Untitled

126 Antonio J. Gil
e3
R0
e2
e1
X
pret
X
x
Rt
Deformation path
and their respective jacobians are summarized as:
xi,A = xi,jX pret
Xj,A ⇒ J ∗ = J ′ J (1)
where the implied summation convention for repeated indices as well as the comma
differentiation symbol xi,j = ∂xi/∂X pret
Xj have been introduced to simplify thealgebra.
The term J above represents the jacobian at the primary state, J ∗ stands for
the jacobian at the end of the secondary state and J ′
symbolizes the jacobian as a
consequence of the incremental deformation.
The description of the deformation and the measure of strain are essential parts
of nonlinear continuum mechanics. From a kinematically point of view, a material
particle position in the primary and secondary states may be expressed in terms
of the incremental displacement field. Analogously, the deformation gradient tensor
may be introduced to characterize adequately the deformation path as:
where:
xi = X pret
i + ui ⇒ xi,A = X pret
ui,j = ∂ui
X
i,A + ui,A = X pret
i,A
∂X pret
j
u
Rt*
pret
+ ui,jXX j,A (2)
In contrast to linear elasticity, many different measures of strain may be used in
nonlinear continuum mechanics. Nevertheless, the Green-Lagrange strain tensor is
considered to be the most appropriate measure specially when dealing with moderate
strains. The Green-Lagrange strain tensor for the secondary state with respect to
the initial nominally stressed configuration is defined in terms of the deformation
gradient tensor and the Delta-Kronecker tensor as:
By substituting equation (2) into equation (4):
E ∗ AB = 1
2 (Xpret
i,A Xpret
i,B
(3)
E ∗ AB = 1
2 (xi,Axi,B − δAB) (4)
− δAB)+ 1
2 (2Xpret
i,A Xpret
j,B
X eij + ui,Aui,B) (5)