Abstract
It is shown that the fourth-order homogeneous polynomial yield function first considered by Gotoh (Int J Mech Sci 19:505–512, 1977) for sheet metals can be recast into a particular general form in intrinsic variables. It leads to a method of generating many specific representations of a fourth-order Hill’s 1979 yield function (in Math Proc Camb Philos Soc 85:179–191, 1979) generalized for off-axis loading. One may use the generalized fourth-order Hill’s yield function for modeling sheet metals of various degrees of planar anisotropy with ease. Furthermore, a sufficient condition on the positivity and convexity of a calibrated Hill’s yield function can also be verified straightforwardly. For six sheet metals whose nine material constants in a Gotoh’s yield function have been reported in the literature, their generalized Hill’s 1979 yield functions are obtained subsequently. The proposed sufficient condition on the positivity and convexity of Hill’s yield function is found to be satisfied by five of these six sheet metals. An approximate Hill’s 1979 yield function for the sixth sheet metal has also been identified, so the proposed sufficient condition on its positivity and convexity is also met. Generalizing fourth-order Hill’s 1979 yield functions may thus be used as an alternative approach for formulating a class of positive and convex yield functions with up to nine material constants for various orthotropic sheet metals.
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Abbreviations
- x, y, z :
-
The orthotropic material symmetry axes corresponding to the rolling (RD), transverse (TD) and normal (ND) directions of a thin sheet metal
- \(\sigma _x,\sigma _y,\tau _{xy}\) :
-
Three in-plane Cartesian (two normal and one shear) components of an applied Cauchy stress in the orthotropic coordinate system of the sheet metal
- \({\varPhi }_{2}, F,G,H,N\) :
-
Hill’s 1948 quadratic anisotropic yield function in Cartesian plane stress components (\(\sigma _x,\sigma _y,\tau _{xy}\)) and its four material constants
- \({\varPhi }_4, A_1,\ldots ,A_9\) :
-
Gotoh’s 1977 fourth-order plane stress anisotropic yield function in Cartesian stress components (\(\sigma _x,\sigma _y,\tau _{xy}\)) and its nine material constants
- \(\sigma _{\theta },r_{\theta },\sigma _b,r_b\) :
-
Yield stress and plastic strain ratio under uniaxial tension at the loading orientation angle \(\theta \) and under equal biaxial tension
- \(\sigma _1,\sigma _2,\theta \) :
-
The so-called intrinsic variables according to Hill [9, 10], namely the in-plane principal stresses \(\sigma _1\) and \(\sigma _2\) and the loading orientation angle \(\theta \) between \(\sigma _1\) and the RD of the sheet metal
- \(\phi _{m}, f,g,h,a,b,c,m\) :
-
Hill’s 1979 3D non-quadratic anisotropic yield function in three principal stresses and its seven material constants
- \(\phi _{h}, f(\theta ),g(\theta ),h(\theta ), a(\theta ),b(\theta ),c(\theta )\) :
-
Plane stress Hill’s 1979 non-quadratic yield function of even-order 2k in intrinsic variables (\(\sigma _1,\sigma _2,\theta \)) generalized for any off-axis loading and its six in-plane anisotropic material functions
- \(\phi _{4}, F(\theta ),G(\theta ),H(\theta ), B(\theta ),C(\theta )\) :
-
Gotoh’s yield function recast in intrinsic variables in terms of five homogeneous stress terms and its five in-plane anisotropic functions, whose Fourier series coefficients are \(F_0,\ldots ,F_4\), and so forth
- \(\phi _p, q_1(\theta ),\ldots ,q_5(\theta )\) :
-
The generalized fourth-order Hill’s 1979 yield function in intrinsic variables and its five in-plane anisotropic functions
- \((I_1,J_1),\ldots ,(I_5,J_5)\) :
-
Five irreducible and independent integer pairs used to define the newly proposed yield function formulation \(\phi _p\)
- \(\phi _{h1}, \phi _{h2}, f(\theta ),g(\theta ),h(\theta ), a(\theta ),b(\theta ),c(\theta )\) :
-
Two specific examples of the generalized fourth-order Hill’s 1979 yield function \(\phi _{h}\) and their in-plane anisotropic functions, whose Fourier series coefficients are \(f_0,\ldots ,f_4\), and so forth
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Tong, W. Generalized fourth-order Hill’s 1979 yield function for modeling sheet metals in plane stress. Acta Mech 227, 2719–2733 (2016). https://doi.org/10.1007/s00707-016-1659-5
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DOI: https://doi.org/10.1007/s00707-016-1659-5