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1.
We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic, isotropic, incompressible
and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order
problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ, the hoop stress for the linear elastic (or the first-order) problem at the point R=(R
ou
R
in
(R
ou
+R
in
)/2)1/3 is independent of the slope of the μ vs. R line. Here R
in
and R
ou
equal, respectively, the inner and the outer radius of the sphere in the reference configuration. For μ(R)∝R
n
, for the linear problem, the hoop stress is constant in the sphere for n=1. However, no such results are found for the second-order (i.e., materially nonlinear) problem. Whereas for the first-order
problem the shear modulus influences only the radial displacement and not the stresses, for the second-order problem the two
elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected
only to pressure on the outer surface, the hoop stress at a point on the inner surface depends upon values of the two elastic
moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed
form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be
used to verify solutions of the problem obtained by using numerical methods. 相似文献
2.
Martin A. Eisenberg Chong-Won Lee Aris Phillips 《International Journal of Solids and Structures》1977,13(12):1239-1255
The thermodynamics of materials with internal state variables has been employed to study the properties of a class of thermoplastic materials in which the evolution equation for the internal variables is given by equation k(i) = g(i)(σkl, 'kl, k(i), θ, · 'kl) where g(i) is homogeneous of order one in · 'kl. The most general form of the Helmholtz potential consistent with the assumption of insensitivity of the elastic relations to inelastic deformation has been derived and a geometric interpretation of the Clausius-Duhem restriction has been made employing the concept of a thermodynamic reference stress. Experimental results of one of the authors have been correlated with the theory. 相似文献
3.
For axially symmetric deformations of the perfectly elastic neo-Hookean and Mooney materials, formal series solutions are
determined in terms of expansions in appropriate powers of 1/R, where R is the cylindrical polar coordinate for the material coordinates. Remarkably, for both the neo-Hookean and Mooney materials,
the first three terms of such expansions can be completely determined analytically in terms of elementary integrals. From
the incompressibility condition and the equilibrium equations, the six unknown deformation functions, appearing in the first
three terms can be reduced to five formal integrations involving in total seven arbitrary constants A, B, C, D, E, H and k
2, and a further five integration constants, making a total of 12 integration constants for the deformation field. The solutions
obtained for the neo-Hookean material are applied to the problem of the axial compression of a cylindrical rubber tube which
has bonded metal end-plates. The solution so determined is approximate in two senses; namely as an approximate solution of
the governing equations and for which the stress free and displacement boundary conditions are satisfied in an average manner
only. The resulting load-deflection relation is shown graphically. The solution so determined, although approximate, attempts
to solve a problem not previously tackled in the literature.
相似文献
4.
H. Irago 《Journal of Elasticity》1999,57(1):55-83
Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ωε having cross section with diameter of order ε, and let u
(0) –Bernoulli–Navier displacement – and u
(2) be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u
(0) + ε2
u
(2)) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the
rod.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
5.
Hermano Frid 《Archive for Rational Mechanics and Analysis》2006,181(1):177-199
We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems
of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems
whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which
are hyperplanes. In particular, we obtain the uniqueness of the self-similar L∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is
in the sense of the convergence as t→∞ in Lloc1 of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in Lloc1(ℝn), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem. 相似文献
6.
Antonios Charalambopoulos 《Journal of Elasticity》1993,31(1):47-69
In this paper, the behavior of the solution of the time-dependent linearized equation of dynamic elasticity is examined.For the homogeneous problem, it is proved that in the exterior of a star-shaped body on the surface of which the displacement field is zero, the solution decays at the rate t
-1 as the time t tends to infinity.For the non-homogeneous problem with a harmonic forcing term, it is proved that for large times, the elastic material in the exterior of the body, tends to a harmonic motion, with the period of the external force.The convergence to the steady harmonic state solution is at the rate t
-1/2 as t tends to infinity, and is uniform on bounded sets. 相似文献
7.
Shaohua Chen Cong Yan Peng Zhang Huajian Gao 《Journal of the mechanics and physics of solids》2009,57(9):1437-1448
We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E0(z/c0)k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of Pcr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off. 相似文献
8.
