# A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes

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## Abstract

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## 1. Introduction

- What is an enhanced continuum mechanics model?
- Why such models are required for the simulation of manufacturing operations?
- Which enhanced models of continuum mechanics can be used to simulate manufacturing operations?
- Which contributions can be found in literature that already used enhanced model to simulate manufacturing operations?

## 2. Challenges in Manufacturing Processes Simulations

- Strain localization in a length scale whose order of magnitude is the same as grain size, therefore violating the limit of validity of material homogeneity;
- Mesh-size hypersensitivity when introducing material softening in the behavior law during the analysis;
- Proper material characterization at high temperatures and high strain rates developing during manufacturing processes;
- The complex material behavior at the tool-workpiece contact location.

#### 2.1. Strain Localization and Mesh-Size Dependency

#### 2.2. Material Characterization at High Temperatures and High Strain Rates

#### 2.3. Tool-Workpiece Contact

## 3. Historical Excursus of SGPTs in Literature

#### 3.1. Aifantis’ Theory

#### 3.2. Gradient of the Local Spin Vector—Fleck and Hutchinson 1993

#### 3.3. Second Gradient of Displacement—Fleck and Hutchinson 1997

#### 3.4. Gradient of the Cumulative Plastic Strain—Fleck and Hutchinson 2001

#### 3.5. Irrotational Plastic Flow and Burgers Tensor—Gurtin and Anand 2005

#### 3.6. The Common Framework—Gudmundson 2004

#### 3.7. Dislocations-Enriched SGT

#### 3.8. Gradient of Micro-Structure Rotation—Cosserat Media

#### 3.9. Gradient of Micro-Structure Deformation—Micromorphic Media

## 4. Reported Applications of SGPTs in Manufacturing Processes Simulation

#### 4.1. Scalar SGPT Applied to Flat Punch Molding

#### 4.2. SGPT Used to Model Rolling at Small Scale

#### 4.3. MSGT Applied to Orthogonal Cutting

^{®}Implicit through a UMAT FORTRAN subroutine, and they used this model to study the effect of the radius of the cutting tool during orthogonal micro-cutting [8], and to verify the increments in material hardening and cutting energy by using the SGPT [116]. In Figure 10a a comparison is reported from the work of Liu et al. in terms of Specific Cutting energy while using different tool radii. In Figure 10b instead, the effect of using the SGPT is reported again in terms of Specific Cutting Energy. They used a sticking-sliding friction model as the one developed by Zorev [70]. The thermal fields during simulations have been solved through the classical heat equation. Heat was assumed to generate at the tool-workpiece contact location due to friction and at the plastic zones due to plastic work; the Taylor–Quinney constant was set to 0.9 (90% of the plastic work would transform into heat) [119]. An adaptive remeshing algorithm was used to obtain chip separation. The minimum mesh size they used measured 0.06 $\mathsf{\mu}$m in the cutting zone, with a chip thickness of 0.5 $\mathsf{\mu}$m.

^{®}Explicit VUMAT subroutine to implement the model. In Figure 11 the difference between using this variety of MSGT and the classical continuum mechanics description is reported for different cutting speeds.

#### 4.4. Micromorphic Media Applied to Forming

^{®}Explicit, through a VUEL subroutine. The introduction of an additional degree of freedom implies a modification of the system of PDEs to be solved, thereby compelling the modification of the Finite Element description rather than an higher-level modification of the material module.

## 5. Conclusions

- The forces/moments used to shear/shape the continuum localize in areas that are comparable to the grain size of the metal; in this case a further hardening due to the dislocations movement must be properly captured by adopting a SGPT;
- The velocity with which the continuum is deformed might induce material to behave viscously, and this must be covered by a proper design of the material behavior;
- The temperature developed during the process might be relatively high, and this imposes to considerate the thermal behavior (often adiabatic) that are taken into account by the definition of a thermodynamically-consistent description of the continuum;
- Implementation-wise if the plastic deformation is expected to occur at the boundaries, a higher degree theory is favorable.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Notation

## Appendix B. Methodology Used to Prepare the Review

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**Figure 1.**Adiabatic Shear Band formation during orthogonal cutting Titanium alloy [12].

