Von Mises and Tresca Plasticity
Tresca
The general concept of plasticity is described in Plasticity.
The yield condition of Tresca is a maximum shear stress condition which can be expressed in the principal stress space ( \(\sigma _{1} \geq \sigma _{2} \geq \sigma _{3}\) ) [Figure 1, “Tresca and Von Mises yield condition (in π-and rendulic plane)”a]:
Equation 1.
\begin{equation} f( \boldsymbol{\sigma} , \kappa ) = | \sigma_{1} - \sigma_{3} | - \bar{\sigma}(\kappa) \end{equation}
with \(\bar {\sigma }(\kappa )\) the uniaxial yield strength as a function of the internal state variable κ.
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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
Equation 2.
\begin{equation} \dot{\boldsymbol{\varepsilon}}^{\mathrm{p}} = \dot{\lambda} \left\{ \negthickspace \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \negthickspace \right\} \end{equation}
Hardening
The relation between the internal state variable κ 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
Equation 3.
\begin{equation} \dot{\kappa} = \sqrt{ \tfrac{2}{3} \left( \dot{\varepsilon}_{1}^{\mathrm{p}} \dot{\varepsilon}_{1}^{\mathrm{p}} + \dot{\varepsilon}_{2}^{\mathrm{p}} \dot{\varepsilon}_{2}^{\mathrm{p}} + \dot{\varepsilon}_{3}^{\mathrm{p}} \dot{\varepsilon}_{3}^{\mathrm{p}} \right) } \end{equation}
which can be elaborated to
Equation 4.
\begin{equation} \dot{\kappa} = \frac{2}{\sqrt{3}} ~ \dot{\lambda} \end{equation}
Work hardening.
For work hardening the basic assumption is
Equation 5.
\begin{equation} \dot{W}^{\mathrm{p}} = \boldsymbol{\sigma}^{\mathrm{T}} \dot{\boldsymbol{\varepsilon}}^{\mathrm{p}} \equiv \bar{\sigma}( \kappa ) \dot{\kappa} \end{equation}
which can be elaborated to
Equation 6.
\begin{equation} \dot{\kappa} = \dot{\lambda} \end{equation}
Relation \(\bar {\sigma }\)-κ.
For the Tresca yield condition the translation of uniaxial experimental data to the equivalent stress-internal state variable, the \(\bar {\sigma }\)-κ relation, is independent upon the hardening hypothesis as shown in the example of Figure 2, “Derivation of hardening diagram for Tresca: (a) uniaxial stress-strain, (b) uniaxial stress-plastic strain, (c) strain-hardening, (d) work-hardening.”.
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Consider the uniaxial stress-strain diagram of Figure 2, “Derivation of hardening diagram for Tresca: (a) uniaxial stress-strain, (b) uniaxial stress-plastic strain, (c) strain-hardening, (d) work-hardening.”a. The plastic strain \(\varepsilon _{1}^{\mathrm {p}}\) is assumed to be given by \(\varepsilon _{1} - \varepsilon _{1}^{\mathrm {e}}\). Figure 2, “Derivation of hardening diagram for Tresca: (a) uniaxial stress-strain, (b) uniaxial stress-plastic strain, (c) strain-hardening, (d) work-hardening.”b 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
Equation 7.
\begin{equation} \dot{\boldsymbol{\varepsilon}}^{\mathrm{p}} = \left\{ \negthickspace \begin{array}{c} \dot{\varepsilon}_{1}^{\mathrm{p}} \\ \dot{\varepsilon}_{2}^{\mathrm{p}} \\ \dot{\varepsilon}_{3}^{\mathrm{p}} \end{array} \negthickspace \right\} = \dot{\lambda} \left\{ \negthickspace \begin{array}{c} 1 \\ - \tfrac{1}{2} \\ - \tfrac{1}{2} \end{array} \negthickspace \right\} \end{equation}
With the relation derived previously, we find for the relation between the uniaxial plastic strain and the internal state variable
Equation 8.
\begin{equation} \dot{\kappa} = \dot{\varepsilon}_{1}^{\mathrm{p}} \end{equation}
for both a strain hardening and a work hardening hypothesis. The relation between the uniaxial stress and the equivalent stress is simply given by
Equation 9.
\begin{equation} \bar{\sigma} = \sigma_{1} \end{equation}
Note that when the hardening is specified in terms of uniaxial stress-strain, the following conditions must be met. Otherwise it will result into fatal errors or unpredictable results:
The first point of the diagram (\(\varepsilon _{1}\),\(\sigma _{1}\)) refers to the initial yielding point. This is checked using the Young’s modulus: \( 0.999 \varepsilon _{1}E \le \sigma _{1} \le 1.001 \varepsilon _{1} E \)
The slope of each interval must be equal or smaller than the Young’s modulus E.
