17.1.1 Tresca

The yield condition of Tresca is a maximum shear stress condition which can be expressed in the principal stress space ( $\sigma_{{1}}^{}$ $\geq$ $\sigma_{{2}}^{}$ $\geq$ $\sigma_{{3}}^{}$ ) [Fig.17.2a]:

$$\boldsymbol\sigma \kappa \sigma_{{1}}^{} \sigma_{{3}}^{} \bar{{\sigma }} \kappa$$ (17.19)

with $\bar{{\sigma }}$($\kappa$) the uniaxial yield strength as a function of the internal state variable $\kappa$ .

Figure 17.2: Tresca and Von Mises yield condition (in $\pi$ -and rendulic plane)

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

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \dot{{\lambda}} \left\{\vphantom{ \negthickspace \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \negthickspace }\right. \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \negthickspace }\right\}$$ (17.20)

17.1.1.1 Hardening

The relation between the internal state variable $\kappa$ and the plastic process is given by the hardening hypothesis. For the Tresca 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

$$\dot{{\kappa}} \sqrt{{ \tfrac{2}{3}\left( \dot{\varepsilon }_{1}^{\mathrm{p}} \d... ...t{\varepsilon }_{3}^{\mathrm{p}} \dot{\varepsilon }_{3}^{\mathrm{p}} \right) }}$$ (17.21)

which can be elaborated to

$$\dot{{\kappa}} {\frac{{2}}{{\sqrt{3}}}} \dot{{\lambda}}$$ (17.22)

Work hardening.

For work hardening the basic assumption is

$$\dot{{W}}^{{\mathrm{p}}}_{} \boldsymbol\sigma \dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \equiv \bar{{\sigma }} \kappa \dot{{\kappa}}$$ (17.23)

which can be elaborated to

$$\dot{{\kappa}} \dot{{\lambda}}$$ (17.24)

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

For the Tresca 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 17.3.

Figure 17.3: Derivation of hardening diagram for Tresca

Consider the uniaxial stress-strain diagram of Figure 17.3a. The plastic strain $\varepsilon_{{1}}^{{\mathrm{p}}}$ is assumed to be given by $\varepsilon_{{1}}^{}$ - $\varepsilon_{{1}}^{{\mathrm{e}}}$ . Figure 17.3b shows the uniaxial stress-plastic strain diagram. For uniaxial stressing, ($\sigma_{{1}}^{}$,$\sigma_{{2}}^{}$,$\sigma_{{3}}^{}$) = ($\sigma_{{1}}^{}$, 0, 0) , plastic flow occurs at a vertex of the yield surface. Symmetry conditions dictate that the two possible yield directions contribute equally to the plastic strain rate vector

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \left\{\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilo... ...amount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right. \begin{array}{c} \dot{\varepsilon }_{1}^{\mathrm{p}} \\ [\smalls... ...thrm{p}} \\ [\smallskipamount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilon... ...mount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right\} \dot{{\lambda}} \left\{\vphantom{ \negthickspace \begin{array}{c} 1 \\ [\smallsk... ...{1}{2}\\ [\smallskipamount] - \tfrac{1}{2} \end{array} \negthickspace }\right. \begin{array}{c} 1 \\ [\smallskipamount] - \tfrac{1}{2}\\ [\smallskipamount] - \tfrac{1}{2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 1 \\ [\smallski... ...1}{2}\\ [\smallskipamount] - \tfrac{1}{2} \end{array} \negthickspace }\right\}$$ (17.25)

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

$$\dot{{\kappa}} \dot{{\varepsilon }}_{{1}}^{{\mathrm{p}}}$$ (17.26)

for both a strain hardening and a work hardening hypothesis. The relation between the uniaxial stress and the equivalent stress is simply given by

$$\bar{{\sigma }} \sigma_{{1}}^{}$$ (17.27)

17.1.1.2 Ambient Influence

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

$$\boldsymbol\sigma \kappa \sigma_{{1}}^{} \sigma_{{3}}^{} {\frac{{ \bar{\sigma }(\kappa) }}{{ \bar{\sigma }(0) }}}$$ (17.28)

with f (T) the temperature dependent tensile strength.