On determining the mechanical properties of nonlinear-elastic composite materials

Kazan' State Architect-Building Academy, Kazan' 420043. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 37, No. 6, pp. 170–180, November–December, 1996.     

Exhaustion of a thin film of a nonlinear-viscous fluid from a slot with slipping relative to the underlying surface

The problem of exhaustion of a thin film of a non-Newtonian fluid with a power rheological law from a slotted orifice is solved with account of film slipping relative to the underlying surface. By the method of group analysis with transformation of the parameters entering the problem, an asymptotic formula for the film profile is obtained and a law of motion of the film edge with small slipping is derived. Kazan' State University, 420008 Kazan'. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 71–76, March–April, 2000.     

Time-dependent interaction of two fluid flows separated by a semi-infinite non-rigid plate

In this paper the following problem is solved in the linear approximation. Let a flat plate separate two uniform inviscid fluid flows with different steady-state densities and velocities. These flows are subject to small time-dependent disturbances due to plate deformation. This problem is solved for arbitrary deformations as well as in the case of the angular harmonic oscillations of a flapping mover. The time-dependent forces acting on the plate are determined, together with the dynamic characteristics of the mover and the position of the fluid-fluid interface. Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 55–64, January–February, 1994.     

Longitudinal nonlinear oscillations of a gas in a closed pipe

Results of an experimental study of longitudinal nonlinear oscillations of a gas in a closed pipe are reported. Pressure waves in a broad range of excitation amplitudes and frequencies are studied. Strong nonlinear oscillations at a frequency thrice as low as the first natural frequency of the gas column are discovered. Institute of Mechanics and Mechanical Engineering, Kazan' Scientific Center, Russian Academy of Sciences, Kazan' 420503. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 60–62, November–December, 1999.     

Plane configurations in a flow of a perfect gas with a maximum critical Mach number

The problems in the construction of bodies which, satisfying some geometric limitations, are exposed to a plane symmetric flow of a perfect (inviscid and heat-nonconducting) gas with a maximum critical Mach numberM* are considered. Solutions are found by a numerical-analytical method with the use of the variables of the velocity hodograph. The Mach numberM* is found as a function of the geometric characteristics of the sought bodies on the basis of approximation of numerical data. Institute of Mathematics and Mechanics at the Kazan' State University, Kazan' 420008. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 106–115, September–October, 1998.     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

A Finite Strain Analysis of Cavity Formation at a Rigid Inhomogeneity

The formation of a cavity by inclusion-matrix interfacial separation is examined by analyzing the response of a plane rigid inclusion embedded in an unbounded incompressible matrix subject to remote equibiaxial dead load traction. A vanishingly thin interfacial cohesive zone, characterized by normal and tangential interface force-separation constitutive relations, is assumed to govern separation behavior. Rotationally symmetric cavity shapes (circles) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation and the remote loading. Nonsymmetrical cavity formation, under rotationally symmetric conditions of geometry and loading, is treated within the theory of infinitesimal strain superimposed on a given finite strain state. Rotationally symmetric and nonsymmetric bifurcations are analyzed and detailed results, for the Mooney–Rivlin strain energy density and for an exponential interface force-separation law, are presented. For the nonsymmetric rigid body displacement mode, a simple formula for the critical load is presented. The effect on bifurcation behavior of interfacial shear stiffness and other interface parameters is treated as well. In particular we demonstrate that (i) for the smooth interface nonsymmetric bifurcation always precedes rotationally symmetric bifurcation, (ii) unlike rotationally symmetric bifurcation, there is no threshold value of interface parameter for which nonsymmetric bifurcation will not occur and (iii) interfacial shear may significantly delay the onset of nonsymmetric bifurcation. Also discussed is the range of validity of a nonlinear infinitesimal strain theory previously presented by the author (Levy [1]). This revised version was published online in July 2006 with corrections to the Cover Date.     

