kinematics.hpp

Overview

Functions for finite strain kinematics, including deformation gradients, stretch tensors (left and right), rotation tensors, and various strain measures (Green-Lagrange, Euler-Almansi, logarithmic).

API Reference

arma::mat ER_to_F(const arma::mat &E, const arma::mat &R)

Provides the transformation gradient, from the Green-Lagrange strain and the rotation.

The transformation gradient \(\mathbf{F}\) is related to the Green-Lagrange strain \(\mathbf{E}\) and the rotation \(\mathbf{R}\) by the following equations:

\[ \mathbf{F} = \mathbf{R} \cdot \mathbf{U} \quad \mathbf{E} = \frac{1}{2} \left( \sqrt{\mathbf{U}^2} - \mathbf{I} \right) \]

where \(\mathbf{U}\) is the right stretch tensor 3x3 matrix and I is the 3x3 identity matrix.

Example:

mat E = randu(3,3);
mat R = eye(3,3);
mat F = ER_to_F(E, R);

Parameters:

• E – 3x3 matrix representing the Green-Lagrange strain tensor

• R – 3x3 matrix representing the rotation tensor

Returns:

a 3x3 matrix representing the transformation gradient \(\mathbf{F}\)

arma::mat eR_to_F(const arma::mat &e, const arma::mat &R)

Provides the transformation gradient, from the logarithmic strain and the rotation.

The transformation gradient \(\mathbf{F}\) is related to the logarithmic strain \(\mathbf{e}\) and the rotation \(\mathbf{R}\) by the following equations:

\[ \mathbf{F} = \mathbf{V} \cdot \mathbf{R} \quad \mathbf{e} = \textrm{ln} \mathbf{V} \]

where \(\mathbf{V}\) the right stretch tensor 3x3 matrix

Example:

mat e = randu(3,3);
mat R = eye(3,3);
mat F = eR_to_F(e, R);

Parameters:

• e – 3x3 matrix representing the logarithmic strain tensor

• R – 3x3 matrix representing the rotation tensor

Returns:

a 3x3 matrix representing the transformation gradient

arma::mat G_UdX(const arma::mat &F)

Provides the gradient of the displacement (Lagrangian) from the transformation gradient.

The gradient of the displacement (Lagrangian) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \nabla_X \mathbf{U} = \mathbf{F} - \mathbf{I} \]

where \(\mathbf{I}\) is the 3x3 identity matrix.

Example:

mat F = randu(3,3);
mat GradU = G_UdX(F);

Parameters:

F – 3x3 matrix representing the transformation gradient

Returns:

a 3x3 matrix representing the gradient of the displacement (Lagrangian)

arma::mat G_Udx(const arma::mat &F)

Provides the gradient of the displacement (Eulerian) from the transformation gradient.

The gradient of the displacement (Eulerian) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \nabla_x \mathbf{U} = \mathbf{I} - \mathbf{F}^{-1} \]

where \(\mathbf{I}\) is the 3x3 identity matrix.

Example:

mat F = randu(3,3);
mat gradU = G_Udx(F);

Parameters:

F – 3x3 matrix representing the transformation gradient

Returns:

3x3 matrix representing the gradient of the displacement (Eulerian)

arma::mat R_Cauchy_Green(const arma::mat &F)

Provides the Right Cauchy-Green tensor from the transformation gradient.

The Right Cauchy-Green tensor \(\mathbf{C}\) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \mathbf{C} = \mathbf{F}^T \cdot \mathbf{F} \]

Example:

mat F = randu(3,3);
mat C = R_Cauchy_Green(F);

Parameters:

F – 3x3 matrix representing the transformation gradient

Returns:

3x3 matrix representing the Right Cauchy-Green tensor

arma::mat L_Cauchy_Green(const arma::mat &F)

Provides the Left Cauchy-Green tensor from the transformation gradient.

The Left Cauchy-Green tensor \(\mathbf{B}\) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \mathbf{B} = \mathbf{F} \cdot \mathbf{F}^T \]

Example:

mat F = randu(3,3);
mat B = L_Cauchy_Green(F);

Parameters:

F – 3x3 matrix representing the transformation gradient

Returns:

3x3 matrix representing the Left Cauchy-Green tensor

void RU_decomposition(arma::mat &R, arma::mat &U, const arma::mat &F)

Provides the RU decomposition of the transformation gradient.

