Nonlinear behavior DIS_ECRO_TRAC#

Definition#

Behavior DIS_ECRO_TRAC is a non-linear behavior that applies to either:

to the local DX degree of freedom of discrete elements with two nodes (mesh SEG2) or and discrete elements with one node (mesh POI1). The behavior is of the « isotropic work hardening » type.
in the local YZ tangential plane, discrete elements with two nodes (mesh SEG2) or and discrete elements with one node (mesh POI1). The behavior is of the « isotropic work hardening » or « kinematic work hardening » type.

The non-linear behavior is given by a \(F=\mathit{fonction}(\mathrm{\Delta }U)\) curve:

for a SEG2, \(\mathrm{\Delta }U\) represents the relative displacement of the 2 nodes in the local coordinate system of the element.
for a POI1, \(\mathrm{\Delta }U\) represents the absolute movement of the node in the element’s local coordinate system.
for a SEG2 or a POI1, \(F\) represents the effort expressed in the local coordinate system of the element.

Data setting#

The only data needed is the function describing the non-linear behavior. This function must meet the following criteria:

It is a function in the sense of code_aster: defined with the operator DEFI_FONCTION,
The interpolations on the x-axis and the y-axis are linear,
The name of the abscissa when defining the function is DX, or DTAN,
Extensions to the left and right of the function are excluded,
The function must be defined by at least 3 points, for an isotropic work hardening function,
The function must be defined by exactly 3 points, for a kinematic work hardening function,
The first point is \((\mathrm{0.0,}0.0)\) and must be given,
The function must be strictly increasing,
The derivative of the function must be less than or equal to its derivative at point \((\mathrm{0.0,0}.0)\).

The first two points of the function are used to define the elastic slope to the behavior. The x-axis and ordinate units should be consistent with those of the problem:

the abscissa must be homogeneous when displaced,
the ordinate (value of the function) must be homogeneous to an effort.

Formulation of behavior#

The formulation is identical to that used in isotropic or kinematic plasticity:

The threshold is represented by the function describing the non-linear behavior
Elastic stiffness \({k}_{\mathit{elas}}\) is defined by the initial slope of the curve.

Under these conditions: \(f=\Vert \mathrm{F}-\mathrm{X}\Vert -R(p)\)

Where:

\(\mathrm{F}\): is the \(({F}_{x},{F}_{y},{F}_{z})\) effort vector.
\(\mathrm{X}\): is the kinematic work hardening vector \(({X}_{x},{X}_{y},{X}_{z})\).
\(p\): plastic flow.
\(R\): the isotropic or kinematic work hardening function.

The behavior is described by the following equations, with \(i\) the local \((x,y,z)\) axis:

\(\dot{{F}_{i}}={K}_{\mathit{elas}}.\left(\dot{{U}_{i}}–\dot{{U}_{i}^{p}}\right)\)

if \(f\le 0\):: \(\dot{\mathrm{\lambda }}=0\)
if \(f=0\) and \(\dot{f}=0\):: \(\dot{{U}_{i}^{p}}=\dot{\mathrm{\lambda }}.\frac{\partial f}{\partial {F}_{i}}\)

case of kinematic work hardening:

\(\dot{{\mathrm{\alpha }}_{i}}=-\dot{\mathrm{\lambda }}\frac{\partial f}{\partial {X}_{i}}\) with \(\dot{{X}_{i}}=-C.\dot{{\mathrm{\alpha }}_{i}}\) and \(C\) kinematic work hardening slope.
case of isotropic work hardening

\(\dot{{p}_{i}}=-\dot{\mathrm{\lambda }}\frac{\partial f}{\partial R}\)

The dissipation increment is given by: \(\dot{W}=\mathrm{F}\mathrm{.}{\dot{\mathrm{U}}}^{P}\)

Local integration of behavior#

The schema used is that of RUNGE - KUTTA of order 5 with error control. Figure RK5 is a combination of the 4th order diagram and the 6th order diagram [bib9], for which the coefficients are determined so as to minimize the error between the order 4 and the order 6.

In*code_aster*, the local integration of this model uses the 5th order RUNGE - KUTTA schema.

The error is known, so it is possible to trigger local subdivisions of the time step if the convergence criterion is not met.

Internal variables#

Behavior DIS_ECRO_TRAC has 14 internal variables:

	Variable name
V1	FORCEX	Force along the element’s local \(x\) axis.
V2	FORCEY	Force along the element’s local \(y\) axis.
V3	FORCEZ	Force along the element’s local \(z\) axis.
V4	DEPLX	Move along the element’s local \(x\) axis.
V5	DEPLY	Move along the element’s local \(y\) axis.
V6	DEPLZ	Move along the element’s local \(z\) axis.
V7	DISSTHER	Dissipation.
V8	PCUM	Plastic indicator.
V9	DEPLPX	Anelastic displacement along the element’s local \(x\) axis.
V10	DEPLPY	Anelastic displacement along the element’s local \(y\) axis.
V11	DEPLPZ	Anelastic displacement along the element’s local \(z\) axis.
V12	FORCXX	Kinematic work hardening along the local \(x\) axis of the element.
V13	FORCXY	Kinematic work-hardening along the local \(y\) axis of the element.
V14	FORCXZ	Kinematic work-hardening along the local \(z\) axis of the element.