# Influence of the Non-Schmid Effects on the Ductility Limit of Polycrystalline Sheet Metals

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## Abstract

**:**

## 1. Introduction

- ▪
- First, second, or fourth-order tensors are represented by bold-face letters and symbols (the order of which is indicated by the context).
- ▪
- Scalar parameters and variables are designated by thin letters and symbols.
- ▪
- Macroscopic (resp. microscopic) fields are designated by capital (resp. small) letters and symbols.
- $\dot{\u2022}$
- time derivative of $\u2022$.
- ${\u2022}^{\nabla}$
- co-rotational derivative of $\u2022$.
- ${\u2022}^{\mathrm{T}}$
- transpose of $\u2022$.
- $\u2022.\u2022$
- inner product.
- $\u2022:\u2022$
- double contraction product ($={\u2022}_{\mathrm{i}\mathrm{j}}{\u2022}_{\mathrm{i}\mathrm{j}}$ for the product between two second-order tensors, and
- ${\u2022}_{\mathrm{i}\mathrm{j}\mathrm{k}\mathrm{l}}{\u2022}_{\mathrm{k}\mathrm{l}}$
- for the product between a fourth-order tensor and a second-order tensor).
- $\u2022\times \u2022$
- vector product.
- $\mathrm{d}\mathrm{e}\mathrm{t}(\u2022)$
- determinant of tensor $\u2022$.
- $\mathrm{s}\mathrm{g}\mathrm{n}(\u2022)$
- sign of $\u2022$.

## 2. Theoretical Framework

- ▪
- ${\mathrm{N}}_{\mathrm{s}}$ is the total number of slip systems.
- ▪
- ${\dot{\gamma}}^{\alpha}$ is the absolute value of the slip rate of the αth slip system.
- ▪
- ${\mathbf{R}}^{\alpha}$ and ${\mathbf{S}}^{\alpha}$ are respectively the symmetric and skew-symmetric part of the Schmid orientation tensor, which is defined as the tensor product ${\overrightarrow{\mathbf{m}}}^{\alpha}\otimes {\overrightarrow{\mathbf{n}}}^{\alpha}$. Vectors ${\overrightarrow{\mathbf{m}}}^{\alpha}$ and ${\overrightarrow{\mathbf{n}}}^{\alpha}$, corresponding to BCC single crystals that we have used in the current work are listed in Appendix A.
- ▪
- ${\mathsf{\tau}}^{\alpha}$ is the resolved shear stress of the αth slip system, which is equal to ${\mathbf{R}}^{\alpha}:\mathsf{\sigma}$.

- ■
- ${\mathsf{\tau}}_{c}^{\alpha}$ is the critical shear stress of the αth slip system.
- ■
- $\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{n}\mathrm{s}}}{\mathrm{a}}_{\mathrm{i}}^{\alpha}{\mathsf{\tau}}_{\mathrm{i}}^{\alpha}$ is an additional term (compared to the classical Schmid law) used to capture the non-Schmid effects. Here, ${\mathsf{\tau}}_{\mathrm{i}}^{\alpha}$ and ${\mathrm{N}}_{\mathrm{n}\mathrm{s}}$ denote the non-Schmid shear stresses and their number, respectively. As to ${\mathrm{a}}_{\mathrm{i}}^{\alpha}$, they represent material parameters, which can be determined by experimental tests or atomistic simulations [28]. For simplicity, we assume in the current contribution that ${\mathrm{a}}_{\mathrm{i}}^{\alpha}$ are the same for all of the slip systems, and we choose the following expansion for the term $\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{n}\mathrm{s}}}{\mathrm{a}}_{\mathrm{i}}^{\alpha}{\mathsf{\tau}}_{\mathrm{i}}^{\alpha}$ [28]:$$\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{n}\mathrm{s}}}{\mathrm{a}}_{\mathrm{i}}^{\alpha}{\mathsf{\tau}}_{\mathrm{i}}^{\alpha}=}\text{\hspace{0.17em}}{\mathrm{a}}_{1}\mathsf{\sigma}:({\overrightarrow{\mathbf{m}}}^{\alpha}\otimes {\overrightarrow{\mathbf{n}}}_{1}^{\alpha})+{\mathrm{a}}_{2}\mathsf{\sigma}:[({\overrightarrow{\mathbf{n}}}^{\alpha}\times {\overrightarrow{\mathbf{m}}}^{\alpha})\otimes {\overrightarrow{\mathbf{n}}}^{\alpha}]+{\mathrm{a}}_{3}\mathsf{\sigma}:[({\overrightarrow{\mathbf{n}}}_{1}^{\alpha}\times {\overrightarrow{\mathbf{m}}}^{\alpha})\otimes {\overrightarrow{\mathbf{n}}}_{1}^{\alpha}]$$