The RU decomposition of the transformation gradient \(\mathbf{F}\) is related to the matrices \(\mathbf{R}\) and \(\mathbf{U}\) by the following equations:

\[ \mathbf{F} = \mathbf{R} \cdot \mathbf{U} \quad \mathbf{U} = \sqrt{\mathbf{F}^T \cdot \mathbf{F}} \quad \mathbf{R} = \mathbf{F} \cdot \mathbf{U}^{-1} \]

Example:

mat F = randu(3,3);
mat R = zeros(3,3);
mat U = zeros(3,3);
RU_decomposition(R, U, F);

Parameters:

• R – 3x3 matrix that is the rotation matrix \(\mathbf{R}\)

• U – 3x3 matrix that is the right stretch tensor \(\mathbf{U}\)

• F – 3x3 matrix representing the transformation gradient

void VR_decomposition(arma::mat &V, arma::mat &R, const arma::mat &F)

Provides the VR decomposition of the transformation gradient.

The VR decomposition of the transformation gradient \(\mathbf{F}\) is related to the matrices \(\mathbf{R}\) and \(\mathbf{V}\) by the following equations:

\[ \mathbf{F} = \mathbf{V} \cdot \mathbf{R} \quad \mathbf{V} = \sqrt{\mathbf{F} \cdot \mathbf{F}^T} \quad \mathbf{R} = \mathbf{V}^{-1} \cdot \mathbf{F} \]

Example:

mat F = randu(3,3);
mat V = zeros(3,3);
mat R = zeros(3,3);
VR_decomposition(V, R, F);

Parameters:

• R – 3x3 matrix that is the rotation matrix \(\mathbf{R}\)

• V – 3x3 matrix that is the left stretch tensor \(\mathbf{V}\)

• F – 3x3 matrix representing the transformation gradient

arma::vec Inv_X(const arma::mat &X)

Provides the invariants of a symmetric tensor.

The invariants of the symmetric tensor \(\mathbf{X}\) are defined as (note that this definition is not unique):

\[ \mathbf{I}_1 = \textrm{trace} \left( X \right) \quad \mathbf{I}_2 = \frac{1}{2} \left( \textrm{trace} \left( X \right)^2 - \textrm{trace} \left( X^2 \right) \right) \quad \mathbf{I}_3 = \textrm{det} \left( X \right) \]

Example:

mat F = randu(3,3);
mat C = R_Cauchy_Green(F);
vec I = Inv_X(F);

Parameters:

X – 3x3 matrix representing the symmetric tensor

Returns:

Vector containing the three invariants of the tensor

arma::mat Cauchy(const arma::mat &F)

Provides the Cauchy tensor from the transformation gradient.

The Cauchy tensor \(\mathbf{b}\) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \mathbf{b} = \left( \mathbf{F} \cdot \mathbf{F}^T \right)^{-1} \]

Note that this tensor \(\mathbf{b}\) is the inverse of the Left Cauchy-Green tensor \(\mathbf{B}\).

Example:

mat F = randu(3,3);
mat b = Cauchy(F);

Parameters:

F – 3x3 matrix representing the transformation gradient

Returns:

3x3 matrix representing the Cauchy tensor

arma::mat Green_Lagrange(const arma::mat &F)

Provides the Green-Lagrange tensor from the transformation gradient.

The Green-Lagrange tensor \(\mathbf{E}\) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \mathbf{E} = \frac{1}{2} \left( \mathbf{F}^T \cdot \mathbf{F} - \mathbf{I} \right) \]

Example:

mat F = randu(3,3);
mat E = Green_Lagrange(F);

Parameters:

F – 3x3 matrix representing the transformation gradient \(\mathbf{F}\)

Returns:

3x3 matrix representing the Green-Lagrange tensor \(\mathbf{E}\)

arma::mat Euler_Almansi(const arma::mat &F)

Provides the Euler-Almansi tensor from the transformation gradient.

