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1.
Earlier it was shown in [1, 2] that the equations of classical nonlinear elasticity constructed for the case of small strains
and arbitrary displacements are ill posed, because their use in specific problems may result in the appearance of “spurious”
bifurcation points. A detailed analysis of these equations and the construction, in their stead, of consistent equations of
geometrically nonlinear theory of elasticity can be found in [3]. Certain steps in this direction were also made in [4, 5].
In [3], it was also stated that the methods and applied program packages (APPs) based on the use of the classical relations
of nonlinear elasticity require some revision and correction. In the present paper, this conclusion is justified and confirmed
by numerical finite-element solutions of several three-dimensional geometrically nonlinear deformation problems and linearized
problems on the stability of equilibrium of a rectilinear beam. These solutions were obtained by using two APPs developed
by the authors and the well-known APP “ANSYS.” It is shown that the classical equations of the geometrically nonlinear theory
of elasticity, which underly the first of the developed APP and the well-known APP “ANSYS,” often lead to overestimated buckling
loads for structural members as compared with the consistent equations proposed in [1–3]. 相似文献
2.
A refined Timoshenko-type model based on the straight-line hypothesis is used to develop an approach to analyzing the stress
state of longitudinally corrugated cylindrical shells with elliptic cross-section. The approach is to reduce the two-dimensional
boundary-value problem that describes the stress–strain state of the shell to a one-dimensional one and to solve it with the
stable numerical discrete-orthogonalization method. The solutions obtained using the straight-line hypothesis and the equations
of three-dimensional elasticity are compared. The dependence of the stress–strain state of the shell on the number and amplitude
of corrugations and the aspect ratio of the cross-section is analyzed 相似文献
3.
Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced
systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber
solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined
theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented
by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact
solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic
equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all
the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed
by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification.
It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous
boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order
terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised.
In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein. 相似文献
4.
Yu. E. Ivanova V. E. Ragozina 《Journal of Applied Mechanics and Technical Physics》2006,47(6):892-898
The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value
problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible
medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based
on solutions of evolution equations different from the equations for quasi-simple waves.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 144–151, November–December, 2006. 相似文献
5.
The paper presents results of applying a heterogeneous mathematical model “elastic body–Timoshenko shell” to design shells
with massive ribs. Numerical results are obtained for a cylindrical shell with ribs. They are compared with results obtained
using the theory of elasticity and the theory of Timoshenko shells with piecewise-constant thickness
Published in Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 132–142, November 2008. 相似文献
6.
V. D. Bondar' 《Journal of Applied Mechanics and Technical Physics》2000,41(1):120-130
For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and
complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value
problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's
formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear
theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter
method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds
to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of
the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated
by graphs.
Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41,
No. 1, pp. 133–143, January–February, 2000. 相似文献
7.
This paper deals with spatial axisymmetric boundary-value problems of the physically nonlinear theory of elasticity for piecewise-homogeneous
spherical bodies. The passage to dimensionless characteristics of the stress-strain state allows us to extract a physical
dimensionless small parameter in the nonlinear state equations. The solution of nonlinear equilibrium equations and boundary-value
problems is searched for in the form of series in positive degrees of the small parameter. This approach allows reducing the
stated physically nonlinear boundary-value problem to a sequence of corresponding linear nonhomogeneous problems. A specific
analytical solution and numerical results are obtained for a two-layer nonlinearly elastic spherical shell under bilateral
pressure.
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika,
Vol. 35, No. 12, pp. 26–32, December, 1999. 相似文献
8.
Adair R. Aguiar 《Journal of Elasticity》2006,84(2):99-129
There are problems in the classical linear theory of elasticity whose closed form solutions, while satisfying the governing equations of equilibrium together with well-posed boundary conditions, predict the existence of regions, often quite small, inside the body where material overlaps. Of course, material overlapping is not physically realistic, and one possible way to prevent it combines linear theory with the requirement that the deformation field be injective. A formulation of minimization problems in classical linear elasticity proposed by Fosdick and Royer [3] imposes this requirement through a Lagrange multiplier technique. An existence theorem for minimizers of plane problems is also presented. In general, however, it is not certain that such minimizers exist. Here, the Euler–Lagrange equations corresponding to a family of three-dimensional problems is investigated. In classical linear elasticity, these problems do not have bounded solutions inside a body of anisotropic material for a range of material parameters. For another range of parameters, bounded solutions do exist but yield stresses that are infinite at a point inside the body. In addition, these solutions are not injective in a region surrounding this point, yielding unrealistic behavior such as overlapping of material. Applying the formulation of Fosdick and Royer on this family of problems, it is shown that both the displacements and the constitutive part of the stresses are bounded for all values of the material parameters and that the injectivity constraint is preserved. In addition, a penalty functional formulation of the constrained elastic problems is proposed, which allows to devise a numerical approach to compute the solutions of these problems. The approach consists of finding the displacement field that minimizes an augmented potential energy functional. This augmented functional is composed of the potential energy of linear elasticity theory and of a penalty functional divided by a penalty parameter. A sequence of solutions is then constructed, parameterized by the penalty parameter, that converges to a function that satisfies the first variation conditions for a minimizer of the constrained minimization problem when this parameter tends to infinity. This approach has the advantages of being mathematically appealling and computationally simple to implement. 相似文献
9.
