Spectral method of fatigue life calculation under random multiaxial loading

We develop a new spectral method for the evaluation of fatigue life under random multiaxial loading. This method is a generalization of the known formulas of Miles, Kowalewski, Raikher, and Bolotin based on the power spectral density function of stresses under uniaxial random loads. The power spectral density function of the equivalent stress determined according to a linear criterion of multiaxial random fatigue is introduced in the indicated formulas. It is shown that, in reducing the multiaxial state of loading to the uniaxial state according to linear criteria, the frequency bands of the components of the stress state are transformed into the frequency band of the equivalent stress without increasing its width. This favorable result cannot be obtained if the equivalent stress is calculated according to nonlinear multiaxial fatigue criteria. Technical University of Opole, Opole, Poland, Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 3, pp. 86–96, May–June, 1996.     

Detailed study of nonlinear wave front distortion of focused sound in superfluid4He

We have investigated a nonlinear phenomenon which appears in a focused sound in superfluid⁴He under pressure higher than 18 atm. Wave front distortion of the focused ultrasound by nonlinear effect was obtained by the Fourier transform of the transducer output as a function of the defocusing length. The wave was found to suffer discontinuous wave front distortion for the input power above a certain value. This distortion is well represented by the picture that a second wave whose phase is shifted by approx. develops, and interferes with the original wave. The amplitude of this second wave decreases suddenly as the pressure is lowered below 18 atm and the nonlinear wave front distortion also disappears. The possible mechanism of this second wave generation are discussed.     

Two-stage temperature rise in a thermoviscoplastic half-plane under impact

Summary A thermoviscoplastic constitutive model is used to treat uniaxial shock wave propagation in a material exhibiting rate-sensitive plasticity and thermal softening. Simple relations are obtained for the pressure jump as a function of the density jump, shock speed as a function of particle velocity, and pressure decay behind the shock front. Also, the model predicts a two-stage temperature rise, involving a jump at the shock front followed by a rise due to plastic dissipation (stress decay). The temperature rise and other quantities are predicted in the case of impact of titanium plates.With 4 Figures     

A closed form solution for linear and nonlinear free vibrations of composite and fiber metal laminated rectangular plates

In this paper, the nonlinear equations of motion for laminated composite rectangular plates based on first order shear deformation theory, which include shear deformation and rotary inertia, have been derived. Then, through introducing a force function, these equations reduced to a set of coupled nonlinear partial differential equations and a compatibility equation. By using the Galerkin method, for the first time, a nonlinear ordinary differential equation is obtained, which includes nonlinear inertia and stiffness terms. By using the multiple time scales method, analytical relations for nonlinear frequency and transverse displacement have been obtained. Results are compared with the literature and good agreement is achieved for both linear and nonlinear frequencies. After proving the validity of our work for isotropic rectangular plates and laminated rectangular plates, linear and nonlinear free vibration of a Fiber Metal Laminate panel have been investigated. Also the effects of some system parameters on the nonlinear frequency have been investigated.     

Recent mathematical resuits in the nonlinear theory of flat and curved elastic membranes of revolution

The present article reviews some recent developments in nonlinear elastic membrane theory with special emphasis on axisymmetric deformation of flat circular and annular membranes subjected to a vertical surface load and with prescribed radial stresses or radial displacements at the edges. The nonlinear Föppl membrane theory of small finite deflections as well as a simplified version of Reissner's finite-rotation theory is employed, assuming linear stress-strain relations. The main analytical techniques are reported which have been applied recently in order to determine the ranges of those boundary parameters for which solutions of the relevant nonlinear boundary value problems exist, and ranges of parameters for which the principal stresses are nonnegative everywhere. Concerning plane membranes, it is shown how the mathematical theory of existence and uniqueness was nearly completed in recent works in contrast to curved membranes where references can be given to rather few results.(Deceased)     

Strength of a Composite Sheet Material in Impact Tension

A procedure and results of studies on the strength of a 2-mm PA6 shock-resistant composite sheet in impact tension are briefly outlined. Experimental investigations include static and impact tensile tests of specimens with a short test portion and sharp symmetrical edge notches. Acoustic emission methods used in the tests of specimens in static tension revealed acoustic signals that point to the development of damages in the material under loads much lower than the ultimate one. Test procedures are similar to those used earlier for testing sheet metals in tension. The wave process in the specimens upon impact tension was analyzed using the viscoelastic model. The influence of viscosity on the width of the wave front, determining stress growth rates upon its propagation, was established. According to the test results, the change from quasistatic to impact tension slightly changes maximum stress levels. Impact tension of notched specimens in the first wave of loading lower than the ultimate one produces a decrease in stresses with time that can be determined by viscosity and damage effects near sharp notches.     

Dispersion relations for testing the validity of linear and nonlinear optical spectra

We exploit efficient dispersion relations, which were developed for terahertz spectroscopy, to show their validity for testing linear and nonlinear optical spectra. As an example, we deal with the measured data for complex reflectivity of a KCl crystal and complex nonlinear susceptibility of a polysilane. It is suggested that the spectral data presented in the literature both for the KCl and the polysilane are consistent with the presented spectra analysis method.     

