1. Reference problem#

1.1. Geometry#

A material point is defined, represented in the modeling A (\(\mathrm{3D}\)) by a volume element \(\mathrm{MA1}\), containing the nodes \(\mathrm{P1}\), \(\mathrm{P2}\), \(\mathrm{P3}\), \(\mathrm{P4}\), \(\mathrm{P5}\), \(\mathrm{P6}\),,,, \(\mathrm{P7}\) and \(\mathit{P8}\).

1.2. Material properties#

Elastic behavior with:	Young’s modulus: \(E\mathrm{=}145200\mathit{MPa}\)
	Poisson’s ratio: \(\nu \mathrm{=}0.3\)

1.2.1. Benchmark calculation: behavior VISC_CIN1_CHAB#

This is the reference solution used in models A and B. The material parameters are:

CIN1_CHAB =_F (R_0=75.5 MPa

R_I=85.27 MPa

B=19.34,

C_I=10.0 MPa

K=1.0,

W=0.0,

G_0=36.68,

A_I=1.0,),

LEMAITRE =_F (N=10.0,

UN_SUR_K =0.025 Mpa-1

UN_SUR_M =0.0,),

1.2.2. Monocrystalline behavior type 1, with sliding system UNIAXIAL#

The parameters used here, in uniaxial, correspond to those used for VISC_CIN1_CHAB, just noting that \(Q\mathrm{=}{R}_{I}\mathrm{-}{R}_{0}\). So we need to get the same results as VISC_CIN1_CHAB.

These behaviors are tested in \(\mathrm{3D}\) (in modeling A) and in \(\mathrm{2D}\) plane constraints (in modeling B).

Type of flow:* MONO_VISC1 **** whose parameters are:

\(c\mathrm{=}10\mathit{MPa}\), \(n\mathrm{=}10\), \(K\mathrm{=}40\mathit{MPa}\)

Isotropic work hardening type:* MONO_ISOT1 **** whose parameters are:

\({R}_{0}\mathrm{=}75.5\mathit{MPa}\) \(b\mathrm{=}19.34\) \(Q\mathrm{=}9.77\mathit{MPa}\) \(h\mathrm{=}0\)

Kinematic work hardening type:* MONO_CINE1 **** whose parameters are:

\(d\mathrm{=}36.68\)

The family of sliding systems is: UNIAXIAL

Two calculations are performed: one with implicit local integration, the other with explicit local integration. It is verified that these two calculations provide identical results (with the exception of time discretization).

Monocrystalline type 2 behavior, comparable to type 1, with sliding system UNIAXIAL ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~

The behavior of the mono-crystal is defined in such a way that one comes back to the behavior of type 1. So the results should match. The settings are:

Type of flow:* MONO_VISC2 **** whose parameters are:

\(n\mathrm{=}10\), \(k=40\mathrm{MPa}\), \(c=10\mathrm{MPa}\), \(d=0\), \(a=0\)

Isotropic work hardening type:* MONO_ISOT2 **** whose parameters are:

\({R}_{0}=75.5\) \({b}_{1}=19.34\) \({b}_{2}=0\) \({Q}_{1}=9.77\mathrm{MPa}\) \({Q}_{2}=0\)

Kinematic work hardening type:* MONO_CINE2 **** whose parameters are:

\(d=36.68\) \(M=0\) \(m=0\) \(c=0\)

The family of sliding systems is: UNIAXIAL

Two calculations are performed: one with implicit local integration, the other with explicit local integration. It is verified that these two calculations provide results identical to those of monocrystalline type 1 behavior.