_{(1)}:

_{(1)}and the definition of ${\mathsf{\tau}}^{*\text{\hspace{0.17em}}\alpha}$. Indeed, we have:

## 3. Algorithmic Aspects

- For each strain-path ratio $\mathsf{\rho}$ comprised between $-1/2$ and $1$ (with typical intervals of $0.1$).
- ■
- For each time increment ${\mathrm{I}}^{\Delta}=\left[{\mathrm{t}}_{0},{\mathrm{t}}_{0}+\Delta \mathrm{t}\right]$:
- ✓
- Compute the plane-stress tangent modulus ${\mathbf{L}}^{\mathrm{PS}}$ from the 3D tangent modulus $\mathbf{L}$ by using an iterative procedure similar to the one developed in [16]. On the other hand, $\mathbf{L}$ is determined from the microscopic tangent moduli $\mathbf{l}$ of the different single crystals by Equation (7). Some indications on the method used to compute $\mathbf{l}$ are given after this algorithm.
- ✓
- For $\mathsf{\theta}=0\xb0$ to $90\xb0$, at user-defined intervals (with typical increments of $1\xb0$):
- -
- compute the determinant of the acoustic tensor $\mathrm{det}(\overrightarrow{\U0001d4dd}.{\mathbf{L}}^{\mathrm{PS}}.\overrightarrow{\U0001d4dd})=0,$. We recall that the components of $\overrightarrow{\U0001d4dd}$ are equal to $(\mathrm{cos}\mathsf{\theta},\mathrm{sin}\mathsf{\theta})$.

- ✓
- Search for the orientation that minimizes $\mathrm{det}(\overrightarrow{\U0001d4dd}.{\mathbf{L}}^{\mathrm{PS}}.\overrightarrow{\U0001d4dd})=0,$ over the different values of θ. If $\mathrm{det}({\overrightarrow{\U0001d4dd}}_{\mathrm{min}}.{\mathbf{L}}^{\mathrm{PS}}.{\overrightarrow{\U0001d4dd}}_{\mathrm{min}})\le 0,$, then localized necking is reached. The corresponding angle θ is the orientation of the localization band, while the corresponding limit strain ${\mathrm{E}}_{11}$ is equal to ${\int}_{0}^{{\mathrm{t}}_{0}+\Delta \mathrm{t}}{\mathrm{G}}_{11}\mathrm{d}\mathrm{t}}={\mathrm{t}}_{0}+\Delta \mathrm{t$ (as ${\mathrm{G}}_{11}$ is equal to 1). The computation is then stopped. Otherwise, the integration is continued for the next time increment.

## 4. Numerical Results

#### 4.1. Material Data

#### 4.2. Significance of the Non-Schmid Effects on the Microscale Constitutive Response

#### 4.3. Influence of the Non-Schmid Effects on Localized Necking

_{11}. The results of this figure clearly show that the non-Schmid effects tend to precipitate the occurrence of localized necking. Indeed, the limit strains predicted when the non-Schmid effects are considered are lower than those determined by the classical Schmid law, by about 3% of deformation.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Slip Systems for the BCC Single Crystals Used in the Non-Schmid Crystal Plasticity Model

**Table A1.**List of slip systems for a BCC crystallographic structure [28].

$\alpha $ | 1 | 2 | 3 | 4 | 5 | 6 |

$\sqrt{3}\text{\hspace{0.17em}}{\overrightarrow{\mathbf{m}}}_{{}^{0}}^{\alpha}$ | $[1\overline{1}1]$ | $[\overline{1}\overline{1}1]$ | $[111]$ | $[\overline{1}11]$ | $[\overline{1}11]$ | $[\overline{1}\overline{1}1]$ |

