3. Digital integration#
3.1. 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
. In the case where both mechanisms are active, i.e.
and
at the end of the elastic prediction phase (@Predictor), the equations constitutive of the non-linear system to be solved at the moment
during the correction phase are,
(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.
or
. In the rest of the document, we will detail only the terms deriving from the system (); the particular cases for which
and
can be easily deduced from () by taking some null terms.
Note: if the user takes
, the viscoplastic mechanism is deactivated.
3.2. Flow directions#
The flow directions associated with
,
,
, and
are respectively given by,
(3.2)
with,
(3.3)
3.3. Tangent operator#
3.3.1. Tangent operator expression#
The stress tensor
unto
is assumed to be a function of
and
,
(3.4)
The coherent tangent operator is given by,
(3.5)
Furthermore, through differentiation,
(3.6)
is the Jacobian matrix of the nonlinear system to be solved. Finally, the tangent operator is expressed,
(3.7)
Equation () can be simplified in the particular case where, 1 —
only appears in
(deformation tensor partition), and 2 —
only depends on
via Hooke’s law. In this case,
takes the following form,
(3.8)
And the product
only involves the first six columns of
; sub-matrix that we note
. Furthermore, Hooke’s law gives,
(3.9)
is obtained from
using function MFrontgetPartialJacobianInvert.
3.3.2. Expression of the Jacobian matrix of the system#
The Jacobian matrix of the system
,
, is given by,
(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
are detailed below in case both mechanisms are activated.
3.3.2.1. First line of the Jacobienne#
(3.11)
Expressions for derivatives of
are,
(3.12)
with,
(3.13)
Expressions for derivatives of
are,
(3.14)
with,
(3.15)
Expressions for derivatives of
are,
(3.16)
with,
(3.17)
3.3.2.2. Second line of the Jacobian#
(3.18)
Expressions for derivatives of
are,
(3.19)
3.3.2.3. Third line of the Jacobian#
(3.20)
Expressions for derivatives of
are,
(3.21)
3.3.2.4. Fourth line of the Jacobian#
(3.22)
Expressions for derivatives of
are,
(3.23)
3.3.2.5. Fifth line of the Jacobian#
(3.24)
Expressions for derivatives of
are,
(3.25)
3.3.2.6. Sixth line of the Jacobian#
(3.26)
Expressions of derivatives of
are,
(3.27)