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.

1. 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.

2. Deformation

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:

4. Stresses

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.

5. Closing

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.

Sources:

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