20,345 research outputs found
The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations
In this paper we consider the equilibrium problem in the relaxed linear model
of micromorphic elastic materials. The basic kinematical fields of this
extended continuum model are the displacement and the
non-symmetric micro-distortion density tensor . In
this relaxed theory a symmetric force-stress tensor arises despite the presence
of microstructure and the curvature contribution depends solely on the
micro-dislocation tensor . However, the relaxed model is able
to fully describe rotations of the microstructure and to predict non-polar
size-effects. In contrast to classical linear micromorphic models, we allow the
usual elasticity tensors to become positive-semidefinite. We prove that,
nevertheless, the equilibrium problem has a unique weak solution in a suitable
Hilbert space. The mathematical framework also settles the question of which
boundary conditions to take for the micro-distortion. Similarities and
differences between linear micromorphic elasticity and dislocation gauge theory
are discussed and pointed out.Comment: arXiv admin note: substantial text overlap with arXiv:1308.376
Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics
We construct identities of Pohozhaev type, in the context of elastostatics
and elastodynamics, by using the Noetherian approach. As an application, a
non-existence result for forced semi-linear isotropic and anisotropic elastic
systems is established
Solutions of Navier Equations and Their Representation Structure
Navier equations are used to describe the deformation of a homogeneous,
isotropic and linear elastic medium in the absence of body forces.
Mathematically, the system is a natural vector (field) O(n,\mbb{R})-invariant
generalization of the classical Laplace equation, which physically describes
the vibration of a string. In this paper, we decompose the space of polynomial
solutions of Navier equations into a direct sum of irreducible
O(n,\mbb{R})-submodules and construct an explicit basis for each irreducible
summand. Moreover, we explicitly solve the initial value problems for Navier
equations and their wave-type extension--Lam\'e equations by Fourier expansion
and Xu's method of solving flag partial differential equations.Comment: 44 page
Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
In 1922, Cartan introduced in differential geometry, besides the Riemannian
curvature, the new concept of torsion. He visualized a homogeneous and
isotropic distribution of torsion in three dimensions (3d) by the "helical
staircase", which he constructed by starting from a 3d Euclidean space and by
defining a new connection via helical motions. We describe this geometric
procedure in detail and define the corresponding connection and the torsion.
The interdisciplinary nature of this subject is already evident from Cartan's
discussion, since he argued - but never proved - that the helical staircase
should correspond to a continuum with constant pressure and constant internal
torque. We discuss where in physics the helical staircase is realized: (i) In
the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d
theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's
case of constant pressure and constant intrinsic torque - and b) in 3d Poincare
gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the
gauge field theory of dislocations of Lazar et al., as we prove for the first
time by arranging a suitable distribution of screw dislocations. Our main
emphasis is on the discussion of dislocation field theory.Comment: 31 pages, 8 figure
On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics
Using the theory of hyperbolic systems we put in perspective the
mathematical and geometrical structure of the celebrated circularly polarized
waves solutions for isotropic hyperelastic materials determined by Carroll in
Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this
class of solutions yields an infinite family of \emph{linear} solutions for the
equations of isotropic elastodynamics. Moreover, we determine a huge class of
hyperbolic partial differential equations having the same property of the shear
wave system. Restricting the attention to the usual first order asymptotic
approximation of the equations determining transverse waves we provide the
complete integration of this system using generalized symmetries.Comment: 19 page
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