$\sqrt{2}\text{\hspace{0.17em}}{\overrightarrow{\mathbf{n}}}_{{}^{0}}^{\alpha}$ | $[011]$ | $[011]$ | $[0\overline{1}1]$ | $[0\overline{1}1]$ | $[101]$ | $[101]$ |

$\sqrt{2}\text{\hspace{0.17em}}{\overrightarrow{\mathbf{n}}}_{{}^{10}}^{\alpha}$ | $[\overline{1}10]$ | $[0\overline{1}1]$ | $[10\overline{1}]$ | $[\overline{1}\overline{1}0]$ | $[101]$ | $[01\overline{1}]$ |

$\alpha $ | 7 | 8 | 9 | 10 | 11 | 12 |

$\sqrt{3}\text{\hspace{0.17em}}{\overrightarrow{\mathbf{m}}}_{{}^{0}}^{\alpha}$ | $[111]$ | $[1\overline{1}1]$ | $[\overline{1}11]$ | $[\overline{1}1\overline{1}]$ | $[111]$ | $[11\overline{1}]$ |

$\sqrt{2}\text{\hspace{0.17em}}{\overrightarrow{\mathbf{n}}}_{{}^{0}}^{\alpha}$ | $[\overline{1}01]$ | $[\overline{1}01]$ | $[110]$ | $[110]$ | $[\overline{1}10]$ | $[\overline{1}10]$ |

$\sqrt{2}\text{\hspace{0.17em}}{\overrightarrow{\mathbf{n}}}_{{}^{10}}^{\alpha}$ | $[1\overline{1}0]$ | $[011]$ | $[\overline{1}0\overline{1}]$ | $[110]$ | $[\overline{1}01]$ | $[0\overline{1}\overline{1}]$ |

