Game Development Reference
In-Depth Information
of the tetrahedron element, which generalizes the stiffness of Hooke's law to a
matrix.
Let us now explain how to compute the stiffness matrix that arises from as-
suming that the work done by the applied forces is transformed into strain (poten-
tial) energy and that it is completely recoverable. The strain energy in the form
of elastic deformation is mostly recoverable in the form of mechanical work. The
definition of the elastic potential energy is given by
Π= U
W,
where U is the strain energy and W is the work done by external forces. The
object is in a stable equilibrium when the potential energy reaches a minimum,
which happens when
Π
∂u e
=0 .
(10.11)
The work W is defined as
W = u e f ext ,
and the strain energy of the elastic linear body is defined as
1
2
T
U =
vol e ε
σ
dV.
(10.12)
Replacing the values of the strain with ε = B e u e and
σ
ε
= E
,weget
Π= 1
2
u e B e EB e u e dV + u e f ext .
vol e
Noting that none of the quantities inside the integral terms depend on the position
of the coordinates, we have for the total potential
Π= 1
2 vol e u e B e EB e u e + u e f e ext ,
and taking the derivative with respect to u e to seek the static equilibrium, as re-
quired in Equation (10.11), we obtain:
f ext =vol e B e EB e u e ,
hence
K =vol e B e E B e .