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| [[Rodney Hill]] has developed several yield criteria for anisotropic plastic deformations. The earliest version was a straightforward extension of the [[von Mises yield criterion]] and had a quadratic form. This model was later generalized by allowing for an exponent ''m''. Variations of these criteria are in wide use for metals, polymers, and certain composites.
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| == Quadratic Hill yield criterion ==
| | Selling advertising is most likely the easiest technique to make money, however most businesses won't advertise with your site until you have a involving quality visitors coming on the consistent essence. There is another way to sell advertising naturally is could would call the best make money easy belief.<br><br>There are many pros to day laboring. First off, you can figure to make over $20 one hour if you see the right employer. Furthermore, if you do a good job, working day laboring may cause full time employment. Also you can choose what days you need to work. Sounds pretty good right?<br><br>Well, actually, if had been tricky, you will find lot of part time work does not need lots of time and power how you'll do it. We only had to spare just a little amount of time, which happens to be it, we all paid. And a lot of it we locate on the online market place or method . online purpose.<br><br>Many people make the mistake of buying a market and signing on with product provides a very small audience or has hardly anything demand. So you have to find out if you have a strong demand for the product likely are looking at their.<br><br>However, day laboring are very few easy task. You need to do great with both hands. You also need to have correct safety equipment which includes steal toed boots while a helmet. Day laboring hard work that can also also be dangerous. Because there is usually no such thing as workers compensation when you day labor you are putting yourself at risk every particular date.<br><br>There is without a doubt something wanting to learn be in deep trouble that, pertaining to instance giving small service for anyone who are interested or 3 step a part time work. With something like part time work, we can get money, aside from my [http://Search.huffingtonpost.com/search?q=main+perform&s_it=header_form_v1 main perform]. Maybe you feel as if part time work is tiring, concerning are far more of an individual must do in in your free time work, immediately after which it the main job again.<br><br>So, a great deal more hear someone say they are fully aware how you may make $5 in 10 minutes, don't automatically think detrimentally. Ask them how! They may know that might just work anyone. They may even give you $5 for merely music playing. At the least, you'll have $5 that you didn't have before!<br><br>If you enjoyed this post and you would like to obtain additional facts pertaining to [http://come-fare-soldi-online.tumblr.com/ investire soldi] kindly browse through our own web page. |
| The quadratic Hill yield criterion<ref>R. Hill. (1948). ''A theory of the yielding and plastic flow of anisotropic metals.'' Proc. Roy. Soc. London, 193:281–297</ref> has the form
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| :<math>
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| F(\sigma_{22}-\sigma_{33})^2 + G(\sigma_{33}-\sigma_{11})^2 + H(\sigma_{11}-\sigma_{22})^2 + 2L\sigma_{23}^2 + 2M\sigma_{31}^2 + 2N\sigma_{12}^2 = 1 ~.
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| </math>
| |
| Here ''F, G, H, L, M, N'' are constants that have to be determined experimentally and <math>\sigma_{ij}</math> are the stresses. The quadratic Hill yield criterion depends only on the deviatoric stresses and is pressure independent. It predicts the same yield stress in tension and in compression.
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| | |
| === Expressions for F, G, H, L, M, N ===
| |
| If the axes of material anisotropy are assumed to be orthogonal, we can write
| |
| :<math>
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| (G + H)~(\sigma_1^y)^2 = 1 ~;~~ (F + H)~(\sigma_2^y)^2 = 1 ~;~~ (F + G)~(\sigma_3^y)^2 = 1
| |
| </math>
| |
| where <math>\sigma_1^y, \sigma_2^y, \sigma_3^y</math> are the normal yield stresses with respect to the axes of anisotropy. Therefore we have
| |
| :<math>
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| F = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_2^y)^2} + \cfrac{1}{(\sigma_3^y)^2} - \cfrac{1}{(\sigma_1^y)^2}\right]
| |
| </math>
| |
| :<math>
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| G = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_3^y)^2} + \cfrac{1}{(\sigma_1^y)^2} - \cfrac{1}{(\sigma_2^y)^2}\right]
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| </math>
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| :<math>
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| H = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_1^y)^2} + \cfrac{1}{(\sigma_2^y)^2} - \cfrac{1}{(\sigma_3^y)^2}\right]
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| </math>
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| Similarly, if <math>\tau_{12}^y, \tau_{23}^y, \tau_{31}^y</math> are the yield stresses in shear (with respect to the axes of anisotropy), we have
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| :<math>
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| L = \cfrac{1}{2~(\tau_{23}^y)^2} ~;~~ M = \cfrac{1}{2~(\tau_{31}^y)^2} ~;~~ N = \cfrac{1}{2~(\tau_{12}^y)^2}
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| </math>
| |
| | |
| === Quadratic Hill yield criterion for plane stress ===
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| The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as
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| :<math>
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| \sigma_1^2 + \cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\sigma_2^2 - \cfrac{2~R_0}{1+R_0}~\sigma_1\sigma_2 = (\sigma_1^y)^2
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| </math>
| |
| where the principal stresses <math>\sigma_1, \sigma_2</math> are assumed to be aligned with the axes of anisotropy with <math>\sigma_1</math> in the rolling direction and <math>\sigma_2</math> perpendicular to the rolling direction, <math>\sigma_3 = 0 </math>, <math>R_0</math> is the [[Lankford coefficient|R-value]] in the rolling direction, and <math>R_{90}</math> is the [[Lankford coefficient|R-value]] perpendicular to the rolling direction.