C. Eck S. A. Nazarov W. L. Wendland 《Archive for Rational Mechanics and Analysis》2001,156(4):275-316
The variational solution of the nonlinear Signorini contact problem determines also the active contact zone Γ
c
. If the latter is known, then the elastic field is a solution of a linear mixed boundary value problem in which on Γ
c
the normal displacement and tangential traction are given, while on the non-contact part the total traction is zero. Such
mixed boundary conditions in general generate singularities of the solution's stress field at the points P
(
k
) where the boundary conditions change. For smooth data, however, the variational solution of the Signorini contact problem
actually belongs to H
2(Ω)2, which implies the disappearance of these singularities, i.e., that the corresponding stress intensity factors vanish.
This paper is devoted to the characterization of the active contact zone Γ
c
by the vanishing stress intensity factors including their sensitivity with respect to varying Γ
c
for two-dimensional problems provided that Γ
c
consists of a finite number of intervals. We use the method of asymptotic expansions and derive an explicit formula for the
sensitivity, which is rigorously justified by employing weighted Sobolev spaces with detached asymptotics. These results can
be used to determine the points P
(
k
) with a corresponding Newton iteration.
Accepted July 6, 2000?Published online January 22, 2001 相似文献
9.
Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source 总被引:1,自引:0,他引:1
M. Lewandowska 《Heat and Mass Transfer》2001,37(4-5):333-342
The Cattaneo hyperbolic and classical parabolic models of heat conduction in the laser irradiated materials are compared.
Laser heating is modelled as an internal heat source, whose capacity is given by g(x,t)= I(t)(1−R)μexp(−μx). Analytical solution for the one-dimensional, semi-infinite body with the insulated boundary is obtained using Laplace transforms
and the discussion of solutions for different time characteristics of the heat source capacity (constant, instantaneous, exponential,
pulsed and periodic) is presented.
Received on 18 May 1999 相似文献
10.
Equivalent lagrangians and the solution of some classes of non-linear equations
(1K)
The second-order ordinary differential equation
, where μ ≠ 1 is linearizable(sl(3, R) algebra) via a point transformation if and only if n = μ or n = 1. We construct a quadratic Lagrangian
, which determines the point transformation Q = F(t,q) and = G(t,q) that maps the Lagrangian to the simple completely integrable Lagrangian
. For n = 4μ − 3 the equation admits the sl(2, R) algebra. In this case we again construct a quadratic Lagrangian and then obtain the corresponding point transformation that reduces the original Lagrangian to the representative Lagrangian
. For both cases, sl(2,R) and sl(3,R), we obtain complete solutions (cf. [1,2]). 相似文献
Full-size image (1K)
11.
Jeyabal Sivaloganathan Scott J. Spector 《Archive for Rational Mechanics and Analysis》2010,196(2):363-394
Let ${A=\{{\bf x} \in \mathbb{R}^n : a < |{\bf x}| < b\}, n \geqq 2, a > 0}Consider a homogeneous, isotropic, hyperelastic body occupying the region
A = {x ? \mathbbRn : a < |x | < b }{A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}} in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions.
In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to
incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity
it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this
paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation
procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration
of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism
u : A → A* between spherical shells A and A* is the deformation
urad(x)=\fracr(R)Rx, R=|x|, (0.1){\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1) 相似文献
12.
Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions
are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating
and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution
of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence
and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical
Lebesgue spaces Lp(R3), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established. 相似文献
13.
We consider a spherically symmetric static problem of general relativity whose solution was obtained in 1916 by Schwarzschild for a metric form of a special type. This solution determines the metric coefficients of the exterior and interior Riemannian spaces generated by a gravitating solid ball of constant density and includes the so-called gravitational radius r g. For a ball of outer radius R=r g, the metric coefficients are singular, and hence the radius r g is traditionally assumed to be the radius of the event horizon of an object called a black hole. The solution of the interior problem obtained for an incompressible ideal fluid shows that the pressure at the ball center increases without bound for R=9/8r g, which is traditionally used for the physical justification of the existence of black holes. The discussion of Schwarzschild’s traditional solution carried out in this paper shows that it should be generalized with respect to both the geometry of the Riemannian space and the elastic medium model. In this connection, we consider the general metric form of a spherically symmetric Riemannian space and prove that the solution of the corresponding static problem exists for a broad class of metric forms. A special metric form based on the assumption that the gravitation generating the Riemannian space inside a fluid ball or an elastic ball does not change the ball mass is singled out from this class. The solution obtained for the special metric form is singular with respect to neither the metric coefficients nor the pressure in the fluid ball and the stresses in the elastic ball. The obtained solution is compared with Schwarzschild’s traditional solution. 相似文献
14.
