17.1.5 Rankine Principal Stress Model

The yield condition of Rankine is a maximum principal stress criterion and can be used to simulate cracking of brittle materials, such as concrete and rock. See also Feenstra [27]. The formulation of the Rankine yield condition reads

$$\boldsymbol\sigma \kappa_{{1}}^{} \sigma_{{1}}^{} \bar{{\sigma }}_{{1}}^{} \kappa_{{1}}^{} \sqrt{{ \tfrac{1}{2}{\boldsymbol{\sigma}}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}} {\tfrac{{1}}{{2}}} \boldsymbol\pi \boldsymbol\sigma \bar{{\sigma }}_{{1}}^{} \kappa_{{1}}^{}$$ (17.68)

with the projection matrix P given by

$$\left[\vphantom{ \negthickspace \begin{array}{cccccc} \tfrac{1}{2... ...& 0 & 0 & 0 \\ [1ex] 0 & 0 & 0 & 0 & 0 & 0 \end{array} \negthickspace }\right. \begin{array}{cccccc} \tfrac{1}{2}& -\tfrac{1}{2}& 0 & 0 & 0 & 0 ... ...& 0 \\ [1ex] 0 & 0 & 0 & 0 & 0 & 0 \\ [1ex] 0 & 0 & 0 & 0 & 0 & 0 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} \tfrac{1}{2... ...& 0 & 0 & 0 \\ [1ex] 0 & 0 & 0 & 0 & 0 & 0 \end{array} \negthickspace }\right]$$ (17.69)

and the projection vector $\boldsymbol\pi$₁ is given by

$$\boldsymbol\pi \left\{\vphantom{ \negthickspace \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \negthickspace }\right. \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \negthickspace }\right\}$$ (17.70)

The formulation shows the basic assumption that the yield condition is formulated for a plane stress situation only. For plane strain and axisymmetric stress situations, the Rankine yield condition is complemented with a tension cut-off criterion in the out-of-plane direction, which can be formulated as

$$\boldsymbol\sigma \kappa_{{3}}^{} \sigma_{{3}}^{} \bar{{\sigma }}_{{3}}^{} \kappa_{{3}}^{} \boldsymbol\pi \boldsymbol\sigma \bar{{\sigma }}_{{3}}^{} \kappa_{{3}}^{}$$ (17.71)

with the projection vector $\boldsymbol\pi$₃ given by

$$\boldsymbol\pi \left\{\vphantom{ \negthickspace \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \negthickspace }\right. \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \negthickspace }\right\}$$ (17.72)

The flow rule is in general given by the associated flow rule g₁ $\equiv$ f₁ and g₃ $\equiv$ f₃ . With the use of Koiter's rule [55], the plastic strain rate vector is determined by

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \dot{{\lambda}}_{{1}}^{} \left\{\vphantom{ \frac{ \mathbf{P} \boldsymbol{\sigma}}{ 2 \Psi } + \alpha_{1} \boldsymbol{\pi}_{1} }\right. {\frac{{ \mathbf{P} \boldsymbol{\sigma}}}{{ 2 \Psi }}} \alpha_{{1}}^{} \boldsymbol\pi \left.\vphantom{ \frac{ \mathbf{P} \boldsymbol{\sigma}}{ 2 \Psi } + \alpha_{1} \boldsymbol{\pi}_{1} }\right\} \dot{{\lambda}}_{{3}}^{} \boldsymbol\pi$$ (17.73)

where the scalar $\Psi$ is given by Equation (17.56).

17.1.5.1 Hardening

The relation between the internal state variables $\kappa_{{1}}^{}$ and $\kappa_{{3}}^{}$ , and the plastic process is given by the hardening hypothesis. For the Rankine 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

$$\begin{split}\dot{\kappa}_{1} &= \sqrt{ \dot{\varepsilon }_{1... ...^{\mathrm{p}} \dot{\varepsilon }_{3}^{\mathrm{p}} } \end{split}$$ (17.74)

which can be elaborated to

$$\begin{split}\dot{\kappa}_{1} &= \dot{\lambda}_{1} \\ [1ex] \dot{\kappa}_{3} &= \dot{\lambda}_{3} \end{split}$$ (17.75)

Work hardening.

For work hardening the basic assumption is

$$\begin{split}\dot{ W}_{1}^{\mathrm{p}} &= {\boldsymbol{\sigma... ...iv \bar{\sigma }_{3}( \kappa_{3} ) \dot{\kappa}_{3} \end{split}$$ (17.76)

which can be elaborated to

$$\begin{split}\dot{\kappa}_{1} &= \dot{\lambda}_{1} \\ [\smallskipamount] \dot{\kappa}_{3} &= \dot{\lambda}_{3} \end{split}$$ (17.77)

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

For the Rankine yield condition, the translation of uniaxial experimental data to the equivalent stress-internal state variable, the $\bar{{\sigma }}$ -$\kappa$ relation, does not depend of the hardening hypothesis as shown in the following example.

Figure 17.9: Derivation of hardening diagram for Rankine

Consider the uniaxial stress-strain diagram of Figure 17.9a. The plastic strain $\varepsilon_{{1}}^{{\mathrm{p}}}$ is assumed to be given by $\varepsilon_{{1}}^{}$ - $\varepsilon_{{1}}^{{\mathrm{e}}}$ . Figure 17.9b shows the uniaxial stress-plastic strain diagram. The uniaxial plastic strain rate is given by

$$\dot{{\varepsilon }}_{{1}}^{{\mathrm{p}}} \dot{{\lambda}}_{{1}}^{}$$ (17.78)

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

$$\bar{{\sigma }}_{{1}}^{} \sigma_{{1}}^{}$$ (17.79)

so the following relation can be derived

$$\dot{{\varepsilon }}_{{1}}^{{\mathrm{p}}} \dot{{\lambda}}_{{1}}^{}$$ (17.80)

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

$$\dot{{\kappa}}_{{1}}^{} \dot{{\varepsilon }}_{{1}}^{{\mathrm{p}}}$$ (17.81)

for both a strain hardening and a work hardening hypothesis. The same procedure holds for the derivation of the $\bar{{\sigma }}_{{3}}^{}$ - $\kappa_{{3}}^{}$ relation.

17.1.5.2 Tensile/Compression Combinations

The principal stress criterion of Rankine describes the tensile cracking of a material like concrete. However, the stress state in structures is often a biaxial stress state, i.e., a combination of tension and compression. These stress states can be modeled by a combination of the yield condition of Rankine to describe the tensile regime and another yield condition to describe the compressive regime. This combined yield condition is then treated as a multi-surface plasticity model which can be solved with stable algorithms. DIANA offers two such combinations: Rankine/Von Mises and Rankine/Drucker-Prager, described by Feenstra [27].

Rankine/Von Mises.

The biaxial stress state in a material can be modeled by a combination of the yield conditions of Rankine and Von Mises [§17.1.2]. The first to describe the tensile regime, the latter to describe the compressive regime. This combined yield surface is especially applicable in plane stress situations.

$$\begin{split}f_{1} ( \boldsymbol{\sigma}, \kappa_{1} ) &= \sq... ...ldsymbol{\sigma} - \bar{\sigma }_{3} ( \kappa_{3} ) \end{split}$$ (17.82)

Rankine/Drucker-Prager.

If the material properties depend on the pressure, the Rankine/Von Mises yield condition is not longer applicable. In these situations the yield condition of Rankine can be combined with the yield condition of Drucker-Prager [§17.1.4]. The combined yield condition is now given by

$$\begin{split}f_{1} ( \boldsymbol{\sigma}, \kappa_{1} ) &= \sq... ...ldsymbol{\sigma} - \bar{\sigma }_{3} ( \kappa_{3} ) \end{split}$$ (17.83)