# Continuum with Singularity

• L. R. Rakotomanana
Part of the Progress in Mathematical Physics book series (PMP, volume 31)

## Abstract

Constitutive relations are mathematical relations between dual variables and primal variables. The essence of constitutive relations is that they define idealized materials. The histories of dual variables at any point M and any time t are determined by the histories of the primal variables in all points of the body up to the time t. When only the nth order time derivatives of primal variables are involved instead of their entire histories, materials are said to be of the rate type n, e.g., [194]. The goal of this chapter is to propose the basic framework for continuum with singularity distribution. The essential points of this chapter were developed in a previous paper [163].

## Keywords

Dual Variable Entropy Inequality Singularity Distribution Torsion Tensor Affine Connection

## References

1. 1.
From relation (4.1), the expansion of the objective time derivative of the free energy with respect to the continuum reads: (fy59-1).Google Scholar
2. 2.
For crystalline solids, e.g., [201], the triplet (u1, u2, u3) is called the crystallographic basis and the constants of structure characterize the crystallographic nonhomogeneity of the solid.Google Scholar
3. 3-In fact, this is a particular case of the localization theorem of integral invariance (3.26). Here, we have sketched an elementary proof.Google Scholar
4. 4.
Eshelby (1951), e.g., [54] first introduced the concept of force acting on a singularity or defect in continuum mechanics. A review of such material forces has been conducted in [133]. Although using a classical nondistorted Riemannian manifold, in [133] a unified concept for different singularity forces has been proposed, such as: material forces, e.g., [132], configurational forces, e.g., [74], [143], force on a singularity, e.g., [54], force on a elastic defect, e.g., [10], [213], force on a dislocation [152], inhomogeneity forces, e.g., [148], [194], g-invariant integral [26], and J-integral fracture [167]. The present work extended the concept of material forces to affinely connected manifolds (continuum with singularity [163]). Indeed, the material forces in (4.34) capture distribution of singularity fields in a continuum. Cartan (1923) [21] propounded the requirement of a change in the affine structure of spacetime to incorporate gravity forces in the framework of non relativistic space-time. His proposition was independent from relativization of time since the phenomena in question did not necessarily involve large speeds of matter, e.g., [48].According to Einstein-Cartan’s theory of gravitation, spacetime corresponding to a distribution of matter should be represented by an affinely connected manifold with nonvanishing torsion and curvature, the torsion field being related to the density of spin (intrinsic angular momentum ).Google Scholar