This work considers the generalized plane problem of a moving dislocation in an anisotropic elastic medium with piezoelectric, piezomagnetic and magnetoelectric effects. The closed-form expressions for the elastic, electric and magnetic fields are obtained using the extended Stroh formalism for steady-state motion. The radial components, Erand Hr, of the electric and magnetic fields as well as the hoop components, Dθ and Bθ, of electric displacement and magnetic flux density are found to be independent of θ in a polar coordinate system. This interesting phenomenon is proven to be is a consequence of the electric and magnetic fields, electric displacement and magnetic flux density that exhibit the singularity r−1 near the dislocation core. As an illustrative example, the more explicit results for a moving dislocation in a transversely isotropic magneto–electro-elastic medium are provided and the behavior of the coupled fields is analyzed in detail. 相似文献
15.
Eiji Yanagida 《Archive for Rational Mechanics and Analysis》1991,115(3):257-274
Positive radial solutions of a semilinear elliptic equation u+g(r)u+h(r)u
p
=0, where r=|x|, xR
n
, and p>1, are studied in balls with zero Dirichlet boundary condition. By means of a generalized Pohoaev identity which includes a real parameter, the uniqueness of the solution is established under quite general assumptions on g(r) and h(r). This result applies to Matukuma's equation and the scalar field equation and is shown to be sharp for these equations. 相似文献
16.
The paper gives a simple derivation of the relaxed energy W
qc
for the quadratic double-well material with equal elastic moduli and analyzes W
qc
in the transversely isotropic case. We observe that the energy W is a sum of a degenerate quadratic quasiconvex function and a function that depends on the strain only through a scalar variable.
For such a W, the relaxation reduces to a one-dimensional convexification. W
qc
depends on a constant g defined by a three-dimensional maximum problem. It is shown that in the transversely isotropic case the problem reduces to
a maximization of a fraction of two quadratic polynomials over [0,1]. The maximization reveals several regimes and explicit
formulas are given in the case of a transversely isotropic, positive definite displacement of the wells.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
17.
In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}
18.
Summary This paper deals with the theoretical treatment of a three-dimensional elastic problem governed by a cylindrical coordinate
system (r,θ,z) for a medium with nonhomogeneous material property. This property is defined by the relation G(z)=G
0(1+z/a)
m
where G
0,a and m are constants, i.e., shear modulus of elasticity G varies arbitrarily with the axial coordinate z by the power product form. We propose a fundamental equation system for such nonhomogeneous medium by using three kinds of
displacement functions and, as an illustrative example, we apply them to an nonhomogeneous thick plate (layer) subjected to
an arbitrarily distributed load (not necessarily axisymmetric) on its surfaces. Numerical calculations are carried out for
several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacement and stress components are shown graphically.
Received 10 May 1999; accepted for publication 15 August 1999 相似文献
19.
The elastostatic plane problem of a layered composite containing an internal or edge crack perpendicular to its boundaries in its lower layer is considered. The layered composite consists of two elastic layers having different elastic constants and heights and rests on two simple supports. Solution of the problem is obtained by superposition of solutions for the following two problems: The layered composite subjected to a concentrated load through a rigid rectangular stamp without a crack and the layered composite having a crack whose surface is subjected to the opposite of the stress distribution obtained from the solution of the first problem. Using theory of Elasticity and Fourier transform technique, the problem is formulated in terms of two singular integral equations. Solving these integral equations numerically by making use of Gauss–Chebyshev integration, numerical results related to the normal stress σx(0,y), the stress-intensity factors, and the crack opening displacements are presented and shown graphically for various dimensionless quantities. 相似文献
20.
We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here Rin and Rou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration. 相似文献
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