Objective stress rates in finite strain inelasticity Objective stress rate

1 objective stress rates in finite strain inelasticity

1.1 incremental loading procedure
1.2 energy-consistent objective stress rates

1.2.1 variation of work done
1.2.2 time derivatives
1.2.3 work-conjugate stress rates
1.2.4 non work-conjugate stress rates
1.2.5 objective rates , lie derivatives

objective stress rates in finite strain inelasticity

many materials undergo inelastic deformations caused plasticity , damage. these material behaviors cannot described in terms of potential. case no memory of initial virgin state exists, particularly when large deformations involved. constitutive relation typically defined in incremental form in such cases make computation of stresses , deformations easier.

the incremental loading procedure

for small enough load step, material deformation can characterized small (or linearized) strain increment tensor

e

=



1
2




[

∇


u

+
(

∇


u


)

t


]


≡


e

i
j


=



1
2



(

u

i
,
j


+

u

j
,
i


)


{\displaystyle {\boldsymbol {e}}={\tfrac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{t}\right]\quad \equiv \quad e_{ij}={\tfrac {1}{2}}(u_{i,j}+u_{j,i})}

where




u



{\displaystyle \mathbf {u} }

displacement increment of continuum points. time derivative

∂

e



∂
t



=



e
˙



=



1
2




[

∇


v

+
(

∇


v


)

t


]


≡





e
˙




i
j


=



1
2



(

v

i
,
j


+

v

j
,
i


)


{\displaystyle {\frac {\partial {\boldsymbol {e}}}{\partial t}}={\dot {\boldsymbol {e}}}={\tfrac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {v} +({\boldsymbol {\nabla }}\mathbf {v} )^{t}\right]\quad \equiv \quad {\dot {e}}_{ij}={\tfrac {1}{2}}(v_{i,j}+v_{j,i})}

is strain rate tensor (also called velocity strain) ,




v

=




u

˙





{\displaystyle \mathbf {v} ={\dot {\mathbf {u} }}}

material point velocity or displacement rate. finite strains, measures seth–hill family (also called doyle–ericksen tensors) can used:

e


(
m
)


=


1

2
m



(


u


2
m


−

i

)


{\displaystyle \mathbf {e} _{(m)}={\frac {1}{2m}}(\mathbf {u} ^{2m}-\mathbf {i} )}

where




u



{\displaystyle \mathbf {u} }

right stretch. second-order approximation of these tensors is

e


(
m
)


≈

e

+



1
2



(
∇

u


)

t


⋅
∇

u

−
(
1
−
m
)

e

⋅

e



{\displaystyle \mathbf {e} _{(m)}\approx {\boldsymbol {e}}+{\tfrac {1}{2}}(\nabla \mathbf {u} )^{t}\cdot \nabla \mathbf {u} -(1-m){\boldsymbol {e}}\cdot {\boldsymbol {e}}}



energy-consistent objective stress rates

consider material element of unit initial volume, starting initial state under initial cauchy (or true) stress





σ


0




{\displaystyle {\boldsymbol {\sigma }}_{0}}

, let




σ



{\displaystyle {\boldsymbol {\sigma }}}

cauchy stress in final configuration. let



w


{\displaystyle w}

work done (per unit initial volume) internal forces during incremental deformation initial state. variation



δ
w


{\displaystyle \delta w}

corresponds variation in work done due variation in displacement



δ

u



{\displaystyle \delta \mathbf {u} }

. displacement variation has satisfy displacement boundary conditions.

let





s


(
m
)




{\displaystyle {\boldsymbol {s}}_{(m)}}

objective stress tensor in initial configuration. define stress increment respect initial configuration




s

=


s


(
m
)


−


σ


0




{\displaystyle {\boldsymbol {s}}={\boldsymbol {s}}_{(m)}-{\boldsymbol {\sigma }}_{0}}

. alternatively, if




p



{\displaystyle {\boldsymbol {p}}}

unsymmetric first piola–kirchhoff stress referred initial configuration, increment in stress can expressed




t

=

p

−


σ


0




{\displaystyle {\boldsymbol {t}}={\boldsymbol {p}}-{\boldsymbol {\sigma }}_{0}}

.

variation of work done

then variation in work done can expressed as

δ
w
=


s


(
m
)


:
δ


e


(
m
)


=

p

:
δ
∇

u



{\displaystyle \delta w={\boldsymbol {s}}_{(m)}:\delta {\boldsymbol {e}}_{(m)}={\boldsymbol {p}}:\delta \nabla \mathbf {u} }

where finite strain measure





e


(
m
)




{\displaystyle {\boldsymbol {e}}_{(m)}}

energy conjugate stress measure





σ


(
m
)




{\displaystyle {\boldsymbol {\sigma }}^{(m)}}

. expanded out,

δ
w
=

(

s

+


σ


0


)

:
δ


e


(
m
)


=

(

t

+


σ


0


)

:
δ
∇

u


.


