Acceleration waves and ellipticity in thermoelastic micropolar media

Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave’s propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. As an example, a quadratic form for the specific free energy is considered and the solutions of the corresponding spectral problem are presented.     

General hoyle–youngdahl and love solutions in the linear inhomogeneous theory of elasticity

The general Hoyle–Youngdahl and Love solutions in the three-dimensional theory of inhomogeneous linear elastic materials are proposed. Following a brief historical outline of various general solutions existing in the classical linear elasticity of homogeneous isotropic media, key steps of the derivation of the Hoyle–Youngdahl and Love solutions are presented. The procedure is then generalized to the case of inhomogeneous elastic materials with elastic constants depending on the z-coordinate. The significance of the solutions and their relevance to modeling of functionally graded materials is discussed in brief     

Elastic properties of Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–NiAl: a modified version of Hashin–Shtrikman bounds

The modified nonlinear relations for the estimation of elastic constants of Al₂O₃–NiAl composite material are developed. The concept of microstructure and interconnectivity of phases at the interface is used. Hashin–Shtrikman relations are described in their actual form and modified version of Hashin–Shtrikman relations for bulk and shear moduli are discussed. These relations for elastic and mechanical properties are applied mainly for Al₂O₃–NiAl composite material. Theoretical predictions using modified relations are compared with Hashin–Shtrikman bounds and experimental results of elastic properties for Al₂O₃–NiAl matrix-inclusion-based composite. It is found that the predicted values of elastic and mechanical properties using modified relations are quite close to the experimental results.     

Spectral decomposition of the compliance fourth-rank tensor for orthotropic materials

Summary The compliance tensor related to orthotropic media is spectrally decomposed and its characteristic values are determined. Further, its idempotent tensors are estimated, giving rise to energy orthogonal states of stress and strain, thus decomposing the elastic potential in discrete elements. It is proven that the essential parameters, required for a complete characterisation of the elastic properties of an orthotropic medium, are the six eigenvalues of the compliance tensor, together with a set of three dimensionless parameters, the eigenangles θ, ϕ and ω. In addition, the intervals of variation of these eigenangles with respect to different values of the elastic constants are presented. Furthermore, bounds on Poisson's ratios are obtained by imposing the thermodynamical constraint on the eigenvalues to be strictly positive, as specified from the positive-definite character of the elastic potential. Finally, the conditions are investigated under which a family of orthotropic media behaves like a transversely isotropic or an isotropic one. Received 5 January 1999; accepted for publication 22 June 1999     

Comment on “Influence of elastic material compressibility on parameters of an expanding spherical stress wave” by E. Włodarczyk, M. Zielenkiewicz

A wide spread error related to the compressibility of elastic media is reproduced in the paper by Włodarczyk and Zielenkiewicz (Shock Waves 18:465–473, 2009) leading to erroneous results and conclusions.     

Inverse doubly periodic problem of the theory of bending of a plate with elastic inclusions

Based on the balanced strength principle, a problem of determining the optimal interference for fitting elastic inclusions into holes of an isotropic elastic plate weakened by a doubly periodic system of circular holes is solved. A closed system of algebraic equations is derived, which allows solving this problem. The resultant interference increases the load-carrying capacity of the composite plate being bent. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 153–161, July–August, 2006.     

Critical states in the interaction of discontinuous waves with interfaces between elastic media

The ray method is used to develop a procedure for studying the kinematics and dynamics of the interaction of discontinuous waves with interfaces between elastic media. The critical states of such interactions are analyzed. For the cases of a free surface and two surfaces contacting with and without sliding, it is established that the kinematic and dynamic conditions for critical states to occur are satisfied simultaneously Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 79–90, February 2009.     

