Reference problem ===================== Geometry --------- The geometry (generated automatically in the macro command SIMU_POINT_MAT [U4.51.12]) is unique and simple: in :math:`\mathrm{3D}` it is a tetrahedron with side 1, and in :math:`\mathrm{2D}` it is a triangle with side 1, at whose nodes linear relationships are applied to obtain a state of homogeneous stress and deformation. Material properties ----------------------- Material characteristics are defined for each behavior using the DEFI_MATERIAU command. The elastic characteristics and the elastic limit selected are those of standard 16 MND5 steel: * :math:`E=200000\mathrm{MPa}`, :math:`\nu =0.3`, :math:`{\sigma }_{y}=437\mathrm{MPa.}` The other parameters describing the laws were chosen from Code_Aster test cases. The following two tables summarize all the Code_Aster laws considered and the associated parameters +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |**Model.**|**viscoplastic laws of ASTER** |**parameters retained** |**test selected for the choice of parameters** | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |A |LEMAITRE | .. code:: |Test ASTER SSNA01A | + + + + + | | | m = 5.6 | | + + + + + | | | Kinv=1/K= 3.2841e-4 | | + + + + + | | | n = 11 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |B |VISC_CIN1_CHAB | .. code:: |work hardening: data 16 MND5 other parameters: ssnv101c | + + + + + | | | SY = 437.0; | | + + + + + | | | Rinf = 758.0; | | + + + + + | | | b = 2.3; | | + + + + + | | | Cinf = 63767.0 | | + + + + + | | | Gamma0 = 341.0 | | + + + + + | | | 1/m = 0 | | + + + Kinv=1/K= 3.2841e-4 + + | | | | | + + + .. code:: + + | | | | | + + + n = 11 + + | | | | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |C |VISC_CIN2_CHAB | .. code:: |Crouching data 16 MND5 other parameters ssnv101c | + + + + + | | | SY = 437.0; | Kinematic choice X1+X2= Xde VMIS_CIN1_CHAB | + + + + + | | | Rinf = 758.0; | | + + + + + | | | b = 2.3; | | + + + + + | | | C1inf = 63767.0/2.0 | | + + + + + | | | C2inf = 63767.0/2.0 | | + + + + + | | | Gam1 = 341.0 | | + + + + + | | | Gam2 = 341.0 | | + + + + + | | | 1/m = 0 | | + + + + + | | | 1/K= 3.2841e-4 | | + + + + + | | | n = 11 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |D |VISC_ENDO_LEMA | .. code:: | | + + + + + | | | SY=0.0 | | + + + + + | | | N=12.0 | | + + + + + | | | UN_SUR_M=1/9.0 | | + + + + + | | | UN_SUR_K=1/2110.0 | | + + + + + | | | R_D=6.3 | | + + + + + | | | A_D=3191.0 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |E |VISC_TAHERI | .. code:: |Test ASTER SSNP101B | + + + + + | | | SY = 437.0; Sinf = 758.0; | | + + + + + | | | alpha = 0.3; | | + + + + + | | | m = 0.1; | | + + + + + | | | a = 312.0; | | + + + + + | | | b = 30.0; | | + + + + + | | | c1 = -0.012; | | + + + + + | | | cinf = 0.065 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |F |VISC_ISOT_LINE |SY=437 MPa, DSY =2024 Mpa |SSNL129 test for part VISC_SINH | + + + + + | | | SIGM_0 =6176. EPSI_0 =3.31131121483e13 M=6.76 | Material data 16 MND5pour work-hardening | + + + + + | | | | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |G |VISC_ISOT_TRAC |tensile curve at 100°C of 16 MND5 |SSNL129 test for part VISC_SINH | + + + + + | | | SIGM_0 =6176. EPSI_0 =3.31131121483e13 M=6.76 | Material data 16 MND5pour work hardening | + + + + + | | | | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |H |VISC_CIN2_MEMO | .. code:: |Cinematic choice X1+X2= Xde VMIS_CIN1_CHAB. | + + + + + | | | R0=SY = 437.0; | Memory effect. | + + + + + | | | Q0 = 758.0-437.0; | | + + + + + | | | Qm=Q0+100 | | + + + + + | | | Mu=10 | | + + + + + | | | Eta=0.5 | | + + + + + | | | b = 2.3; | | + + + + + | | | C1inf = 63767.0/2.0 | | + + + + + | | | C2inf = 63767.0/2.0 | | + + + + + | | | Gam1 = 341.0 | | + + + + + | | | Gam2 = 341.0 | | + + + + + | | | 1/m =0 | | + + + + + | | | 1/K= 3.2841e-4 | | + + + + + | | | n = 11 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |I |VISCOCHAB | .. code:: |Work harden:Data 16 MND5 other parameters ssnv101c | + + + + + | | | SY = 437.0; | Kinematic choice X1+X2= Xde VMIS_CIN1_CHAB Memory effect. | + + + + + | | | Rinf = 758.0; | | + + + + + | | | b = 2.3; | | + + + + + | | | C1inf = 63767.0/2.0 | | + + + + + | | | C2inf = 63767.0/2.0 | | + + + + + | | | Gam1 = 341.0 | | + + + + + | | | Gam2 = 341.0 | | + + + + + | | | 1/m = 0 | | + + + + + | | | 1/K= 3.2841e-4 | | + + + + + | | | n = 11 | | + + + + + | | | Q0 = 758.0-437.0; | | + + + + + | | | Qm=Q0+100 | | + + + + + | | | Mu=10 | | + + + + + | | | Eta=0.5 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |J |MONOCRISTAL |viscoplastic laws of ASTER |Plastic parameters from SSNV171.Orthotropic parameters from SSLV120 | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |K | .. code:: |YOUNG = 124000.e6; | | + + + + + | | VMIS_JOHN_COOK | POISSON = 0.34; | | + + + + + | | | A=0.34; A=90.e6 | | + + + + + | | | B=292.e6 | | + + + + + | | | C=0.025 | | + + + + + | | | N_ PUIS =0.31 | | + + + + + | | | M_ PUIS =0.31 M_ =0.34; A=0.34; A=90.e6 B=292.e6 B=292.e6 C=0.025 N_ =0.31 M_ =0.31 | | + + + + + | | | EPSI0 | | + + + + + | | | TROOM =298.0 | | + + + + + | | | TMELT =1083.0 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ |L | .. code:: |YOUNG = 145000. ; | | + + + + + | | HAYHURST | POISSON = 0.34; | | + + + + + | | | BIGA =9.7E-8 | | + + + + + | | | DELTA1 =1.0, DELTA2 =0.0, BIGA =9.707593E-08, H1ST=0.33, H2ST=1.0, K=9.69 | | + + + + + | | | H1=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=3.E4, H2=-3.E4, H2=-3.E4, H2=-3.E4 SIG0 ALPHAD EPS0 11 | | +----------+--------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+ Boundary conditions and loads ------------------------------------- Characteristics of loading paths ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Two loading paths have been defined to deal with cases :math:`\mathrm{3D}` and :math:`\mathrm{2D}` plan. They are common to all laws of behavior. Each of them meets the following criteria: * an accumulated plastic deformation, :math:`p`, from 4 to :math:`\text{5\%}` over the entire path, * an increase of :math:`1\text{\%}` in the cumulative plastic deformation :math:`p` during a portion of the trip, * in the presence of viscosity, a deformation stress rate of 10-3, 10-4 and :math:`{10}^{-5}{\text{s}}^{-1}` respectively. These were evaluated approximately by considering an equivalent deformation of :math:`5\text{\%}` over the entire journey: i.e. travel times of 50, 500 and 5000 seconds respectively for :math:`\text{v1}`, :math:`\text{v2}` and :math:`\text{v3}`. The tests returned correspond to a speed of :math:`{10}^{-5}{\text{s}}^{-1}`. This calibration was carried out on law VMIS_ISOT_LINE, then carried over to the other laws. The proposed loading causes each component of the deformation tensor to vary in a decoupled manner by successive step. A cyclic load-discharge path is proposed by covering the states of traction and compression as well as an inversion of the signs of shear in order to test a wide range of values. Schematically, it follows a course on 8 segments :math:`\text{[O-A-B-C-O-C’-B’-A’-O]}` where the second part of the path :math:`\text{[O-C’-B’-A’-O]}` is symmetric with respect to the origin of the first :math:`\text{[O-A-B-C-O]}`. Application of requests ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We come back to the study of a material point (using the macro-command SIMU_POINT_MAT [:ref:`U4.51.12 `]) by soliciting an element in a homogeneous manner by imposing: * in :math:`\mathrm{3D}`, the 6 components of the deformation tensor: :math:`\stackrel{ˉ}{\varepsilon }=\left[\begin{array}{ccc}{\varepsilon }_{\mathrm{xx}}& {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{xz}}\\ {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{yy}}& {\varepsilon }_{\mathrm{yz}}\\ {\varepsilon }_{\mathrm{xz}}& {\varepsilon }_{\mathrm{yz}}& {\varepsilon }_{\mathrm{zz}}\end{array}\right]` * in :math:`\mathrm{2D}` the three components of the tensor :math:`\stackrel{ˉ}{\varepsilon }=\left[\begin{array}{cc}{\varepsilon }_{\mathrm{xx}}& {\varepsilon }_{\mathrm{xy}}\\ {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{yy}}\end{array}\right]` For a more general description, the imposed deformation tensor