The effective thermomechanical behavior of inelastic fiber-reinforced materials

A continuum theory is presented for the prediction of the average behavior of unidirectional fiber-reinforced materials in which both constituents are thermoelastic in the linear region and thermo-inelastic in the nonlinear region. The resulting effective constitutive equations are given by a set of temperature dependent incremental relations from which the response of the composite to a given mechanical and thermal loading can be determined. The derived theory is applied to investigate the overall behavior of unidirectional graphite fibers reinforcing an aluminum alloy matrix under various types of applied stresses and temperature changes. In particular, the effect of residual stresses developed when the composite is cooled is investigated.     

Linear stability analysis of a sharp-interface model for dewetting thin films

The stability of the receding front at the growing rim of a thin liquid film dewetting from a substrate is studied. The underlying forces that drive the dewetting motion are given by the intermolecular potential between the liquid film and the substrate. The role of slippage in the emerging instability is studied via a sharp-interface model for the dewetting thin film, which is derived from the lubrication model via matched asymptotic expansions. Using the separation of the time-scale for the slow growth of the rim and the time-scale on which the rim destabilises, the sharp-interface results are compared to earlier results for the lubrication model and good agreement for the unstable modes is obtained. The main advantage of the sharp-interface model is that it allows for the derivation of traveling solutions for the base state and subsequently a systematic linear stability analysis via normal modes. Interestingly, unlike the dispersion relations that are typically encountered for the well-known finger-instability in thin-film flows, where the dependence of the growth rate on the wave number is quadratic, here it is linear.     

Appearance of singularities of optical fields under torsion of crystals containing threefold symmetry axes

We present an analysis of the effect of torsion stresses on the spatial distribution of optical birefringence in crystals of different point symmetry groups. The symmetry requirements needed so that the optical beam carries dislocations of the phase front are evaluated for the case when the crystals are twisted and the beam closely corresponds to a plane wave. It is shown that the torsion stresses can produce screw-edge, pure screw, or pure edge dislocations of the phase front in the crystals belonging to cubic and trigonal systems. The conditions for appearance of canonical and noncanonical vortices in the conditions of crystal torsion are analyzed.     

Bending and creep buckling response of viscoelastic functionally graded beam-columns

The viscoelastic creep response of flexural beams and beam-columns made with functionally graded materials is numerically investigated. The paper highlights the challenges associated with the modeling and analysis of such structures, and presents a nonlinear theoretical model for their bending and creep buckling analysis. The model accounts for the viscoelasticity of the materials using differential-type constitutive relations that are based on the linear Boltzmann’s principle of superposition. The model is general in terms of its ability to deal with any material volume faction distribution through the depth of the beam, and with different linear viscoelastic laws, boundary conditions, and loading schemes. The governing equations are solved through time stepping numerical integration, which yields an exponential algorithm following the expansion of the relaxation function into a Dirichlet series. A numerical study that examines the capabilities of the model and quantifies the creep response of functionally graded beam-columns is presented, with special focus on the stresses and strains redistribution over time and on the creep buckling response. The results show that the creep response of such structures can be strongly nonlinear due to the variation of the viscoelastic properties through the depth, along with unique phenomena that are not observed in homogenous structures.     

Nonlinear steady two-layer interfacial flow about a submerged point vortex

Two-dimensional, two-layer steady interfacial flow about a point vortex is studied in a uniform stream for each layer. The upper layer is of finite depth with a rigid lid on the upper surface, and the depth of the lower layer is assumed infinite. The point vortex is located in lower-layer fluid. We study this problem using not only a linear analytical method but also a nonlinear numerical method. A linear solution is derived in terms of a complex exponential integral function. The fully nonlinear problem is formulated by an integro-differential equation system. The equation system is solved using Newton’s method to determine the unknown steady interfacial surface. The numerical results of the downstream wave are provided by a linear solution and fully nonlinear solution. A comparison between linear solutions and nonlinear solutions shows that the nonlinear effect is apparent when the vortex strength increases. The effects of point vortex strengths, Froude numbers, and density ratios on the amplitudes of the downstream waves are studied. We analyze the effects of point vortex strengths, Froude numbers, and density ratios on the wavelengths of the downstream waves.     

弹性压应力波作用下矩形薄板动力屈曲解析解

根据板的非线性动力平衡方程和压缩波前附加约束方程,基于双特征参数法和应力波理论,求解了矩形薄板在面内轴向冲击载荷作用下动力屈曲位移的解析解。揭示了矩形薄板动力屈曲过程中板的厚宽比、屈曲模态、冲击载荷大小和临界屈曲长度之间的关系。求得的屈曲模态与之前文献中用差分解得出的结果吻合良好。     

Green’s function for torsional waves in a cylindrically monoclinic material

Two exact Green’s functions for impulsive and time-harmonic torsional waves in a monoclinic material are presented. The impulsive Green’s function is expressed in the closed form of simple algebraic functions and its wave front shape is a torus with inclined elliptic cross section. The time-harmonic Green’s function is also obtained exactly, but in the form of definite integral. Time development of the wave front for the impulsive wave and amplitude contours for the time-harmonic wave are illustrated.     

