Cinematics =========== Deformations ------------ Continuous medium ~~~~~~~~~~~~~~~ In this case, the displacements of the structure are represented by a vector field :math:`u` with three components in general. The deformation (in the hypothesis of small disturbances) is defined by the deformation tensor :math:`\varepsilon` by (options EPSI_ELGA and EPSI_ELNO): :math:`{\varepsilon }_{\text{ij}}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i})` We may want to calculate the "mechanical" deformation, that is to say by subtracting the thermal expansions (options EPME_ELGA and EPME_ELNO): :math:`{\mathrm{\epsilon }}_{\text{ij}}^{m}\left(u\right)=\frac{1}{2}\left({u}_{i,j}+{u}_{j,i}\right)-{\mathrm{\epsilon }}^{\text{th}}` For non-linear calculations, it is sometimes interesting to know the plastic deformation (options EPSP_ELGA and EPSP_ELNO) noted :math:`{\mathrm{\epsilon }}^{\mathit{pl}}`. In the case of large displacements, the Green-Lagrange deformations are (options EPSG_ELGA and EPSG_ELNO): :math:`{E}_{\text{ij}}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i}+{u}_{k,i}{u}_{k,j})` From which we may want to subtract thermal deformations (options EPMG_ELGA and EPMG_ELNO): :math:`{E}_{\text{ij}}^{m}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i}+{u}_{k,i}{u}_{k,j})-{\varepsilon }^{\text{th}}` For the hypothesis of plane deformations (D_ PLAN), it is important to note that the plane deformation condition is written on the *total* deformation: :math:`{\mathrm{\epsilon }}_{33}\left(u\right)=0` If we write the total deformation as the sum of a purely mechanical deformation and the deformation resulting from the control variables (such as the thermal expansion deformation): :math:`{\mathrm{\epsilon }}_{33}\left(u\right)={\mathrm{\epsilon }}_{33}^{m}+{\mathrm{\epsilon }}_{33}^{\mathit{th}}=0` So it is natural that in plane deformations, the *mechanic* deformation out of plane is not zero in the presence of thermal expansion. :math:`{\mathrm{\epsilon }}_{33}^{m}\left(u\right)\ne 0` Case of beams ~~~~~~~~~~~~~~~~ In traditional beam theories, each :math:`P` point on the beam represents a straight section. It is therefore the elements of reduction of the torsor :math:`(T(s),\Omega (s))` and the displacement of the supposed rigid straight section that characterize the displacement of the point :math:`P` to the curvilinear abscissa :math:`s`. :math:`T` is the translation of the center of inertia of the section, :math:`\Omega (s)` the rotation vector of the section at this point. The application of the virtual work theorem (cf. [bib2] _) naturally leads to defining as deformation the torsor :math:`(\varepsilon ,\chi )` derived from :math:`(T(s),\Omega (s))` with respect to the curvilinear abscissa :math:`s`: :math:`\begin{array}{}\varepsilon =\frac{\mathrm{dT}}{\text{ds}}+\tau \wedge \Omega \\ \chi =\frac{d\Omega }{\text{ds}}\end{array}` .. image:: images/100003960000149100000B5EED36AAA3CEB1754C.svg :width: 265 :height: 146 .. _RefImage_100003960000149100000B5EED36AAA3CEB1754C.svg: So let's say: :math:`\begin{array}{}\varepsilon ={\varepsilon }_{L}\tau +{\gamma }_{T}\\ \chi ={\gamma }_{t}\tau +\text{K}\end{array}` :math:`{\varepsilon }_{L}` is the longitudinal deformation, :math:`{\gamma }_{T}` is the vector of the distortion deformations (zero in the Navier-Bernoulli hypothesis), :math:`{\gamma }_{t}` is the torsional deformation of the section, :math:`\text{K}` is the bending strain. Note: For beam models taking into account warpage, kinematics is more complicated to describe, but they nevertheless lead to concepts similar to those presented above. Case of shells ~~~~~~~~~~~~~~~ Here we will limit ourselves to the cases of plaques. In fact, in the general case of shells: • spatial derivations use mathematical concepts that are too complicated for the purposes of this document, [:external:ref:`R3.07.04 `], • shells are very often modelled by elements of assembled shell elements. In this case, only material normals are assumed to be rigid. The displacement of these normals is therefore represented by the elements