45 research outputs found
Geometric Analysis of Hyper-Stresses
A geometric analysis of high order stresses in continuum mechanics is
presented. Virtual velocity fields take their values in a vector bundle \vbts
over the n-dimensional space manifold. A stress field of order k is represented
mathematically by an n-form valued in the dual of the vector bundle of k-jets
of \vbts. While only limited analysis can be performed on high order stresses
as such, they may be represented by non-holonomic hyper-stresses, n-forms
valued in the duals of iterated jet bundles. For non-holonomic hyper-stresses,
the analysis that applies to first order stresses may be iterated. In order to
determine a unique value for the tangent surface stress field on the boundary
of a body and the corresponding edge interactions, additional geometric
structure should be specified, that of a vector field transversal to the
boundary
Generalized Stress Concentration Factors for Equilibrated Forces and Stresses
As a sequel to a recent work we consider the generalized stress concentration
factor, a purely geometric property of a body that for the various external
loading fields indicates the worst ratio between the maximum of the optimal
stress and maximum of the external loading. The optimal stress concentration
factor pertains to a stress field that satisfies the principle of virtual work
and for which the stress concentration factor is minimal. Unlike the previous
work, we require that the external loading be equilibrated and that the stress
field be a symmetric tensor field
Metric Independent Analysis of Second Order Stresses
A metric independent geometric analysis of second order stresses in continuum
mechanics is presented. For a vector bundle over the -dimensional space
manifold, the value of a second order stress at a point in space is
represented mathematically by a linear mapping between the second jet space of
at and the space of -alternating tensors at . While only limited
analysis can be performed on second order stresses as such, they may be
represented by non-holonomic stresses, whose values are linear mapping defined
on the iterated jet bundle, , and for which an iterated analysis
for first order stresses may be performed. As expected, we obtain the surface
interactions on the boundaries of regions in space
Optimal Stresses in Structures
For a given external loading on a structure we consider the optimal stresses.
Ignoring the material properties the structure may have, we look for the
distribution of internal forces or stresses that is in equilibrium with the
external loading and whose maximal component is the least. We present an
expression for this optimal value in terms of the external loading and the
matrix relating the external degrees of freedom and the internal degrees of
freedom. The implementation to finite element models consisting of elements of
uniform stress distributions is presented. Finally, we give an example of
stress optimization for of a two-element model of a cylinder under external
traction.Comment: 15 pages, 2 figure
Geometric Aspects of Singular Dislocations
The theory of singular dislocations is placed within the framework of the
theory of continuous dislocations using de Rham currents. For a general
-dimensional manifold, an -current describes a local layering
structure and its boundary in the sense of currents represents the structure of
the dislocations. Frank's rules for dislocations follow naturally from the
nilpotency of the boundary operator
Reynolds Transport Theorem for Smooth Deformations of Currents on Manifolds
The Reynolds transport theorem for the rate of change of an integral over an
evolving domain is generalized. For a manifold , a differentiable motion
of in the manifold , an -current in , and the
sequence of images of the current under the motion, we
consider the rate of change of the action of the images on a smooth -form in
. The essence of the resulting computations is that the derivative
operator is represented by the dual of the Lie derivative operation on smooth
forms.Comment: uncertainty has risen on the changes made in Version
De Donder Construction for Higher Jets
In this paper, we generalize De Donder approach to construct boundary forms
that depend on the adapted coordinate system used. In continuum mechanics, use
of boundary forms leads to splitting of the total force acting on the body into
body force and surface traction. Moreover, this splitting is independent of the
choice of the boundary form used. In calculus of variations, use of boundary
forms leads to equations in exterior differential forms that are equivalent to
the Euler-Lagrange equations. Infinitesimal symmetries of the theory lead to
conservation laws valid for any choice of the boundary form used. In an
example, we show that the boundary conditions lead to independence of constants
of motion of the choice of the boundary form
Hyper-Stresses in -Jet Field Theories
For high-order continuum mechanics and classical field theories
configurations are modeled as sections of general fiber bundles and generalized
velocities are modeled as variations thereof. Smooth stress fields are
considered and it is shown that three distinct mathematical stress objects play
the roles of the traditional stress tensor of continuum mechanics in Euclidean
spaces. These objects are referred to as the variational hyper-stress, the
traction hyper-stress and the non-holonomic stress. The properties of these
three stress objects and the relations between them are studied
On jets, almost symmetric tensors, and traction hyper-stresses
The paper considers the formulation of higher-order continuum mechanics on
differentiable manifolds devoid of any metric or parallelism structure. For
generalized velocities modeled as sections of some vector bundle, a variational
kth order hyper-stress is an object that acts on jets of generalized velocities
to produce power densities. The traction hyper-stress is introduced as an
object that induces hyper-traction fields on the boundaries of subbodies.
Additional aspects of multilinear algebra relevant to the analysis of these
objects are reviewed.Comment: 23 page
A unified geometric treatment of material defects
A unified theory of material defects, incorporating both the smooth and the
singular descriptions, is presented based upon the theory of currents of
Georges de Rham. The fundamental geometric entity of discourse is assumed to be
represented by a single differential form or current, whose boundary is
identified with the defect itself. The possibility of defining a less
restrictive dislocation structure is explored in terms of a plausible weak
formulation of the theorem of Frobenius. Several examples are presented and
discussed.Comment: 6 pages, CanCNSM2013, Montrea