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| | |
| For the special case of transverse isotropy we have <math>R=R_0 = R_{90}</math> and we get
| |
| :<math>
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| \sigma_1^2 + \sigma_2^2 - \cfrac{2~R}{1+R}~\sigma_1\sigma_2 = (\sigma_1^y)^2
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| </math>
| |
| | |
| :{| class="toccolours jy
| |
| collapsible collapsed "width="80%" style="text-align:left"
| |
| !Derivation of Hill's criterion for plane stress
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| |-
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| | For the situation where the principal stresses are aligned with the directions of anisotropy we have
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| :<math>
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| f := F(\sigma_2-\sigma_3)^2 + G(\sigma_3-\sigma_1)^2 + H(\sigma_1-\sigma_2)^2 - 1 = 0 \,
| |
| </math>
| |
| where <math>\sigma_1, \sigma_2, \sigma_3</math> are the principal stresses. If we assume an associated flow rule we have
| |
| :<math>
| |
| \epsilon^p_i = \lambda~\cfrac{\partial f}{\partial \sigma_i} \qquad \implies \qquad
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| \cfrac{d\epsilon^p_i}{d\lambda} = \cfrac{\partial f}{\partial \sigma_i} ~.
| |
| </math>
| |
| This implies that
| |
| :<math>
| |
| \begin{align}
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| \cfrac{d\epsilon^p_1}{d\lambda} &= 2(G+H)\sigma_1 - 2H\sigma_2 - 2G\sigma_3 \\
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| \cfrac{d\epsilon^p_2}{d\lambda} &= 2(F+H)\sigma_2 - 2H\sigma_1 - 2F\sigma_3 \\
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| \cfrac{d\epsilon^p_3}{d\lambda} &= 2(F+G)\sigma_3 - 2G\sigma_1 - 2F\sigma_2 ~.
| |
| \end{align}
| |
| </math>
| |
| For plane stress <math>\sigma_3 = 0</math>, which gives
| |
| :<math>
| |
| \begin{align}
| |
| \cfrac{d\epsilon^p_1}{d\lambda} &= 2(G+H)\sigma_1 - 2H\sigma_2\\
| |
| \cfrac{d\epsilon^p_2}{d\lambda} &= 2(F+H)\sigma_2 - 2H\sigma_1\\
| |
| \cfrac{d\epsilon^p_3}{d\lambda} &= - 2G\sigma_1 - 2F\sigma_2 ~.
| |
| \end{align}
| |
| </math>
| |
| The [[Lankford coefficient|R-value]] <math>R_0</math> is defined as the ratio of the in-plane and out-of-plane plastic strains under uniaxial stress <math>\sigma_1</math>. The quantity <math>R_{90}</math> is the plastic strain ratio under uniaxial stress <math>\sigma_2</math>. Therefore, we have
| |
| :<math>
| |
| R_0 = \cfrac{d\epsilon^p_2}{d\epsilon^p_3} = \cfrac{H}{G} ~;~~
| |
| R_{90} = \cfrac{d\epsilon^p_1}{d\epsilon^p_3} = \cfrac{H}{F} ~.
| |
| </math>
| |
| Then, using <math>H=R_0 G</math> and <math>\sigma_3=0</math>, the yield condition can be written as
| |
| :<math>
| |
| f := F \sigma_2^2 + G \sigma_1^2 + R_0 G(\sigma_1-\sigma_2)^2 - 1 = 0 \,
| |
| </math>
| |
| which in turn may be expressed as
| |
| :<math>
| |
| \sigma_1^2 + \cfrac{F+R_0 G}{G(1+R_0)}~\sigma_2^2 - \cfrac{2R_0}{1+R_0}~\sigma_1\sigma_2 = \cfrac{1}{(1+R_0)G}~.