{\displaystyle \delta w=\left({\boldsymbol {s}}+{\boldsymbol {\sigma }}_{0}\right):\delta {\boldsymbol {e}}_{(m)}=\left({\boldsymbol {t}}+{\boldsymbol {\sigma }}_{0}\right):\delta \nabla \mathbf {u} \,.}

the objectivity of stress tensor





s


(
m
)




{\displaystyle {\boldsymbol {s}}_{(m)}}

ensured transformation second-order tensor under coordinate rotations (which causes principal stresses independent coordinate rotations) , correctness of





s


(
m
)


:
δ


e


(
m
)




{\displaystyle {\boldsymbol {s}}_{(m)}:\delta {\boldsymbol {e}}_{(m)}}

second-order energy expression.

from symmetry of cauchy stress, have

σ


0


:
δ
∇

u

=


σ


0


:
δ

e


.


{\displaystyle {\boldsymbol {\sigma }}_{0}:\delta \nabla \mathbf {u} ={\boldsymbol {\sigma }}_{0}:\delta {\boldsymbol {e}}\,.}

for small variations in strain, using approximation

s

:
δ


e


(
m
)


≈

s

:
δ
∇

u



{\displaystyle {\boldsymbol {s}}:\delta {\boldsymbol {e}}_{(m)}\approx {\boldsymbol {s}}:\delta \nabla \mathbf {u} }

and expansions

σ


0


:
δ


e


(
m
)


=


σ


0


:

[



∂


e


(
m
)




∂
∇

u




:
δ
∇

u

]

 
,
 
 


σ


0


:
δ

e

=


σ


0


:

[



∂

e



∂
∇

u




:
δ
∇

u

]



{\displaystyle {\boldsymbol {\sigma }}_{0}:\delta {\boldsymbol {e}}_{(m)}={\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}_{(m)}}{\partial \nabla \mathbf {u} }}:\delta \nabla \mathbf {u} \right]~,~~{\boldsymbol {\sigma }}_{0}:\delta {\boldsymbol {e}}={\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}}{\partial \nabla \mathbf {u} }}:\delta \nabla \mathbf {u} \right]}

we equation

σ


0


:

[



∂


e


(
m
)




∂
∇

u




:
δ
∇

u

]

+

s

:
δ
∇

u

=


σ


0


:

[



∂

e



∂
∇

u




:
δ
∇

u

]

+

t

:
δ
∇

u


.


{\displaystyle {\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}_{(m)}}{\partial \nabla \mathbf {u} }}:\delta \nabla \mathbf {u} \right]+{\boldsymbol {s}}:\delta \nabla \mathbf {u} ={\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}}{\partial \nabla \mathbf {u} }}:\delta \nabla \mathbf {u} \right]+{\boldsymbol {t}}:\delta \nabla \mathbf {u} \,.}

imposing variational condition resulting equation must valid strain gradient



δ
∇

u



{\displaystyle \delta \nabla \mathbf {u} }

, have

(
1
)


s

=

t

−


σ


0


:

[



∂


e


(
m
)




∂
∇

u




−



∂

e



∂
∇

u




]



{\displaystyle (1)\qquad {\boldsymbol {s}}={\boldsymbol {t}}-{\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}_{(m)}}{\partial \nabla \mathbf {u} }}-{\frac {\partial {\boldsymbol {e}}}{\partial \nabla \mathbf {u} }}\right]}

we can write above equation as

(
2
)



s


(
m
)


=

p

−


σ


0


:


∂

∂
∇

u





[


e


(
m
)


−

e

]


.


{\displaystyle (2)\qquad {\boldsymbol {s}}_{(m)}={\boldsymbol {p}}-{\boldsymbol {\sigma }}_{0}:{\frac {\partial }{\partial \nabla \mathbf {u} }}\left[{\boldsymbol {e}}_{(m)}-{\boldsymbol {e}}\right]\,.}



time derivatives

the cauchy stress , first piola-kirchhoff stress related (see stress measures)

σ

=

p

⋅


f


t



j

−
1


=
(

p

+

p

⋅
∇


u


t


)

j

−
1



.