## References

- Keeler, S.P.; Backofen, W.A. Plastic instability and fracture in sheets stretched over rigid punches. Trans. ASM
**1963**, 56, 25–48. [Google Scholar] - Goodwin, G.M. Application of strain analysis to sheet metal forming problems in press shop. Metall. Ital.
**1968**, 60, 767–774. [Google Scholar] - Marciniak, Z.; Kuczynski, K. Limit strains in processes of stretch-forming sheet metal. Int. J. Mech. Sci.
**1967**, 9, 609–620. [Google Scholar] [CrossRef] - Stören, S.; Rice, J.R. Localized necking in thin sheets. J. Mech. Phys. Solids
**1975**, 23, 421–441. [Google Scholar] [CrossRef] [Green Version] - Rice, J.R. The localization of plastic deformation. In Proceedings of the 14th International Congress of Theoretical and Applied Mechanics, Delft, The Netherlands, 30 August–4 September 1976; pp. 207–220. [Google Scholar]
- Petryk, H. On energy criteria of plastic instability. In Plastic Instability, Proc. Considere Memorial; Ecole Nationale des Ponts et Chaussees: Paris, France, 1985; pp. 215–226. [Google Scholar]
- Petryk, H.; Thermann, K. Post-critical plastic deformation of biaxially stretched sheets. Int. J. Solids Struct.
**1996**, 33, 689–705. [Google Scholar] [CrossRef] - Petryk, H.; Thermann, K. Post-critical plastic deformation in incrementally nonlinear materials. J. Mech. Phys. Solids
**2002**, 50, 925–954. [Google Scholar] [CrossRef] - Haddag, B.; Abed-Meraim, F.; Balan, T. Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage. Int. J. Plasticity
**2009**, 25, 1970–1996. [Google Scholar] [CrossRef] [Green Version] - Mansouri, L.Z.; Chalal, H.; Abed-Meraim, F. Ductility limit prediction using a GTN damage model coupled with localization bifurcation analysis. Mech. Mater.
**2014**, 76, 64–92. [Google Scholar] [CrossRef] - Kuroda, M.; Tvergaard, V. A phenomenological plasticity model with non-normality effects representing observations in crystal plasticity. J. Mech. Phys. Solids
**2004**, 49, 1239–1263. [Google Scholar] [CrossRef] - Ben Bettaieb, M.; Abed-Meraim, F. Investigation of localized necking in substrate-supported metal layers: Comparison of bifurcation and imperfection analyses. Int. J. Plasticity
**2016**, 65, 168–190. [Google Scholar] [CrossRef] - Signorelli, J.W.; Bertinetti, M.A.; Turner, P.A. Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model. Int. J. Plasticity
**2009**, 25, 1–25. [Google Scholar] [CrossRef] - Schwindt, C.; Schlosser, F.; Bertinetti, M.A.; Signorelli, J.W. Experimental and Visco-Plastic Self-consistent evaluation of forming limit diagrams for anisotropic sheet metals: An efficient and robust implementation of the M-K model. Int. J. Plasticity
**2015**, 73, 62–99. [Google Scholar] [CrossRef] - Franz, G.; Abed-Meraim, F.; Berveiller, M. Strain localization analysis for single crystals and polycrystals: Towards microstructure-ductility linkage. Int. J. Plasticity
**2013**, 48, 1–33. [Google Scholar] [CrossRef] [Green Version] - Akpama, H.K.; Ben Bettaieb, M.; Abed-Meraim, F. Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: Bifurcation analysis versus imperfection approach. Int. J. Plasticity
**2017**, 91, 205–237. [Google Scholar] [CrossRef] [Green Version] - Miehe, C.; Schröder, J.; Schotte, J. Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials. Comput. Method Appl. Mech. Eng.
**1999**, 171, 387–418. [Google Scholar] [CrossRef] - Watanabe, I.; Setoyama, D.; Nagasako, N.; Iwata, N.; Nakanishi, K. Multiscale prediction of mechanical behavior of Ferrite-Pearlite steel with Numerical Material Testing. Int. J. Numer. Meth. Eng.
**2012**, 89, 829–845. [Google Scholar] [CrossRef] - Schmid, E.; Boas, W. Plasticity of Crystals; Chapman and Hall: London, UK, 1935. [Google Scholar]
- Akpama, H.K.; Ben Bettaieb, M.; Abed-Meraim, F. Influence of the yield surface curvature on the forming limit diagrams predicted by crystal plasticity theory. Lat. Am. J. Solids Struct.
**2016**, 13, 1250–1269. [Google Scholar] [CrossRef] - Bassani, J.; Ito, K.; Vitek, V. Complex macroscopic plastic flow arising from non-planar dislocation core structures. Mater. Sci. Eng. A
**2001**, 319–321, 97–101. [Google Scholar] [CrossRef] - Gröger, R.; Vitek, V. Multiscale modeling of plastic deformation of molybdenum and tungsten. III. Effects of temperature and plastic strain rate. Acta Mater.
**2008**, 56, 5426–5439. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Beyerlein, I. An atomistically-informed dislocation dynamics model for the plastic anisotropy and tension-compression asymmetry of BCC metals. Int. J. Plasticity
**2011**, 27, 1471–1484. [Google Scholar] [CrossRef] - Mapar, A.; Ghassemi-Armaki, H.; Pourboghrat, F.; Kumar, K.S. A differential-exponential hardening law for non-Schmid crystal plasticity finite element modeling of ferrite single crystals. Int. J. Plasticity
**2017**, 91, 268–299. [Google Scholar] [CrossRef] - Chen, P.; Ghassemi-Armaki, H.; Kumar, S.; Bower, A.; Bhat, S.; Sadagopan, S. Microscale-calibrated modeling of the deformation response of dual-phase steels. Acta Mater.
**2014**, 65, 133–149. [Google Scholar] [CrossRef] - Srivastava, A.; Ghassemi-Armaki, H.; Sung, H.; Chen, P.; Kumar, S.; Bower, A.F. Micromechanics of plastic deformation and phase transformation in a three-phase TRIP-assisted advanced high strength steel: Experiments and modeling. J. Mech. Phys. Solids
**2015**, 78, 46–69. [Google Scholar] [CrossRef] [Green Version] - Qin, Q.; Bassani, J.-L. Non-associated plastic flow in single crystals. J. Mech. Phys. Solids
**1992**, 40, 835–862. [Google Scholar] [CrossRef] - Gröger, R.; Bailey, A.; Vitek, V. Multiscale modeling of plastic deformation of molybdenum and tungsten: I. Atomistic studies of the core structure and glide of 1/2111 screw dislocations at 0K. Acta Mater.
**2008**, 56, 5401–5411. [Google Scholar] [CrossRef] - Qin, Q.; Bassani, J.-L. Non-Schmid yield behavior in single crystals. J. Mech. Phys. Solids
**1992**, 40, 813–833. [Google Scholar] [CrossRef] - Koester, A.; Ma, A.; Hartmaier, A. Atomistically informed crystal plasticity model for body-centered cubic iron. Acta Mater.
**2012**, 60, 3894–3901. [Google Scholar] [CrossRef] - Kuroda, M. A phenomenological plasticity model accounting for hydrostatic stress-sensitivity and vertex-type of effect. Mech. Mater.
**2004**, 36, 285–297. [Google Scholar] [CrossRef] - Akpama, H.K.; Ben Bettaieb, M.; Abed-Meraim, F. Numerical integration of rate-independent BCC single crystal plasticity models: Comparative study of two classes of numerical algorithms. Int. J. Numer. Meth. Eng.
**2016**, 108, 363–422. [Google Scholar] [CrossRef] - Chang, Y.W.; Asaro, R.J. An experimental study of shear localization in aluminum-copper single crystals. Acta Metall.
**1981**, 29, 241–257. [Google Scholar] [CrossRef] - Barlat, F. Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals. Mater. Sci. Eng.
**1987**, 91, 55–72. [Google Scholar] [CrossRef]

