite anisotropic body is studied. A boundary-value problem is formulated in terms of the components of the covariant displacement vector for small strains. Particular attention is given to the case of plain strain. In this case, a numerical solution is found for a long rectangular body with a central crack under tension. As a result, conditions for the occurrence and development of a plastic zone at the crack tip are established. A plastic zone on the lateral surface of the body is discovered. How both zones extend and coalesce is elucidated. The effect of anisotropy on the occurrence of a plastic zone is evaluated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 29–44, July 2006.     

On indentation of a pair of rigid punches connected by an elastic beam into an elastic half-plane with regard to the friction and adhesion forces in the contact region

The contact problem of indentation of a pair of rigid punches with plane bases connected by an elastic beam into the boundary of an elastic half-plane is considered under the conditions of plane strain state. The external load is generated by lumped forces applied to the punches and a uniformly distributed normal load acting on the beam.It is assumed that the contact between the punch and the elastic half-plane can be described by L. A. Galin’s statement, i.e., it is assumed that the adhesion acts in the interior part of each of the contact regions and the tangential stresses obeying the Coulomb law act on their boundaries.With the symmetry taken into account, the problem is stated only for a single punch, and solving this problem is reduced to a system of four singular integral equations for the tangential and normal stresses in the adhesion region and the contact pressure in the sliding zones. The solution of the constitutive system together with three conditions of equilibrium of the system of punches connected by a beam is constructed by direct numerical integration by the method of mechanical quadratures.As a result of the numerical analysis, the contact stress distribution functions were constructed and the values of the sliding zones and the punch rotation angle were determined for various values of the geometric, elastic, and force characteristics.     

Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading

The behaviour of a bi-piezoelectric ceramic layer with a centre interfacial crack subjected to anti-plane shear and in-plane electric loading has been studied. The dislocation density functions and the Fourier integral transform method have been employed to eliminate the problem of singular integral equations. The normalized energy release rate, stress and electrical displacement intensity factors, G/G₀,K_III/K_III0 and K_D/K_D0, respectively, were determined for different geometric and property parameters by use of two different crack surface electric boundary conditions, i.e. impermeable and permeable. It has been shown that the effects of the thickness and material constants of the piezoelectric layer on all the three parameters, i.e. G/G₀,K_III/K_III0 and K_D/K_D0 were significant.     

负压爆炸载荷和数据采集间隔对弹性开孔板动应力集中的影响

基于ANSYS 7.0/LS-DYNA程序,对3 m3 m四边简支,厚度0.025 m,中心开有0.3 m0.3 m方孔的弹性板,在正压和负压三角爆炸载荷作用下的应力响应进行了分析,利用能量密度时间分布函数（TDFED）确定了动应力集中因子,并给出其计算步骤。计算结果表明计算总时间和数据采集时间间隔对动应力集中因子影响较大,而负压荷载影响较小。     

Adhesion in the contact of a spherical indenter with a layered elastic half-space

With the emergence of micro- and nano-technology, the contact mechanics of MEMS and NEMS devices and components is becoming more important. Thus it is important to gain a better understanding of the role of coatings and thin films on micro- and nano-scale contact phenomena, and to understand the interactions of measurement devices, such as an atomic force microscope (AFM), with layered media.More specifically, in this work the frictionless contact, with adhesion, between a spherical indenter and an elastic-layered medium is investigated. This configuration can be viewed as either a single contact model or as a building block of a multi-asperity rough surface contact model. As the scale decreases to the nano level, adhesion becomes an important issue. The presence of adhesion affects the relationships among the applied force, the penetration of the indenter, and the size of the contact area. This axisymmetric problem includes the effect of adhesion using a Maugis type of adhesion model. This model spans the range of the Tabor parameter between the JKR and DMT regions. The key parameters in this analysis are the elastic moduli ratio of the layer and the substrate, the dimensionless layer thickness, and the Maugis adhesion parameter. The results can be applied to a rigid or to an elastic indenter.     

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.     

非平稳随机激励作用下多自由度体系系统动力可靠性研究

首先应用Ｐｒｉｅｓｔｌｅｙ的演变谱理论，推求了多自由度线性体系在非平稳随机激励作用下的响应演变谱密度和时变均方响应。同时结合首次超越理论，分析了在非平稳随机激励作用下，多自由度体系系统动力可靠性问题。最后讨论了确定多自由度体系系统动力可靠性上下限的问题。     

A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion

In this work, we study a one-dimensional problem in a generalized thermoelastic diffusion in infinite medium with a spherical cavity subjected to a time dependent thermal shock of its internal boundary which is assumed to be traction free. The chemical potential is also assumed to be a known function of time on the bounding cavity. Laplace transform techniques are used. The solution of the problem in the transformed domain is obtained by using a direct approach without the customary use of potential functions. By means of numerical Laplace inversion, the problem is solved in the physical domain. Numerical results predict finite speeds of propagation for thermoelastic and diffusive waves. To investigate the diffusions effects, a comparison is made with the results obtained in the thermoelastic problem.     

应用混合变量余能原理求解不同载荷作用下四边简支矩形板的弯曲

由功的互等定理导出了弯曲矩形板混合变量的总余能,应用混合变量余能原理求解四边简支矩形板弯曲问题,给出了四边简支矩形板的混合变量余能表达式,求解了不同载荷作用下四边简支矩形板的弯曲问题,并将计算的不同载荷作用下的挠度和弯矩值与ANSYS有限元的计算值比较,给出了一种求解四边简支矩形板的新方法。     

SH 波入射时半空间界面裂纹与圆形衬砌的相互作用

利用复变函数和Green函数法研究了双相介质半空间界面裂纹及界面附近圆形衬砌对SH 波的散射与动应力集中问题。首先,采用映像思想构造满足自由边界条件的散射波表达式,进而求解所需的Green函数;其次,采用裂纹切割技术构造裂纹,并根据连续性条件建立了求解该问题的无穷代数方程组;最后,给出了不同入射波数时界面裂纹与衬砌的相互作用。结果表明,裂纹的存在显著放大了衬砌界面的动应力集中。     

Inertial motion of a body in an ideal fluid from the state at rest

The motion of a body in an ideal incompressible fluid flow without vortices in the absence of external forces is considered. It is demonstrated that the body can move inertially from the state at rest if its shape satisfies certain conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 214–219, July–August, 2008.     

Asymptotics of stresses and strain rates near the tip of a transverse shear crack in a material whose behavior is described by a fractional-linear law

An approximate solution of the problem of determining the fields of stresses and strain rates due to creep near the tip of a transverse shear crack in a material whose behavior is described by a fractional-linear law of the theory of steady-state creep is given. It is shown that the strain rates have a singularity of the type ∼ r^−α near the crack tip; the order of singularity α changes discretely, depending on the polar angle, and takes the values 1, 2/3, and 1/2. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 165–176, January–February, 2009.