| |
| </math>
| |
| This is of the same form as the required expression. All we have to do is to express <math>F,G</math> in terms of <math>\sigma_1^y</math>. Recall that,
| |
| :<math>
| |
| \begin{align}
| |
| F & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_2^y)^2} + \cfrac{1}{(\sigma_3^y)^2} - \cfrac{1}{(\sigma_1^y)^2}
| |
| \right] \\
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| G & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_3^y)^2} + \cfrac{1}{(\sigma_1^y)^2} - \cfrac{1}{(\sigma_2^y)^2}
| |
| \right] \\
| |
| H & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_1^y)^2} + \cfrac{1}{(\sigma_2^y)^2} - \cfrac{1}{(\sigma_3^y)^2}
| |
| \right]
| |
| \end{align}
| |
| </math>
| |
| We can use these to obtain
| |
| :<math>
| |
| \begin{align}
| |
| R_0 = \cfrac{H}{G} & \implies
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| (1+R_0)\cfrac{1}{(\sigma_3^y)^2} - (1+R_0)\cfrac{1}{(\sigma_2^y)^2} = (1-R_0)\cfrac{1}{(\sigma_1^y)^2} \\
| |
| R_{90} = \cfrac{H}{F} & \implies
| |
| (1+R_{90})\cfrac{1}{(\sigma_3^y)^2} - (1-R_{90})\cfrac{1}{(\sigma_2^y)^2} = (1+R_{90})\cfrac{1}{(\sigma_1^y)^2}
| |
| \end{align}
| |
| </math>
| |
| Solving for <math>\cfrac{1}{(\sigma_3^y)^2}, \cfrac{1}{(\sigma_2^y)^2}</math> gives us
| |
| :<math>
| |
| \cfrac{1}{(\sigma_3^y)^2} = \cfrac{R_0+R_{90}}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2} ~;~~
| |
| \cfrac{1}{(\sigma_2^y)^2} = \cfrac{R_0(1+R_{90})}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2}
| |
| </math>
| |
| Plugging back into the expressions for <math>F,G</math> leads to
| |
| :<math>
| |
| F = \cfrac{R_0}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2} ~;~~
| |
| G = \cfrac{1}{1+R_0}~\cfrac{1}{(\sigma_1^y)^2}
| |
| </math>
| |
| which implies that | |
| :<math>
| |
| \cfrac{1}{G(1+R_0)} = (\sigma_1^y)^2 ~;~~ \cfrac{F+R_0 G}{G(1+R_0)} = \cfrac{R_0(1+R_{90})}{R_{90}(1+R_0)} ~.
| |
| </math>
| |
| Therefore the plane stress form of the quadratic Hill yield criterion can be expressed as
| |
| :<math>
| |
| \sigma_1^2 + \cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\sigma_2^2 - \cfrac{2~R_0}{1+R_0}~\sigma_1\sigma_2 = (\sigma_1^y)^2
| |
| </math>
| |
| |}
| |
| | |
| == Generalized Hill yield criterion ==
| |
| The generalized Hill yield criterion<ref>R. Hill. (1979). '' Theoretical plasticity of textured aggregates. '' Math. Proc. Camb. Phil. Soc., 85(1):179–191.</ref> has the form
| |
| :<math>
| |
| \begin{align}
| |
| F|\sigma_{2}-\sigma_{3}|^m & + G|\sigma_{3}-\sigma_{1}|^m + H|\sigma_{1}-\sigma_{2}|^m + L|2\sigma_1 - \sigma_2 - \sigma_3|^m \\
| |
| & + M|2\sigma_2 - \sigma_3 - \sigma_1|^m + N|2\sigma_3 - \sigma_1 - \sigma_2|^m = \sigma_y^m ~.
| |
| \end{align}
| |
| </math>
| |
| where <math>\sigma_i</math> are the principal stresses (which are aligned with the directions of anisotropy), <math>\sigma_y</math> is the yield stress, and ''F, G, H, L, M, N'' are constants. The value of ''m'' is determined by the degree of anisotropy of the material and must be greater than 1 to ensure convexity of the yield surface.
