1. Using 1D behavioral relationships#
1.1. 1D behavior relationships in Code_Aster#
The relationships covered in this document are:
VMIS_ISOT_LINE |
Von Mises with symmetric linear isotropic work hardening |
VMIS_ISOT_TRAC |
Von Mises with any isotropic work hardening |
VMIS_CINE_LINE |
Von Mises with symmetric linear kinematic work hardening. |
ECRO_CINE_1D |
Von Mises with symmetric linear kinematic work hardening. |
GRILLE_ISOT_LINE |
Von Mises with symmetric linear isotropic work hardening |
GRILLE_CINE_LINE |
Von Mises with symmetric linear kinematic work hardening |
PINTO_MENEGOTTO |
Behaviour of reinforced concrete reinforcements |
GRILLE_PINTO_MEN |
Behaviour of reinforced concrete reinforcements |
VMIS_ASYM_LINE |
Von Mises with asymmetric linear work hardening and restoration |
VISC_IRRA_LOG, GRAN_IRRA_LOG |
Viscoplastic behavior of fuel assemblies: Models from tests REFLETet FLETAN |
MAZARS |
Behavior of MAZARSdans its version \(\mathrm{1D}\). |
RELAX_ACIER |
Behaviour to model the relaxation of pre-stressed cables. |
These (incremental) behavior relationships are given in the operator STAT_NON_LINE [U4.51.03] under the keyword factor COMPORTEMENT, by the keyword RELATION [U4.51.03]. They are only valid for small deformations. for each behavioral relationship, we describe the calculation of the stress field from a given deformation increment (cf. Newton’s algorithm [R5.03.01]), the calculation of nodal forces \(R\) and the tangent matrix \({K}_{i}^{n}\).
1.2. General ratings#
All quantities evaluated at the previous moment are indexed by \(\text{-}\).
The quantities evaluated at moment \(t+\Delta t\) are not indexed.
Increments are designated by \(\Delta\). We thus have:
\(Q\text{=}Q(t+\Delta t)\text{=}{Q}^{\text{-}}(t)+\Delta Q\text{=}{Q}^{\text{-}}+\Delta Q\)
\(\sigma\) |
stress tensor (in 1D, we are only interested in the single uniaxial non-zero component). |
\(\tilde{\sigma }\) |
diverter operator: \(\tilde{{\sigma }_{\mathrm{ij}}}={\sigma }_{\mathrm{ij}}-\frac{1}{3}{\sigma }_{\mathrm{kk}}{\delta }_{\mathrm{ij}}\). |
\((\text{})\mathrm{eq}\) |
Von Mises equivalent value, equal in 1D to the absolute value |
\(\Delta \varepsilon\) |
deformation increment. |
\(A\) |
elasticity tensor, equal in 1D to Young’s modulus E |
\(\lambda ,\mu ,E,K\) |
isotropic elasticity coefficients. |
\(\alpha\) |
secant thermal expansion coefficient. |
\(T\) |
temperature. |
\({(\text{})}_{\text{+}}\) |
positive part. |
\(P\) |
cumulative plastic deformation |
\({\varepsilon }^{p}\) |
plastic deformation |
1.3. Change of variables#
Regardless of the type of finite element referring to a 1D law of behavior, a change in variables must be made to pass from elementary quantities (forces, displacements) to stresses and deformations.
1.3.1. Calculation of deformations (small deformations)#
For each of the finite elements of Code_Aster, in STAT_NON_LINE, the global algorithm (Newton) provides the elementary routine, which integrates the behavior, with an increase in the field of movement.
For member elements, the deformation (a single axial component) is calculated by:
\(\varepsilon =\frac{u(l)-u(0)}{l}\),
and the increase in deformation by:
\(\Delta \varepsilon =\frac{\Delta u(l)-\Delta u(0)}{l}\),
For grid elements (models GRILLE and GRILLE_MEMBRANE), membrane deformation is calculated as for shell elements DKT. Simply, only one direction physically corresponds to the reinforcement directions. We therefore find ourselves in the presence of 1D behavior.
On the other hand, in small deformations, for all the models described in this document, the partition of the deformations is written at all times in the form of an elastic contribution, thermal expansion, and plastic deformation:
\(\varepsilon (t)={\varepsilon }^{e}(t)+{\varepsilon }^{\mathrm{th}}(t)+{\varepsilon }^{p}(t)\) with
\(\begin{array}{}{\varepsilon }^{e}(t)={A}^{-1}(T(t))\sigma (t)=\frac{1}{E(T)}\sigma (t)\\ {\varepsilon }^{\mathrm{th}}(t)=\alpha (T(t))(T(t)-{T}_{\mathrm{ref}})\mathrm{Id}\end{array}\)
1.3.2. Calculation of generalized efforts (integrated constraints)#
After integrating the 1D behavior, it is necessary to integrate the constraint component obtained, to provide the global algorithm (Newton) with a vector containing the generalized efforts.
For member elements, we calculate the force (uniform in the element, assuming that the cross section is constant) by: \(N=S\sigma ,\)
and the equivalent nodal force vector (as for the beam elements, [R3.08.01]) by:
\(F=\left[\begin{array}{}-N\\ N\end{array}\right]\)
For the elements of GRILLE, the forces are calculated as for the elements of shells DKT (membrane forces) by integrating the constraints into the thickness (a single layer and a single integration point).