On the basic laws of anisotropic viscoelasticity

The main aim of this work is to develop a consistent formulation of the rheological behavior for different anisotropic polymer systems. The unified theory of anisotropic viscoelasticity is developed based on the symmetry principles. The Maxwell rheological equation is extended to nonsymmetric anisotropic liquids. Transitions from the most general anisotropy to particular cases of anisotropy are established. It appears that the coupled relaxation of symmetric and antisymmetric stresses is a natural phenomenon in nonsymmetric viscoelasticity. Within the concept of an internal state variable, a stress–order relation is derived for a fully nonlinear case. The order tensor dynamics is also considered. A simple method of deriving the equation of the internal rotational motion is developed for the general macroscopic anisotropy. This paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece     

Stress state of a cylindrical composite panel weakened by a circular hole

The paper studies the stress-strain state of a deep cylindrical panel weakened by a hole and subjected to a tensile load at the outer boundary. A variational difference method is used. A numerical analysis is carried out for an orthotropic panel with low shear stiffness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 73–78, May 2006.     

A Combined Static/Dynamic Technique for Damage Detection of Laminated Composite Plates

There is a growing demand to develop viable techniques for effective damage detection of composite structures, and the dynamics-based approach has been broadly used in structural health monitoring. A new combined static/dynamic technique for improved damage detection of laminated composite plates is presented. The promise of the technique is that under the sustaining static load, the abnormality of dynamic response due to damage may become more pronounced and easy to be detected. The experimental program consists of testing an E-glass/epoxy composite plate with an embedded delamination under a pre-set static compressive force, and the dynamic response of laminated composite plates is measured using two different actuator–sensor systems: (1) PZT (lead–zirconate–titanate) actuators and scanning laser vibrometer (SLV) sensing system (PZT–SLV), and (2) PZT actuators and Polyvinylidenefluoride (PVDF) sensors (PZT–PVDF). The influence of sustaining static forces to dynamic response of delaminated composite plates is evaluated. The numerical finite element (FE) analysis is also conducted to verify the effectiveness of this technique. The experimental and numerical mode shapes are used to detect the presence, location, and size of the delamination and to study the effect of static load on dynamic response. Two relatively new damage detection algorithms (i.e., Simplified Gapped Smoothing Method (GSM) and Generalized Fractal Dimension (GFD)) are employed to analyze the Uniform Load Surface (ULS) calculated from the experimental and numerical data. From the dynamic response and analysis results using the damage detection algorithms, it is observed that as the sustaining static load increases, the delamination is much easier to be identified through the enlarged damage parameters. The present combined static/dynamic technique is capable of magnifying the effect of damage, thus improving the effectiveness of damage detection.     

Radial electroelastic nonstationary vibration of a hollow piezoceramic cylinder subject to electric excitation

The paper studies the radial nonstationary vibration of a piezoceramic cylinder polarized throughout the thickness and subjected to a dynamic electric load. A numerical algorithm for solving an initial–boundary-value problem using mesh-based approximations and difference schemes is developed. The dynamic electroelastic state of the cylinder subjected to a constant potential difference applied instantaneously is analyzed Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 30–35, February 2009.     

Stability of annular orthotropic plates

It is assumed that the orthotropy of the plate material is rectangular or polar, and a uniformly distributed, external compressive or tensile load is applied to the internal boundary of the plate. Stability is analyzed by the Ritz method with the use of Alfutov-Balabukh and Bryan energy criteria. Diagrams of the critical external load and buckling modes as a function of the plate dimensions are given. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 166–170, March–April, 2000.     

Numerical solution of the problem of an ice sheet under a moving load

A method for analyzing the bending of an ice sheet subjected to a moving load is proposed. The problem is solved in a dynamic formulation. The algorithm of solution is based on the finiteelement method and the finite-difference method. The method proposed allows one to determine the stress-strain state of an ice sheet for any law of motion of a load over ice. Two versions of initial conditions are considered. Examples of calculations are given. Komsomol’sk-on-Amur State Technical University, Komsomol’sk-on-Amur 681013. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 243–248, July–August, 1999.     

