Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

COUPLING OF PERIDYNAMIC THEORY AND THE FINITE ELEMENT METHOD 719
in which ρ is the mass density of the e-th element, g is the body force density, and I indicates the I -th
node of the e-th element. Hence, f (e)
i indicates the external force acting on the I -th node. The body
force density is only known at the collocation points. Fortunately, the collocation points are placed by
using a Gaussian integration technique. This allows the numerical evaluation of (26) by using collocation
points as integration points for the e-th element. Furthermore, Fn f is constructed by assembling the nodal
forces given by (26). Finally, the coupled system of equations can be expressed as
in which
˙U
� = { ˙U n p
˙U n f
n
Ü + cn
�
˙U
n −1 n
= M� F� , (27)
�
˙U n f }T
and Ü � = {Ü n p
are the first and second time derivatives of the displacements, the matrix M is given by
�
⎡ ⎤
� 0 0
M = ⎣ 0 M
�
0 0
0 ⎦ ,
M
(29)
and the vector F by
�
n n
F = {F p
�
F n f Fnf }T . (30)
As suggested in [Underwood 1983], the damping coefficient cn can be written as
�
�(U�
cn = 2 n T
) 1K nU� n � � � (U n T
) U� n
�
�
(31)
in which 1 K n is the diagonal “local” stiffness matrix expressed as
1 n
Kii = − � n
Fi �
/m � ii n−1
− F
�
i /mii �
Ü n f
�� ˙U n−1/2
i
(see [Underwood 1983]). The time integration is performed using central-difference explicit integration,
with a time step size of unity; explicitly,
Ü n f }T
(28)
(32)
�
˙U
n+1/2
= (2 − cn) ˙U
n−1/2 −1 n
+ 2M� F�
�
�
�� (2 + cn), (33)
U
�
n+1 n
= U� + ˙U
n+1/2
. (34)
�
However, the integration algorithm given by (33) and (34) cannot be used to start the integration due to
0 = 0:
an unknown velocity field at t−1/2 . Integration can be started by assuming that U 0 �= 0 and ˙U
�
�
˙U
1/2 1
=
�
2 M −1 1/2
F� . (35)
�
4. Numerical results
The present approach is demonstrated by considering a bar subjected to tension and a plate with a circular
cutout. The bar under tension is considered to compare the displacements of the present coupled analysis,
the peridynamic theory, and finite element methods while not allowing for failure to occur. The damage
prediction capability of the coupled analysis is then illustrated by considering a plate with a circular
cutout. Although the numerical values of the material properties are representative of engineering materials,
they are not specific to a particular material. Also, the effects of the peridynamic parameters