Reference problem ===================== Geometry --------- The geometry generated automatically in the macro command SIMU_POINT_MAT [U4.51.12] is unique and simple: it is a tetrahedron with side 1, at whose nodes linear relationships are applied to obtain a homogeneous state of stress and deformation. Material properties ---------------------- Material characteristics are defined using the DEFI_MATERIAU command. The elastic characteristics are: * :math:`E=32000\mathrm{MPa}` (except for D-modeling, see table below) * :math:`\nu =0.2`, The other parameters describing the laws were chosen from Code_Aster test cases. The following table summarizes all the Code_Aster laws considered and the associated parameters: +--------+-------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------+ |Modeling|*Code_Aster* laws of behavior |parameters retained |test used for the choice of parameters | +--------+-------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------+ |A |BETON_RAG |COMP_BETON = 'ENDO_FLUA', |**Unrealistic values**, for the purposes of the computer verification test.| + + + + + | | | ENDO_MC = 1.95, ENDO_MT = 1.95, = 2.00, = 2.00, = 2.00, ENDO_SIGUC = 35.00 MPa, ENDO_SIGUT = 3.18 MPa, 1891, p = 0.15, # Units: 0.15, #, # Units,, and .Day ENDO_DRUPRA MPa MPa FLUA_SPH_KR = 200000.0, FLUA_SPH_KI = 20000.0, FLUA_SPH_NR = 20000.0, = 350000.0, = 350000.0, = 350000.0, = 350000.0, = 350000.0, = 350000.0, | | + + + FLUA_SPH_NI FLUA_DEV_KR FLUA_DEV_KI FLUA_DEV_NR FLUA_DEV_NI + + | | | | | +--------+-------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------+ |B | .. code:: |K_RS = 2.0E5 (MPa) |Parameters identical to test SSNV163A | + + + + + | | BETON_UMLV | ETA_RS = 4.0E10 (MPa /s) | | + + + + + | | | K_IS = 5.0E4 (MPa) | | + + + + + | | | ETA_IS = 1.0E11 (MPa /s) | | + + + + + | | | K_RD = 5.0E4 (/s) K_RS = 1.0 MPa | | + + + + + | | | ETA_RD E10 (MPa /s) | | + + + + + | | | ETA_ID = 1.0E11 (MPa /s) | | +--------+-------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------+ |C | .. code:: |K_RS = 2.0E5 (MPa) |Parameters identical to test SSNV163D | + + + + + | | BETON_BURGER | ETA_RS = 4.0E10 (MPa /s) | | + + + + + | | | ETA_IS = 1.0E11 (MPa /s) | | + + + + + | | | K_RD = 5.0E4 (MPa) | | + + + + + | | | ETA_RD = 1.0E10 (/s) MPa | | + + + + + | | | ETA_ID = 1.0E11 (MPa /s) | | + + + + + | | | ETA_FD = 0. (Mpa/s) | | + + + + + | | | KAPPA = 3.0E-3 | | +--------+-------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------+ |D | .. code:: |E= 34129 (Mpa) |Parameters identical to test SSNV261A | + + + + + | | ENDO_LOCA_EXP | KAPPA = 5.84194 | | + + + + + | | | P = 5.84194 P = 2.0 | | + + + + + | | | SIGC = 3.033958 (Mpa) | | + + + + + | | | SIG0 = 0.827102 (Mpa) | | + + + + + | | | REST_RIGIDITE = 8532.28 | | +--------+-------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------+ Boundary conditions and loads ------------------------------------- Characteristics of the loading path ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The proposed loading causes each component of the tensor to vary in a decoupled manner from the deformations in successive steps. A cyclic charge-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:`\mathrm{[}O\mathrm{-}A\mathrm{-}B\mathrm{-}C\mathrm{-}O\mathrm{-}C’\mathrm{-}B’\mathrm{-}A’\mathrm{-}O\mathrm{]}` where the second part of the path :math:`\mathrm{[}O\mathrm{-}C’\mathrm{-}B’\mathrm{-}A’\mathrm{-}O\mathrm{]}` is symmetric with respect to the origin of the first :math:`\mathrm{[}O\mathrm{-}A\mathrm{-}B\mathrm{-}C\mathrm{-}O\mathrm{]}`. 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 stressing an element in a homogeneous manner by imposing in :math:`\mathrm{3D}`, the 6 components of the deformation tensor: :math:`\stackrel{ˉ}{\varepsilon }\mathrm{=}\left[\begin{array}{ccc}{\varepsilon }_{\mathit{xx}}& {\varepsilon }_{\mathit{xy}}& {\varepsilon }_{\mathit{xz}}\\ {\varepsilon }_{\mathit{xy}}& {\varepsilon }_{\mathit{yy}}& {\varepsilon }_{\mathit{yz}}\\ {\varepsilon }_{\mathit{xz}}& {\varepsilon }_{\mathit{yz}}& {\varepsilon }_{\mathit{zz}}\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 }\mathrm{=}\left[\begin{array}{ccc}{\varepsilon }_{\mathit{xx}}& {\varepsilon }_{\mathit{xy}}& {\varepsilon }_{\mathit{xz}}\\ {\varepsilon }_{\mathit{xy}}& {\varepsilon }_{\mathit{yy}}& {\varepsilon }_{\mathit{yz}}\\ {\varepsilon }_{\mathit{xz}}& {\varepsilon }_{\mathit{yz}}& {\varepsilon }_{\mathit{zz}}\end{array}\right]\mathrm{=}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& \mathrm{-}1& 0\\ 0& 0& 1\end{array}\right]+{d}_{2}\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& \mathrm{-}1\end{array}\right]+\left[\begin{array}{ccc}0& {\varepsilon }_{\mathit{xy}}& {\varepsilon }_{\mathit{xz}}\\ {\varepsilon }_{\mathit{xy}}& 0& {\varepsilon }_{\mathit{yz}}\\ {\varepsilon }_{\mathit{xz}}& {\varepsilon }_{\mathit{yz}}& 0\end{array}\right]` in :math:`\mathrm{3D}` Description of the imposed deformation path ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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:`O-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: 5.6091in :height: 4.3909in .. _RefImage_1000000000000318000002630F3FC811B946DA5C.png: Initial conditions -------------------- Zero stresses and deformations.