Tensor Functions


Functions

void Tensors::Mult (Tensor2 const &S, Tensor2 const &T, Tensor2 &R)
 Tensor multiplication $ \TeSe{R} \gets \TeSe{S}\bullet\TeSe{T} $.
REAL Tensors::Norm (Tensor2 const &T)
 Euclidian norm of a symmetric 2nd order tensor.
REAL Tensors::Det (Tensor2 const &T)
 Determinant of T.
bool Tensors::Inv (Tensor2 const &T, Tensor2 &R)
 Inverse of a tensor T $ \TeSe{R} \gets \TeSe{T}^{-1} $.
bool Tensors::Eigenvals (Tensor2 const &T, REAL L[3])
 Eigenvalues (L) of a tensor T $ L_{i=1\cdots3} \gets eigenvalues(\TeSe{T}) $.
bool Tensors::Eigenvp (Tensor2 const &T, REAL L[3], Tensor2 P[3])
 Eigenvalues (L) and Eigenprojectors (P) of a tensor T $ L_{i=1\cdots3} \gets eigenvalues(\TeSe{T}) \quad \TeSe{P}_{i=1\cdots3} \gets eigenprojectors(\TeSe{T}) $.
bool Tensors::Sqrt (Tensor2 const &T, Tensor2 &R)
 Square root of a tensor T $ \TeSe{R} \gets \sqrt{\TeSe{T}} $.
void Tensors::CharInvs (Tensor2 const &T, REAL I[3])
 Characteristic invariants of a symmetric second order tensor.
void Tensors::Strain_Ev_Ed (Tensor2 const &Eps, REAL &Ev, REAL &Ed)
 Strain Invariants.
void Tensors::Stress_p_q (Tensor2 const &Sig, REAL &p, REAL &q)
 Stress Invariants (Cambridge).
REAL Tensors::Stress_q (Tensor2 const &Sig)
 Cambridge's q deviatoric stress invariant.
REAL Tensors::Sin3ThDev (Tensor2 const &S)
 Sin3Th given deviator (S) of the stress tensor.
void Tensors::Stress_p_q_S_sin3th (Tensor2 const &Sig, REAL &p, REAL &q, Tensor2 &S, REAL &sin3th)
 Stress Invariants (Cambridge) + deviator of Sigma.
bool Tensors::Stress_tn_ts (Tensor2 const &Sig, REAL &tn, REAL &ts)
 Stress Invariants (Professor Nakai's invariants).
void Tensors::Stress_P_Q (Tensor2 const &Sig, REAL &P, REAL &Q)
 Stress Invariants (Professor Brannon's isomorphic invariants).
void Tensors::Stress_P_Q_S_sin3th (Tensor2 const &Sig, REAL &P, REAL &Q, Tensor2 &S, REAL &sin3th)
 Stress Invariants (Professor Brannon's isomorphic invariants) + deviator of Sigma.
REAL Tensors::Sin3Th (REAL SI, REAL SII, REAL SIII)
 Returns Sin3Th, given three principal values, which are not necessary sorted.
void Tensors::Hid2Sig (REAL const &p, REAL const &q, REAL const &th, REAL &Sig1, REAL &Sig2, REAL &Sig3)
 Converts hidrostatic coordinates to sigma (principal) coord (T in radians).
void Tensors::Hid2Sig (REAL const *P, REAL const *Q, REAL const *T, REAL *SI, REAL *SII, REAL *SIII, int Size)
 Converts hidrostatic coordinates to sigma (principal) coord (T in radians).
void Tensors::Hid2Sig_ (REAL const &SX, REAL const &SY, REAL const &p, REAL &SI, REAL &SII, REAL &SIII)
 Converts hidrostatic coordinates (Sx, Sy, Sz) to sigma (principal) coord (T in radians).

Function Documentation

void Tensors::Mult ( Tensor2 const &  S,
Tensor2 const &  T,
Tensor2 &  R 
) [inline]

Tensor multiplication $ \TeSe{R} \gets \TeSe{S}\bullet\TeSe{T} $.

-*- IMPORTANT -*-

This function is valid only if the result R=S*T would be symmetric, however this is NOT always true, even if S and T are both symmetric !!!

