Digital integration
=====================


Unknowns and equations of the nonlinear system
----------------------------------------------

The model is integrated implicitly (@ DSL Implicit) via the Mfront tool. The "numerical" internal variables make up the vector

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. In the case where both mechanisms are active, i.e.

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and

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at the end of the elastic prediction phase (@Predictor), the equations constitutive of the non-linear system to be solved at the moment

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during the correction phase are,



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 (3.1)

The non-linear system is also solved in the case where only one of the two mechanisms is activated at the end of the prediction phase, i.e.

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or

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. In the rest of the document, we will detail only the terms deriving from the system (); the particular cases for which

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and

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can be easily deduced from () by taking some null terms.

**Note:** if the user takes

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, the viscoplastic mechanism is deactivated.


Flow directions
------------------------

The flow directions associated with

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,

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,

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, and

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are respectively given by,



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 (3.2)

with,



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 (3.3)


Tangent operator
-----------------


Tangent operator expression
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The stress tensor

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unto

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is assumed to be a function of

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and

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,



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 (3.4)

The coherent tangent operator is given by,



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 (3.5)

Furthermore, through differentiation,



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 (3.6)


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is the Jacobian matrix of the nonlinear system to be solved. Finally, the tangent operator is expressed,



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 (3.7)

Equation () can be simplified in the particular case where, 1 —

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only appears in

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(deformation tensor partition), and 2 —

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only depends on

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via Hooke's law. In this case,

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takes the following form,



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 (3.8)

And the product

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only involves the first six columns of

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; sub-matrix that we note

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. Furthermore, Hooke's law gives,



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 (3.9)


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is obtained from

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using function MFrontgetPartialJacobianInvert.


Expression of the Jacobian matrix of the system
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The Jacobian matrix of the system

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,

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, is given by,



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 (3.10)

In MFront, this matrix can be obtained by numerical disturbance — @Algorithm NewtonRaphson_NumericalJacobian — or analytically, as is the case here. The components of

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are detailed below in case both mechanisms are activated.


First line of the Jacobienne
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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 (3.11)

Expressions for derivatives of

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are,



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 (3.12)

with,



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 (3.13)

Expressions for derivatives of

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are,



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 (3.14)

with,



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 (3.15)

Expressions for derivatives of

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are,



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 (3.16)

with,



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 (3.17)


Second line of the Jacobian
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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 (3.18)

Expressions for derivatives of

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are,



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 (3.19)


Third line of the Jacobian
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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 (3.20)

Expressions for derivatives of

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are,



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 (3.21)


Fourth line of the Jacobian
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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 (3.22)

Expressions for derivatives of

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are,



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 (3.23)


Fifth line of the Jacobian
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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 (3.24)

Expressions for derivatives of

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are,



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 (3.25)


Sixth line of the Jacobian
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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 (3.26)

Expressions of derivatives of

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are,



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 (3.27)