Spontaneous acoustic emission from strong shocks in solids

Summary  The role of free electrons in the stability of strong shock waves in metals under spontaneous acoustic emission is investigated. For that purpose, a three-term form of the equation of state is employed in order to describe the cold pressure, the thermal atomic pressure and the thermal pressure of free electrons. The equation of state enables the calculation of the sound velocity behind the shock, which in turn is utilized in the Dyakov–Kontorovich criteria for the shock stability. The integral over the Fermi–Dirac distribution function that describes the specific internal energy of free electrons is replaced by a model algebraic function that possesses correct asymptotic limits at low and high temperatures. It is shown that strong shock waves in all metals are prone to instability under spontaneous emission. However, the threshold for that instability is shifted to higher Mach numbers if free electrons are taken into account. It is further shown that the stabilizing effect of free electrons is vastly overestimated if the expressions for degenerate electron gas are employed for temperatures that are larger than the Fermi temperature. Received 22 November 1999; accepted for publication 12 July 2000     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.     

Specific features of the nonlinearly elastic behavior of cylindrical compressible bodies in torsion

The torsion problem of a cylinder with a circular transverse cross section twisted by end moments that are equal in magnitude and opposite in direction is considered for various models of nonlinearly elastic compressible media. The problem is solved by the semi-inverse method of elasticity theory. The Poynting effect, which consists of variation in the length of a shaft in torsion, is treated qualitatively and quantitatively. The results of the numerical and asymptotic (only terms that are quadratic relative to the displacement gradient are conserved) solutions for various models of the nonlinearly elastic behavior of materials are compared. An analysis of the results shows that in some cases, the quasilinear model is not applicable for studying the behavior of nonlinearly elastic compressible media. Rostov State Construction University, Rostov-on-Don 344022. Rostov State University, Rostov-on-Don 344090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 188–193, March–April, 2000.     

Seismic Wave Propagation in Composite Elastic Media

It has been known since the time of Biot–Gassman theory (Biot, J Acoust Soc Am 28:168–178, 1956, Gassmann, Naturf Ges Zurich 96:1–24, 1951) that additional seismic waves are predicted by a multicomponent theory. It is shown in this article that if the second or third phase is also an elastic medium then multiple p and s waves are predicted. Futhermore, since viscous dissipation no longer appears as an attenuation mechanism and the media are perfectly elastic, these waves propagate without attenuation. As well, these additional elastic waves contain information about the coupling of the elastic solids at the pore scale. Attempts to model such a medium as a single elastic solid causes this additional information to be misinterpreted. In the limit as the shear modulus of one of the solids tends to zero, it is shown that the equations of motion become identical to the equations of motion for a fluid filled porous medium when the viscosity of the fluid becomes zero. In this limit, an additional dilatational wave is predicted, which moves the fluid though the porous matrix much similar to a heart pumping blood through a body. This allows for a connection with studies which have been done on fluid-filled porous media (Spanos, 2002).     

Anisotropy of elastic properties of materials

Papers dealing with the generalized Hooke’s law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin’s approach disclosing the structure of the generalized Hooke’s law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 131–151, November–December, 2008.     

On determining stresses in rigid inclusions. Plane problems

Plane problems of determining the stress-strain state of an isotropic elastic domain with a rigid inclusion are considered. It is shown that the stress field in the inclusion is uniquely determined. This field is uniform for a plane with an elliptic inclusion, and the stresses at infinity and in the inclusion are related by mutually single-valued formulas. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 183–186, July–August, 2009.     

On universal deformations in analysis of Signorini’s nonlinear theory of hyperelastic media

It is shown that uniform compression/tension and simple shear as universal deformations are quite useful in studying Signorini’s nonlinear theory of hyperelastic materials. They make it possible to formulate restrictions for the elastic constants of the theory and to explain the Poynting effect __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 54–60, December 2007.     

