Biomedical Engineering Reference
In-Depth Information
T ¼ JF 1 R ð F T Þ 1 ;
ð 19 Þ
where J is the Jacobian matrix required for the transformation from undeformed to
deformed coordinates. The Jacobian matrix is the determinant of the deformation
gradient tensor,
J ¼ det ð F Þ¼ k 1 k 2 k 3 :
ð 20 Þ
The second Piola-Kirchhoff stress tensor is particularly useful as it directly relates
to the strain energy function used in formulating the constitutive models that
specify passive material properties.
In order to specify anisotropic material properties, these tensors are additionally
referred to the locally varying fiber coordinate system, m a (i.e., the fiber (f ), sheet
(s) and sheet-normal (n) directions). By applying the principle of virtual work,
stress equilibrium of the system, in the Einstein notation, yields,
Z
T ab F b du j ; a dV ¼ Z
q ð b j f j Þ du j dV þ Z
s j du j dS ;
ð 21 Þ
V
V
S
T ab and F b are the second Piola-Kirchhoff stress tensor and deformation gradient
tensor respectively, expressed with respect to the fiber coordinate system, so that
the virtual displacements are expressed in the reference coordinate system and
differentiated with respect to the m a coordinate system [ 41 ]. The density of the
material (q) is in the reference system; b j and f j are the body force and accel-
eration vectors over the body volume; V, and s j are the surface traction vectors
representing external surface forces acting on the deformed surface, S [ 41 ]. For the
finite deformation problem, the finite element basis functions are used to inter-
polate the virtual displacement yields, du j in Eq. ( 21 ). The integrals in Eq. ( 21 ) are
subsequently transformed to the local coordinate space. Due to the nonlinearity of
the formulation, the resulting integrals form can be evaluated numerically using,
for example, the Gaussian quadrature scheme, which approximates the integrals
using a weighted sum of integrand evaluations,
Z 1
f ð n Þ dn ¼ X
I
w i f ð n i Þþ E i ;
ð 22 Þ
0
i ¼ 1
where w i are the weighting factors, often specified using the interpolating basis
functions, and n i are the locations of the solution points (gauss points) where the
integrand is to be evaluated. E i is the error in the approximation and I is the order
of the Gaussian quadrature scheme. By incorporating the passive constitutive
equations (described in Sect. 4.2 ), second Piola-Kirchhoff stress and Lagrangian
strains may be calculated at the gauss points. These integral approximations are
subsequently combined, together with boundary constraints, to form a global system
of nonlinear equations in terms of the unknown displacements of each node in the
mesh. The system can then be solved using an appropriate numerical method.
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