Reference problem ===================== Geometry --------- The geometry (generated automatically in the macro command SIMU_POINT_MAT [U4.51.12]) is unique and simple: in 3D it is a tetrahedron on side 1, and in 2D it is a triangle on 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 and isotropic work-hardening characteristics selected are those of standard :math:`\mathrm{16MND5}` steel: * :math:`E=200000\mathrm{MPa}`, * :math:`\nu =0.3`, * :math:`{\sigma }_{y}=437\mathrm{MPa}`. The other parameters describing the laws were chosen based on the ASTER test cases. The following two tables summarize all the laws of Code ASTER considered and the associated parameters: +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |Fashionable.|Lois Code_Aster |Parameters retained |Criteria used for the choice of parameters | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |A | .. code:: |:math:`\mathrm{SY}=437\mathrm{MPa}`, :math:`\mathrm{DSY}=\mathrm{2024MPa}` |Material data :math:`\mathrm{16MND5}` | + + + + + | | VMIS_ISOT_LINE | | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |B | .. code:: |Traction curve at :math:`100°C` from :math:`16\mathrm{MND5}` |Material data :math:`\mathrm{16MND5}` | + + + + + | | VMIS_ISOT_TRAC | | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |C | .. code:: |:math:`\mathrm{SY}=437\mathrm{MPa}`, :math:`\mathrm{DSY}=\mathrm{2024MPa}` |Material data :math:`\mathrm{16MND5}` | + + + + + | | VMIS_CINE_LINE | | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |D | .. code:: |:math:`\mathrm{SY}=437\mathrm{MPa}`, :math:`\mathrm{DSY}=\mathrm{2024MPa}` :math:`{C}_{\mathrm{PRAG}}=1486.9`. |Material data :math:`\mathrm{16MND5}` | + + + + + | | VMIS_ECMI_LINE | | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |E | .. code:: |Traction curve at :math:`100°C` from :math:`16\mathrm{MND5}` :math:`{C}_{\mathrm{PRAG}}=1486.9`. |Material data :math:`\mathrm{16MND5}` | + + + + + | | VMIS_ECMI_TRAC | | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |F | .. code:: |:math:`\mathrm{SY}=437.0`; |work hardening: :math:`\mathrm{données16MND5}` other parameters: ssnv101c | + + + + + | | VMIS_CIN1_CHAB | :math:`\mathrm{Rinf}=758.0`; | | + + + + + | | | :math:`b=2.3`; | | + + + + + | | | :math:`\mathrm{Cinf}=63767.0` | | + + + + + | | | :math:`\mathrm{Gamma0}=341.0` | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |G | .. code:: |:math:`\mathrm{SY}=437.0`; |Wrenchingdata | + + + + :math:`\mathrm{16MND5}` + | | VMIS_CIN2_CHAB | :math:`\mathrm{Rinf}=758.0`; | other parameters ssnv101c | + + + + + | | | :math:`b=2.3`; | Kinematic choice :math:`\mathrm{X1}+\mathrm{X2}=X` from VMIS_CIN1_CHAB | + + + + + | | | :math:`\mathrm{C1inf}=63767.0/2.0` | | + + + + + | | | :math:`\mathrm{C2inf}=63767.0/2.0` | | + + + + + | | | :math:`\mathrm{Gam1}=341.0` | | + + + + + | | | :math:`\mathrm{Gam2}=341.0` | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |H | .. code:: |:math:`\mathrm{SY}=437.0`; | | + + + + + | | VMIS_ISOT_PUIS | :math:`\mathrm{APUI}=1.3` | | + + + + + | | | :math:`\mathrm{NPUI}=3.5` | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |J | .. code:: |:math:`E=\mathrm{619,3}\mathit{MPa}` |Hostun sand | + + + + + | | MOHR_COULOMB | :math:`\nu =\mathrm{0,3}` | | + + + + + | | | :math:`\varphi =33°` | | + + + + + | | | :math:`\psi =27°` | | + + + + + | | | :math:`{c}_{0}=1\mathit{MPa}` | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |K | .. code:: |:math:`E=619.3\mathit{MPa}` |Hostun sand | + + + + + | | MohrCoulombAS | :math:`\mathrm{\nu }=0.3` | | + + + + + | | | :math:`C=1\mathit{kPa}` | | + + + + + | | | :math:`\mathrm{\varphi }=33°` | | + + + + + | | | :math:`\mathrm{\psi }=27°` | | + + + + + | | | :math:`{\mathrm{\theta }}_{T}=20°` | | + + + + + | | | :math:`a=0.25C/\mathrm{tan}(\mathrm{\varphi })` | | + + + + + | | | :math:`{h}_{C}=0` | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ |L | .. code:: |YoungModulus=7.0E9 |Fictional | + + + + + | | NLH_CSRM | PoissonRatio=0.3 IsocompLaslim=50.0E6 IsotenseLaslim=0.1E6 MCCSlopeCSL =0.5 NLHIndex =1.0 MBigocritCoef=10.0 AbigocritCoef=10.0 AbigocritCoef=0.75 IncompIndex=0.75 IncompIndex=15.0 Tau=2.0e2 PerzynaExpo=2.0 NLHModulusP =7.0e9/2.5 NLHModulusV =0.01*7.0e9 | | + + + + + | | | | | +------------+--------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+ Boundary conditions and loads ------------------------------------- Characteristics of loading paths ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Two loading paths have been defined to deal with 3D and 2D plane cases. They are common to all laws of behavior. Each of them meets the following criteria: * an accumulated plastic deformation, :math:`p`, of 4 to 5% over the entire path, * a 1% increase in :math:`p` during a portion of the trip, 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 over 8 segments :math:`[O-A-B-C-O-C’-B’-A’-O]` where the second part of the path [O-C'-B'-A'-O] is symmetric with respect to the origin of the first :math:`[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) by soliciting an element in a homogeneous manner by imposing: * in 3D, the 6 components of the deformation tensor: .. image:: images/Object_1.svg :width: 134 :height: 77 .. _RefImage_Object_1.svg: * in 2D the three components of the tensor: .. image:: images/Object_2.svg :width: 134 :height: 77 .. _RefImage_Object_2.svg: For a more general description, the imposed deformation tensor will be decomposed into a hydrostatic and deviatoric part on shear bases: * .. image:: images/Object_3.svg :width: 134 :height: 77 .. _RefImage_Object_3.svg: in 2D, * .. image:: images/Object_4.svg :width: 134 :height: 77 .. _RefImage_Object_4.svg: 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:`{\varepsilon }_{\mathit{xx}}\mathrm{\times }E` ", "675", "1350", "1350", "1350", "0", "-1350", "-1350", "-675", "0" ":math:`{\varepsilon }_{\mathit{yy}}\mathrm{\times }E` ", "675", "450", "450", "1350", "1350", "0", "-1350", "-450", "-675", "0" ":math:`{\varepsilon }_{\mathit{xy}}\mathrm{\times }E\mathrm{/}(1+\nu )` ", "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: 100 :height: 50 .. _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:`{\varepsilon }_{\mathit{xx}}\mathrm{\times }E` ", "787.5", "1050", "1050", "350", "350", "0", "-350", "-1050", "-787.5", "0" ":math:`{\varepsilon }_{\mathit{yy}}\mathrm{\times }E` ", "525.0", "-175", "-175", "-350", "-350", "175", "525", "0" ":math:`{\varepsilon }_{\mathit{zz}}\mathrm{\times }E` ", "262.5", "700", "700", "-525", "-525", "525", "-700", "-262.5", "0" ":math:`{\varepsilon }_{\mathit{xy}}\mathrm{\times }E\mathrm{/}(1+\nu )` ", "700", "350", "350", "1050", "1050", "-1050", "-350", "-700", "0" ":math:`{\varepsilon }_{\mathit{xz}}\mathrm{\times }E\mathrm{/}(1+\nu )` ", "-350", "350", "350", "700", "700", "0", "-700", "700", "0" ":math:`{\varepsilon }_{\mathit{yz}}\mathrm{\times }E\mathrm{/}(1+\nu )` ", "0", "700", "-350", "-350", "0", "350", "-700", "0", "0" ":math:`P` ", "525", "525", "525", "-175", "-175", "-525", "-525", "0" ":math:`\mathrm{d1}` ", "262.5", "525", "525", "525", "0", "-525", "-525", "-262.5", "0" ":math:`\mathrm{d2}` ", "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: 330 :height: 50 .. _RefImage_1000000000000318000002630F3FC811B946DA5C.png: Initial conditions -------------------- Zero stresses and deformations.