17.1.2 Von Mises

The yield condition of Von Mises is a smooth approximation of the Tresca yield condition: a circular cylinder in the principal stress space [Fig.17.2b]. The yield function of Von Mises is given by the square root formulation

$$\boldsymbol\sigma \kappa \sqrt{{ 3 J_{2} }} \bar{{\sigma }} \kappa \sqrt{{ \tfrac{1}{2}\boldsymbol{\sigma}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}} \bar{{\sigma }} \kappa$$ (17.29)

where $\bar{{\sigma }}$($\kappa$) is the uniaxial yield strength as a function of the internal state variable $\kappa$ . The projection matrix P is given by

$$\left[\vphantom{ \negthickspace \begin{array}{cccccc} 2 & -1 & -1... ...0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace }\right. \begin{array}{cccccc} 2 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0... ...& 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} 2 & -1 & -1... ...0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace }\right]$$ (17.30)

The flow rule is generally given by the associated flow rule g $\equiv$ f , which results for the plastic strain rate vector in

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \dot{{\lambda}} {\frac{{\mathbf{P}\boldsymbol{\sigma}}}{{2 \bar{\sigma}}}}$$ (17.31)

17.1.2.1 Hardening

The relation between the internal state variable $\kappa$ and the plastic process is given by the hardening hypothesis. For the Von Mises yield condition we consider two different hypotheses: strain hardening and work hardening.

Strain hardening.

In the case of strain hardening the relation is given in the principal space by

$$\dot{{\kappa}} \sqrt{{ \tfrac{2}{3}\left( \dot{\varepsilon }_{1}^{\mathrm{p}} \d... ...t{\varepsilon }_{3}^{\mathrm{p}} \dot{\varepsilon }_{3}^{\mathrm{p}} \right) }}$$ (17.32)

which can be elaborated to

$$\dot{{\kappa}} \dot{{\lambda}}$$ (17.33)

Work hardening.

For work hardening the basic assumption is

$$\dot{{W}}^{{\mathrm{p}}}_{} \boldsymbol\sigma \dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \equiv \bar{{\sigma }} \kappa \dot{{\kappa}}$$ (17.34)

with

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \left\{\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilo... ...amount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right. \begin{array}{c} \dot{\varepsilon }_{1}^{\mathrm{p}} \\ [\medski... ...mathrm{p}} \\ [\medskipamount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilon... ...mount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right\} \dot{{\lambda}} {\frac{{ 1 }}{{ 2 \bar{\sigma } }}} \left\{\vphantom{ \negthickspace \begin{array}{c} 2 \sigma _{1} -... ...a _{2} + 2 \sigma _{3} \\ [\medskipamount] \end{array} \negthickspace }\right. \begin{array}{c} 2 \sigma _{1} - \sigma _{2} - \sigma _{3} \\ [\... ...t] - \sigma _{1} - \sigma _{2} + 2 \sigma _{3} \\ [\medskipamount] \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 2 \sigma _{1} - ... ... _{2} + 2 \sigma _{3} \\ [\medskipamount] \end{array} \negthickspace }\right\}$$ (17.35)

Equation (17.34) can be elaborated to

$$\dot{{\kappa}} \dot{{\lambda}}$$ (17.36)

Relation $\bar{{\sigma }}$ -$\kappa$ .

For the Von Mises yield condition the translation of uniaxial experimental data to the equivalent stress-internal state variable, the $\bar{{\sigma}}$ -$\kappa$ relation, is independent upon the hardening hypothesis as shown in the example of Figure 17.4.

Figure 17.4: Derivation of hardening diagram for Von Mises

Consider the uniaxial stress-strain diagram of Figure 17.4a. The plastic strain $\varepsilon_{{1}}^{{\mathrm{p}}}$ is assumed to be given by $\varepsilon_{{1}}^{}$ - $\varepsilon_{{1}}^{{\mathrm{e}}}$ . Figure 17.4b shows the uniaxial stress-plastic strain diagram. The uniaxial plastic strain rate is given by

$$\dot{{\varepsilon }}_{{1}}^{{\mathrm{p}}} \dot{{\lambda}} {\frac{{ \sigma _{1} }}{{ \bar{\sigma } }}}$$ (17.37)

The relation between the uniaxial stress and the equivalent stress is simply

$$\bar{{\sigma }} \sigma_{{1}}^{}$$ (17.38)

The following relation can be derived

$$\dot{\varepsilon}_{{1}}^{{\mathrm{p}}} \dot{{\lambda}}$$ (17.39)

With the relation derived previously, we find for the relation between the uniaxial plastic strain and the internal state variable

$$\dot{{\kappa}} \dot{{\varepsilon }}_{{1}}^{{\mathrm{p}}}$$ (17.40)

for both a strain hardening and a work hardening hypothesis.

17.1.2.2 Ambient Influence

DIANA can handle the influence of temperature, concentration (e.g. moisture content in concrete) or maturity on the Von Mises yield condition. For temperature dependency, the yield condition is given by

$$\boldsymbol\sigma \kappa \sqrt{{ 3 J_{2} }} {\frac{{ \bar{\sigma }( \kappa ) }}{{ \bar{\sigma }(0) }}} \sqrt{{ \tfrac{1}{2}\boldsymbol{\sigma}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}} {\frac{{ \bar{\sigma }( \kappa ) }}{{ \bar{\sigma }(0) }}}$$ (17.41)

with f (T) the temperature dependent tensile strength.