The Euler-Almansi tensor \(\mathbf{A}\) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \mathbf{A} = \frac{1}{2} \left( \mathbf{I} - \left( \mathbf{F} \cdot \mathbf{F}^T \right)^T \right) \]

Example:

mat F = randu(3,3);
mat A = Euler_Almansi(F);

Parameters:

F – 3x3 matrix representing the transformation gradient \(\mathbf{F}\)

Returns:

3x3 matrix representing the Euler-Almansi tensor \(\mathbf{A}\)

arma::mat Log_strain(const arma::mat &F)

Provides the logarithmic strain tensor from the transformation gradient.

The logarithmic strain tensor \(\mathbf{e}\) is related to the transformation gradient \(\mathbf{F}\) by the following equation:

\[ \mathbf{e} = \textrm{ln} \left( V \right) = \frac{1}{2} \textrm{ln} \left( V^2 \right) = \frac{1}{2} \textrm{ln} \left( \mathbf{F} \cdot \mathbf{F}^T \right) \]

Note that the log function is here the matrix logarithm of symmetric/hermitian positive definite matrix V^2. TSee the armadillo documentation for more details of the log matrix implementation.

Example:

mat F = randu(3,3);
mat e = Log_strain(F);

Parameters:

F – 3x3 matrix representing the transformation gradient \(\mathbf{F}\)

Returns:

3x3 matrix representing the logarithmic strain tensor \(\mathbf{e}\)

arma::mat finite_L(const arma::mat &F0, const arma::mat &F1, const double &DTime)

Provides the approximation of the Eulerian velocity tensor from the transformation gradient at two different times.

The Eulerian velocity tensor \(\mathbf{L}\) is related to the transformation gradients \(\mathbf{F_0}\) at time \(t_0\), \(\mathbf{F_1}\) at time \(t_1\), and the time difference \(\Delta t = t_1 - t_0\) by the following equation:

\[ \mathbf{L} = \frac{1}{\Delta t} \left( \mathbf{F}_1 - \mathbf{F}_0 \right) \cdot \mathbf{F}_1^{-1} \]

Example:

mat F0 = randu(3,3);
mat F1 = randu(3,3);
double DTime = 0.1;
mat L = finite_L(F0, F1, DTime);

Parameters:

• F0 – 3x3 matrix representing the transformation gradient \(\mathbf{F_0}\) at time \(t_0\)

• F1 – 3x3 matrix representing the transformation gradient \(\mathbf{F_1}\) at time \(t_1\)

• DTime – double representing the time difference Delta \(\Delta t = t_1 - t_0\)

Returns:

3x3 matrix representing the approximation of the Eulerian velocity tensor

arma::mat finite_D(const arma::mat &F0, const arma::mat &F1, const double &DTime)

Provides the approximation of the Eulerian symmetric rate tensor from the transformation gradient at time \(t_0\), the transformation gradient at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\).

The Eulerian symmetric rate tensor \(\mathbf{D}\) is related to the transformation gradient \(\mathbf{F}_0\) at time \(t_0\), the transformation gradient \(\mathbf{F}_1\) at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\) by the following equation:

\[ \mathbf{D} = \frac{1}{2} \left( \mathbf{L} + \mathbf{L}^T \right) \]

where \(\mathbf{L}\) is the Eulerian velocity tensor. \(\mathbf{D}\) is commonly referred as the rate of deformation (this necessitates although a specific discussion).

Example:

mat F0 = randu(3,3);
mat F1 = randu(3,3);
double DTime = 0.1;
mat D = finite_D(F0, F1, DTime);

Parameters:

• F0 – 3x3 matrix representing the transformation gradient \(\mathbf{F}_0\) at time \(t_0\)

• F1 – 3x3 matrix representing the transformation gradient \(\mathbf{F}_1\) at time \(t_1\)

• DTime – time difference \(\Delta t = t_1 - t_0\)

Returns:

3x3 matrix representing the approximation of the Eulerian symmetric rate tensor \(\mathbf{D}\)

arma::mat finite_W(const arma::mat &F0, const arma::mat &F1, const double &DTime)

Provides the approximation of the Eulerian antisymmetric spin tensor from the transformation gradient at time \(t_0\), the transformation gradient at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\).