Stress-strain state of an ice sheet subjected to a moving load under shallow-water conditions

The following two classes of problems of determining the stress-strain state of an ice sheet under a moving load are considered: determination of the resonant velocity for a load moving over a continuous ice field and calculation of the deflections of an ice field with a bounded ice-free zone subjected to a moving load. The problems are solved in a dynamic formulation. The algorithm of solution is based on the finite-clement method and finite-difference methods. Examples of calculations are given. Komsomol'sk-on-Amur State Technical University, Komsomol'sk-on-Amur 681013. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol 41, No. 4, pp. 206–210, July–August, 2000.     

Dynamic Deformation of a Curved Plate with a Rigid Insert

A general solution is obtained for dynamic bending of ideal rigid-plastic plates with a clamped or simply supported curved contour containing an absolutely rigid insert of an arbitrary shape. The plate is affected by a short-time high-intensity explosive dynamic load uniformly distributed over the surface. It is shown that there are several mechanisms of plate deformation. Equations for dynamic deformation are derived for each mechanism, and conditions of occurrence are analyzed. Examples of numerical solutions are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 126–138, March–April, 2006.     

Motion of a system of seismic sources over ice on a body of water under the action of a pulse

The vertical motion of a system of two identical seismic sources and a spring truck tractor on ice under the action of a shock pulse from the seismic sources is studied to estimate the strength of ice. It is shown that during the pulse time, the interaction of the masses of the seismic sources and the tractor is small and the compressibility effect of the liquid can be ignored. Calculations show that for the seismic sources, the dynamic load far exceeds the static load and for the tractor, the static load is maximal. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 147–155, March–April, 2007.     

The plane problem on vertical impact of a rotating circular cylindrical shell against the surface of a liquid

The plane nonsymmetric problem on impact against and immersion into a compressible fluid of a thin electic cylindrical shell is considered. The shell rotates about its axis with a given angular velocity. The boundary-value problem is reduced to an infinite system of integral Volterra equations of the second kind. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 103–113, May, 2000.     

Two-dimensional problem of periodic loading of an elastic plate floating on the surface of an infinitely deep fluid

The action of external periodic pressure on an elastic plate floating on the surface of a fluid assumed to be ideal and incompressible is examined by the method of normal modes in the linear formulation. The behavior of the matrix of coefficients of the hydrodynamic load on the plate is considered in detail for different frequencies. The behavior of the plate under localized periodic loading is compared for the cases of a heavy fluid with a finite or infinite depth and for a weightless infinite-depth fluid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 61–72, May–June, 2005.     

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

The anti-plane dynamic problem of a functionally graded piezoelectric plane containing a hole–crack system is treated by a non-hypersingular traction-based boundary integral equation method. The material parameters vary exponentially in the same manner in an arbitrary direction. The system is loaded by an incident SH-type wave, and impermeable boundary conditions are assumed. Using a frequency-dependent fundamental solution of the wave equation, the boundary value problem is transformed into a system of integro-differential equations along the boundary of the hole and on the crack line. Its numerical solution yields the dynamic stress intensity factors and stress concentration factors. A parametric study reveals their dependence on the hole–crack scenario and its geometry, characteristics of the dynamic load and magnitude and direction of material inhomogeneity.     

Dynamic behavior of braced thin-walled beams

The present paper considers the dynamic behavior of beams with open thin-walled cross-sections along their length with bracings (connecting beams and truss). The effect of the constrained torsion warping, rotary inertia, and flexural-torsion coupling due to nonsymmetric cross-sections is included. In the case of simply supported beams, closed-form solutions for determining coupled natural frequencies and corresponding mode shapes are newly derived. The frequency equation, given in a determinantal form, is expanded in an explicit analytical form and then solved using the Mathcad 2001 Professional symbolic computing package. Some illustrative examples on the application of the present theory are given for coupled bending-torsion vibrations of braced thin-walled beams. As compared with FEM, numerical results demonstrate the accuracy and effectiveness of the proposed method Published in Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 129–143, November 2007.