Showing papers in "Acta Mechanica in 1997"
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TL;DR: In this article, an advanced lumped parameter joint model is developed and identified by experimental investigations for an isolated bolted joint, which is implemented in a Finite Element program for calculating the dynamic response of assembled structures incorporating the influence of micro-and macroslip of several bolted joints.
Abstract: The nonlinear transfer behaviour of an assembled structure such as a large lightweight space structure is caused by the nonlinear influence of structural connections. Bolted or riveted joints are the primary source of damping compared to material damping, if no special damping treatment is added to the structure. Simulation of this damping amount is very important in the design phase of a structure. Several well known lumped parameter joint models used in the past to describe the dynamic transfer behaviour of isolated joints by Coulomb friction elements are capable of describing global states of slip and stick only. The present paper investigates the influence of joints by a mixed experimental and numerical strategy. A detailed Finite Element model is established to provide understanding of different slip-stick mechanisms in the contact area. An advanced lumped parameter model is developed and identified by experimental investigations for an isolated bolted joint. This model is implemented in a Finite Element program for calculating the dynamic response of assembled structures incorporating the influence of micro- and macroslip of several bolted joints.
348 citations
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TL;DR: In this paper, a new spin tensor and a new objective tensor-rate for the Eulerian logarithmic strain in V and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed.
Abstract: Two yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. A new spin tensor and a new objective tensor-rate are accordingly introduced. Further, new rate-form constitutive models based on this objective tensor-rate are established. It is proved that
(i).
an objective corotational rate of the logarithmic strain inV can be exactly identical with the stretching and in all strain measures only inV enjoys this property, and
(ii).
InV and the Cauchy stress σ form a work-conjugate pair of strain and stress.
302 citations
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TL;DR: In this article, the applicability of aluminium foam as filler material in tubes made of mild steel having square or circular cross sections, which are crushed axially at low loading velocities was investigated.
Abstract: This study, with the emphasis on experiments, investigates the applicability of aluminium foam as filler material in tubes made of mild steel having square or circular cross sections, which are crushed axially at low loading velocities. In addition to the experiments finite element studies are performed to simulate the crushing behaviour of the tested square tubes, were a crushable foam material model is shown to be suitable for describing the inelastic response of aluminium foam with respect to the considered problems. The experimental results for the square tubes reveal efficiency improvements with respect to energy absorption of up to 60%, resulting from changed buckling modes of the tubes and energy dissipation during the compression of the foam material itself. The principal features as well as the changes of the crushing process due to filling can also be studied by the numerical simulations. A global failure mechanism due to a high foam density can be observed for filled circular tubes. Aluminium foam is shown to be a suitable material for filling thin-walled tubular steel structures, holding the potential of enhancing the energy absorption capacity considerably, provided the plastic buckling remains characterized by local modes.
216 citations
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TL;DR: In this paper, a geometric nonlinear and thermodynamical consistent constitutive theory is proposed, which allows the representation of the thermomechanical behavior of carbon black filled rubber, which is based on a simple spring dashpot system of viscoplasticity.
Abstract: A geometric nonlinear and thermodynamical consistent constitutive theory is proposed, which allows the representation of the thermomechanical behaviour of carbon black filled rubber. In a recent paper [1] it was shown that the mechanical behaviour of this material is mainly influenced by nonlinear elasticity coupled with some inelastic effects, in particular the Mullins-effect, nonlinear rate dependence and a weak equilibrium hysteresis. In the present paper, the Mullins-effect is not taken into consideration. At first we discuss a uniaxial approach, based on a simple spring dashpot system of viscoplasticity. The essential feature of this model is a decomposition of the total stress into a rate independent equilibrium stress and a nonlinear rate dependent overstress. The equilibrium stress is decomposed into a sum of two terms as well: The first term, the elastic part of the equilibrium stress, is a nonlinear function of the total strain, and the second term, the so-called hysteretic part, depends in a rate independent manner on the strain history. Both the overstress and the hysteretic part of the equilibrium stress are determined by nonlinear elasticity relations which depend on internal variables. These internal variables are inelastic strains, and the corresponding evolution equations are developed in consideration of the second law of thermodynamics. Accordingly, we demonstrate that the principle of non-negative dissipation is satisfied for arbitrary deformation processes. In a further step, we transfer the structure of this model to the three-dimensional and geometric nonlinear case. In a certain sense similar to finite deformation elasto-plasticity, we introduce two multiplicative decompositions of the deformation gradient into elastic and inelastic parts. The first decomposition is defined with respect to the overstress and the second one with respect to the hysteretic part of the equilibrium stress. Consequently, two intermediate configurations are induced, which lead two different decompositions of the Green's strain tensor into elastic and inelastic parts. The latter are the internal variables of the model. For physical reasons, we define the corresponding stress tensors and derivatives in the sense of the concept of dual variables [7], [39]. Theconstitutive equations for the overstress and for the hysteretic part of the equilibrium stress are specified by nonlinear elasticity relations, formulated with respect to the different intermediate configurations. In order to facilitate a separate description of inelastic bulk and distortional effects, we introduce kinematic decompositions of the deformation gradient into volumetric and distortional parts. Numerical simulations demonstrate that the developed theory represents the mechanical behaviour of a tread compound at room temperature very well. Thermomechanical heating effects, which are caused by inelastic deformations are also described by the theory. The method proposed in this paper can be utilised to generalise uniaxial rheological models to three-dimensional finite strain theories, which are admissible in the sense of the second law of thermodynamics.