The Eulerian antisymmetric spin tensor \(\mathbf{W}\) is related to the transformation gradient \(\mathbf{F}_0\) at time \(t_0\), the transformation gradient \(\mathbf{F}_1\) at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\) by the following equation:

\[ \mathbf{W} = \frac{1}{2} \left( \mathbf{L} - \mathbf{L}^T \right) \]

where \(\mathbf{L}\) is the Eulerian velocity tensor. \(\mathbf{W}\) corresponds to the Jaumann corotationnal rate.

Example:

mat F0 = randu(3,3);
mat F1 = randu(3,3);
mat DTime = 0.1;
mat W = finite_W(F0, F1, DTime);

Parameters:

• F0 – 3x3 matrix representing the transformation gradient \(\mathbf{F_0}\) at time \(t_0\)

• F1 – 3x3 matrix representing the transformation gradient \(\mathbf{F_1}\) at time \(t_1\)

• DTime – time difference \(\Delta t = t_1 - t_0\)

Returns:

3x3 matrix representing the approximation of the Eulerian antisymmetric spin tensor \(\mathbf{W}\)

arma::mat finite_Omega(const arma::mat &F0, const arma::mat &F1, const double &DTime)

Provides the approximation of the Eulerian rigid-body rotation spin tensor from the transformation gradient at time \(t_0\), the transformation gradient at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\).

The Eulerian rigid-body rotation spin tensor \(\mathbf{\Omega}\) is related to the transformation gradient \(\mathbf{F}_0\) at time \(t_0\), the transformation gradient \(\mathbf{F}_1\) at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\) by the following equation:

\[ \mathbf{\Omega} = \frac{1}{\Delta t} \left( \mathbf{R}_1 - \mathbf{R}_0 \right) \cdot \mathbf{R}_1^{T} \]

where \(\mathbf{R}_0\) and \(\mathbf{R}_1\) are the rotation matrices obtained from RU decompositions of the transformation gradients \(\mathbf{F}_0\) and \(\mathbf{F}_1\), respectively. This corresponds to the Green-Naghdi corotationnal rate.

Example:

mat F0 = randu(3,3);
mat F1 = randu(3,3);
mat DTime = 0.1;
mat Omega = finite_Omega(F0, F1, DTime);

Parameters:

• F0 – 3x3 matrix representing the transformation gradient \(\mathbf{F_0}\) at time \(t_0\)

• F1 – 3x3 matrix representing the transformation gradient \(\mathbf{F_1}\) at time \(t_1\)

• DTime – time difference \(\Delta t = t_1 - t_0\)

Returns:

3x3 matrix representing the approximation of the Eulerian rigid-body rotation spin tensor (Green-Naghdi corotational rate).

arma::mat finite_DQ(const arma::mat &Omega0, const arma::mat &Omega1, const double &DTime)

Provides the Hughes-Winget approximation of a increment of rotation or transformation from the spin/velocity at time \(t_0\), the spin/velocity at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\).

The Hughes-Winget approximation of a increment of rotation or transformation \(\mathbf{\Delta Q}\) is related to the spin/velocity \(\mathbf{\Omega}_0\) at time \(t_0\), the spin/velocity \(\mathbf{\Omega}_1\) at time \(t_1\) and the time difference \(\Delta t = t_1 - t_0\) by the following equation:

\[ \mathbf{\Delta Q} = \left( \mathbf{I} + \frac{1}{2} \Delta t \, \mathbf{\Omega}_0 \right) \cdot \left( \mathbf{I} - \frac{1}{2} \Delta t \, \mathbf{\Omega}_1 \right)^{-1} \]

where \(\mathbf{I}\) is the 3x3 identity matrix.

Example:

mat Omega0 = randu(3,3);
mat Omega1 = randu(3,3);
mat DTime = 0.1;
mat DQ = finite_DQ(Omega0, Omega1, DTime);

Parameters:

• Omega0 – spin/velocity at time \(t_0\)

• Omega1 – spin/velocity at time \(t_1\)

• DTime – time difference \(\Delta t = t_1 - t_0\)

Returns:

3x3 matrix representing the Hughes-Winget approximation of a increment of rotation or transformation