2,687 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
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
Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics
Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations
We study the propagation of two-dimensional finite-amplitude shear waves in a
nonlinear pre-strained incompressible solid, and derive several asymptotic
amplitude equations in a simple, consistent, and rigorous manner. The scalar
Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations
of motion for all elastic generalized neo-Hookean solids (with strain energy
depending only on the first principal invariant of Cauchy-Green strain).
However, we show that the Z equation cannot be a scalar equation for the
propagation of two-dimensional shear waves in general elastic materials (with
strain energy depending on the first and second principal invariants of
strain). Then we introduce dispersive and dissipative terms to deduce the
scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and
Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid
mechanics.Comment: 15 page
The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity
We investigate a family of isotropic volumetric-isochoric decoupled strain
energies based on the Hencky-logarithmic (true, natural)
strain tensor , where is the infinitesimal shear modulus,
is the infinitesimal bulk modulus with
the first Lam\'{e} constant, are dimensionless
parameters, is the gradient of deformation,
is the right stretch tensor and is the deviatoric part of the strain tensor . For small elastic strains, approximates the classical
quadratic Hencky strain energy which is not everywhere
rank-one convex. In plane elastostatics, i.e. , we prove the everywhere
rank-one convexity of the proposed family , for and . Moreover, we show that the
corresponding Cauchy (true)-stress-true-strain relation is invertible for
and we show the monotonicity of the Cauchy (true) stress tensor as a
function of the true strain tensor in a domain of bounded distortions. We also
prove that the rank-one convexity of the energies belonging to the family
is not preserved in dimension
Geometry of logarithmic strain measures in solid mechanics
We consider the two logarithmic strain measureswhich are isotropic invariants of the
Hencky strain tensor , and show that they can be uniquely characterized
by purely geometric methods based on the geodesic distance on the general
linear group . Here, is the deformation gradient,
is the right Biot-stretch tensor, denotes the principal
matrix logarithm, is the Frobenius matrix norm, is the
trace operator and is the -dimensional deviator of
. This characterization identifies the Hencky (or
true) strain tensor as the natural nonlinear extension of the linear
(infinitesimal) strain tensor , which is the
symmetric part of the displacement gradient , and reveals a close
geometric relation between the classical quadratic isotropic energy potential
in
linear elasticity and the geometrically nonlinear quadratic isotropic Hencky
energywhere
is the shear modulus and denotes the bulk modulus. Our deduction
involves a new fundamental logarithmic minimization property of the orthogonal
polar factor , where is the polar decomposition of . We also
contrast our approach with prior attempts to establish the logarithmic Hencky
strain tensor directly as the preferred strain tensor in nonlinear isotropic
elasticity
On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush
In a series of papers which are either published [A.R. Hadjesfandiari and
G.F. Dargush, Couple stress theory for solids, Int. J. Solids Struct. 48,
2496-2510, 2011; A.R. Hadjesfandiari and G.F. Dargush, Fundamental solutions
for isotropic size-dependent couple stress elasticity, Int. J. Solids Struct.
50, 1253-1265, 2013] or available as preprints Hadjesfandiari and Dargush have
reconsidered the linear indeterminate couple stress model. They are postulating
a certain physically plausible split in the virtual work principle. Based on
this postulate they claim that the second-order couple stress tensor must
always be skew-symmetric. Since they use an incomplete set of boundary
conditions in their virtual work principle their statement contains
unrecoverable errors. This is shown by specifying their development to the
isotropic case. However, their choice of constitutive parameters is
mathematically possible and still yields a well-posed boundary value problem.Comment: arXiv admin note: text overlap with arXiv:1504.0086
Theorems on existence and global dynamics for the Einstein equations
This article is a guide to theorems on existence and global dynamics of
solutions of the Einstein equations. It draws attention to open questions in
the field. The local-in-time Cauchy problem, which is relatively well
understood, is surveyed. Global results for solutions with various types of
symmetry are discussed. A selection of results from Newtonian theory and
special relativity that offer useful comparisons is presented. Treatments of
global results in the case of small data and results on constructing spacetimes
with prescribed singularity structure or late-time asymptotics are given. A
conjectural picture of the asymptotic behaviour of general cosmological
solutions of the Einstein equations is built up. Some miscellaneous topics
connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living
Rev. Rel. 5 (2002)
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