Internally, the uniaxial stress-strain diagram is converted to a uniaxial stress-plastic strain diagram. The conversion from the uniaxial strain to the uniaxial plastic strain is done based on \(\kappa = \varepsilon - \sigma /E\). Once the uniaxial stress-plastic strain hardening diagram is obtained the rest of the calculation is done based on the standard Tresca elasto-plastic model where the Young’s modulus E is used as usual.
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
Equation 10.
\begin{equation} f ( \boldsymbol{\sigma} , \kappa ) = | \sigma_{1} - \sigma_{3} | - f ( T ) \frac{ \bar{\sigma}(\kappa) }{ \bar{\sigma}(0) } \end{equation}
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 [Figure 1, “Tresca and Von Mises yield condition (in π-and rendulic plane)”b]. The yield function of Von Mises is given by the square root formulation
Equation 11.
\begin{equation} f( \boldsymbol{\sigma} , \boldsymbol{\eta}, \kappa ) = \sqrt{ 3 J_{2} } - \bar{\sigma}( \kappa ) = \sqrt{ \tfrac{1}{2} (\boldsymbol{\sigma}-\boldsymbol{\eta})^{\mathrm{T}} \mathbf{P} (\boldsymbol{\sigma}-\boldsymbol{\eta}) } - \bar{\sigma}( \kappa ) \end{equation}
where \(\bar {\sigma }(\kappa )\) is the uniaxial yield strength as a function of the internal state variable κ, and η is the back stress or the center of the yield circle on the π plane [Figure 1, “Tresca and Von Mises yield condition (in π-and rendulic plane)”b], which moves in the direction of the plastic flow if kinematic hardening effect is taken into account. The projection matrix P is given by
Equation 12.
\begin{equation} \label{eq:steel:plastic:projmatVM} \mathbf{P} = \left[ \negthickspace \begin{array}{cccccc} 2 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ -1 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace \right] \end{equation}
The flow rule is generally given by the associated flow rule \(g \equiv f\), which results for the plastic strain rate vector in
Equation 13.
\begin{equation} \dot{\boldsymbol{\varepsilon}}^{\mathrm{p}} = \dot{\lambda} ~ \frac{\mathbf{P}(\boldsymbol{\sigma}-\boldsymbol{\eta})}{2 \bar{\sigma}} \end{equation}
The evolution of back stress η is given as
Equation 14.
\begin{equation} \label{eq:steel:backstress} \dot{\boldsymbol{\eta}}=\frac{2}{3}(1-\gamma) ~ \frac{\partial\bar{\sigma}}{\partial\kappa} ~ \dot{\lambda} ~ \frac{\mathbf{P}(\boldsymbol{\sigma}-\boldsymbol{\eta})}{2 \bar{\sigma}} \end{equation}
where γ is a scalar parameter (\(0 \le \gamma \le 1\)). The parameter γ controls the contribution of the isotropic and kinematic hardening effect. If \(\gamma =1\) than only the isotropic hardening effect is taken into account i.e. \(\dot {\boldsymbol{\eta} }=0\). If \(\gamma =0\) then only the kinematic hardening effect is taken into account. For intermediate value of γ, e.g. \(\gamma =0.5\), mixed isotropic and kinematic hardening effects are considered. Diana also offers the possibility to specify a constant η i.e. the yield surface is moved by a constant stress shift. In this case η does not evolve according to equation Equation 14 50.34, but remains constant as defined by the user.
Hardening
The relation between the internal state variable κ 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
Equation 15.
\begin{equation} \dot{\kappa} = \sqrt{ \tfrac{2}{3} \left( \dot{\varepsilon}_{1}^{\mathrm{p}} \dot{\varepsilon}_{1}^{\mathrm{p}} + \dot{\varepsilon}_{2}^{\mathrm{p}} \dot{\varepsilon}_{2}^{\mathrm{p}} + \dot{\varepsilon}_{3}^{\mathrm{p}} \dot{\varepsilon}_{3}^{\mathrm{p}} \right) } \end{equation}
which can be elaborated to
Equation 16.
\begin{equation} \dot{\kappa} = \dot{\lambda} \end{equation}
Work hardening.
For work hardening the basic assumption is
Equation 17.
\begin{equation} \label{eq:steel:VM:wp} \dot{W}^{\mathrm{p}} = \boldsymbol{\sigma}^{\mathrm{T}}\dot{\boldsymbol{\varepsilon}}^{\mathrm{p}} \equiv \bar{\sigma}( \kappa ) \dot{\kappa} \end{equation}
with
Equation 18.
\begin{equation} \dot{\boldsymbol{\varepsilon}}^{\mathrm{p}} = \left\{ \negthickspace \begin{array}{c} \dot{\varepsilon}_{1}^{\mathrm{p}} \\ \dot{\varepsilon}_{2}^{\mathrm{p}} \\ \dot{\varepsilon}_{3}^{\mathrm{p}} \end{array} \negthickspace \right\} = \dot{\lambda} \frac{ 1 }{ 2 \bar{\sigma} } \left\{ \negthickspace \begin{array}{c} 2 \sigma_{1} - \sigma_{2} - \sigma_{3} \\ - \sigma_{1} + 2 \sigma_{2} - \sigma_{3} \\ - \sigma_{1} - \sigma_{2} + 2 \sigma_{3} \\ \end{array} \negthickspace \right\} \end{equation}
Equation 17 can be elaborated to
Equation 19.
\begin{equation} \dot{\kappa} = \dot{\lambda} \end{equation}
Relation \(\bar {\sigma }\)-κ.
For the Von Mises yield condition the translation of uniaxial experimental data to the equivalent stress-internal state variable, the \(\bar {\sigma }\)-κ relation, is independent upon the hardening hypothesis as shown in the example of Figure 3, “Derivation of hardening diagram for Von Mises: (a) uniaxial stress-strain, (b) uniaxial stress-plastic strain, (c) strain-hardening, (d) work-hardening.”.
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Consider the uniaxial stress-strain diagram of Figure 3, “Derivation of hardening diagram for Von Mises: (a) uniaxial stress-strain, (b) uniaxial stress-plastic strain, (c) strain-hardening, (d) work-hardening.”a. The plastic strain \(\varepsilon _{1}^{\mathrm {p}}\) is assumed to be given by \(\varepsilon _{1} - \varepsilon _{1}^{\mathrm {e}}\). Figure 3, “Derivation of hardening diagram for Von Mises: (a) uniaxial stress-strain, (b) uniaxial stress-plastic strain, (c) strain-hardening, (d) work-hardening.”b shows the uniaxial stress-plastic strain diagram. The uniaxial plastic strain rate is given by
Equation 20.
\begin{equation} \dot{\varepsilon}_{1}^{\mathrm{p}} = \dot{\lambda} ~ \frac{ \sigma_{1} }{ \bar{\sigma} } \end{equation}
The relation between the uniaxial stress and the equivalent stress is simply
Equation 21.
\begin{equation} \bar{\sigma} = \sigma_{1} \end{equation}
The following relation can be derived
Equation 22.
\begin{equation} \dot\varepsilon_{1}^{\mathrm{p}} = \dot{\lambda} \end{equation}
With the relation derived previously, we find for the relation between the uniaxial plastic strain and the internal state variable
Equation 23.
\begin{equation} \dot{\kappa} = \dot{\varepsilon}_{1}^{\mathrm{p}} \end{equation}
for both a strain hardening and a work hardening hypothesis.
Note that when the hardening is specified in terms of uniaxial stress-strain, the following conditions must be met. Otherwise it will result into fatal errors or unpredictable results:
The first point of the diagram (\(\varepsilon _{1}\),\(\sigma _{1}\)) refers to the initial yielding point. This is checked using the Young’s modulus: \( 0.999 \varepsilon _{1}E \le \sigma _{1} \le 1.001 \varepsilon _{1} E \)
The slope of each interval must be equal or smaller than the Young’s modulus E.
Internally, the uniaxial stress-strain diagram is converted to a uniaxial stress-plastic strain diagram. The conversion from the uniaxial strain to the uniaxial plastic strain is done based on \(\kappa = \varepsilon - \sigma /E\). Once the uniaxial stress-plastic strain hardening diagram is obtained the rest of the calculation is done based on the standard Von Mises elasto-plastic model where the Young’s modulus E is used as usual.
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
Equation 24.
\begin{equation} f( \boldsymbol{\sigma} , \boldsymbol{\eta}, \kappa ) = \sqrt{ 3 J_{2} } - f(T) \frac{ \bar{\sigma}( \kappa ) }{ \bar{\sigma}(0) } = \sqrt{ \tfrac{1}{2} (\boldsymbol{\sigma}-\boldsymbol{\eta})^{\mathrm{T}} \mathbf{P} (\boldsymbol{\sigma}-\boldsymbol{\eta})} - f( T ) \frac{ \bar{\sigma}( \kappa ) }{ \bar{\sigma}(0) } \end{equation}
with \(f(T)\) the temperature dependent tensile strength.
Batch syntax — Appendix Tresca or Von Mises Plasticity
Aspects
Table 1, “Aspects for Von Mises and Tresca plasticity” lists the availability of aspects for Von Mises and Tresca plasticity in Diana .
Aspect | Availability |
Thermal effects [Ambient and Time Dependency] | ✓ |
Heat flow [Heat Flow Properties] | ✓ |
Rayleigh damping [Viscous Damping] | ✓ |
Composite failure criteria [Composite Failure Criteria] | |
Wöhler diagram [Wöhler Diagrams] | |
Material safety factors [Material Safety Factors] | ✓ |
Additional dynamic surface mass [Distributed Mass] | ✓ |
Additional dynamic 3D line mass [Distributed Mass] | ✓ |