共查询到20条相似文献,搜索用时 15 毫秒
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A generalization of the Cauchy theory of forces and stresses to the geometry of differentiable manifolds is presented using
the language of differential forms. Body forces and surface forces are defined in terms of the power densities they produce
when acting on generalized velocity fields. The normal to the boundary is replaced by the tangent space equipped with the
outer orientation induced by outward pointing vectors. Assuming that the dimension of the material manifold is m, stresses are modelled as m − 1 covector valued forms. Cauchy's formula is replaced by the restriction of the stress form to the tangent space of the
boundary while the outer orientation of the tangent space is taken into account. The special cases of volume manifolds and
Riemannian manifolds are discussed.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
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Stephen C. Preston 《Journal of Mathematical Fluid Mechanics》2013,15(4):807-821
In this paper we construct a new noninvariant Riemannian metric on the semidirect product of the diffeomorphism group of a manifold and the space of positive functions on that manifold, which has the property that certain geodesics give the equations of barotropic fluid mechanics. We compute a formula for its curvature, analyze the sign of the curvature, and determine directly the growth of Jacobi fields in a few special cases. 相似文献
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This work presents a formulation of Cauchy's flux theory of continuum mechanics in the framework of geometric integration theory as formulated by H. Whitney and extended recently by J. Harrison. Starting with convex polygons, one constructs a formal vector space of polyhedral chains. A Banach space of chains is obtained by a completion process of this vector space with respect to a norm. Then, integration operators, cochains, are defined as elements of the dual space to the space of chains. Thus, the approach links the analytical properties of cochains with the corresponding properties of the domains in an optimal way. The basic representation theorem shows that cochains may be represented by forms. The form representing a cochain is a geometric analog of a flux field in continuum mechanics. 相似文献
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M. Šilhavý 《Archive for Rational Mechanics and Analysis》1991,116(3):223-255
Communicated by D. Owen 相似文献
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M. Šilhavý 《Archive for Rational Mechanics and Analysis》1989,107(1):1-22
Communicated by D. Owen 相似文献
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R. Segev 《Journal of Elasticity》2001,63(1):55-59
The Cauchy postulates are required for the formulation and proof of Cauchy's theorem for the existence of stress. The generalized postulates and theorem in the geometric setting of differentiable manifolds was considered in a previous paper. This note presents an inconsistency in one of the proposed postulates, the boundedness postulate, and corrects it by specifying a weaker requirement. 相似文献
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Bernt Wennberg 《Archive for Rational Mechanics and Analysis》1997,139(3):291-302
Elastic collisions are characterized by the conservation of momentum and energy. We consider some geometrical aspects of
such collisions, when the energy of one particle can be expressed in terms of the moments as . The geometry of elastic collisions is essential for the regularizing property of the gain term in the Boltzmann equation,
which was proved by P.-L. Lions. We show how such results can be deduced from a regularity theorem for generalized Radon transforms by Sogge & Stein. This is possible for and for ; we also show that the same technique cannot be used with other choices of .
(Accepted May 20, 1996) 相似文献
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In this paper we consider the asymptotic expression of the solution of the Cauchy's problem for a higher order equation when
the limit equation has singularity. In order to construct the asymptotic expression of the solution, the region is divided
into three sub-areas. In every small region, the solution of the differential equation is different.
Project supported by the National Natural Science Foundation of China 相似文献
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In an optimal control problem one seeks a time-varying input to a dynamical systems in order to stabilize a given target trajectory, such that a particular cost function is minimized. That is, for any initial condition, one tries to find a control that drives the point to this target trajectory in the cheapest way. We consider the inverted pendulum on a moving cart as an ideal example to investigate the solution structure of a nonlinear optimal control problem. Since the dimension of the pendulum system is small, it is possible to use illustrations that enhance the understanding of the geometry of the solution set. We are interested in the value function, that is, the optimal cost associated with each initial condition, as well as the control input that achieves this optimum. We consider different representations of the value function by including both globally and locally optimal solutions. Via Pontryagin’s maximum principle, we can relate the optimal control inputs to trajectories on the smooth stable manifold of a Hamiltonian system. By combining the results we can make some firm statements regarding the existence and smoothness of the solution set. 相似文献
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Wensen Liu Wensen Liu Xanthippi Markenscoff Michael Paukshto 《Journal of Elasticity》1999,54(2):113-128
We construct the orthogonal bases of the Cosserat eigenvectors ũ(−1) for the first boundary value problem of an elastic solid sphere and an infinite elastic space containing a spherical rigid
inclusion. These orthogonal bases are expressed in terms of the Jacobi and Legendre polynomials. An example of a nonharmonic
heat source shows the convergence of the sequence of the eigenvectors ũ(−1).
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
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John S. Selker Michael Niemet Norton G. Mcduffie Steven M. Gorelick Jean-Yves Parlange 《Transport in Porous Media》2007,68(1):107-127
The injection of gases into liquid saturated porous media is of theoretical and practical interest (e.g., air sparging for
the removal of volatile organic compounds from contaminated aquifer sediments). The influence of the rate of gas delivery
and the vertical distance from the source are developed. The concept of a “near-injection region” is presented in which the
pressure gradients exceed buoyant gradients and thus exhibits largely radial flow. The near-injection size is shown to have
an area required to carry the injected gas flow under unit gradient. The parabolic movement of gas outside of this area which
has often been observed is explained as reflecting the sum of many realizations of gas channels following random lateral movements
as they precede upward independent of flux. These concepts are confirmed through comparison with published and experimental
data of air injection into slabs consisting of saturated sands of a range of textures. 相似文献
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Paula Balseiro 《Archive for Rational Mechanics and Analysis》2014,214(2):453-501
In this paper, we consider the Hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these Poisson structures. We illustrate our results with mechanical examples, and in particular show how to recover several well-known facts in the special case of Chaplygin symmetries. 相似文献