of reduction of a :math:`(T,\Omega )` torsor. :math:`T` is the translation of the point located on the middle sheet, :math:`\Omega` the rotation vector of the normal at this point. It is clear that the normal component of :math:`\Omega` is zero (in the case of non-micropolar media). The vector :math:`\text{I}` is introduced into the tangential plane defined by: :math:`\text{I}=\Omega \wedge \text{n}` where :math:`n` is the normal vector orienting the surface. .. image:: images/100004C0000022D40000151688DCFC184B81DAB9.svg :width: 265 :height: 146 .. _RefImage_100004C0000022D40000151688DCFC184B81DAB9.svg: That is, the decomposition: :math:`T=w\text{n}+{\text{u}}_{T}` :math:`{\text{u}}_{T}` is the tangential displacement, :math:`w` is the arrow. In the same way as for beams, the application of the virtual work theorem (cf. [bib2] _) leads to defining as deformation the set formed by the tensors :math:`E` and :math:`K` and the vector :math:`\gamma`, all these quantities being defined in the tangential plane by: :math:`\begin{array}{ccc}{E}_{\alpha \beta }& \text{=}& \frac{1}{2}({u}_{\alpha ,\beta }+{u}_{\beta ,\alpha })\\ {K}_{\alpha \beta }& \text{=}& \frac{1}{2}({l}_{\alpha ,\beta }+{l}_{\beta ,\alpha })\\ {\gamma }_{\alpha }& \text{=}& {l}_{\alpha }+{w}_{,\alpha }\end{array}` The deformation is therefore defined by 7 real numbers. :math:`{E}_{\alpha \beta }` are membrane strains, :math:`{K}_{\alpha \beta }` are the inverses of the curvatures of the deformed middle sheet, :math:`{\gamma }_{\alpha }` is the warp deformation vector. Note: Again, there is no universal convention and the disparity between conventions is even greater than for the tensors of effort. **Link with the three-dimensional deformation field** Under these conditions, we have: :math:`\begin{array}{cc}{\varepsilon }_{\alpha \beta }& \text{=}{E}_{\alpha \beta }+{x}_{3}{K}_{\alpha \beta }\\ {\varepsilon }_{\alpha 3}& \text{=}{\gamma }_{\alpha }\\ {\varepsilon }_{\text{33}}& \text{=}0\end{array}` Related quantities in Code_Aster ----------------------------------- DEPL_R and DEPL_C ~~~~~~~~~~~~~~~~~ The quantities DEPL_R and DEPL_C have as components the degrees of freedom of finite element modeling and therefore do not necessarily have only the components of the displacement fields, which are: DX, DY, DZ To whom must be added for beams or shells: DRX, DRY, DRZ For shells, we need all three components of the rotation vector, because the finite element equation can only be expressed in a global Cartesian coordinate system. EPSI_R ~~~~~~ The quantity EPSI_R represents the deformations of the structure, so it must have, at a minimum, the components: - deformation fields :math:`\varepsilon` of continuous media (as a global reference frame): EPXX, EPYY, EPZZ, EPXY,, EPXZ, EPYZ - beam deformation fields (in the "user" coordinate system for the beam): EPXX, GAXY, GAXZ, KY, KZ, GAT - fields of shell deformations (necessarily as a "user" coordinate system on the surface) EXX, EYY, EXY, KXX, KYY,, KXY, GAX, GAY Calculation options ----------------- Fields EPSI_ELGA, EPME_ELGA, EPSG_ELGA, EPMG_ELGA, and EPSP_ELGA ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ These are fields containing the deformations at the Gauss points and possibly at the sub-points of the elements. .. csv-table:: "Calculation option", "Symbolic concept name RESULTAT ", "Calculation done", "3D", "Pipes, Multifiber beams", "Shells, plates (except DKTG and Q4GG)" "EPSI_ELGA "," EPSI_ELGA ", "from a field of displacement in small deformations", ":math:`\varepsilon` "," :math:`\mathrm{\epsilon }` In "user" coordinate system 6 components", ":math:`\varepsilon` in "user" coordinate system" "EPSG_ELGA "," EPSG_ELGA ", "Green-Lagrange tensor from a displacement field", ":math:`E` ", "not available", "not available", "not available" "EPME_ELGA "," EPME_ELGA ", "from a displacement field and a temperature field in small deformations", ":math:`{\mathrm{\epsilon }}^{m}` "," :math:`{\mathrm{\epsilon }}^{m}` In "user" coordinate system 6 components", "not available" "EPMG_ELGA "," EPMG_ELGA ", "Green-Lagrange