will be decomposed into a hydrostatic and deviatoric part on shear bases: :math:`\stackrel{ˉ}{\varepsilon }=\left[\begin{array}{cc}{\varepsilon }_{\mathrm{xx}}& {\varepsilon }_{\mathrm{xy}}\\ {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{yy}}\end{array}\right]=p\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+d\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]+{\varepsilon }_{\mathrm{xy}}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]` in :math:`\mathrm{2D}`, :math:`\stackrel{ˉ}{\varepsilon }=\left[\begin{array}{ccc}{\varepsilon }_{\mathrm{xx}}& {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{xz}}\\ {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{yy}}& {\varepsilon }_{\mathrm{yz}}\\ {\varepsilon }_{\mathrm{xz}}& {\varepsilon }_{\mathrm{yz}}& {\varepsilon }_{\mathrm{zz}}\end{array}\right]=p\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+{d}_{1}\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right]+{d}_{2}\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]+\left[\begin{array}{ccc}0& {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{xz}}\\ {\varepsilon }_{\mathrm{xy}}& 0& {\varepsilon }_{\mathrm{yz}}\\ {\varepsilon }_{\mathrm{xz}}& {\varepsilon }_{\mathrm{yz}}& 0\end{array}\right]` in 3D. Description of the imposed deformation path in 2D ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The path applied is described in the table below, the deformation values are calibrated with respect to the elastic modulus: .. csv-table:: "time", "1", "2", "3", "3", "3", "4", "5", "6", "7", "8" "Charging point", ":math:`A` "," :math:`B` "," "," :math:`C` "," "," :math:`O` "," :math:`C’` "," :math:`B’` "," :math:`A’` "," :math:`O`" ":math:`E\mathrm{.}{\varepsilon }_{\mathrm{xx}}` ", "675", "1350", "1350", "1350", "0", "-1350", "-1350", "-675", "0" ":math:`E\mathrm{.}{\varepsilon }_{\mathrm{yy}}` ", "675", "450", "450", "1350", "1350", "0", "-1350", "-450", "-675", "0" ":math:`\frac{E}{(1+\nu )}{\varepsilon }_{\mathrm{xy}}` ", "450", "180", "180", "0", "0", "0", "-180", "-450", "0" ":math:`p` ", "675", "900", "900", "", "1350", "", "-1350", "-900", "-675", "0" ":math:`d` ", "0", "0", "0", "450", "450", "0", "0", "0" This path is illustrated by the following graph: .. image:: images/1000000000000318000002632D218E4097047B7B.png :width: 5.7744in :height: 4.4543in .. _RefImage_1000000000000318000002632D218E4097047B7B.png: Description of the imposed deformation path in 3D ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The path applied is described in the table below, the deformation values applied are calibrated with respect to the elastic modulus: .. csv-table:: "Segment number", "1", "2", "2", "3", "3", "3", "4", "5", "6", "7", "8" "Segment", ":math:`0-A` "," :math:`A-B` "," "," :math:`B-C` "," "," :math:`O` "," :math:`C’` "," :math:`B’` "," :math:`A’` "," :math:`O`" ":math:`E\mathrm{.}{\varepsilon }_{\mathrm{xx}}` ", "787.5", "1050", "1050", "350", "350", "0", "-350", "-1050", "-787.5", "0" ":math:`E\mathrm{.}{\varepsilon }_{\mathrm{yy}}` ", "525.0", "-175", "-175", "-350", "-350", "175", "525", "0" ":math:`E\mathrm{.}{\varepsilon }_{\mathrm{zz}}` ", "262.5", "700", "700", "-525", "-525", "525", "-700", "-262.5", "0" ":math:`\frac{E}{(1+\nu )}{\varepsilon }_{\mathrm{xy}}` ", "700", "350", "350", "1050", "1050", "-1050", "-350", "-700", "0" ":math:`\frac{E}{(1+\nu )}{\varepsilon }_{\mathrm{xz}}` ", "-350", "350", "350", "700", "700", "0", "-700", "700", "0" ":math:`\frac{E}{(1+\nu )}{\varepsilon }_{\mathrm{yz}}` ", "0", "700", "-350", "-350", "0", "350", "-700", "0", "0" ":math:`p` ", "525", "525", "525", "-175", "-175", "-525", "-525", "0" ":math:`{d}_{1}` ", "262.5", "525", "525", "525", "0", "-525", "-525", "-262.5", "0" ":math:`{d}_{2}` ", "262.5", "-175", "-175", "350", "350", "0", "-350", "175", "-262.5", "0" This path is illustrated by the following graph: .. image:: images/1000000000000318000002630F3FC811B946DA5C.png :width: 5.6091in :height: 4.3909in .. _RefImage_1000000000000318000002630F3FC811B946DA5C.png: Initial conditions -------------------- Zero stresses and deformations.