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In the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain and a displacement field by = (+) where ,. Barré de Saint-Venant derived the compatibility condition for an arbitrary symmetric second rank tensor field to be of this form, this has now been generalized to higher rank symmetric tensor fields on spaces of dimension
For two-dimensional, plane strain problems the strain-displacement relations are = ; = [+] ; = Repeated differentiation of these relations, in order to remove the displacements and , gives us the two-dimensional compatibility condition for strains
A descent datum consists roughly of a covering of an object V of C by a family V i, elements x i in the fiber over V i, and morphisms f ji between the restrictions of x i and x j to V ij =V i × V V j satisfying the compatibility condition f ki = f kj f ji. The descent datum is called effective if the elements x i are essentially the pullbacks ...
Derivation of the Lagrangian and Eulerian finite strain tensors. A measure of deformation is the difference between the squares of the differential line element , in the undeformed configuration, and , in the deformed configuration (Figure 2). Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred.
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. [1]
For symmetric tensors, these definitions are reduced. [2] The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that + =
The constitutive relations between the and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are: = = where u is the four-velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), and denotes the ...
Given two tensor bundles E → M and F → M, a linear map A: Γ(E) → Γ(F) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A(fs) = fA(s), for each section s in Γ(E) and each smooth function f on M.