The Kinematics Library

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

Provides the transformation gradient \(\mathbf{F}\), from the Green-Lagrange strain \(\mathbf{E}\) and the rotation \(\mathbf{R}\):

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

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

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

Provides the transformation gradient \(\mathbf{F}\), from the logarithmic strain \(\mathbf{e}\) and the rotation \(\mathbf{R}\):

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

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

mat G_UdX(const mat &F)

Provides the gradient of the displacement (Lagrangian) from the transformation gradient \(\mathbf{F}\):

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

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

mat G_Udx(const mat &F)

Provides the gradient of the displacement (Eulerian) from the transformation gradient \(\mathbf{F}\):

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

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

mat R_Cauchy_Green(const mat &F)

Provides the Right Cauchy-Green tensor \(\mathbf{C}\): from the transformation gradient \(\mathbf{F}\):

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

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

mat L_Cauchy_Green(const mat &F)

Provides the Left Cauchy-Green tensor \(\mathbf{B}\): from the transformation gradient \(\mathbf{F}\):

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

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

RU_decomposition(mat &R, mat &U, const mat &F)

Provides the RU decomposition of the transformation gradient \(\mathbf{F}\):

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

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

VR_decomposition(mat &R, mat &V, const mat &F)

Provides the VR decomposition of the transformation gradient \(\mathbf{F}\):

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

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

vec Inv_X(const mat &X)

Provides the invariants of a symmetric tensor \(\mathbf{X}\):

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

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

mat Cauchy(const mat &F)

Provides the Cauchy tensor \(\mathbf{b}\): from the transformation gradient \(\mathbf{F}\):

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

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

mat Green_Lagrange(const mat &F)

Provides the Green-Lagrange tensor \(\mathbf{E}\): from the transformation gradient \(\mathbf{F}\):

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

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

mat Euler_Almansi(const mat &F)

Provides the Euler_Almansi tensor \(\mathbf{e}\): from the transformation gradient \(\mathbf{F}\):

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

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

mat Log_strain(const mat &F)

Provides the logarithmic strain tensor \(\mathbf{h}\): from the transformation gradient \(\mathbf{F}\):

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

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

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

Provides the approximation of the Eulerian velocity tensor \(\mathbf{L}\): from the transformation gradient \(\mathbf{F}_0\): at time \(t_0\):, \(\mathbf{F}_1\): at time \(t_1\): and the time difference \(\Delta t = t_1 - t_0\):

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

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

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

Provides the approximation of the Eulerian symmetric rate tensor \(\mathbf{D}\): from the transformation gradient \(\mathbf{F}_0\): at time \(t_0\):, \(\mathbf{F}_1\): at time \(t_1\): and the time difference \(\Delta t = t_1 - t_0\): This is commonly referred as the rate of deformation (this necessitates although a specific discussion)

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

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

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

Provides the approximation of the Eulerian antisymmetric spin tensor \(\mathbf{W}\): from the transformation gradient \(\mathbf{F}_0\): at time \(t_0\):, \(\mathbf{F}_1\): at time \(t_1\): and the time difference \(\Delta t = t_1 - t_0\): . This correspond to the Jaumann corotationnal rate:

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

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

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

Provides the approximation of the Eulerian rigid-body rotation spin tensor \(\mathbf{\Omega}\): from the transformation gradient \(\mathbf{F}_0\): at time \(t_0\):, \(\mathbf{F}_1\): at time \(t_1\): and the time difference \(\Delta t = t_1 - t_0\): . This correspond to the Green-Naghdi corotationnal rate:

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

Note that the rotation matrix is obtained from a RU decomposition of the transformation gradient \(\mathbf{F}\):

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

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

Provides the Hughes-Winget approximation of a increment of rotation or transformation from the spin/velocity \(\mathbf{\Omega}_0\): at time \(t_0\):, \(\mathbf{\Omega}_1\): at time \(t_1\): and the time difference \(\Delta t = t_1 - t_0\):

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

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