Mircopolar Model of Fracture for Composite Material

Failure of a composite is a complex process accompanied by irreversible changes in the microstructure of the material. Microscopic mechanisms are known of the accumulation of damage and failure of the type of localized and multiple ruptures of the fibers delamination along interphase boundaries, and also mechanisms associated with fracture of fibers. In this work, we propose a mathematical model of the local mechanism of failure of a composite material randomly reinforced with a system of short fibers. We implement the Cosserat moment model of crack tip for filament material, reinforced with whiskers or in fiber- reinforced polycrystalline materials. It is assumed that the angular distribution of the fibers is isotropic and the elastic characteristics of the fibers are considerably higher than the elastic constants of the matrix. We implement the homogenization procedure for the effective Cosserat constants similarly to the effective elastic constants. The singular solution in the vicinity of the crack tip in the Cosserat moment model is found. Using this solution, we examine the bending stresses in the filaments due to effective moment stresses in the material. The constructed model describes the phenomenon of fracture of the fibers occurring during crack propagation in those composites. The following assumptions are used as the main hypotheses for the micromechanical model. The matrix contains a nucleation crack. When the load is increased the crack grows and its boundary comes into contact with the reinforcing fibers. A further increase of the stress causes bending of the fiber. When~the fiber curvature reaches a specific critical value, the fiber ruptures. If the stress at infinity is given, the fibers no longer delay the development of failure during crack propagation The degree of bending distortion of the fiber in the vicinity of the boundary of the crack is determined by the moment model of the material. The necessity to take into account the moment stresses in the failure theory of the reinforced material was stressed in [Muki and Sternberg (1965) Zeitschrift f angew Math und Phys 16:611–615; Garajeu and Soos (2003) Math Mech Solids 8(2):189–218; Ostoja-Starzewski et al (1999) Mech Res Commun 26:387–396]. The moment Cosserat stresses were accounted also for inhomogeneous biomechanical materials by Buechner and Lakes (2003) Bio Mech Model Mechanobiol 1: 295–301. We should also mention the important methodological studies [Sternberg and Muki (1967) J Solids Struct 1:69–95; Atkinson and Leppington (1977) Int J Solids Struct 13: 1103–1122] concerned with the moment stresses in homogeneous fracture mechanics.     

Antiplane stress singularities in a bonded bimaterial piezoelectric wedge

Summary  In this paper, the eigen-equations governing antiplane stress singularities in a bonded piezoelectric wedge are derived analytically. Boundary conditions are set as various combinations of traction-free, clamped, electrically open and electrically closed ones. Application of the Mellin transform to the stress/electric displacement function or displacement/electric potential function and particular boundary and continuity conditions yields identical eigen-equations. All of the analytical results are tabulated. It is found that the singularity orders of a bonded bimaterial piezoelectric wedge may be complex, as opposed to those of the antiplane elastic bonded wedge, which are always real. For a single piezoelectric wedge, the eigen-equations are independent of material constants, and the eigenvalues are all real, except in the case of the combination C–D. In this special case, C–D, the real part of the complex eigenvalues is not dependent on material constants, while the imaginary part is. Received 26 March 2002; accepted for publication 2 July 2002     

Vibration of beams with general boundary conditions due to a moving random load

Summary  The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail. Received 3 December 2001; accepted for publication 30 April 2002     

Vibrations of a corrugated orthotropic cylindrical shell with free edges

The natural vibrations of a corrugated elastic orthotropic cylindrical shell with a directrix perpendicular to its edges that are free are examined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 97–114, December, 2006.     

A colorimetric reaction to quantify fluid mixing

We found the colorimetric reaction of Tiron (1,2-dihydroxybenzene-3,5-disulfonic acid) and molybdate suitable for optical quantification of chemical reaction during fluid–fluid mixing in laboratory chambers. This reaction consists of two colorless reagents that mix to rapidly form colored, stable, soluble products. These products can be digitally imaged and quantified using light absorbance to study fluid–fluid mixing. Here we provide a model and equilibrium constants for the relevant complexation reactions. We also provide methods for relating light absorbance to product concentrations. Practical implementation issues of this reaction are discussed and an example of imaged absorbances for fluid–fluid mixing in heterogeneous porous media is given.