2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The reference solution is calculated analytically, knowing that in traction, only the traction criterion is activated, and that in the case of hydrostatic loading, we project ourselves to the top of the traction cone. It is therefore necessary to solve a linear system of an equation with one unknown, which makes it possible to obtain the cumulative plastic deformation under tension. This makes it possible to then calculate deformations and stresses.

2.2. Calculation of the reference reference solution#

For more details on the notations and the equation, refer to the reference document [R7.01.03]. Only the main equations are mentioned here.

We note « \(a\) « , the displacement imposed in the directions \(x\), \(y\) and \(z\). The strain tensor is of the form \((a,a,a,0.,0.,0.)\) using the usual Code_Aster notations (three main components, three shear components).

The stress tensor is of the form \((\sigma ,\sigma ,\sigma ,0.,0.,0.)\), in modeling A.

General model equations:

The equations constituting the model are written by distinguishing the isotropic part from the deviatoric part of the stress and strain tensors.

and

The equivalent stress is then written as:

In the case of an incremental formulation, and of a variable law of behavior, by noting with an exponent « e » the elastic components of stress and deformation, we obtain:

and

Compression criteria

And in traction

are expressed in the following way:

: plastic multiplier in compression

: plastic multiplier in traction

and

The coefficients of the model

Plastic deformations in tension and compression are expressed:

For the constraint, we obtain:









for the equivalent stress:

The two criteria then lead to a system of two equations with two unknowns.

and

to be solved:

Analogously, in the case of the only activated traction criterion, configuration of the test case, we obtain a system of one equation with one unknown

to be solved:

Resolution with projection at the top of the traction cone:

We therefore seek to solve this system, by using the particular shape of the stress and deformation tensors, which are uniform on the structure.

Starting with \(\varepsilon \mathrm{=}(a,a,a,0.,0.,0.)\) and \(\sigma \mathrm{=}(\sigma ,\sigma ,\sigma ,0.,0.,0.)\), we get:

The elastic stress tensor

The elastic stress diverter

Elastic hydrostatic stress

The equivalent elastic stress

In the case of a post-peak linear strain hardening curve under tension, the expression of the work hardening parameter is as follows:

with

where

refers to the maximum temperature during the loading history,

tensile strength.

The equation characterizing the projection at the top of the traction cone is as follows:

Eq 2.2-1

being the tensile rupture energy (characteristic of the material).

What makes it possible to obtain the plastic multiplier:

Then the constraint:

Knowing a, the imposed displacement, we get all the unknowns of the problem.

2.3. Uncertainty about the solution#

Since the solution is analytical, the uncertainty is negligible, in the order of the precision of the machine.

2.4. Bibliographical references#

The model was defined based on the following theses and is described in the specification report:

1. Heinfling, during his thesis « Contribution to the numerical modeling of the behavior of concrete and reinforced concrete structures under thermo-mechanical stresses at high temperature »,
1. 1. Georgin, during his thesis « Contribution to the modeling of concrete under rapid dynamic stress. Taking into account the speed effect through viscoplasticity ».

SCSA /128 IQ1/RAP /00.034 Version 1.2, Development of a 3D concrete behavior model with double plasticity criteria in the*Code_Aster - Specifications ».