Parameters:
S  In: Left operand
T  In: Right operand
R  Out: R=S*T

Definition at line 233 of file functions.h.

REAL Tensors::Norm ( Tensor2 const &  T  )  [inline]

Euclidian norm of a symmetric 2nd order tensor.

Returns:
Euclidian norm of Tensor T
Parameters:
T  In: Tensor T

Definition at line 257 of file functions.h.

bool Tensors::Inv ( Tensor2 const &  T,
Tensor2 &  R 
) [inline]

Inverse of a tensor T $ \TeSe{R} \gets \TeSe{T}^{-1} $.

Returns:
true if success, otherwise return false
Parameters:
T  In: Tensor T
R  Out: R=inv(T)

Definition at line 269 of file functions.h.

bool Tensors::Eigenvals ( Tensor2 const &  T,
REAL  L[3] 
) [inline]

Eigenvalues (L) of a tensor T $ L_{i=1\cdots3} \gets eigenvalues(\TeSe{T}) $.

Returns:
true if success, otherwise return false
Parameters:
T  In: Tensor T
L  Out: Eigenvalues of T

Definition at line 288 of file functions.h.

bool Tensors::Eigenvp ( Tensor2 const &  T,
REAL  L[3],
Tensor2  P[3] 
) [inline]

Eigenvalues (L) and Eigenprojectors (P) of a tensor T $ L_{i=1\cdots3} \gets eigenvalues(\TeSe{T}) \quad \TeSe{P}_{i=1\cdots3} \gets eigenprojectors(\TeSe{T}) $.

Any symmetric second order tensor $\TeSe{T}$ may be calculated according to its Spectral Decomposition:

\[ \TeSe{T} = L_0\TeFi{V}_0\otimes\TeFi{V}_0 + L_1\TeFi{V}_1\otimes\TeFi{V}_1 + L_2\TeFi{V}_2\otimes\TeFi{V}_2 \]

in which $L_k$ and $\TeFi{V}_k$ are the eigenvalues and eigenvectors of tensor $\TeSe{T}$, respectively

The eigenprojector is defined according to:

\[ \TeSe{P}_k = \TeFi{V}_{(k)} \otimes \TeFi{V}_{(k)} \qquad no \quad sum \quad on \quad (k) \]

Therefore, the spectral representation can be re-written as:

\[ \TeSe{T} = L_0\TeSe{P}_0 + L_1\TeSe{P}_1 + L_2\TeSe{P}_2 \]

Or, considering Einstein's summation rule (over index k, from 1 to 3)

\[ \TeSe{T} = L_k \TeSe{P}_k \]

The following properties hold for the eigenprojectors, in fact, for any projector (Brannon, 2000),

\[ \TeSe{P}_i \bullet \TeSe{P}_j = \TeSe{P}_i \quad if \quad i = j \]

\[ \TeSe{P}_i \bullet \TeSe{P}_j = \TeSe{0} \quad if \quad i \ne j \]

\[ \TeSe{P}_1 + \TeSe{P}_2 + \TeSe{P}_3 = \TeSe{I} \]

Returns:
true if success, otherwise return false
Parameters:
T  In: Tensor T
L  Out: Eigenvalues of T
P  Out: Igenprojectors of T

Definition at line 297 of file functions.h.

bool Tensors::Sqrt ( Tensor2 const &  T,
Tensor2 &  R 
) [inline]

Square root of a tensor T $ \TeSe{R} \gets \sqrt{\TeSe{T}} $.

Returns:
true if success, otherwise return false
Parameters:
T  In: Tensor T
R  Out: R=sqrt(T)

Definition at line 335 of file functions.h.

void Tensors::CharInvs ( Tensor2 const &  T,
REAL  I[3] 
) [inline]

Characteristic invariants of a symmetric second order tensor.