188 citations
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TL;DR: In this paper, the Laplace transform method is used to find the roots of algebraic equations with fractional exponents, which allows one to investigate the roots behavior in a wide range of single-mass system parameters.
Abstract: Free damped vibrations of an oscillator, whose viscoelastic properties are described in terms of the fractional calculus Kelvin-Voight model, Maxwell model, and standard linear solid model are determined. The problem is solved by the Laplace transform method. When passing from image to pre-image one is led to find the roots of an algebraic equation with fractional exponents. The method for solving such equations is proposed which allows one to investigate the roots behaviour in a wide range of single-mass system parameters. A comparison between the results obtained on the basis of the three models has been carried out. It has been shown that for all models the characteristic equations do not possess real roots, but have one pair of complex conjugates, i.e. the test single-mass systems subjected to the impulse excitation do not pass into an aperiodic regime in none of magnitudes of the relaxation and creep times. Main characteristics of vibratory motions of the single-mass system as functions of the relaxation time or creep time, which are equivalent to the temperature dependencies, are constructed and analyzed for all three models.
177 citations
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TL;DR: In this article, the authors considered the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear.
Abstract: Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.
125 citations
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TL;DR: In this article, a constitutive modeling for the particle size effect on the strength of metal matrix composites is investigated, based on a gradient-dependent theory of plasticity that incorporates strain gradients into the expression of the flow stress of matrix materials, and a finite unit cell technique is used to calculate the overall flow properties of composites.
Abstract: Constitutive modeling for the particle size effect on the strength of particulate-reinforced metal matrix composites is investigated. The approach is based on a gradient-dependent theory of plasticity that incorporates strain gradients into the expression of the flow stress of matrix materials, and a finite unit cell technique that is used to calculate the overall flow properties of composites. It is shown that the strain gradient term introduces a spatial length scale in the constitutive equations for composites, and the dependence of the flow stress on the particle size/spacing can be obtained. Moreover, a nondimensional analysis along with the numerical result yields an explicit relation for the strain gradient coefficient in terms of particle size, strain, and yield stress. Typical results for aluminum matrix composites with ellipsoidal particles are calculated and compare well with data measured experimentally.
120 citations
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TL;DR: In this paper, the energy equation for the boundary layer flow of an electrically conducting fluid under the influence of a constant transverse magnetic field over a linearly stretching non-isothermal flat sheet is considered.
Abstract: This paper presents solutions of the energy equation for the boundary layer flow of an electrically conducting fluid under the influence of a constant transverse magnetic field over a linearly stretching non-isothermal flat sheet. Effects due to dissipation, stress work and heat generation are considered. Analytical solutions of the resulting linear nonhomogeneous boundary value problems, expressed in terms of Kummer's functions, are presented for the case of prescribed surface temperature as well as the case of prescribed wall heat flux, both of which are assumed to be quadratic functions of distance. The boundary value problems are also solved by direct numerical integration yielding results in excellent agreement with the analytical solutions.
86 citations
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TL;DR: In this article, the authors prove that the kinetic and strain energies of a solution with finite energy become asymptotically equal as time tends to infinity, and they also show that the same holds for the case of dipolar bodies.