| |
| | |
| === Generalized Hill yield criterion for plane stress ===
| |
| For transversely isotropic materials with <math>1-2</math> being the plane of symmetry, the generalized Hill yield criterion reduces to (with <math>F=G</math> and <math>L=M</math>)
| |
| :<math>
| |
| \begin{align}
| |
| f := & F|\sigma_2-\sigma_3|^m + F|\sigma_3-\sigma_1|^m + H|\sigma_1-\sigma_2|^m + L|2\sigma_1 - \sigma_2 - \sigma_3|^m \\
| |
| & + L|2\sigma_2-\sigma_3-\sigma_1|^m + N|2\sigma_3-\sigma_1-\sigma_2|^m - \sigma_y^m \le 0
| |
| \end{align}
| |
| </math>
| |
| The [[R-value (plasticity)|R-value]] or [[Lankford coefficient]] can be determined by considering the situation where <math>\sigma_1 > (\sigma_2 = \sigma_3 = 0)</math>. The R-value is then given by
| |
| :<math>
| |
| R = \cfrac{(2^{m-1}+2) L - N + H}{(2^{m-1} - 1) L + 2 N + F} ~.
| |
| </math>
| |
| Under [[plane stress]] conditions and with some assumptions, the generalized Hill criterion can take several forms.<ref>Chu, E. (1995). ''Generalization of Hill's 1979 anisotropic yield criteria''. Journal of Materials Processing Technology, vol. 50, pp. 207-215.</ref>
| |
| * '''Case 1:''' <math>L = 0, H = 0.</math>
| |
| :<math>
| |
| f:= \cfrac{1+2R}{1+R}(|\sigma_1|^m + |\sigma_2|^m) - \cfrac{R}{1+R} |\sigma_1 + \sigma_2|^m - \sigma_y^m \le 0
| |
| </math>
| |
| * '''Case 2:''' <math>N = 0, F = 0.</math>
| |
| :<math>
| |
| f:= \cfrac{2^{m-1}(1-R)+(R+2)}{(1-2^{m-1})(1+R)}|\sigma_1 -\sigma_2|^m - \cfrac{1}{(1-2^{m-1})(1+R)} (|2\sigma_1 - \sigma_2|^m + |2\sigma_2-\sigma_1|^m)- \sigma_y^m \le 0
| |
| </math>
| |
| * '''Case 3:''' <math>N = 0, H = 0.</math>
| |
| :<math>
| |
| f:= \cfrac{2^{m-1}(1-R)+(R+2)}{(2+2^{m-1})(1+R)}(|\sigma_1|^m -|\sigma_2|^m) + \cfrac{R}{(2+2^{m-1})(1+R)} (|2\sigma_1 - \sigma_2|^m + |2\sigma_2-\sigma_1|^m)- \sigma_y^m \le 0
| |
| </math>
| |
| * '''Case 4:''' <math>L = 0, F = 0.</math>
| |
| :<math>
| |
| f:= \cfrac{1+2R}{2(1+R)}|\sigma_1 - \sigma_2|^m + \cfrac{1}{2(1+R)} |\sigma_1 + \sigma_2|^m - \sigma_y^m \le 0
| |
| </math>
| |
| * '''Case 5:''' <math>L = 0, N = 0.</math>. This is the [[Hosford yield criterion]].
| |
| :<math>
| |
| f := \cfrac{1}{1+R}(|\sigma_1|^m + |\sigma_2|^m) + \cfrac{R}{1+R}|\sigma_1-\sigma_2|^m - \sigma_y^m \le 0
| |
| </math>
| |
| : ''Care must be exercised in using these forms of the generalized Hill yield criterion because the yield surfaces become concave (sometimes even unbounded) for certain combinations of'' <math>R</math> and <math>m</math>.<ref>Zhu, Y., Dodd, B., Caddell, R. M. and Hosford, W. F. (1987). ''Limitations of Hill's 1979 anisotropic yield criterion.'' International Journal of Mechanical Sciences, vol. 29, pp. 733.</ref>
| |
| | |
| == Hill 1993 yield criterion ==
| |
| In 1993, Hill proposed another yield criterion <ref>Hill. R. (1993). ''User-friendly theory of orthotropic plasticity in sheet metals.'' International Journal of Mechanical Sciences, vol. 35, no. 1, pp. 19–25.</ref> for plane stress problems with planar anisotropy. The Hill93 criterion has the form
| |
| :<math>
| |
| \left(\cfrac{\sigma_6}{\sigma_0}\right)^2 + \left(\cfrac{\sigma_2}{\sigma_{90}}\right)^2 + \left[ (p + q - c) - \cfrac{p\sigma_1+q\sigma_2}{\sigma_b}\right]\left(\cfrac{\sigma_1\sigma_2}{\sigma_0\sigma_{90}}\right) = 1
| |
| </math>
| |
