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

FiniteDeformation-simple

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

FiniteDeformation-full

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.

 

Sources:

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