We present the theory of space–time elasticity and demonstrate that it is the extended reversible thermodynamics and gives the coupled model of thermoelasticity and heat conductivity and involves traditional thermoelasticity. We formulate the generally covariant variational model’s dynamic thermoelasticity and heat conductivity in which the basic kinematic and static variables are unified tensor objects (subject, matter). Variation statement defines the whole set of the initial-boundary problems for the 4D vector governing equation (Euler equation), the spatial projections of which define motion equations and the time projection gives the heat conductivity equation. We show that space–time elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, space–time elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity. Moreover, we establish that the Maxwell–Cattaneo law and Fourier law can be defined for the reversible processes as compatibility equations without introducing dissipation. We argue that the present framework of space–time elasticity should prove adequate to describe the thermoelastic phenomena at low temperatures for interpreting the results of molecular simulations of heat conduction in solids and for the optimal heat and stress management in the microelectronic components and the thermoelectric devices. 相似文献
10.
A simplified model of bending dynamics of a hydrostatically compressed thin shell near the threshold of stability of its form
is constructed within the framework of the nonlinear theory of elasticity. Conditions of existence and explicit expressions
for spatially localized perturbations and patterns composed of dents on the shell surface, which are “precursors” of subsequent
changes in the shell form, are found.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 124–134, November–December, 2007. 相似文献
11.
Shahid Hussain Arshad Muhammad Nawaz Naeem Nazra Sultana Abdul Ghafar Shah Zafar Iqbal 《Archive of Applied Mechanics (Ingenieur Archiv)》2011,81(3):319-343
In the present study, a vibration frequency analysis of a bi-layered cylindrical shell composed of two independent functionally
graded layers is presented. The thickness of the shell layers is assumed to be equal and constant. Material properties of
the constituents of bi-layered functionally graded cylindrical shell are assumed to vary smoothly and continuously through
the thickness of the layers of the shell and are controlled by volume fraction power law distribution. The expressions for
strain–displacement and curvature–displacement relationships are utilized from Love’s first approximation linear thin shell
theory. The versatile Rayleigh–Ritz approach is employed to formulate the frequency equations in the form of eigenvalue problem.
Influence of material distribution in the two functionally graded layers of the cylindrical shells is investigated on shell
natural frequencies for various shell parameters with simply supported end conditions. To check the validity, accuracy and
efficiency of the present methodology, results obtained are compared with those available in the literature. 相似文献
12.
Some possible modifications of the governing equations of the linear theory of elasticity are considered. The stress–strain relation is specified by an integral equality instead of the local Hooke’s law. The modified integrodifferential boundary value problem is reduced to the minimization of a nonnegative functional under differential constraints. A numerical algorithm based on polynomial approximations of unknown functions (stresses and displacements) is developed and applied to linear elasticity problems. The bilateral estimation criteria of solution errors are proposed in order to analyze the algorithm convergence rate. The numerical results obtained by applying the integrodifferential relation method and the conventional variational method are compared and discussed. 相似文献
13.
In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described
by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable
first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used
to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification
of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the
parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary
constitutive function that are likely to yield such solutions are also indicated. 相似文献
14.
This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations
in the form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated and
the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions
of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems.
Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological
singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended
here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory
cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal.
11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient)
additional to the standard Lamé constants (Georgiadis et al., J. Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy
density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution
method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity
solutions (Flamant–Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted
at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the
singular behavior present in the classical solutions.
相似文献
15.
