Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 101

will be denoted by F Dr'. Namely,

@' i
@x j D I Crd D I C

@d i
@x j :

F Dr' D (2.53)

F plays a key role in specifying the relationship between the first and second Piola-
Kirchhoff stress tensors, and in the relationship between strain and displacement.
The first and second Piola-Kirchhoff stress tensors are related through the gradient
of deformation as follows:

S D F…: (2.54)

While the first Piola-Kirchhoff stress tensor is not generally symmetric, the
second Piola-Kirchoff stress tensor is, and is, therefore, more suited for the
description of physical properties of materials in terms of constitutive relations.
Constitutive relations , which specify the material properties of a structure, typi-
cally express a relationship between stress and strain, more precisely, between the
second Piola-Kirchhoff stress tensor … and the Green-Lagrange strain tensor E:

… D ….E/;

where the Green-Lagrange strain tensor is defined via deformation gradient as

2 F T F I :

1

E WD (2.55)

A calculation shows that in terms of the displacement gradient, E is given by:

2 rd Crd T

Crdrd T :

1

E WD (2.56)

Therefore, a general relationship between strain and displacement gradient is
quadratic. For small displacement gradients, the quadratic term can be neglected,
and the relationship becomes linear:

2 rd Crd T D D.d/;

1

E " WD (2.57)

where D is known as the symmetrized gradient of displacement.
Therefore, in summary, the elastodynamics of elastic structures is described by
the second Newton's law of motion

s @ tt d Dr S in S .0;T/;

where

• S D F…is the first Piola-Kirchhoff stress tensor,
• … is the second Piola-Kirchhoff stress tensor,