5. Identifying the characteristics of the material

Consider a uniaxial tensile compression test, [Figure]. It is proposed to show how it makes it possible to identify the Prager constant and the isotropic work hardening function. In such a test, the various tensors have fixed directions, that is to say that:

\(\tilde{\sigma }\text{=}\sigma \Delta X\text{=}X\Delta {\varepsilon }^{p}=\frac{3}{2}{\varepsilon }^{p}D\) [éq5-1]

with \(D\text{=}\left[\begin{array}{ccc}2/3& & \\ & -1/3& \\ & & -1/3\end{array}\right]\)

As long as the load is monotonic, so in the traction phase, the following relationships are immediately obtained:

\(p\text{=}{\varepsilon }^{p}X\text{=}\frac{3}{2}C{\varepsilon }^{p}{s}^{t}\text{=}\frac{3}{2}C{\varepsilon }^{p}\text{+}R({\varepsilon }^{p})\) [éq5-2]

The Prager constant is determined by the first compression plasticization, since we have:

\(\{\begin{array}{c}{\sigma }_{A}^{t}\text{=}\frac{3}{2}C{\varepsilon }_{A}^{p}\text{+}R({\varepsilon }_{A}^{p})\\ {\sigma }_{A}^{c}\text{=}\frac{3}{2}C{\varepsilon }_{A}^{p}\text{-}R({\varepsilon }_{A}^{p})\end{array}\Rightarrow C\text{=}\frac{{\sigma }_{A}^{t}\text{+}{\sigma }_{A}^{c}}{3{\varepsilon }_{A}^{p}}\) [éq5-3]

Figure 5-a :: ** Uniaxial compression tensile test

The work-hardening curve \({\sigma }^{t}\text{=}F({\varepsilon }^{p})\) is deduced from the traction curve \({\sigma }^{t}\text{=}F(\varepsilon )\) provided by the user under the keywords ECRO_LINE (SYet D_ SIGM_EPSI (linear work-hardening)) or else TRACTION (work-hardening of any kind). Finally, it makes it possible to obtain the isotropic work hardening function by [eq]:

\(R({\varepsilon }^{p})\text{=}{s}^{t}({\varepsilon }^{p})-\frac{3}{2}C{\varepsilon }^{p}\)

For the effective calculation of R (p), according to document R5.03.02, we take advantage of the linearity (ECMI_LINE) or the piecewise linearity of the traction curve (ECMI_TRAC):

ECMI_LINE:

\(\begin{array}{}{\sigma }^{t}\text{=}F({\varepsilon }^{p})\text{=}{\sigma }_{y}+\frac{E\text{.}{E}_{T}}{E\text{-}{E}_{T}}p\\ R(p)\text{=}{\sigma }_{y}+(\frac{E\text{.}{E}_{T}}{E\text{-}{E}_{T}}\text{-}\frac{3}{2}C)p\text{=}{\sigma }_{y}\text{+}{R}^{\text{'}}\text{.}p\\ \end{array}\) [éq5-4]

The equation [eq] then becomes:

\(\frac{3}{2}(2\mu \text{+}C)\Deltap \text{+}{\sigma }_{y}+{R}^{\text{'}}\text{.}(p\text{+}\Deltap )\text{=}{s}_{\text{eq}}^{e}\) [éq5-5]

ECMI_TRAC:

\(\begin{array}{}{\sigma }^{t}\text{=}F({\varepsilon }^{p})\text{=}{\sigma }_{i}\text{+}\frac{{\sigma }_{i\text{+}1}\text{-}{s}_{i}}{{p}_{i\text{-}1}\text{-}{p}_{i}}(p\text{-}{p}_{i}),\text{pour}{p}_{i}\text{<=}p\text{<=}{p}_{i\text{+}1}\\ R(p)\text{=}{\sigma }_{i}\text{+}\frac{{\sigma }_{i\text{+}1}\text{-}{\sigma }_{i}}{{p}_{i\text{-}1}\text{-}{p}_{i}}(p\text{-}{p}_{i})\text{-}\frac{3}{2}\text{Cp}\text{=}{\sigma }_{i}\text{-}\frac{{\sigma }_{i\text{+}1}\text{-}{\sigma }_{i}}{{p}_{i\text{-}1}\text{-}{p}_{i}}{p}_{i}\text{+}{R}^{\text{'}}\text{.}p\end{array}\) [éq5-6]

Note:

For use: the correspondence between the behavior model VMIS_CINE_LINE and the behavior VMIS_ECMI_LINE is as follows:

• With VMIS_CINE_LINE, it is necessary to introduce in DEFI_MATERIAU a linear work hardening with a slope And by:

D_ SIGM_EPSI: And

• With VMIS_ECMI_LINE, to reproduce the same linear kinematic work hardening behavior, you must give in DEFI_MATERIAU.

linear work hardening \({E}_{T}\) : D_ SIGM_EPSI: and

The Prager constant \(C\) : * PRAGER: C

\(C\) is determined by: \(C\text{=}\frac{2}{3}\frac{{\text{EE}}_{T}}{E\text{-}{E}_{T}}\)

It should be noted that the identification of \(C\) and of \(R({\varepsilon }^{p})\) only make sense in a limited field of deformations (small deformations). In particular, if \({\sigma }^{t}({\varepsilon }^{p})\) has an asymptote \({\sigma }_{\text{max}}^{t}\) for \({\varepsilon }^{p}\) large enough, then the kinematic contribution of work hardening is no longer meaningful. It is therefore advisable to restrict yourself to the area where work hardening is strictly positive.

• Withvec VMIS_CINE_GC, linear work hardening with a slope must be introduced in DEFI_MATERIAU and by:

D_ SIGM_EPSI*: And*

This behavior is available in plane stress. It is not necessary to give the Prager constant. The correspondence between ETet is made by the code and taken into account in particular when calculating \(\mathrm{\Delta }P\), chapter 6.