Multimodal nonlinear spectral response of a beam with impact under random input

A linearization procedure for estimating the spectral response of a randomly excited beam—stop system is proposed. The elastic stop is replaced by a spring with a stiffness depending on the amplitude of the deflection at the impact location. The probability density function of the amplitude is obtained using the stochastic averaging principle. Next, an estimate of the nonlinear response spectrum is derived providing the expectation of the spectral density function of the random spring linear system with respect to the probability density function of the amplitude response (assumed to be a random variable). The efficiency of the method is checked by comparing results with those of numerical simulations.     

On the characterization of fluxes in nonlinear irreversible thermodynamics

The linear Onsager theory of irreversible thermodynamics is extended to include nonlinear phenomenological relations by means of Onsager fluxes. Such fluxes satisfy a full system of reciprocity relations, vanish in thermodynamic equilibrium, and give a non-negative production of entropy. A complete characterization of Onsager fluxes is obtained in terms of non-negative scalar valued functions which vanish in thermodynamic equilibrium. These same functions are also shown to characterize all C² fluxes which satisfy the second law of thermodynamics. Each system of Onsager fluxes is shown to derive from a dissipation function which attains its absolute minimum in thermodynamic equilibrium. The reaction rates given by reaction kinetics are shown to be Onsager fluxes and their dissipation functions are explicitly calculated.     

On a geometrically exact curved/twisted beam theory under rigid cross-section assumption

 A geometrically exact curved/ twisted beam theory, that assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal frames of reference starting from a 3-D beam theory. The relevant engineering strain measures with an initial curvature correction term at any material point on the current beam cross-section, that are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also summarized in a manner that is easy to understand. Received: 17 June 2002 / Accepted: 21 January 2003 The work was partly sponsored by a grant (CDAAH04-95-1-0175) from the Army Research Office with Dr. Gary Anderson as the grant monitor. We would also like to thank Prof. Raymond Plaut of Dept. of Civil and Environmental Engineering at Virginia Polytechnic Institute and State University for his technical help.     

Symmetrical indentation over a uniformly expanding contact region—II. Perfect adhesion

A two-dimensional wave propagation analysis of half-space indentation by a rigid indentor of a general symmetric shape at arbitrary speeds is considered under conditions of frictionless contact and perfect adhesion. The contact region is assumed to expand symmetrically at a constant rate and the indentor profile and time history are approximated by polynomial curves. Homogeneous function techniques reduce the mathematical analysis to Hilbert problems and the field variables are written as single integrals of the solutions. For a wedge traveling at a constant speed the contact stresses and resultant forces on the half-space are plotted. Wedge angle-indentor velocity-contact area expansion rate relations required for smooth separation from the indentor are also graphed.     

Nonlinear adhesive behavior effects in a cracked orthotropic sheet stiffened by a semi-infinite orthotropic sheet

The stress-intensity factors are determined for a cracked orthotropic sheet adhesively bonded to an orthotropic stringer where the adhesive layer is modeled with a nonlinear stress-strain curve. Since the stringer is modeled as a semi-infinite sheet, the solution is most appropriate for a crack tip located near a stringer edge. By the use of Green's functions and the complex variable theory of orthotropic elasticity developed by Lekhnitskii, a set of integral equations is obtained. The integral equations are replaced by an equivalent set of algebraic equations, which are solved to obtain the shear stress distribution in the adhesive layer. With these adhesive stresses, the crack-tip stress-intensity factors are found.When the adhesive was modeled with a nonlinear stress-strain curve, the peak shear stresses in the adhesive were considerably reduced in comparison to the solution for the linear elastic adhesive. This resulted in increases in the stress-intensity factors for the nonlinear adhesive solution compared to the linear adhesive solution. The nonlinear adhesive did not have a significant effect on the stress-intensity factor unless the near crack tip was beneath the stringer. The present investigation assumes that the adhesive bond remains intact. Onset of adhesive failure is predicted to occur at decreasing levels of applied stress as the crack propagates beneath the stringer.     

Chemical reactions in spherically symmetric problems of mechanochemistry

A stress-assisted chemical reaction front propagation in a linear elastic solid is considered. The reaction between gas and solid constituents is sustained by the diffusion of the gas through the transformed material. The consideration is based on the kinetic equation in a form of the dependence of the reaction front velocity on the normal component of the chemical affinity tensor that in turn depends on stresses and gas concentration. Spherically symmetric problems of mechanochemistry are solved for the reaction front propagation in a sphere, in a body with a spherical hole and in an inclusion placed into an infinite medium. It is demonstrated how stresses can enhance, retard and even lock the reaction. The effects of the sign and value of the reaction front curvature are also examined.