A method of studying the natural vibrations of highly inhomogeneous shells of revolution is developed. The method is based
on a nonclassical theory of shells that allows for transverse shear and reduction. By separating variables, the two-dimensional
problem is reduced to a sequence of one-dimensional eigenvalue problems. The inverse iteration method is used to reduce these
problems to a sequence of inhomogeneous boundary-value problems solved by the orthogonal sweep method. The capabilities of
the method are illustrated by solving certain representative problems and comparing their solutions with those obtained using
the three-dimensional theory of elasticity, the classical theory of shells, and the refined Timoshenko model
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Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 38–47, September 2007. 相似文献
16.
O. V. Voinov 《Journal of Applied Mechanics and Technical Physics》2007,48(3):385-392
The motion of thin films of a viscous incompressible liquid in a gas under the action of capillary forces is studied. The
surface tension depends on the surfactant concentration, and the liquid is nonvolatile. The motion is described by the well-known
model of quasi-steady-state viscous film flow. The linear-wave solutions are compared with the solution using the Navier-Stokes
equations. Situations are studied where a solution close to the inviscid two-dimensional solutions exists and in the case
of long wavelength, the occurrence of sound waves in the film due to the Gibbs surface elasticity is possible. The behavior
of the exact solutions near the region of applicability of asymptotic equations is studied, and nonmonotonic dependences of
the wave characteristics on wavenumber are obtained.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 103–111, May–June, 2007. 相似文献
17.
The problems of nonlinear deformation of a thin current-carrying shell under the coupled action of an unsteady electromagnetic
field and a mechanical field are studied. The nonlinear magneto-elastic kinetic equations, the physical equations, the geometric
equations, the electrodynamics equations, and the expressions of Lorentz force of a thin current-carrying shell under the
action of a coupled field are given. Normal Cauchy form nonlinear differential equations, which include ten basic unknown
functions in all, are obtained by the variable replacement method. Using the difference and quasilinearization methods, the
nonlinear magneto-elastic equations are reduced to a sequence of quasilinear differential equations, which can be solved by
the method of discrete orthogonalization. Numerical solutions for the stresses and deformations in the thin current-carrying
strip-shell with two simply supported edges are obtained by considering a specific example. The results that the stresses
and deformations in a thin current-carrying strip-shell with two simply supported edges change with variation of the electromagnetic
parameters are discussed, through a special case. It is shown that the deformations of the shell can be controlled by changing
the electromagnetic parameters
Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 130–144, September 2007. 相似文献
18.
N. I. Ostrosablin 《Journal of Applied Mechanics and Technical Physics》2006,47(4):564-572
This paper deals with the general formulas of affine transformations that preserve invariance of the static equations of the
linear theory of elasticity in the case of arbitrary anisotropic materials. The invariance of the equations with respect to
affine transformations allows one to model a given anisotropic material by another material. All anisotropic materials are
divided into classes of mutually congruent materials. The congruency conditions are obtained for orthotropic and isotropic
materials and for orthotropic and transversely isotropic materials.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 124–134, July–August, 2006. 相似文献
19.
The elastostatic problem for cracked shallow spherical shell resting on linear elastic foundation is considered. The problem is formulated for a homogeneous isotropic material within the confines of a linearized shallow shell theory. By making use of integral transforms and asymptotic analysis, the problem is reduced to the solution of a pair of singular integral equations. The stress distribution obtained, around the crack tip, is similar to that of the elasticity solutions. The numerical results obtained agree well with those of previous work, where the elastic supports were neglected. The influences of the shell curvature and the modulus of subgrade reaction on the stress intensity factor are given. 相似文献
20.
Constructing an analytical solution for lamb waves using the cosserat continuum approach 总被引:2,自引:0,他引:2
M. A. Kulseh V. P. Matveenko I. N. Shardakov 《Journal of Applied Mechanics and Technical Physics》2007,48(1):119-125
The problem of propagation of a Lamb elastic wave in a thin plate is considered using the Cosserat continuum model. The deformed
state is characterized by independent displacement and rotation vectors. Solutions of the equations of motion are sought in
the form of wave packets specified by a Fourier spectrum of an arbitrary shape for three components of the displacement vector
and three components of the rotation vector which depend on time, depth, and the longitudinal coordinate. The initial system
of equations is split into two systems, one of which describes a Lamb wave and the second corresponds to a transverse wave
whose amplitude depends on depth. Analytical solutions in displacements are obtained for the waves of both types. Unlike the
solution for Lamb waves, the solution obtained for the transverse wave has no analogs in classical elasticity theory. The
solution for the transverse wave is compared with the solution for the Lamb wave.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 143–150, January–February, 2007.
An erratum to this article is available at . 相似文献