Fundamentals - Preventive Biomechanics - page 108

where the first integral term denotes the internal virtual work dW int and the latter
three represent the external virtual work dW ext done on the element and expressed
through discrete work contributions at the element nodes. The virtual work is
composed of force vectors acting at the element nodes and the corresponding
virtual displacement vector

dW int ¼ P int du i dW ext ¼ P ext du i

and ð 3 : 359 Þ

where in the current configuration, P int is the node vector of the integrated force
flow which is given by

P int ¼ Z
X e

S X

n

oN i ð X Þ
ox

dV ð 3 : 360 Þ

i ¼ 1

and P ext is the node vector of the external forces which is given by
P ext ¼ Z
oX e

N i ð X Þ dA þ Z
X e

N i ð X Þ k dV Z
X e

t X

n

X

n

X

n

N i ð X Þ v q dV :

i ¼ 1 i ¼ 1 i ¼ 1

ð 3 : 361 Þ

Rewriting ( 3.358 ), the equilibrium at a finite element given in discrete form is
thus given by

du i ¼ 0

P int P ext P int ¼ P ext :

thus ð 3 : 362 Þ

Assembly of all element contributions i.e. assembly of the contributions at all
inter-element nodes, leads to the global representation of the internal and external
force vectors acting at the discretized body domain

P int ¼ [ P ext ¼ [

n n

P int P ext and ð 3 : 363 Þ

e ¼ 1 e ¼ 1

Equations ( 3.363 ) 1 and ( 3.363 ) 2 together with ( 3.360 ) and ( 3.361 ) represent set
of nonlinear algebraic equations which are generally solved after linearization for
nodal displacement by means of a numerical method, such as the N EWTON -
R APHSON method.