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 $W$ over the $n$-dimensional space manifold, the value of a second order stress at a point $x$ in space is represented mathematically by a linear mapping between the second jet space of $W$ at $x$ and the space of $n$-alternating tensors at $x$. 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, $J^{1}(J^{1}W)$, 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 $n$-dimensional manifold, an $(n-1)$-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 $B$, a differentiable motion $m$ of $B$ in the manifold $\mathcal{S}$, an $r$-current $T$ in $B$, and the sequence of images $m(t)_{\sharp}T$ of the current under the motion, we consider the rate of change of the action of the images on a smooth $r$-form in $\mathcal{S}$. 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 kk-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