Abstract: Our study is concerned with the thermoelasticity of dipolar bodies. We prove that the Cesaro means of the kinetic and strain energies of a solution with finite energy become asymptotically equal as time tends to infinity.
70 citations
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TL;DR: In this paper, the authors analyzed the dynamic flexural behavior of elastic two-layer beams with interlayer slip, where the Bernoulli-Euler hypothesis is assumed to hold for each layer separately and a linear constitutive equation between the horizontal slip and the interlaminar shear force is considered.
Abstract: The objective of the present paper is to analyze the dynamic flexural behavior of elastic two-layer beams with interlayer slip. The Bernoulli-Euler hypothesis is assumed to hold for each layer separately, and a linear constitutive equation between the horizontal slip and the interlaminar shear force is considered. The governing sixth-order initial-boundary value problem is solved by separating the dynamic response in a quasistatic and in a complementary dynamic response. The quasistatic portion that may also contain singularities or discontinuities due to sudden load changes is determined in a closed form. The remaining complementary dynamic part is non-singular and can be approximated by a truncated modal series of fast accelerated convergence. The solution of the resulting generalized decoupled single-degree-of-freedom oscillators is given by means of Duhamel,s convolution integral, whereby the velocity and acceleration of the loads are the driving terms. Light damping is considered via modal damping coefficients. The proposed procedure is illustrated for dynamically loaded layered single-span beams with interlayer slip, and the improvement in comparison to the classical modal analysis is demonstrated.
57 citations
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TL;DR: In this article, a variational formulation for micropolar continua with constrained and non-constrained rotations is presented and the interrelation between the different formulations is highlighted.
Abstract: The objective of this work is to elaborate upon the variational setting for micropolar continua withconstrained andunconstrained rotations. To this end, several mixed variational principles and their regularizations are considered for both the geometrically linear and nonlinear case. The interrelation between the different formulations are highlighted. The most advantageous result is obtained by translating the insight gained for the geometrically linear case to the geometrically nonlinear case involving large strains and large rotations. It turns out that a particular micropolar description involves standard constitutive models for the symmetric stress part together with a nonsymmetric penalty stress thus circumventing to describe the constitutive law in terms of a nonsymmetric strain measure.
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TL;DR: In this article, it was shown both by experiment and also by numerical simulation that for a vertically hanging folded chain the free part, if released, is falling faster than a free falling body under gravitational acceleration.
Abstract: It is shown both by experiment and also by numerical simulation that for a vertically hanging folded chain the free part, if released, is falling faster than a free falling body under gravitational acceleration. A qualitative explanation of thisparadoxical phenomenon is given by showing that a downpulling force at the fold is created. In the simulation this force is also calculated quantitatively.
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TL;DR: In this article, a new class of constitutive models for viscoelastic media with finite strains is derived, which employ the so-called fractional derivatives of tensor functions.
Abstract: A new class of constitutive models is derived for viscoelastic media with finite strains. The models employ the so-called fractional derivatives of tensor functions. We introduce fractional derivatives for an objective tensor which satisfies some natural assumptions. Afterwards, we construct fractional differential analogs of the Kelvin-Voigt, Maxwell, and Maxwell-Weichert constitutive models. The models are verified by comparison with experimental data for viscoelastic solids and fluids. We consider uniaxial tension of a bar and radial oscillations of a thick-walled spherical shell made of the fractional Kelvin-Voigt incompressible material. Explicit solutions to these problems are derived and compared with experimental data for styrene butadiene rubber and synthetic rubber. It is shown that the fractional Kelvin-Voigt model provides excellent prediction of experimental data. For uniaxial tension of a bar and simple shear of an infinite layer made of the fractional Maxwell compressible material, we develop explicit solutions and compare them with experimental data for polyisobutylene specimens. It is shown that the fractional Maxwell model ensures fair agreement between experimental data and results of numerical simulation. This model allows the number of adjustable parameters to be reduced significantly compared with other models which ensure the same level of accuracy in the prediction of experimental data.
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TL;DR: In this paper, a method for substantially improving the performance of the higher-order plate and shell theories in connection with the accurate stress analysis of homogeneous and laminated composite structural elements is presented.
