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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
L. A. F. de Souza | R. D. Machado
are referred to as failure or rupture criterion. The fragile rupture
condition must be considered as a special case in which the failure
criterion due to yielding corresponds to the failure criterion due to
rupture (Gagliardo
et al
. [3]).
Considering totally anisotropic materials, it must be admitted that
the failure modes are conditioned both by the normal and tangen-
tial stresses, considering that the fractures may occur due to differ-
ent sets of stresses that act on the element. In a general way, this
theory may be presented by:
(21)
F
i
σ
i
6
i=1
+
F
ij
σ
i
σ
j
6
j=1
+
6
i=1
F
ijk
σ
i
σ
j
6
k=1
6
j=1
6
i=1
σ
k
+
...
= 1
The coefficients
F
i
,
F
ij
and
F
ijk
are tensor rearranged structures of
the 1
st
, 2
nd
and 3
rd
order, respectively. An advantage of this method
is that it is possible to use as many terms as are necessary for the
approximation of experimental points of a material. However, once
each constant is associated with a unique type of mechanical test
for its determination, Equation 21 is normally restricted to 2
nd
order
terms. If this is not done, the quantity and complexity of the neces-
sary tests for determining the constants would make the method
impracticable. Thus, Equation 21 is reduced to:
(22)
F
i
σ
i
6
i=1
+
F
ij
σ
i
σ
j
6 6
j=1
i=1
=1
Considering a plane state of stresses applied to orthotropic materi-
als, and developing Equation (22), we have:
(23)
F
1
σ
1
+F
2
σ
2
+F
11
σ
1
2
+F
22
σ
2
2
+2F
12
σ
1
σ
2
+ F
44
σ
4
2
=1
Where
σ
i
,
i
= 1,
…
,3, are the main stresses and
σ
4
is the shear
stress. Equation (23) indicates that the state of stresses is at a
critical point (on the verge of the failure). However, if the state of
stresses given by the left side of Equation (23) presents a numeri-
cal result lower than one, we have a safety situation. Different from
other strength criteria, this one takes into consideration the effect
of the hydrostatic components of the stresses.
6. Arc-length method with the modified
Newton-Raphson iterative process
When applying the Newton-Raphson method for limit-point prob-
lems with a load control, the stiffness matrix tends to singularize
around this point in its ascending trajectory. An alternative to de-
tect and surpass the limit point is to use solution methods associat-
ed with the Newton-Raphson method, for instance, the arc-length
method.
The arc-length method is characterized for presenting a concomi-
tant control of the load and displacement. There are two variables:
the increment of the load factor
∆ϕ
and the displacement increment
vector
∆u
. In each step of the solution, the iteration trajectories are
perpendicular to the arcs, which in turn, can be approximated by
tangents to the equilibrium trajectory at the initial points of these
steps (Ramm [15]). Considering the arc-length method with the
modified Newton-Raphson iterative process, the equilibrium equa-
tions for the i-th iteration can be written as:
(24)
K
T
a
u
i
= φ
i
R
0
+ Q
i-1
Where
∆φ
i
is the increment of the load factor of iteration i,
∆u
i
is
the displacement increment vector,
is the reference loads vector,
K
T a
is the updated tangential stiffness matrix only at the beginning
of each loading step, and
∆Q
i-1
is the non-balanced loads vector
given by:
(25)
Q
i-1
=R
ext
i-1
-F
int
i
R
ext i-1
being the external forces vector and is the internal nodal
forces vector. Vector must be written as a function of the load fac-
tor, updated at the end of the previous iteration and the constant
reference loads vector
,
through the following relation:
(26)
R
ext
i-1
=φ
i-1
R
0
For a system of the
n
+1 order,
n
being the number of degrees of
freedom for the structure, we have:
(27)
[
{
{ {
{
[
K
T
a
-R
0
u
1
φ
1
u
i
φ
i
=
Q
i-1
0
Where
∆u
1
is the first displacement increment vector of the solu-
tion step and
∆φ
1
the first increment of the load factor in the given
step. It can be noticed that the resolution of the system given in
(27) creates a system of equations with non-trivial solution even if
the matrix is singular, which is a great advantage for the solution
of problems with limit point.
6.1 Convergence criteria
Aiming to limit the iterative processes, two convergence criteria are
established: one for the displacements and another for the forces.
The convergence criteria for the displacements must obey the fol-
lowing inequality:
(28)
‖
u
i
‖ ‖
u
i
‖
≤
u
tol