2. Reference solution#
2.1. Calculation method used for the reference solution#
The reference solution is calculated semi-analytically, knowing that in traction, only the traction criterion is activated. It is therefore necessary to solve a system of one equation with one unknown, which makes it possible to obtain by dichotomy, for example, the cumulative plastic deformation under traction. This makes it possible to then calculate deformations and stresses. This is possible, knowing the displacement, and therefore the deformation in the two imposed directions. Moving in the third direction is then an unknown part of the problem.
The reference solution is calculated by traction only. The solution is determined by an independent Fortran dichotomy resolution program. In compression, in discharge, the exact solution has not been recalculated, and constitutes a non-regression solution of the code, linked to version 5.02.14.
For modeling B, the results are deduced by rotating the stress tensor from modeling A, from the intrinsic coordinate system of the cube to the user coordinate system, the stress field of the two configurations being identical in the intrinsic coordinate system of the cube.
2.2. Calculation of the reference reference solution#
For more details on the notations and the equation, refer to the reference document. Only the main equations are mentioned here.
We note \(\text{a}\), the displacement imposed in the direction \(x\), and \(\text{2.a}\) the displacement imposed in the direction \(z\). The strain tensor is of the form \((a,{\varepsilon }_{y},\mathrm{2.a},0.,0.,0.)\) using the usual Code_Aster notations (three main components, three shear components).
The stress tensor is of the form \(({\sigma }_{x},0.,{\sigma }_{z},0.,0.,0.)\), in modeling A.
The traction criterion is expressed in the form:
The constitutive equations are written by distinguishing the isotropic part from the deviatoric part of the stress and strain tensors.
The equivalent constraint is then written as: \({\sigma }^{\mathit{eq}}\mathrm{=}\sqrt{\frac{3}{2}\mathit{tr}(s)}\)
In the case of an incremental formulation, and a variable law of behavior, by noting with an exponent \(e\) the elastic components of stress and deformation, we obtain:
and
The compression and traction criteria are expressed in the following way:
Plastic deformations in tension and compression are expressed:
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For the constraint, we obtain:
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for the equivalent stress: |
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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:
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 from
. 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
So we’re trying to solve the equation:
Eq 2.2-1
Knowing that the stress in direction \(y\) is zero, we get a second equation:
From where:
that can be substituted in the expression for the criterion [éq 2.2-1].
Knowing \(a\), the imposed displacement, we obtain a non-linear equation with one unknown, which can be solved simply by dichotomy, and which makes it possible to calculate the deformation \({\varepsilon }_{y}\), then the set of unknowns of the system, to be calculated.
2.3. Uncertainty about the solution#
It is negligible, in terms of machine precision.
2.4. Bibliographical references#
The model was defined based on the following theses:
GEORGIN, during his thesis « Contribution to the numerical modeling of the behavior of concrete and reinforced concrete structures under thermo-mechanical stresses at high temperature »,
HEINFLING, during his thesis « Contribution to the modeling of concrete under rapid dynamic stress. Taking into account the speed effect through viscoplasticity », and is described in the specification report:
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 ».