tensor from a displacement field and a temperature field", ":math:`{E}^{m}` ", "not available", "not available", "not available" "EPSP_ELGA "," EPSP_ELGA ", "from a displacement field, from a stress field, from a temperature field in small deformations", ":math:`{\mathrm{\epsilon }}^{\mathit{pl}}` "," :math:`{\mathrm{\epsilon }}^{\mathit{pl}}` In "user" coordinate system 6 components", "not available" Fields EPSI_ELNO, EPME_ELNO, EPSG_ELNO, EPMG_ELNOet EPSP_ELNO ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ These are fields containing the deformations regardless of the modeling for exploitation purposes (printing or post-processing visualization) at the nodes and possibly at the sub-points of the elements. .. csv-table:: "Calculation option", "Symbolic concept name RESULTAT ", "Calculation done", "3D", "Pipes, Multi_fiber beams", "Shells, plates (except DKTG and Q4GG)" "EPSI_ELNO "," EPSI_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", ":math:`\epsilon` "," :math:`\mathrm{\epsilon }` in "user" frame", ":math:`\epsilon` in "user" frame", "in "user" frame" "EPSG_ELNO "," EPSG_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", ":math:`E` ", "not available", "not available", "not available" "EPME_ELNO "," EPME_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", ":math:`{\mathrm{\epsilon }}^{m}` "," :math:`{\mathrm{\epsilon }}^{m}` in the "user" coordinate system", "not available" "EPMG_ELNO "," EPMG_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", ":math:`{E}^{m}` ", "not available", "not available", "not available" "EPSP_ELNO "," EPSP_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", ":math:`{\mathrm{\epsilon }}^{\mathit{pl}}` "," :math:`{\mathrm{\epsilon }}^{\mathit{pl}}` in the "user" coordinate system", "not available" "*", "EPSI_NOEU ", "by arithmetic mean at the nodes of the quantities at the nodes per element", ":math:`\varepsilon` "," :math:`\varepsilon` in "user" coordinate system", ":math:`\epsilon` in "user" coordinate system" "*", "EPSG_NOEU ", "by arithmetic mean at the nodes of the quantities at the nodes per element", ":math:`E` ", "not available", "not available", "not available" "*", "EPME_NOEU ", "by arithmetic mean at the nodes of the quantities at the nodes per element", ":math:`{\mathrm{\epsilon }}^{m}` "," :math:`{\mathrm{\epsilon }}^{m}` in "user" coordinate system", "not available" "*", "EPMG_NOEU ", "by arithmetic mean at the nodes of the quantities at the nodes per element", ":math:`{E}^{m}` ", "not available", "not available", "not available" "*", "EPSP_NOEU ", "by arithmetic mean at the nodes of the quantities at the nodes per element", ":math:`{\mathrm{\epsilon }}^{\mathit{pl}}` "," :math:`{\mathrm{\epsilon }}^{\mathit{pl}}` in "user" coordinate system", "not available" Fields DEGE_ELGA and DEGE_ELNO ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ These are fields containing the generalized deformations on the beam or shell elements for exploitation purposes (printing or post-processing visualization) at Gauss points or at the nodes of the structure. .. csv-table:: "Calculation option", "Symbolic concept name RESULTAT ", "Calculation performed", "3D", "Beams, multi-fiber beams", "Plates, 1D cases" "DEGE_ELGA "," DEGE_ELGA ", "from a field that moves in small deformations", "not available", "not available", ":math:`(E,K,\gamma )` in "user" coordinate system" "DEGE_ELNO "," DEGE_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", "not available", ":math:`(\varepsilon ,\chi )` in "user" frame", ":math:`(E,K,\gamma )` in "user" coordinate system" "Calculation option", "Symbolic concept name RESULTAT ", "Calculation performed", "3D", "Beams, multi-fiber beams", "Plates, 1D cases" "DEGE_ELGA "," DEGE_ELGA ", "from a field that moves in small deformations", "not available", "not available", ":math:`(E,K,\gamma )` in "user" coordinate system" "DEGE_ELNO "," DEGE_ELNO ", "by extrapolation to the nodes of quantities at Gauss points", "not available", ":math:`(\varepsilon ,\chi )` in "user" frame", ":math:`(E,K,\gamma )` in "user" coordinate system"