{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {p}}\cdot {\boldsymbol {f}}^{t}j^{-1}=({\boldsymbol {p}}+{\boldsymbol {p}}\cdot \nabla \mathbf {u} ^{t})j^{-1}\,.}

for small incremental deformations,

j

−
1


≈
1
−
∇
⋅

u


.


{\displaystyle j^{-1}\approx 1-\nabla \cdot \mathbf {u} \,.}

therefore,

Δ

σ

=

σ

−


σ


0


≈
(

p

+

p

⋅
∇


u


t


)
(
1
−
∇
⋅

u

)
−


σ


0



.


{\displaystyle \delta {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}-{\boldsymbol {\sigma }}_{0}\approx ({\boldsymbol {p}}+{\boldsymbol {p}}\cdot \nabla \mathbf {u} ^{t})(1-\nabla \cdot \mathbf {u} )-{\boldsymbol {\sigma }}_{0}\,.}

substituting




t

+


σ


0


=

p



{\displaystyle {\boldsymbol {t}}+{\boldsymbol {\sigma }}_{0}={\boldsymbol {p}}}

,

Δ

σ

≈
[

t

+


σ


0


+
(

t

+


σ


0


)
⋅
∇


u


t


]
(
1
−
∇
⋅

u

)
−


σ


0



.


{\displaystyle \delta {\boldsymbol {\sigma }}\approx [{\boldsymbol {t}}+{\boldsymbol {\sigma }}_{0}+({\boldsymbol {t}}+{\boldsymbol {\sigma }}_{0})\cdot \nabla \mathbf {u} ^{t}](1-\nabla \cdot \mathbf {u} )-{\boldsymbol {\sigma }}_{0}\,.}

for small increments of stress




t



{\displaystyle {\boldsymbol {t}}}

relative initial stress





σ


0




{\displaystyle {\boldsymbol {\sigma }}_{0}}

, above reduces to

(
3
)

Δ

σ

≈

t

−


σ


0


(
∇
⋅

u

)
+


σ


0


⋅
∇


u


t



.


{\displaystyle (3)\qquad \delta {\boldsymbol {\sigma }}\approx {\boldsymbol {t}}-{\boldsymbol {\sigma }}_{0}(\nabla \cdot \mathbf {u} )+{\boldsymbol {\sigma }}_{0}\cdot \nabla \mathbf {u} ^{t}\,.}

from equations (1) , (3) have

(
4
)


s

=
Δ

σ

+


σ


0


(
∇
⋅

u

)
−


σ


0


⋅
∇


u


t


−


σ


0


:

[



∂


e


(
m
)




∂
∇

u




−



∂

e



∂
∇

u




]



{\displaystyle (4)\qquad {\boldsymbol {s}}=\delta {\boldsymbol {\sigma }}+{\boldsymbol {\sigma }}_{0}(\nabla \cdot \mathbf {u} )-{\boldsymbol {\sigma }}_{0}\cdot \nabla \mathbf {u} ^{t}-{\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}_{(m)}}{\partial \nabla \mathbf {u} }}-{\frac {\partial {\boldsymbol {e}}}{\partial \nabla \mathbf {u} }}\right]}

recall




s



{\displaystyle {\boldsymbol {s}}}

increment of stress tensor measure





s


(
m
)




{\displaystyle {\boldsymbol {s}}_{(m)}}

. defining stress rate

s

=:



s
∘



(
m
)


Δ
t


{\displaystyle {\boldsymbol {s}}=:{\overset {\circ }{\boldsymbol {s}}}_{(m)}\delta t}

and noting that

Δ

σ

=



σ
˙



Δ
t


{\displaystyle \delta {\boldsymbol {\sigma }}={\dot {\boldsymbol {\sigma }}}\delta t}

we can write equation (4) as

(
5
)




s
∘



(
m
)


Δ
t
=



σ
˙



Δ
t
+


σ


0


(
∇
⋅

v

)
Δ
t
−


σ


0


⋅
∇


v


t


Δ
t
−


σ


0


:

[



∂


e


(
m
)




∂
∇

u




−



∂

e



∂
∇

u




]



