# True stress, and relative stress (Finite Deformation)

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$$\mathbf{P} = J\mathbf{F}^{-T} \sigma$$\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.

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