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

COUPLING OF PERIDYNAMIC THEORY AND THE FINITE ELEMENT METHOD 727
subdomain. Under these considerations, the force density in this domain can be approximated as,
g∞ = 2
∞�
i=0
∞�
∞�
j=0 k=0
nx. f (u = 0, u ′ , x ′ , x = 0, t)� 3 ∞ , (A5)
in which nx is the unit vector in the x-direction. The position vector x ′ is defined as
x ′ � 1
= �∞ i + 2
j + 1
2
Finally, the displacement vector u ′ can be written as
k + 1�T
′
2 = {x
y ′
z ′ } T . (A6)
u ′ �
∂u∗ =
∂ x x′ − ν ∂u∗
∂ x y′ − ν ∂u∗
∂ x z′
�T . (A7)
If the material point is surrounded by material points of the same material and there is no interface
or free surfaces within the cutoff radius, which is shown as point 1 in Figure 9, each component of the
vector given in (A4) should be equal to the response given in (A5). However, this does not occur due
to the approximations in the computation of the integral given in (1). Therefore, the present approach
also attempts to correct not only the material stiffness variations due to surface effects but also the
approximations in the numerical integration. Therefore, the correction is applied to all collocation points
regardless of their position.
As illustrated in Figure A3, the scaling constant gi j between the pair of material points located at xi
and x j is calculated by assuming an ellipsoidal variation as
gi j = � (nx/gi jcx) 2 + (n y/gi jcy) 2 + (nz/gi jcz) 2� −1/2 , (A8)
in which nx, n y, nz are the components of the normal vector, n in the undeformed configuration between
the pair of material point; it is defined as
n = ζ/|ζ | = {nx, n y, nz} T . (A9)
Figure A3. Graphical representation of scaling coefficient.