Nonlinear Finite Element Analysis of Concrete Structures

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the material is assumed to be hyperelastic (Green-elastic) possessing
a strain energy function so that D.., , = D, . . . holds.
0 0 y ijkl kliD
The terms e,
kl
, and a.
i j
. in the constitutive equation
^
denote initial
strains due to shrinkage, thermal expansions, etc. and initial
stresses, respectively. More realistic constitutive equations
than the above might well be used in the finite element formulation,
but for the present purpose, eq. (3) suffices.
Having defined the field equations, the boundary conditions will
be set up. For this purpose we divide the total boundary S of
the structure into two regions, a region S
tractions, are specified and a region S
where surface forces,
where displacemants are
prescribed. The static boundary conditions specify that
o. . n. = t. on S. (4.1-4)
ID 1 i t
where n. is the outward unit vector normal to the boundary and
D
t. denotes the given tractions. The kinematic boundary conditions
i
specify that
u ± = u. on S u (4.1-5)
where u. is the prescribed displacements. If the structural response
satisfies the equations (l)-(5) then the true solution
has been established. Let us now consider a reformulation of some
of these equations.
Satisfaction of the equilibrium equations all through the structure
is equivalent to
|u* (o ijfj + b..) dV = 0 (4.1-6)
v
when u. is any arbitrary function that can be considered as a
displacement. The term dV denotes an infinitesimal volume. From
this equation follows
r- - *
j [ ( u i °ij>,j -
u i,j °iji dv + u* b i dV = 0