Cracking of an orthotropic substrate reinforced by an orthotropic plate

A solution method is derived to determine the stress intensity factors for both an internal crack and an edge crack in an orthotropic substrate that is reinforced on its boundary by a finite-length orthotropic plate. The method utilizes the Green’s functions for a pair of dislocations and a concentrated force on the boundary while invoking the concept of superposition. Enforcing the traction-free boundary condition along the crack surfaces and the continuity of displacement gradients along the plate/substrate interface results in a coupled system of singular integral equations. An asymptotic analysis of the kernels in these equations for the region of the junction point between the plate corner and the substrate boundary reveals the strength of the singularity in the case of an edge crack. The numerical solution of the integral equations provides results for the stress intensity factors for both an internal crack and an edge crack perpendicular to the substrate boundary and aligned with one of the corners of the plate. The present results have been validated against previously published stress intensity factors for an internal crack and an edge crack in an isotropic substrate.     

The weight function for cracks in piezoelectrics

The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces.     

On the characterization of the pressure-sinkage relationship of snow covers containing an ice layer

Based on an analysis performed by Meyerhof and on field data obtained in two snow covers, a method for estimating the bearing capacity of an ice layer in a snow cover is proposed. A procedure for characterizing the pressure-sinkage relationship of a snow cover containing an ice layer is suggested. The objective is to provide an improved characterization of the behaviour of snow-covered terrain for the prediction of vehicle mobility.     

The Action of an Arbitrary Load on the Surface of an Orthotropic Half-Space

The exact solution to the first boundary-value problem for a half-space is constructed on the basis of the general solution of the equilibrium equations for an orthotropic medium (nine elastic constants). The stress–strain state of an orthotropic half-space whose surface is under an arbitrarily applied concentrated force is described as an example. The well-known solution for the isotropic case is obtained by the same scheme, which confirms the reliability of the result.     

Energy transfer from an inductive storage by means of an explosive current disconnect

Energy transfer from an inductive storage is considered for two types of systems: a disconnect with an intrinsic parasitic inductance for an inductive load and a purely resistive disconnect for a resistive load. Solutions are obtained for the voltage, power, and energy transferred to the load. The dependence of the efficiency of the device on its parameters is established.     

Solving the Contact Problem for a Rectangular Die on an Elastic Foundation

A method of orthogonal polynomials is proposed to solve the contact problem for a rectangular die on an elastic foundation. For the case of an elastic half-space, an exact formula is derived for the translation of the die under symmetric loading     

Load-bearing capacity of curvilinear ice plates reinforced by a rigid insert

A method is developed for determining the load-bearing capacity of ice plates, which are modeled by an ideal rigid-plastic plate located on an incompressible foundation. The plate with a simply supported or clamped arbitrary smooth curvilinear contour is subjected to a load uniformly distributed over an arbitrarily shaped reinforced local central region. An analytical expression for the ultimate load is derived. A plate shaped as an ellipse is considered as an example.     

On pattern selection in mixed convection in rectangular ducts

Mixed convection in a horizontal rectangular duct has the same critical Rayleigh number as natural convection in a rectangular cavity for the onset of convection. The linear stability analysis predicts either an odd or an even number of convective rolls to appear depending on the aspect ratio of the cross section. However, it has been shown both experimentally and numerically that an even number of convective rolls appears under supercritical conditions for fully developed mixed convection. The paper first presents an analytical solution for the buoyancy-induced mainstream velocity, w ^b , at the onset of buoyancy-induced motion in a forced convective flow. Then, a comparison in the initial growth rate of w ^b is made between the case of an odd and an even number of rolls; which shows the selection of an even number of rolls over an odd number in mixed convection except for low aspect ratio ducts.     

On the Start-Up Behavior and Steady-State Oscillation of Singularly Perturbed Harmonic Oscillators

In this paper we present an analytic methodology for the analysis of a class of electrical harmonic oscillators. We combine geometric methods with the theory of singularly perturbed systems, which we use as tool for reduced order modeling. So we are able to define an easy to use formula in order to reduce an oscillatory system to center manifold. Thus we get a model for the start-up behavior as well as for the steady-state oscillation of sinusoidal oscillators. Furthermore, we demonstrate our technique by means of the Clapp oscillator which is an important member of the Colpitts, Clapp and Pierce oscillator family.     

Exact solution of problem of deflection of an anisotropic plate with an aperture

The exact solution of the problem of the deflection of an anisotropic plate weakened by an aperture is known only for the case in which the aperture has the shape of a circle or an ellipse [1, 2]. An exact solution has not been derived for any other aperture shapes. Approximate methods [3–6] which are widespread for the case of multiply connected anisotropic plates [7] are applied to the determination of the bending moments in an anisotropic plate near an aperture differing little from an elliptical or circular one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 168–177, September–October, 1977.     

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 67–77, November 2005.     