**Figure 2.**Significance of the non-Schmid effects on the slip system activity: (

**a**) Results without non-Schmid effects (${\mathrm{a}}_{1}={\mathrm{a}}_{2}={\mathrm{a}}_{3}=0$); (

**b**) Results with non-Schmid effects (${\mathrm{a}}_{1}=0.2;{\mathrm{a}}_{2}={\mathrm{a}}_{3}=0$).

**Figure 3.**Impact of the non-Schmid effects on the evolution of the in-plane components of the Cauchy stress tensor: (

**a**) Results without non-Schmid effects (${\mathrm{a}}_{1}={\mathrm{a}}_{2}={\mathrm{a}}_{3}=0$); (

**b**) Results with non-Schmid effects (${\mathrm{a}}_{1}=0.2;{\mathrm{a}}_{2}={\mathrm{a}}_{3}=0$).

**Figure 4.**Significance of the non-Schmid effects on the evolution of some representative components of the macroscopic tangent modulus for the plane-strain state: (

**a**) Results without non-Schmid effects (${\mathrm{a}}_{1}={\mathrm{a}}_{2}={\mathrm{a}}_{3}=0$); (

**b**) Results with non-Schmid effects (${\mathrm{a}}_{1}=0.2;{\mathrm{a}}_{2}={\mathrm{a}}_{3}=0$).

**Figure 5.**Impact of the non-Schmid effects on the evolution of the minimum of the determinant of the acoustic tensor as a function of E

_{11}: (

**a**) Uniaxial tensile state ($\mathsf{\rho}=-0.5$); (

**b**) Plane-strain state $(\mathsf{\rho}=0)$.

**Figure 6.**Impact of the non-Schmid effects on: (

**a**) the shape and the level of the forming limit diagrams; (

**b**) the evolution of the necking band orientation as a function of the strain-path ratio.

**Figure 7.**(

**a**) Initial crystallographic texture ({111} pole figure); (

**b**) Predicted forming limit diagrams.

Elasticity | Hardening | ||||
---|---|---|---|---|---|

E | ν | $\mathrm{n}$ | ${\mathsf{\tau}}_{0}$ | $\mathrm{q}$ | ${\mathrm{h}}_{0}$ |

65 GPa | 0.3 | 0.15 | 40 MPa | 1.4 | 390 MPa |

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**MDPI and ACS Style**

Ben Bettaieb, M.; Abed-Meraim, F.
Influence of the Non-Schmid Effects on the Ductility Limit of Polycrystalline Sheet Metals. *Materials* **2018**, *11*, 1386.
https://doi.org/10.3390/ma11081386

**AMA Style**

Ben Bettaieb M, Abed-Meraim F.
Influence of the Non-Schmid Effects on the Ductility Limit of Polycrystalline Sheet Metals. *Materials*. 2018; 11(8):1386.
https://doi.org/10.3390/ma11081386

**Chicago/Turabian Style**

Ben Bettaieb, Mohamed, and Farid Abed-Meraim.
2018. "Influence of the Non-Schmid Effects on the Ductility Limit of Polycrystalline Sheet Metals" *Materials* 11, no. 8: 1386.
https://doi.org/10.3390/ma11081386