I mean, stress tensor.
Finite element analysis was, in its early days, proposed for engineering solutions to dam constructions – by discretising a continuous domain into smaller elements and approximating the displacements of each elements upon loading. A major advantage of FEM over FDM is its ability to handle complex geometry: in FEM, the domain is (usually) approximated as many small triangles; but in FDM, it usually deals with rectangular coordinates.
The above is a classic diagram for the deformation of a continuous body. There are two types of strain tensors descriptions:
- Eulerian is the simplest representation where the spatial coordinate is fixed but the change in material point is updated .
- Lagrangian strain tensor tracks the material point, and updates the spatial coordinates. This relies on the initial/reference configuration at , which is defined with respect to .
We are using the Lagrangian scheme below.
a. Deformation Gradient
is at , and displaces to at ; likewise, is at , and displaces to at .
We assume that the line segments and are very small. Then:
When we establish , and approximate around for , the above equation becomes:
This is the deformation gradient.
b. Deformation Tensor
Right Cauchy – Green Deformation tensor gives us the change in the stretch of the element:
The volume change incurred by a deformation is characterised by the Jacobian of the deformation gradient: . Incompressible materials (whose volumes remain constant) have .
For isotropic problems, the principal invariants of C are:
From the deformation, we infer the change in (squared) length of the element in the continuous body.
3. Green-Lagrangian Strain
Since , the explicit form of is:
a. Vector definition
- is the unit normal and is the area of the element in the reference configuration;
- is the unit normal and is the area of the element in the deformed configuration.
- acts on the surface element on the deformed configuration.
b. Cauchy stress (True stress)
Force per unit deformed area on the deformed body:
c. 1st Piola-Kirchhoff stress
Force per unit undeformed area on the deformed body:
Relationship with : ,
d. 2nd Piola-Kirchhoff stress
Force per unit undeformed area on the undeformed body:
The 2nd P-K stress is usually expressed as a strain energy function – which will be covered in another post on constitutive law formulation.
A number of mechanical models for soft tissues were developed around these principle stress and strain tensors – most notably the Neo-hookean model for collagenous tissue. This has been widely used in evaluating cardiovascular tissue mechanics.