A theory and a simulation capability for the growth of a solid electrolyte interphase layer at an anode particle in a Li-ion battery

A major mechanism for electrochemical aging of Li-ion batteries is the growth of a solid electrolyte interphase (SEI) layer on the surface of anode particles, which leads to capacity fade and also results in a rise in cell resistance. We have formulated a continuum theory for the growth of an SEI layer—a theory which accounts for the generation of the attendant growth stresses. The theory has been numerically implemented in a finite-element program. This simulation capability for SEI growth is coupled with our previously published chemo-mechanical simulation capability for intercalation of Li-ions in electrode particles. Using this new combined capability we have simulated the formation and growth of an SEI layer during cyclic lithiation and delithiation of an anode particle, and predicted the evolution of the growth stresses in the SEI layer. The evolution of the stress state within the SEI layer and at the SEI/anode-particle interface for spherical- and spheroidal-shaped graphite particles is studied. This knowledge of the local interfacial stresses provides a good estimate for the propensity of potential delamination of an SEI layer from an anode particle.     

Domain of convergence of an iterative procedure for an autonomous boundary-value problem

We find an estimate for the range of values of a small parameter for which the convergence of an iterative procedure for the construction of solutions of an autonomous weakly nonlinear Noether boundary-value problem for a system of ordinary differential equations in the critical case is preserved. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 416–432, July–September, 2006.     

Thermostressed state of a piezoelectric bodywith a plane crack under symmetric thermal load

The paper addresses a thermoelectroelastic problem for a piezoelectric body with an arbitrarily shaped plane crack in a plane perpendicular to the polarization axis under a symmetric thermal load. A relationship between the intensity factors for stress (SIF) and electric displacement (EDIF) in an infinite piezoceramic body with a crack under a thermal load and the SIF for a purely elastic body with a crack of the same shape under a mechanical load is established. This makes it possible to find the SIF and EDIF for an electroelastic material from the elastic solution without the need to solve specific problems of thermoelasticity. The SIF and EDIF for a piezoceramic body with an elliptic crack and linear distribution of temperature over the crack surface are found as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 96–108, March 2008.     

Nonlinear surface waves on cubic materials

The effect of nonlinearity on the propagation of surface Rayleigh waves on an elastic medium of cubic symmetry is examined by applying the results of an earlier analysis for general anisotropic materials. Explicit calculations are given of the parameters relating to the generation of higher harmonics in an initially sinusoidal wave for a series of thirty-five materials for which experimental measurements of the third-order moduli are available.     

Constitutive identification of affine rods

We present an identification procedure based on a version of the de Saint-Venant semi-inverse method for transferring constitutive information from a two-dimensional Cauchy continuum model to a one-dimensional affine rod model of a two-dimensional slender body. To test and evaluate the accuracy of the identification procedure proposed, we compare the dispersion and frequency curves predicted by the two models for an isotropic material and for an anisotropic one.     

Upwind adaptive finite element investigations of the two-dimensional reactive interaction of supersonic gaseous jets

We describe an upwind finite element method aimed at numerically simulating the two-dimensional transonic flow of a reactive gaseous mixture. The method uses in particular a triangular finite element mesh, with an adaptive procedure based on mesh refinement by triangle division, and an upwind non-oscillatory scheme based on an approximate Riemann solver for the evaluation of the convective terms for all species. Results concerning the reactive interaction of two supersonic gaseous jets are presented.     

The thermoelastic equilibrium of a transversally isotropic medium with an elliptic crack under symmetric loading

The stress intensity factors (SIFs) are evaluated for flat elliptical cracks located in a transversally isotropic material (cracks are assumed perpendicular to the transtropy axis) under an arbitrary load and symmetric temperature. The SIFs for an elliptical crack in a transversally isotropic medium are determined using the formulas (derived by the author in his previous studies) of transition from an isotropic to transversally isotropic material and the relative problem for an isotropic medium. It is proved that these formulas can be employed for an arbitrary homogeneous transversally isotropic material (no matter whether the roots of some characteristic equation of the material are real or complex) with an arbitrary flat crack or a system of coplanar flat cracks, including elliptical ones, under an arbitrary load and symmetric temperature. A transversally isotropic material with two coplanar elliptical cracks is considered as an illustrative example. The dependences of the SIFs on the parameters of cracks and their arrangement at a decreasing temperature are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 96–105, April, 2000.     

Effect of the physical nonlinearity of a material on the stress state of a medium with an elastic spherical inclusion under uniform loading

We have used the perturbation method as the basis for obtaining an approximate solution of the three-dimensional problem for a physically nonlinear elastic medium with an elastic inclusion under uniform tension— compression. From this solution, we can obtain as a special case a solution for an elastic medium with a stress-free cavity and for an elastic medium with a rigid inclusion. We have plotted the normal and tangential stresses as a function of the radius and the ratio of shear moduli for the inclusion and the medium. We have investigated their behavior under different loading conditions. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 46–51, November, 1998.     

An enhanced beam-theory model of the mixed-mode bending (MMB) test—Part II: Applications and results

The paper presents an enhanced beam-theory (EBT) model of the mixed-mode bending (MMB) test, whereby the specimen is considered as an assemblage of two sublaminates partly connected by an elastic–brittle interface. Analytical expressions for the compliance, energy release rate, and mode mixity are deduced. A compliance calibration strategy enabling numerical or experimental evaluation of the interface elastic constants is also presented. Furthermore, analytical expressions for the crack length correction parameters—analogous to those given by the corrected beam-theory (CBT) model for unidirectional laminated specimens—are furnished for multidirectional laminated specimens, as well. Lastly, an example application to experimental data reduction is presented.