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58
NON–LINEAR ELASTICITY AS A FIELD THEORY
Y.N. Radayev ([email protected]), S.A. Lychev ([email protected])
Non-linear elasticity is considered from the physical viewpoint as a field the-
ory. First, field equations and constitutive relations of the finite-strain elasticity
in their canonical forms are derived in a conventional way. In view of practical
application to fracture, special attention is paid to to the construction and imme-
diate consequences of the canonical equations of energy and pseudomomentum
balance, thus demonstrating the wealth of the framework, and allowing readily
introduce Eshelby's stress tensor, the path-independent integral and Eshelby's
force, acting on an elastic inhomogeneity. Then the field-theoretic concept is
used in order to define the tensors of non-linear elasticity and reveal their natural
co-ordinate representation. The variational formulation and Noether's theorem
are chosen to derive conservation laws and additional path-independent inte-
grals. Inverse-motion description and variational symmetries of the Hamiltonian
action are shown provide a true field theory of non-linear inhomogeneous elas-
ticity.
The null Lagrangian theory for n-dimensional manifold (including 4-
dimensional Minkowski space-time) is developed in an attempt to extend the
canonical formalism of nonlinear field theory. All developments are presented in
the frame of finite strains. By the aid of divergence formula for the null Lagran-
gians regular in n-dimensional star-shaped domains, a general representation of
the null Lagrangian depending as maximum on the first order field gradients is
obtained. A method of systematic derivation of the null Lagrangians in n-
dimensional manifold is proposed. It is shown that in the case of nonlinear elas-
tic field in 3-dimensional space the null Lagrangian is represented via 15 arbi-
trary independent field functions. Null Lagrangians of 4-dimensional material
manifold are then analysed. Invariant under actual placement translations null
Lagrangians are obtained. Material forces acting on an empty 4-dimensional
manifold are finally studied.
