Mechanics

Continuum-mechanics operations: deformation and stress measures, push-forward and pull-back operations. More...

Functions
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::vol (Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::dev (Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::push_forward_covariant (Tensor< T, N > const &F, Vector< T, N > const &u)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::pull_back_covariant (Tensor< T, N > const &F, Vector< T, N > const &u)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::push_forward_contravariant (Tensor< T, N > const &F, Vector< T, N > const &u)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::pull_back_contravariant (Tensor< T, N > const &F, Vector< T, N > const &u)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::push_forward_covariant (Tensor< T, N > const &F, Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::pull_back_covariant (Tensor< T, N > const &F, Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::push_forward_contravariant (Tensor< T, N > const &F, Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::pull_back_contravariant (Tensor< T, N > const &F, Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::piola (Tensor< T, N > const &F, Vector< T, N > const &u)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::piola_inverse (Tensor< T, N > const &F, Vector< T, N > const &u)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::piola (Tensor< T, N > const &F, Tensor< T, N > const &sigma)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::piola_inverse (Tensor< T, N > const &F, Tensor< T, N > const &P)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION T	minitensor::smallest_eigenvalue (Tensor< T, N > const &A)
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION bool	minitensor::check_strict_ellipticity (Tensor4< T, N > const &A)
template<typename T , Index N>
std::pair< bool, Vector< T, N > >	minitensor::check_strong_ellipticity (Tensor4< T, N > const &A)

Detailed Description

Continuum-mechanics operations: deformation and stress measures, push-forward and pull-back operations.

Function Documentation

vol()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::vol

(

Tensor< T, N > const &

A

)

Volumetric part of 2nd-order tensor

Parameters

A

tensor

Returns

\( \frac{1}{3} \mathrm{tr}\:A I \)

Definition at line 216 of file MiniTensor_Mechanics.h.

dev()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::dev

(

Tensor< T, N > const &

A

)

Deviatoric part of 2nd-order tensor

Parameters

A

tensor

Returns

\( A - vol(A) \)

Definition at line 234 of file MiniTensor_Mechanics.h.

push_forward_covariant() [1/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::push_forward_covariant	(	Tensor< T, N > const &	F,
		Vector< T, N > const &	u
	)

Push forward covariant vector

Parameters

F	deformation-gradient-like tensor
u	vector

Returns

\( F^{-T} u \)

Definition at line 250 of file MiniTensor_Mechanics.h.

pull_back_covariant() [1/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::pull_back_covariant	(	Tensor< T, N > const &	F,
		Vector< T, N > const &	u
	)

Pull back covariant vector

Parameters

F	deformation-gradient-like tensor
u	vector

Returns

\( F^T u \)

Definition at line 305 of file MiniTensor_Mechanics.h.

push_forward_contravariant() [1/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::push_forward_contravariant	(	Tensor< T, N > const &	F,
		Vector< T, N > const &	u
	)

Push forward contravariant vector

Parameters

F	deformation-gradient-like tensor
u	vector

Returns

\( F u \)

Definition at line 345 of file MiniTensor_Mechanics.h.

pull_back_contravariant() [1/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::pull_back_contravariant	(	Tensor< T, N > const &	F,
		Vector< T, N > const &	u
	)

Pull back contravariant vector

Parameters

F	deformation-gradient-like tensor
u	vector

Returns

\( F^{-1} u \)

Definition at line 385 of file MiniTensor_Mechanics.h.

push_forward_covariant() [2/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::push_forward_covariant	(	Tensor< T, N > const &	F,
		Tensor< T, N > const &	A
	)

Push forward covariant tensor

Parameters

F	deformation-gradient-like tensor
A	tensor

Returns

\( F^{-T} A F^{-1} \)

Definition at line 440 of file MiniTensor_Mechanics.h.

pull_back_covariant() [2/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::pull_back_covariant	(	Tensor< T, N > const &	F,
		Tensor< T, N > const &	A
	)

Pull back covariant tensor

Parameters

F	deformation-gradient-like tensor
A	tensor

Returns

\( F^T A F\)

Definition at line 494 of file MiniTensor_Mechanics.h.

push_forward_contravariant() [2/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::push_forward_contravariant	(	Tensor< T, N > const &	F,
		Tensor< T, N > const &	A
	)

Push forward contravariant tensor

Parameters

F	deformation-gradient-like tensor
A	tensor

Returns

\( F A F^T \)

Definition at line 507 of file MiniTensor_Mechanics.h.

pull_back_contravariant() [2/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::pull_back_contravariant	(	Tensor< T, N > const &	F,
		Tensor< T, N > const &	A
	)

Pull back contravariant tensor

Parameters

F	deformation-gradient-like tensor
A	tensor

Returns

\( F^{-1} A F^{-T} \)

Definition at line 520 of file MiniTensor_Mechanics.h.

piola() [1/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::piola	(	Tensor< T, N > const &	F,
		Vector< T, N > const &	u
	)

Piola transformation for vector

Parameters

F	deformation-gradient-like tensor
u	vector

Returns

\( \det F F^{-1} u \)

Definition at line 574 of file MiniTensor_Mechanics.h.

piola_inverse() [1/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::piola_inverse	(	Tensor< T, N > const &	F,
		Vector< T, N > const &	u
	)

Inverse Piola transformation for vector

Parameters

F	deformation-gradient-like tensor
u	vector

Returns

\( (\det F)^{-1} F u \)

Definition at line 624 of file MiniTensor_Mechanics.h.

piola() [2/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::piola	(	Tensor< T, N > const &	F,
		Tensor< T, N > const &	sigma
	)

Piola transformation for tensor, applied on second index. Useful for transforming Cauchy stress to 1PK stress.

Parameters

F	deformation-gradient-like tensor
sigma	Cauchy stress tensor

Returns

\( \det F sigma F^{-T} \)

Definition at line 670 of file MiniTensor_Mechanics.h.

piola_inverse() [2/2]

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::piola_inverse	(	Tensor< T, N > const &	F,
		Tensor< T, N > const &	P
	)

Inverse Piola transformation for tensor, applied on second index. Useful for transforming 1PK stress to Cauchy stress.

Parameters

F	deformation-gradient-like tensor
P	first Piola-Kirchhoff stress tensor

Returns

\( (\det F)^{-1} P F^T \)

Definition at line 720 of file MiniTensor_Mechanics.h.

smallest_eigenvalue()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION T minitensor::smallest_eigenvalue

(

Tensor< T, N > const &

A

)

Smallest eigenvalue by inverse iteration.

Definition at line 736 of file MiniTensor_Mechanics.h.

check_strict_ellipticity()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION bool minitensor::check_strict_ellipticity

(

Tensor4< T, N > const &

A

)

Check strict ellipticity condition for 4th-order tensor. Assume A has major symmetries.

Parameters

A	4th-order tensor is transformed into 2nd-order tensor.

Returns

whether the smallest eigenvalue of the 2nd-order tensor is less or equal than zero.

Definition at line 781 of file MiniTensor_Mechanics.h.

check_strong_ellipticity()

template<typename T , Index N>

std::pair< bool, Vector< T, N > > minitensor::check_strong_ellipticity

(

Tensor4< T, N > const &

A

)

Check strong ellipticity condition for 4th-order tensor. Assume A has major and minor symmetries.

Parameters

A	4th-order tensor.

Returns

whether \( (m\odot n):A:(m\odot n) > 0 \forall m,n \neq 0 \).

Definition at line 812 of file MiniTensor_Mechanics.h.