| where <math>\sigma_0</math> is the uniaxial tensile yield stress in the rolling direction, <math>\sigma_{90}</math> is the uniaxial tensile yield stress in the direction normal to the rolling direction, <math>\sigma_b</math> is the yield stress under uniform biaxial tension, and <math>c, p, q</math> are parameters defined as
| |
| :<math>
| |
| \begin{align}
| |
| c & = \cfrac{\sigma_0}{\sigma_{90}} + \cfrac{\sigma_{90}}{\sigma_0} - \cfrac{\sigma_0\sigma_{90}}{\sigma_b^2} \\
| |
| \left(\cfrac{1}{\sigma_0}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{\sigma_b}\right)~p & =
| |
| \cfrac{2 R_0 (\sigma_b-\sigma_{90})}{(1+R_0)\sigma_0^2} - \cfrac{2 R_{90} \sigma_b}{(1+R_{90})\sigma_{90}^2} + \cfrac{c}{\sigma_0} \\
| |
| \left(\cfrac{1}{\sigma_0}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{\sigma_b}\right)~q & =
| |
| \cfrac{2 R_{90} (\sigma_b-\sigma_{0})}{(1+R_{90})\sigma_{90}^2} - \cfrac{2 R_{0} \sigma_b}{(1+R_{0})\sigma_{0}^2} + \cfrac{c}{\sigma_{90}}
| |
| \end{align}
| |
| </math>
| |
| and <math>R_0</math> is the R-value for uniaxial tension in the rolling direction, and <math>R_{90}</math> is the R-value for uniaxial tension in the in-plane direction perpendicular to the rolling direction.
| |
| | |
| == Extensions of Hill's yield criteria ==
| |
| The original versions of Hill's yield criteria were designed for material that did not have pressure-dependent yield surfaces which are needed to model [[polymer]]s and [[foam]]s.
| |
| | |
| === The Caddell-Raghava-Atkins yield criterion ===
| |
| An extension that allows for pressure dependence is Caddell-Raghava-Atkins (CRA) model <ref>Caddell, R. M., Raghava, R. S. and Atkins, A. G., (1973), ''Yield criterion for anisotropic and pressure dependent solids such as oriented polymers.'' Journal of Materials Science, vol. 8, no. 11, pp. 1641-1646.</ref> which has the form
| |
| :<math>
| |
| F (\sigma_{22}-\sigma_{33})^2 + G (\sigma_{33}-\sigma_{11})^2 + H (\sigma_{11}-\sigma_{22})^2 + 2 L \sigma_{23}^2 + 2 M \sigma_{31}^2 + 2 N\sigma_{12}^2 + I \sigma_{11} + J \sigma_{22} + K \sigma_{33} = 1~.
| |
| </math>
| |
| | |
| === The Deshpande-Fleck-Ashby yield criterion ===
| |
| Another pressure-dependent extension of Hill's quadratic yield criterion which has a form similar to the [[Bresler Pister yield criterion]] is the Deshpande, Fleck and Ashby (DFA) yield criterion <ref>Deshpande, V. S., Fleck, N. A. and [[M. F. Ashby|Ashby, M. F.]] (2001). '' Effective properties of the octet-truss lattice material.'' Journal of the Mechanics and Physics of Solids, vol. 49, no. 8, pp. 1747-1769.</ref> for [[honeycomb structures]] (used in [[Sandwich structured composite|sandwich composite]] construction). This yield criterion has the form
| |
| :<math>
| |
| F (\sigma_{22}-\sigma_{33})^2 + G (\sigma_{33}-\sigma_{11})^2 + H (\sigma_{11}-\sigma_{22})^2 + 2 L \sigma_{23}^2 + 2 M \sigma_{31}^2 + 2 N\sigma_{12}^2 + K (\sigma_{11} + \sigma_{22} + \sigma_{33})^2 = 1~.
| |
| </math>
| |
| | |
| == References ==
| |
| <references/>
| |
| | |
| == External links ==
| |
| * [http://aluminium.matter.org.uk/content/html/eng/default.asp?catid=183&pageid=2144416653 Yield criteria for aluminum]
| |
| * [http://www.tecnun.es/Asignaturas/estcompmec/documentos/thinsheets.pdf Yield criteria for thin metal sheets]
| |
| | |
| {{DEFAULTSORT:Hill Yield Criteria}}
| |
| [[Category:Plasticity]]
| |
| [[Category:Solid mechanics]]
| |
| [[Category:Yield criteria]]
| |
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