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
Undeformed body (t=0) and Deformed body (t=T)
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 $P(x_1, x_2, x_3, t)$.
- Lagrangian strain tensor tracks the material point, and updates the spatial coordinates. This relies on the initial/reference configuration at $t=0$, which is defined with respect to $P(a_1, a_2, a_3, t)$.
We are using the Lagrangian scheme below.
2. Deformation
a. Deformation Gradient
Deformation tensor mapping undeformed and deformed continuous body
$ P $ is at $ \mathbf{X}$, and displaces to $ p$ at $ \mathbf{x}$; likewise, $ Q $ is at $ \mathbf{X} + d\mathbf{X} $, and displaces to $ p $ at $ \mathbf{x}+d\mathbf{x} $.
We assume that the line segments $ d\mathbf{X}$ and $ d\mathbf{x}$ are very small. Then:
$$ \mathbf{x} + d\mathbf{x} = \mathbf{X} + d\mathbf{X} + \mathbf{u}(\mathbf{X} + d\mathbf{X}) $$
When we establish $ \mathbf{x} – \mathbf{X} = \mathbf{u}(\mathbf{X})$, and approximate around $ P$ for $ \mathbf{u}(\mathbf{X} + d\mathbf{X}) = \mathbf{u}(\mathbf{X}) + d\mathbf{u} \approx \mathbf{u}(\mathbf{X}) + \nabla_\mathbf{X} \mathbf{u} \cdot d\mathbf{X}$, the above equation becomes:
$$ d\mathbf{x} = d\mathbf{X} + \nabla_\mathbf{X} \mathbf{u} \cdot d\mathbf{X} = (\mathbf{I}+\nabla_\mathbf{X} \mathbf{u})d\mathbf{X} = \mathbf{F} d \mathbf{X} $$
This $ \mathbf{F}$ is the deformation gradient.
$ \mathbf{F} = \frac{\partial\mathbf{x}}{\partial\mathbf{X}} = \nabla \mathbf{x}, \mathit{F_{i,J} = \frac{\partial x_i}{\partial X_J}}$
b. Deformation Tensor
Right Cauchy – Green Deformation tensor gives us the change in the stretch $ \lambda = \frac{d\mathbf{x}}{d\mathbf{X}}$ of the element:
$ \mathbf{C} = \mathbf{F}^T \mathbf{F} = \mathbf{U}^2 or \mathit{C}_{I,,J} =\mathit{F}_{k,I} \mathit{F}_{k,J} = \frac{\partial x_k}{\partial X_I} \frac{\partial x_k}{\partial X_J} $
The volume change incurred by a deformation is characterised by the Jacobian of the deformation gradient: $ J = det(\mathbf{F})$. Incompressible materials (whose volumes remain constant) have $ J=1$.
For isotropic problems, the principal invariants of C are:
$ I_1 = tr(\mathbf{C}) = \mathit{C}_II = \lambda_1^2 + \lambda_2^2 + \lambda_3^2$
$ I_2 = \frac{1}{2} [tr(\mathbf{C})^2 – tr(\mathbf{C}^2)] = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_1^2 \lambda_3^2$
$ I_3 = det(C) = \lambda_1^2 \lambda_2^2 \lambda_3^2$
From the deformation, we infer the change in (squared) length of the element in the continuous body.
3. Green-Lagrangian Strain
$$ \frac{|d\mathbf{x}|^2-|d\mathbf{X}|^2}{2} = \frac{1}{2} {d\mathbf{XC}d\mathbf{X} – d\mathbf{X} \cdot d\mathbf{X}} = \frac{1}{2} {d\mathbf{X}(\mathbf{C}-\mathbf{I}) d\mathbf{X}} $$
$$ \mathbf{E} = \frac{1}{2}(\mathbf{C}-\mathbf{I}) , E_{IJ} = \frac{1}{2}(C_{IJ}-\delta_{IJ}) $$
Since $ \mathbf{F}_{i,j} = \delta_{i,j} + \mathbf{u}_{i,j}$, the explicit form of $ \mathbf{E}$ is:
$ \mathbf{E}_{i,j} = \frac{1}{2}(\frac{\partial u_i}{\partial X_J}+\frac{\partial u_j}{\partial X_I}+\frac{\partial u_k}{\partial X_I} \frac{\partial u_k}{\partial X_J})$
4. Stresses
a. Vector definition
- $ \mathbf{N}$ is the unit normal and $ d\mathit{S}$ is the area of the element in the reference configuration;
- $ \mathbf{n}$ is the unit normal and $ d\mathit{s}$ is the area of the element in the deformed configuration.
- $ d\mathbf{f} $ acts on the surface element on the deformed configuration.
b. Cauchy stress (True stress) $ \sigma$
Force per unit deformed area on the deformed body:
$$ d\mathbf{f} = \mathbf{\sigma n} d\mathit{s} $$
c. 1st Piola-Kirchhoff stress $ \mathbf{P}$
Force per unit undeformed area on the deformed body:
$ d\mathbf{f} = \mathbf{PN} d\mathit{S}$
Relationship with $ \sigma$: $ \mathbf{P} = J\mathbf{F}^{-T} \sigma$, $latex \sigma = J^{-1}\mathbf{PF}^T $
d. 2nd Piola-Kirchhoff stress $ \mathbf{S}$
Force per unit undeformed area on the undeformed body:
$$ \mathbf{S} = \mathit{J}\mathbf{F}^{-1}\mathbf{\sigma F}^{-T} = P\mathbf{F}^{-1} $$
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: