17.2.1 Hill

A relatively simple yield condition that can capture orthotropy in the strength properties has been proposed by Hill [38] as an extension of the Von Mises yield condition [Fig.17.19]:

$$\boldsymbol\sigma \kappa \sqrt{{ \tfrac{3}{2}\boldsymbol{\sigma}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}} \bar{{\sigma }}$$ (17.200)

with $\bar{{\sigma }}$ the reference yield strength.

Figure 17.19: Hill yield condition (in $\pi$ - and rendulic plane)

DIANA supports the Hill yield condition for ideal plasticity only. The projection matrix P is given by

$${\frac{{1}}{{3}}} \left[\vphantom{ \negthickspace \begin{array}{cccccc} \alpha_{12}... ...\ [0.5ex] 0 & 0 & 0 & 0 & 0 & 6\alpha_{66} \end{array} \negthickspace }\right. \begin{array}{cccccc} \alpha_{12} + \alpha_{13} & -\alpha_{12} & ... ...& 0 & 6\alpha_{55} & 0 \\ [0.5ex] 0 & 0 & 0 & 0 & 0 & 6\alpha_{66} \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} \alpha_{12}... ...\ [0.5ex] 0 & 0 & 0 & 0 & 0 & 6\alpha_{66} \end{array} \negthickspace }\right]$$ (17.201)

The parameters of the yield condition are determined from the yield strengths in the material axes which can be determined experimentally. The flow rule is generally given by the associated flow rule g $\equiv$ f , which results for the plastic strain rate vector in

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \dot{{\lambda}} {\frac{{3 \mathbf{P}\boldsymbol{\sigma}}}{{2\bar{\sigma }}}}$$ (17.202)

If the yield strengths in the x , y and z directions are given by $\sigma_{{\mathrm{y}.xx}}^{}$ , $\sigma_{{\mathrm{y}.yy}}^{}$ and $\sigma_{{\mathrm{y}.zz}}^{}$ respectively [Fig.17.20],

Figure 17.20: Orthotropic yield strengths

and the yield strength in shear by $\sigma_{{\mathrm{y}.xy}}^{}$ , $\sigma_{{\mathrm{y}.yz}}^{}$ and $\sigma_{{\mathrm{y}.zx}}^{}$ then the following relations can be given

$$\left[\vphantom{ \negthickspace \begin{array}{cccccc} 1 & 1 & 0 &... ... 0 & 3 & 0\\ [1.0ex] 0 & 0 & 0 & 0 & 0 & 3 \end{array} \negthickspace }\right. \begin{array}{cccccc} 1 & 1 & 0 & 0 & 0 & 0\\ [1.0ex] 1 & 0 & 1 ... ...0\\ [1.0ex] 0 & 0 & 0 & 0 & 3 & 0\\ [1.0ex] 0 & 0 & 0 & 0 & 0 & 3 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} 1 & 1 & 0 &... ... 0 & 3 & 0\\ [1.0ex] 0 & 0 & 0 & 0 & 0 & 3 \end{array} \negthickspace }\right] \left\{\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...[1.0ex] \alpha_{55} \\ [1.0ex] \alpha_{66} \end{array} \negthickspace }\right. \begin{array}{c} \alpha_{12} \\ [1.0ex] \alpha_{13} \\ [1.0ex] ... ...[1.0ex] \alpha_{44} \\ [1.0ex] \alpha_{55} \\ [1.0ex] \alpha_{66} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...1.0ex] \alpha_{55} \\ [1.0ex] \alpha_{66} \end{array} \negthickspace }\right\} \left\{\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{... ...{\mathrm{y}.zx}} \right)^{\negthickspace 2} \end{array} \negthickspace }\right. \begin{array}{c} 2\left( \dfrac{\bar{\sigma }}{\sigma _{\mathrm{y... ...{\bar{\sigma }}{\sigma _{\mathrm{y}.zx}} \right)^{\negthickspace 2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{\... ...\mathrm{y}.zx}} \right)^{\negthickspace 2} \end{array} \negthickspace }\right\}$$ (17.203)

which is easily solved resulting in the parameters

$$\left\{\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...[1.0ex] \alpha_{55} \\ [1.0ex] \alpha_{66} \end{array} \negthickspace }\right. \begin{array}{c} \alpha_{12} \\ [1.0ex] \alpha_{13} \\ [1.0ex] ... ...[1.0ex] \alpha_{44} \\ [1.0ex] \alpha_{55} \\ [1.0ex] \alpha_{66} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...1.0ex] \alpha_{55} \\ [1.0ex] \alpha_{66} \end{array} \negthickspace }\right\} \left[\vphantom{ \negthickspace \begin{array}{cccccc} \tfrac{1}{2... ...\ [1.0ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{3} \end{array} \negthickspace }\right. \begin{array}{cccccc} \tfrac{1}{2}& \tfrac{1}{2}& -\tfrac{1}{2}& ... ...0 & 0 & \tfrac{1}{3}& 0\\ [1.0ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{3} \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} \tfrac{1}{2... ...\ [1.0ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{3} \end{array} \negthickspace }\right] \left\{\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{... ...{\mathrm{y}.zx}} \right)^{\negthickspace 2} \end{array} \negthickspace }\right. \begin{array}{c} 2\left( \dfrac{\bar{{\sigma }}}{\sigma _{\mathrm... ...{\bar{\sigma }}{\sigma _{\mathrm{y}.zx}} \right)^{\negthickspace 2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{\... ...\mathrm{y}.zx}} \right)^{\negthickspace 2} \end{array} \negthickspace }\right\}$$ (17.204)

Application.