Abstract: This paper proposes a method for substantially improving the performance of the higher-order plate and shell theories in connection with the accurate stress analysis of homogeneous and laminated composite structural elements. The presentation of the method is based on the equations of the “general five-degrees-of-freedom” shear deformable plate theory. Since the method is entirely new, it is initially applied to the solution of the problem of simply supported plates deformed by cylindrical bending, for which there exists an exact elasticity solution [12]. Hence, its reliability is substantially validated by means of appropriate comparisons between numerical results based on the present plate theory and this exact elasticity solution. Moreover, the one-dimensional version of the present plate theory, employed for the cylindrical bending of plates, is considered as a general three-degrees-of-freedom shear deformable beam theory. This advanced beam theory is used for an accurate stress analysis of two-layered composite beams having one of their edges rigidly clamped and the other either rigidly clamped, free of tractions or simply supported. This final set of applications can be thought of alternatively as a stress analysis of two-layered plates deformed in cylindrical bending and subjected to several, different sets of edge boundary conditions.
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TL;DR: In this paper, a new thick plate theory is derived by using Papkovich-Neuber solution of three-dimensional elasticity, which is based on a new thin plate theory.
Abstract: In this paper a new thick plate theory is derived by using Papkovich-Neuber solution of three-dimensional elasticity.
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TL;DR: In this paper, the authors deal with free convection heat and mass transfer from a vertical plate embedded in a fluid saturated porous medium with constant wall temperature and concentration and apply integral method of Von-Karman type to obtain the analytical solution of this fundamental problem.
Abstract: This study deals with free convection heat and mass transfer from a vertical plate embedded in a fluid saturated porous medium with constant wall temperature and concentration. The temperature and concentration variations across the boundary layer produce a buoyancy effect which gives rise to flow field. Integral method of Von-Karman type is applied to obtain the analytical solution of this fundamental problem.
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TL;DR: Similarity solutions of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocities were determined for a range of exponents α and amplitudes β.
Abstract: Similarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βyα are determined for a range of exponents α and amplitudes β Asymptotic analysis of the equations shows that for α −2/3 the shear stressf″(0) parameter is determined as a function of α and β Symmetric free-shear flow solutions become singular as α→α0 = −1/2 and no solutions are found in the range −1 −1/2 the centerline velocityf′(0) is determined as a function of α and β An asymptotic analysis of the singular behavior of these two problems as α→α0, given in a separate Appendix, shows excellent comparison with the numerical results Similarity solutions at the critical values α0 have exponential decay in the far field and correspond to the Glauert wall jet for wall-bounded flow and to the Schlichting/Bickley planar jet for symmetric free-shear flow
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TL;DR: In this article, a new analytic finite element method (AFEM) was proposed for solving the governing equations of steady magnetohydrodynamic (MHD) duct flows, which is used to calculate the flow field, induced magnetic field, and the first partial derivatives of these fields.
Abstract: A new analytic finite element method (AFEM) is proposed for solving the governing equations of steady magnetohydrodynamic (MHD) duct flows. By the AFEM code one is able to calculate the flow field, the induced magnetic field, and the first partial derivatives of these fields. The process of the code generation is rather lengthy and complicated, therefore, to save space, the actual formulation is presented only for rectangular ducts. A distinguished feature of the AFEM code is the resolving capability of the high gradients near the walls without use of local mesh refinement. Results of traditional FEM, AFEM and finite difference method (FDM) are compared with analytic results demonstrating the manifest superiority of the AFEM code. The programs for the AFEM codes are implemented in GAUSS using traditional computer arithmetic and work in the range of low and moderate Hartmann numbersM<1000.
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TL;DR: In this article, a microcontinuum approach is used to determine the velocity and pressure distributions in the channel which is taken to diverge exponentially, and analytical expressions for the velocity in the thin layer (interstitial space) as well as corresponding computational results are presented.
Abstract: This paper is concerned with a mathematical analysis of the flow of blood in the lungs. The smallest microscopic blood vessels in human lungs are organized into sheet like networks. Each sheet may be idealized into a channel bounded by two thin layers of porous media. A microcontinuum approach is used to determine the velocity and pressure distributions in the channel which is taken to diverge exponentially. Analytical expressions for the velocity and pressure distributions in the thin layer (interstitial space) as well as the corresponding computational results are presented. The flow velocity in the channel and the cell rotational velocity have also been determined.
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TL;DR: In this paper, the authors proposed the use of nodal sensitivities as error indicators and estimators for numerical analysis in mechanics, defined as rates of change of response quantities with respect to nodal positions.