**Figure 2.**Shear bands produced during FSW of the Aluminum alloy AA2050 with an advancing side on the left.

**Figure 3.**Variation of Torsional stiffness during torsion test on a copper wire (

**a**) and variation of Hardness during microindentation test on tungsten single crystal (

**b**) [20].

**Figure 4.**Variation of the normalized bending moment on thin Nickle foils during micro-bending test [21].

**Figure 5.**Mesh hypersensitivity demonstrated through compression simulation on a Hat-Specimen [65]. Distribution of equivalent plastic strain obtained for different mesh sizes using the Classical CM (

**a**). Distribution of equivalent plastic strain in case a Generalized CM (Cosserat Medium) is used (

**b**). Boundary conditions and geometry of the axial-symmetric cross section of the Hat-Specimen used for simulation (

**c**). The red line in (

**c**) indicates the location where the equivalent plastic strain was sampled.

**Figure 6.**Measured Temperature field in a torsional Kolsky test at a strain rate of 1600 s${}^{-1}$ [66].

**Figure 7.**(

**a**) Correlation between the measured temperature field and the stress-strain curve in a shear test performed at a strain rate of 500 s

^{−1}. (

**b**) Difference between the temperature fields measured during two shear tests at different strain rates [67].

**Figure 8.**Temperature distribution in the chip during orthogonal cutting on AISI 1045 steel obtained with a camera in circumferential (

**a**) and axial (

**b**) directions [69].

**Figure 10.**Predicted effect of the tool radius on the Specific Cutting Energy Variation during orthogonal cutting test (

**a**), and effect of the adoption of a SGT on the predicted Specific Cutting Energy (

**b**) [8].

**Figure 11.**Predicted Specific Cutting Energy Variation during orthogonal cutting test for different cutting speed with and without SGT [122].

Name | # d.o.f. | # Charact. Lengths | Additional Feature | Higher Order/Higher Grade |
---|---|---|---|---|

Aifantis [36,79,80] | 3 | 1/2 | ${\sigma}^{Y}\propto {\nabla}^{2}{\epsilon}_{eq}^{p}$ | Higher Grade |

Fleck1993 [85] | 3 | 2 | ${p}^{\left(i\right)}\propto \mathrm{skew}\left(\dot{\underline{\mathbf{u}}}\otimes \underline{\mathbf{\nabla}}\right)$ | Higher Grade |

Fleck1997 [20] | 3 | 6 | ${p}^{\left(i\right)}\propto \underline{\mathbf{u}}\otimes \underline{\mathbf{\nabla}}\otimes \underline{\mathbf{\nabla}}$ | Higher Grade |

Fleck2001 [42] | 3 | 1 | ${p}^{\left(i\right)}\propto {\mathsf{\epsilon}}^{p}\otimes \underline{\mathbf{\nabla}}$ | Higher Grade |

Gurtin [89] | 3 | 2 | ${p}^{\left(i\right)}\propto ({\underset{\sim}{\mathsf{\epsilon}}}^{p}\otimes \underline{\mathbf{\nabla}},\underset{\sim}{\mathsf{\theta}})$ | Higher Grade |

Gudmundson [81] | 3 | 6 | ${p}^{\left(i\right)}\propto {\mathsf{\epsilon}}^{p}\otimes \underline{\mathbf{\nabla}}$ | Higher Grade |

Gao [90,91] | 3 | 1 | ${\sigma}^{Y}\propto \underline{\mathbf{u}}\otimes \underline{\mathbf{\nabla}}\otimes \underline{\mathbf{\nabla}}$ | Higher Grade |

Cosserat [99] | 6 | 3 | ${p}^{\left(i\right)}\propto (\underset{\sim}{\mathsf{\theta}},\mathsf{\Gamma})$ | Higher Order |

Micromorphic [103] | 12 | 2 | ${p}^{\left(i\right)}\propto (\underset{\sim}{\mathsf{\varphi}},\underset{\sim}{\mathsf{\varphi}}\otimes \underline{\nabla})$ | Higher Order |

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**MDPI and ACS Style**

Russo, R.; Girot Mata, F.A.; Forest, S.; Jacquin, D.
A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes. *J. Manuf. Mater. Process.* **2020**, *4*, 87.
https://doi.org/10.3390/jmmp4030087

**AMA Style**

Russo R, Girot Mata FA, Forest S, Jacquin D.
A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes. *Journal of Manufacturing and Materials Processing*. 2020; 4(3):87.
https://doi.org/10.3390/jmmp4030087

**Chicago/Turabian Style**

Russo, Raffaele, Franck Andrés Girot Mata, Samuel Forest, and Dimitri Jacquin.
2020. "A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes" *Journal of Manufacturing and Materials Processing* 4, no. 3: 87.
https://doi.org/10.3390/jmmp4030087