{\displaystyle (5)\qquad {\overset {\circ }{\boldsymbol {s}}}_{(m)}\delta t={\dot {\boldsymbol {\sigma }}}\delta t+{\boldsymbol {\sigma }}_{0}(\nabla \cdot \mathbf {v} )\delta t-{\boldsymbol {\sigma }}_{0}\cdot \nabla \mathbf {v} ^{t}\delta t-{\boldsymbol {\sigma }}_{0}:\left[{\frac {\partial {\boldsymbol {e}}_{(m)}}{\partial \nabla \mathbf {u} }}-{\frac {\partial {\boldsymbol {e}}}{\partial \nabla \mathbf {u} }}\right]}

taking limit @



Δ
t
→
0


{\displaystyle \delta t\rightarrow 0}

, , noting





σ


0


=

σ



{\displaystyle {\boldsymbol {\sigma }}_{0}={\boldsymbol {\sigma }}}

@ limit, 1 gets following expression objective stress rate associated strain measure





e


(
m
)




{\displaystyle {\boldsymbol {e}}_{(m)}}

:

(
6
)




s
∘



(
m
)


=



σ
˙



+

σ

(
∇
⋅

v

)
−

σ

⋅
∇


v


t


−

σ

:


∂

∂
t




[


∂

∂
∇

u





(


e


(
m
)


−

e

)

]


.


{\displaystyle (6)\qquad {\overset {\circ }{\boldsymbol {s}}}_{(m)}={\dot {\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}(\nabla \cdot \mathbf {v} )-{\boldsymbol {\sigma }}\cdot \nabla \mathbf {v} ^{t}-{\boldsymbol {\sigma }}:{\frac {\partial }{\partial t}}\left[{\frac {\partial }{\partial \nabla \mathbf {u} }}\left({\boldsymbol {e}}_{(m)}-{\boldsymbol {e}}\right)\right]\,.}

here







σ
˙




i
j


=
∂

σ

i
j



/

∂
t


{\displaystyle {\dot {\sigma }}_{ij}=\partial \sigma _{ij}/\partial t}

= material rate of cauchy stress (i.e., rate in lagrangian coordinates of initial stressed state).

work-conjugate stress rates

a rate there exists no legitimate finite strain tensor





e


(
m
)




{\displaystyle {\boldsymbol {e}}_{(m)}}

associated according eq. (6) energetically inconsistent, i.e., use violates energy balance (i.e., first law of thermodynamics).

evaluating eq. (6) general



m


{\displaystyle m}

,



m
=
2


{\displaystyle m=2}

, 1 gets general expression objective stress rate:

(
7
)




s
∘



(
m
)


=



s
∘



(
2
)


+



1
2



(
2
−
m
)
[

σ

⋅



e
˙



+
(

σ

⋅



e
˙




)

t


]


{\displaystyle (7)\qquad {\overset {\circ }{\boldsymbol {s}}}_{(m)}={\overset {\circ }{\boldsymbol {s}}}_{(2)}+{\tfrac {1}{2}}(2-m)[{\boldsymbol {\sigma }}\cdot {\dot {\boldsymbol {e}}}+({\boldsymbol {\sigma }}\cdot {\dot {\boldsymbol {e}}})^{t}]}

where






s
∘



(
2
)




{\displaystyle {\overset {\circ }{\boldsymbol {s}}}_{(2)}}

objective stress rate associated green-lagrangian strain (



m
=
2


{\displaystyle m=2}

).

in particular,

m
=
2


{\displaystyle m=2}

gives truesdell stress rate




m
=
0


{\displaystyle m=0}

gives jaumann rate of kirchhoff stress




m
=
1


{\displaystyle m=1}

gives biot stress rate

(note m = 2 leads engesser s formula critical load in shear buckling, while m = -2 leads haringx s formula can give critical loads differing >100%).

non work-conjugate stress rates

other rates, used in commercial codes, not work-conjugate finite strain tensor are:

the jaumann, or corotational, rate of cauchy stress: differs jaumann rate of kirchhoff stress missing rate of relative volume change of material. lack of work-conjugacy not serious problem since term negligibly small many materials , 0 incompressible materials (but in indentation of sandwich plate foam core, rate can give error of >30% in indentation force).
the cotter–rivlin rate corresponds



m
=
−
2


{\displaystyle m=-2}

again misses volumetric term.
the green–naghdi rate: objective stress rate not work-conjugate finite strain tensor, not because of missing volumetric term because material rotation velocity not equal spin tensor. in vast majority of applications, energy errors caused these differences totally negligible must pointed out large energy error demonstrated case shear strains , rotations exceeding 0.25.
the oldroyd rate.

objective rates , lie derivatives

the objective stress rates regarded lie derivatives of various types of stress tensor (i.e., associated covariant, contravariant , mixed components of cauchy stress) , linear combinations. lie derivative not include concept of work-conjugacy.