Abstract: This paper proposes the use of special sensitivities, called nodal sensitivities, as error indicators and estimators for numerical analysis in mechanics. Nodal sensitivities are defined as rates of change of response quantities with respect to nodal positions. Direct analytical differentiation is used to obtain the sensitivities, and the infinitesimal perturbations of the nodes are forced to lie along the elements. The idea proposed here can be used in conjunction with general purpose computational methods such as the Finite Element Method (FEM), the Boundary Element Method (BEM) or the Finite Difference Method (FDM); however, the BEM is the method of choice in this paper. The performance of the error indicators is evaluated through two numerical examples in linear elasticity.
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TL;DR: In this article, the effect of uniform internal heat generation on the onset of steady Marangoni convection in a horizontal layer of quiescent fluid heated from below is analyzed.
Abstract: In this paper we use a combination of analytical and numerical techniques to analyse the effect of uniform internal heat generation on the onset of steady Marangoni convection in a horizontal layer of quiescent fluid heated from below. We obtain for the first time the closed form analytical solution for the onset of steady Marangoni convection and give a comprehensive description of the stability characteristics of the layer both when the lower boundary is conducting and when it is insulating to temperature perturbations. We also present asymptotically- and numerically-calculated results for the linear growth rates of the steady modes. In particular, we show that the effect of increasing the internal heat generation is always to destabilise the layer and give asymptotic expressions for the critical Marangoni number and critical wavenumber in the limit of large internal heat generation.
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TL;DR: In this article, the authors present a rather detailed discussion of the features of the Eshelby tensor in the context of linearized elasticity, including the integrand of the path-independent J-integral.
Abstract: The Eshelby tensor (also referred to as the Maxwell tensor of elasticity, or the energy momentum tensor of elasticity, or the material momentum tensor), is being widely used in contracted form (e.g., with the unit normal vector) in the study of defect and fracture mechanics, most prominently as the integrand of the path-independent J-integral. However, the properties and the physical interpretation of the components of this tensor itself have remained seemingly unexplored. This contribution attempts to remedy this situation and presents a rather detailed discussion of the features of this tensor in the context of linearized elasticity.
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TL;DR: In this article, a broad class of plane-strain axially-symmetric deformation patterns in geomaterials is studied within the framework of large strain pressure sensitive plasticity.
Abstract: A broad class of plane-strain axially-symmetric deformation patterns in geomaterials is studied within the framework of large strain pressure-sensitive plasticity. Invariant, non-associated deformationtype theories are formulated for the Mohr-Coulomb (M-C) and Drucker-Prager (D-P) solids with arbitrary hardening and accounting for an initial hydrostatic state of stress. With the M-C model we arrive at a single first order differential equation, while for the D-P solid an algebraic constraint supplements the governing differential equation. The analysis centers on the effective stress as the independent variable. A simplified treatment is given for the cavitation limit and some useful relations are derived for thin walled cylinders. The theory is applied to the triaxial calibration test for Castlegate sandstone and then used to simulate the hole closure problem. Numerical examples are provided for the case of a cavity embedded in an infinite medium subjected to external or internal pressure. Results for the D-P inner cone model were found to be in close agreement with those obtained from the M-C model.
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TL;DR: In this paper, the case of flow past an oblate spheroid is considered and the drag experienced by it is evaluated, and the exact solution is obtained to the first order in the small parameter characterizing the deformation.
Abstract: Creeping axisymmetric slip flow past a spheroid whose shape deviates slightly from that of a sphere is investigated. An exact solution is obtained to the first order in the small parameter characterizing the deformation. As an application, the case of flow past an oblate spheroid is considered and the drag experienced by it is evaluated. Special well-known cases are deduced and some observations made.
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TL;DR: In this article, a mathematical model for axisymmetric laminar flow generated by a rotating disk inside a cylinder with an open top, containing a viscous fluid above as layer of fluid-saturated porous medium is presented.
Abstract: A numerical investigation has been undertaken to characterize the axisymmetric laminar flow generated by a rotating disk inside a cylinder with an open top, containing a viscous fluid above as layer of fluid-saturated porous medium. The mathematical model is based on a continuum approach for both fluid and porous regions. Attention is focussed on conditions favouring steady, stable, axisymmetric solutions of the Darcy-Brinkman-Lapwood equation. The accuracy of the method is verified by solving some vortex flow problems in disk-cylinder geometries and comparing the results with: (a) existing numerical solutions and, (b) experimental pressure measurements in a similar geometry. Calculations are performed to investigate the fluid exchange between the porous region (porewater) and the overlying water. Results indicate that flow through composite (fluid-sediment) systems can be handled with good accuracy by the method presented here. With our approach the magnitude of advective porewater transport in sediments may be predicted. This finding is important for improved designs of flux chambers and also for understanding advective transport phenomena.