1.2.3. Monocrystalline type 2 behavior, complete test#

The parameters of the behavior of the mono-crystal of type 2 are all non-zero:

Type of flow:* MONO_VISC2 **** whose parameters are:

\(n\mathrm{=}10\), \(k\mathrm{=}40\mathit{MPa}\), \(c\mathrm{=}10\mathit{MPa}\), \(d=0.1\), \(a=0.5\)

Isotropic work hardening type:* MONO_ISOT2 **** whose parameters are:

\({R}_{0}\mathrm{=}75.5\) \({b}_{1}\mathrm{=}19.34\) \({b}_{2}=10\) \({Q}_{1}\mathrm{=}9.77\mathit{MPa}\) \({Q}_{2}=10\)

Kinematic work hardening type:* MONO_CINE2 **** whose parameters are:

\(d\mathrm{=}36.68\) \(M=10\) \(m=\mathrm{0,1}\) \(c=10\)

The family of sliding systems is: UNIAXIAL . The tests are non-regression.

1.2.4. Calculation with single-crystal type 1 behavior and orthotropic elasticity#

The orthotropy parameters in fact correspond to isotropy:

ELAS_ORTH =_F (E_L = 145200.0,

E_T = 145200.0,

E_N = 14520.0,

NU_LT = 0. ,

NU_LN = 0. ,

NU_TN = 0. ,

G_LT = 72600. ,

G_LN = 72600. ,

G_TN=72600),

The results must therefore correspond to the reference calculation. Two calculations are performed: one with implicit local integration, the other with explicit integration.

1.2.5. Calculation with the single-crystal type 1 behavior and the sliding systems of ZIRCONIUM#

Five calculations are carried out with this family of sliding systems:

a non-regression calculation with the family defined in the code
a comparative calculation to the first, by providing a table containing the interaction matrix
a calculation comparative to the previous ones, with a polycrystal comprising a single grain,
a comparative calculation with the previous ones, by providing five families defined from a table containing sliding systems. All of these systems correspond to the Zirconium family. This tests the possibility of defining different material coefficients according to the sliding systems in question,
a calculation identical to the previous one, with a polycrystal comprising a single grain.

1.2.6. Kocks-Rauch behavior: mono-crystalline, with BCC24 sliding system#

The behavior of the single crystal is defined by the flow: MONO_DD_KR ** whose parameters are:

K = 8.62E-5,

TAUR = 498. ,

TAU0 = 132. ,

GAMMA0 = 1.E6,

DELTAG0 = 0.768,

BSD = 2.514E-5,

GCB = 31.822,

KDCS = 22.9,

P = 0.335,

Q = 1.12,

H1 = 0.25,

H2 = 0.25,

H3 = 0.25,

H4 = 0.25

Three calculations are performed with this behavior:

An implicit MONOCRISTAL calculation
An explicit MONOCRISTAL calculation
An explicit POLYCRISTAL calculation, with only one phase

These three calculations should lead to the same results.

1.3. Boundary conditions and loads#

Knot \(\mathit{P4}\)	: \(\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}0\)
Knot \(\mathit{P8}\)	: \(\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}0\)
Nodes \(\mathit{P2}\) and \(\mathit{P6}\)	: \(\mathit{DX}\mathrm{=}0\)
Nodes \(\mathit{P1}\), \(\mathit{P3}\), \(\mathit{P5}\) and \(\mathit{P7}\)	: either \(\mathit{FX}\mathrm{=}25\) or \(\mathit{DX}\mathrm{=}0.001\)

The imposed force load is increasing from \(\mathit{FX}\mathrm{=}0\) to \(\mathit{FX}\mathrm{=}25\mathrm{\times }0.755N\), in an increment, which leads to a uniaxial stress state of \(75.5\mathit{MPa}\) (linearity limit)

The load then increases to \(\mathit{FX}\mathrm{=}25\mathrm{\times }0.955N\) in \(n\) increments. The reference calculation is obtained with \(n\mathrm{=}100\). Monocrystalline calculations are done with \(n\mathrm{=}20\).

With respect to behaviors MONO_VISC2 and MONO_DD_KR, loading is an imposed displacement varying from 0, at the initial moment, to 0.001 at time 2, in \(m\) increments.

For implicit resolutions, \(m\mathrm{=}20\), and for explicit resolutions, \(m\mathrm{=}100\).

For modeling B, the load is an imposed displacement varying from 0, at the initial instant, to 0.001 at time 3, in 20 increments.