Parameters:
T  In: Tensor T
I  Out: Characterist invariants

Definition at line 357 of file functions.h.

void Tensors::Strain_Ev_Ed ( Tensor2 const &  Eps,
REAL &  Ev,
REAL &  Ed 
) [inline]

Strain Invariants.

\[ Eps=\TeSe{\varepsilon} \]

\[ Ev=\varepsilon_v=tr(\TeSe{\varepsilon}) \]

\[ Ed=\varepsilon_d=\sqrt{\nicefrac{2}{3}\TeSe{e}:\TeSe{e}} \quad \TeSe{e}=dev(\TeSe{\varepsilon})=\TeSe{\varepsilon}-\varepsilon_V\TeSe{I}/3 \]

Parameters:
Eps  In: Strain tensor Epsilon
Ev  Out: Volumetric strain
Ed  Out: Deviatoric strain

Definition at line 365 of file functions.h.

void Tensors::Stress_p_q ( Tensor2 const &  Sig,
REAL &  p,
REAL &  q 
) [inline]

Stress Invariants (Cambridge).

\[ Sig=\TeSe{\sigma} \]

\[ p=tr(\TeSe{\sigma})/3 \]

\[ q=\sqrt{\nicefrac{3}{2}\TeSe{S}:\TeSe{S}} \quad \TeSe{S}=dev(\TeSe{\sigma})=\TeSe{\sigma}-p\TeSe{I} \]

Parameters:
Sig  In: Stress tensor
p  Out: Cambridge mean stress
q  Out: Cambridge deviatoric stress

Definition at line 374 of file functions.h.

REAL Tensors::Stress_q ( Tensor2 const &  Sig  )  [inline]

Cambridge's q deviatoric stress invariant.

Returns:
Cambridge's q deviatoric stress invariant
Parameters:
Sig  In: Stress tensor

Definition at line 383 of file functions.h.

REAL Tensors::Sin3ThDev ( Tensor2 const &  S  )  [inline]

Sin3Th given deviator (S) of the stress tensor.

Parameters:
S Deviator of the stress tensor
Returns:
sin(3*th)
 *
 *            +30 S1
 *         0->  |     
 *     -30  \ th|       th: theta
 *       ',  \  |
 *         '. \ |
 *           '.\|
 *            .' -.                
 *          .'     '.
 *        .'         '.
 *   S2 .'             '. S3
 *
 *

Definition at line 391 of file functions.h.

void Tensors::Stress_p_q_S_sin3th ( Tensor2 const &  Sig,
REAL &  p,
REAL &  q,
Tensor2 &  S,
REAL &  sin3th 
) [inline]

Stress Invariants (Cambridge) + deviator of Sigma.

Parameters:
S  deviator of Sigma
sin3th  theta3 = theta*3

Definition at line 406 of file functions.h.

bool Tensors::Stress_tn_ts ( Tensor2 const &  Sig,
REAL &  tn,
REAL &  ts 
) [inline]

Stress Invariants (Professor Nakai's invariants).

\[ Sig=\TeSe{\sigma} \]

\[ t_N=3\frac{I_3}{I2} \]

\[ t_S=\frac{\sqrt{I_1I_2I_3-9I_3^2}}{I_2} \]

Returns:
true if success, otherwise return false
Parameters:
Sig  In: Stress tensor
tn  Out: mean stress invariant
ts  Out: deviatoric stress invariant

Definition at line 418 of file functions.h.

void Tensors::Stress_P_Q ( Tensor2 const &  Sig,
REAL &  P,
REAL &  Q 
) [inline]

Stress Invariants (Professor Brannon's isomorphic invariants).

\[ Sig=\TeSe{\sigma} \]

\[ P=\sqrt{3}p \]

\[ Q=\sqrt{\TeSe{S}:\TeSe{S}} \quad \TeSe{S}=dev(\TeSe{\sigma})=\TeSe{\sigma}-p\TeSe{I} \]

Parameters:
Sig  In: Stress tensor
P  Out: Isomorphic mean stress
Q  Out: Isomorphic deviatoric stress

Definition at line 438 of file functions.h.

void Tensors::Stress_P_Q_S_sin3th ( Tensor2 const &  Sig,
REAL &  P,
REAL &  Q,
Tensor2 &  S,
REAL &  sin3th 
) [inline]

Stress Invariants (Professor Brannon's isomorphic invariants) + deviator of Sigma.

Parameters:
Sig  In: Stress tensor
P  Out: Isomorphic mean stress
Q  Out: Isomorphic deviatoric stress
S  deviator of Sigma

Definition at line 447 of file functions.h.

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