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TL;DR: In this paper, the deformation of an isotropic structural plate is controlled by applying an electric potential to a piezoelectric ceramic plate of crystal class 6mm.
Abstract: The present paper deals with a thermoelastic problem in an isotropic structural plate to which a piezoelectric ceramic plate of crystal class 6mm is perfectly bonded. It is assumed that the combined plate is subjected to a thermal load and then is deformed. In this case, we try to control the deformation of the isotropic structural plate by applying an electric potential to the piezoelectric ceramic plate. By analyzing the piezothermoelastic problem in the combined plate, we obtain an appropriate applied electric potential which alters the isotropic structural plate to a prescribed deformation. Finally numerical calculations are carried out for an isotropic steel plate to which a cadmium selenide plate is perfectly bonded, and the results are illustrated graphically.
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TL;DR: In this article, a numerical solution for the three dimensional transient motion of a marine cable during installation is presented for the case of a cable laying vessel arbitrarily chaning speed and direction while paying out cable with seabed slack.
Abstract: A numerical solution for the three dimensional transient motion of a marine cable during installation is presented for the case of a cable laying vessel arbitrarily chaning speed and direction while paying out cable with seabed slack. The cable transient behaviour is governed by the numerical solution of a set of non-linear partial differential equations with the solution methodology incorporating both spatial and temporal integration. The space integration is carried out by dividing the cable inton straight elements with equilibrium relationships and geometric compatibility equations satisfied for each element. The position of each element is described by its elevation and azimuth angles and, therefore, a system of 2n non-linear ordinary differential equations is established. The time integration of this set of equations is performed using a high order Runge-Kutta technique. Results are presented fro the cable tension and element elevation and azimuth angles as functions of time and for transient cable geometries when the cable ship executes horizontal planar manoeuvres.
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TL;DR: In this article, the effects of higher-order deformations on natural frequencies and buckling stresses of a thick circular ring with rectangular cross-sections subjected to circumferential tensile and/or compressive stresses are studied.
Abstract: The effects of higher-order deformations on natural frequencies and buckling stresses of a thick circular ring with rectangular cross-sections subjected to circumferential tensile and/or compressive stresses are studied. Based on the power series expansion of displacement components, a set of fundamental dynamic equations of a one-dimensional higher-order ring theory is derived through Hamilton's sprinciple. Several sets of truncated approximate theories which can take into account the complete effects of higher-order deformations such as shear deformation and depth change and rotary inertia are applied to solve the eigenvalue problems of a thick circular ring. In order to assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress for the flexural and extensional displacement modes of thick rings are examined in detail. It seems that the present approximate theories can predict benchmark data of the natural frequency and buckling stress of thick rings more accurately compared to other existing theories.
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TL;DR: In this paper, two degree-of-freedom systems with weak quadratic nonlinearities were studied under weak external and parametric excitations respectively, and the method of averaging was used to obtain a set of four first-order amplitude equations that govern the dynamics of the firstorder asymptotic approximation to the response.
Abstract: Two degree-of-freedom systems with weak quadratic nonlinearities are studied under weak external and parametric excitations respectively. All six possible cases, that arise in the presence of 1∶2 internal resonance, are investigated. The method of averaging is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order asymptotic approximation to the response. An analytical technique, based on Melnikov's method is used to predict the parameter range for which chaotic dynamics exists in the undamped averaged system. Numerical studies show that such chaotic responses are quite common in these quadratic systems, and they seem to persist even in the presence of damping.
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TL;DR: In this paper, simple estimations for the overall conductivity and elastic properties of some isotropic locally-ordered composites are deduced from two different variational approaches, based on limited information about microgeometry of the composites.
Abstract: Simple estimations for the overall conductivity and elastic properties of some isotropic locally-ordered composites are deduced from two different variational approaches. The estimations based on limited information about microgeometry of the composites lie inside the Hashin-Shtrikman bounds over the whole range of parameters.