The major field of application for the Hill yield condition is in the analysis of thin metal sheets where the orthotropy is caused by the rolling direction of the metal. See also De Borst & Feenstra [21]. In shell or plane stress applications, it is not quite natural to provide for a yield stress in the out-of-plane direction. In these cases it is possible to provide a 45° off-axis yield strength [Fig.17.21].

Figure 17.21: Off-axis yield strength

The stresses in the material axes can be determined from the stress in the direction with an angle $\theta$ with the material axes by the standard transformation

$$\begin{split}\sigma _{xx} & = \sigma _{\theta} \cos^{2}\theta... ...ma _{xy} & = \sigma _{\theta} \sin\theta \cos\theta \end{split}$$ (17.205)

Substitution in the yield condition results in the following condition for the off-axis yield strength

$$\begin{split}(\alpha_{12} + \alpha_{13} ) \cos^{4}\theta + (\... ...gma _{\mathrm{y},\theta}}\right)^{\negthickspace 2} \end{split}$$ (17.206)

If the yield strengths are now given in the x , y and $\theta$ = 45° direction by $\sigma_{{\mathrm{y}.xx}}^{}$ , $\sigma_{{\mathrm{y}.yy}}^{}$ and $\sigma_{{\mathrm{y}.45}}^{}$ respectively, and the yield strength in shear by $\sigma_{{\mathrm{y}.xy}}^{}$ then the following relations can be given

$$\left[\vphantom{ \negthickspace \begin{array}{cccc} 1 & 1 & 0 & 0... ...}{4}& \tfrac{3}{2}\\ [1.0ex] 0 & 0 & 0 & 3 \end{array} \negthickspace }\right. \begin{array}{cccc} 1 & 1 & 0 & 0\\ [1.0ex] 1 & 0 & 1 & 0\\ [1.... ...& \tfrac{1}{4}& \tfrac{1}{4}& \tfrac{3}{2}\\ [1.0ex] 0 & 0 & 0 & 3 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccc} 1 & 1 & 0 & 0... ...}{4}& \tfrac{3}{2}\\ [1.0ex] 0 & 0 & 0 & 3 \end{array} \negthickspace }\right] \left\{\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...[1.0ex] \alpha_{23} \\ [1.0ex] \alpha_{44} \end{array} \negthickspace }\right. \begin{array}{c} \alpha_{12} \\ [1.0ex] \alpha_{13} \\ [1.0ex] \alpha_{23} \\ [1.0ex] \alpha_{44} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...1.0ex] \alpha_{23} \\ [1.0ex] \alpha_{44} \end{array} \negthickspace }\right\} \left\{\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{... ..._{\mathrm{y}.xy}}\right)^{\negthickspace 2} \end{array} \negthickspace }\right. \begin{array}{c} 2\left( \dfrac{\bar{\sigma }}{\sigma _{\mathrm{y... ...c{\bar{\sigma }}{\sigma _{\mathrm{y}.xy}}\right)^{\negthickspace 2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{\... ...{\mathrm{y}.xy}}\right)^{\negthickspace 2} \end{array} \negthickspace }\right\}$$ (17.207)

which is easily solved resulting in the parameters

$$\left\{\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...[1.0ex] \alpha_{23} \\ [1.0ex] \alpha_{44} \end{array} \negthickspace }\right. \begin{array}{c} \alpha_{12} \\ [1.0ex] \alpha_{13} \\ [1.0ex] \alpha_{23} \\ [1.0ex] \alpha_{44} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \alpha_{12} \\ ... ...1.0ex] \alpha_{23} \\ [1.0ex] \alpha_{44} \end{array} \negthickspace }\right\} \left[\vphantom{ \negthickspace \begin{array}{cccc} \tfrac{1}{2}&... ... 2 & -1\\ [1.0ex] 0 & 0 & 0 & \tfrac{1}{3} \end{array} \negthickspace }\right. \begin{array}{cccc} \tfrac{1}{2}& \tfrac{1}{2}& -2 & 1\\ [1.0ex]... ...rac{1}{2}& \tfrac{1}{2}& 2 & -1\\ [1.0ex] 0 & 0 & 0 & \tfrac{1}{3} \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccc} \tfrac{1}{2}&... ... 2 & -1\\ [1.0ex] 0 & 0 & 0 & \tfrac{1}{3} \end{array} \negthickspace }\right] \left\{\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{... ..._{\mathrm{y}.xy}}\right)^{\negthickspace 2} \end{array} \negthickspace }\right. \begin{array}{c} 2\left( \dfrac{\bar{\sigma }}{\sigma _{\mathrm{y... ...c{\bar{\sigma }}{\sigma _{\mathrm{y}.xy}}\right)^{\negthickspace 2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 2\left( \dfrac{\... ...{\mathrm{y}.xy}}\right)^{\negthickspace 2} \end{array} \negthickspace }\right\}$$ (17.208)