31 research outputs found
Extension the Noether's theorem to Lagrangian formulation with nonlocality
A Lagrangian formulation with nonlocality is investigated in this paper. The
nonlocality of the Lagrangian is introduced by a new nonlocal argument that is
defined as a nonlocal residual satisfying the zero mean condition. The nonlocal
Euler-Lagrangian equation is derived from the Hamilton's principle. The
Noether's theorem is extended to this Lagrangian formulation with nonlocality.
With the help of the extended Noether's theorem, the conservation laws relevant
to energy, linear momentum, angular momentum and the Eshelby tensor are
determined in the nonlocal elasticity associated with the mechanically based
constitutive model. The results show that the conservation laws exist only in
the form of the integral over the whole domain occupied by body. The
localization of the conservation laws is discussed in detail. We demonstrate
that not every conservation law corresponds to a local equilibrium equation.
Only when the nonlocal residual of conservation current exists, can a
conservation law be transformed into a local equilibrium equation by
localization.Comment: 13 page
The gauge theory of dislocations: static solutions of screw and edge dislocations
We investigate the T(3)-gauge theory of static dislocations in continuous
solids. We use the most general linear constitutive relations bilinear in the
elastic distortion tensor and dislocation density tensor for the force and
pseudomoment stresses of an isotropic solid. The constitutive relations contain
six material parameters. In this theory both the force and pseudomoment
stresses are asymmetric. The theory possesses four characteristic lengths l1,
l2, l3 and l4 which are given explicitely. We first derive the
three-dimensional Green tensor of the master equation for the force stresses in
the translational gauge theory of dislocations. We then investigate the
situation of generalized plane strain (anti-plane strain and plane strain).
Using the stress function method, we find modified stress functions for screw
and edge dislocations. The solution of the screw dislocation is given in terms
of one independent length l1=l4. For the problem of an edge dislocation, only
two characteristic lengths l2 and l3 arise with one of them being the same
l2=l1 as for the screw dislocation. Thus, this theory possesses only two
independent lengths for generalized plane strain. If the two lengths l2 and l3
of an edge dislocation are equal, we obtain an edge dislocation which is the
gauge theoretical version of a modified Volterra edge dislocation. In the case
of symmetric stresses we recover well known results obtained earlier.Comment: 33 pages, 17 figure
New Mechanics of Traumatic Brain Injury
The prediction and prevention of traumatic brain injury is a very important
aspect of preventive medical science. This paper proposes a new coupled
loading-rate hypothesis for the traumatic brain injury (TBI), which states that
the main cause of the TBI is an external Euclidean jolt, or SE(3)-jolt, an
impulsive loading that strikes the head in several coupled degrees-of-freedom
simultaneously. To show this, based on the previously defined covariant force
law, we formulate the coupled Newton-Euler dynamics of brain's micro-motions
within the cerebrospinal fluid and derive from it the coupled SE(3)-jolt
dynamics. The SE(3)-jolt is a cause of the TBI in two forms of brain's rapid
discontinuous deformations: translational dislocations and rotational
disclinations. Brain's dislocations and disclinations, caused by the
SE(3)-jolt, are described using the Cosserat multipolar viscoelastic continuum
brain model.
Keywords: Traumatic brain injuries, coupled loading-rate hypothesis,
Euclidean jolt, coupled Newton-Euler dynamics, brain's dislocations and
disclinationsComment: 18 pages, 1 figure, Late
Gauge theory and defects in solids
This new series Mechanics and Physics of Discrete Systems aims to provide a coherent picture of the modern development of discrete physical systems. Each volume will offer an orderly perspective of disciplines such as molecular dynamics, crystal mechanics and/or physics, dislocation, etc. Emphasized in particular are the fundamentals of mechanics and physics that play an essential role in engineering applications.Volume 1, Gauge Theory and Defects in Solids, presents a detailed development of a rational theory of the dynamics of defects and damage in solids. Solutions to field