# Theory, Method and Practice of Metal Deformation Instability: A Review

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definition of Deformation Instability

**m**is normal vector of the yield surface and h is the hardening rate.

- (1)
- Hart’s instability criterion [60]

- (2)
- Jona’s instability criterion [61]

- (3)
- Semiatin’s instability criterion [62]

- (4)
- Dynamic Material Model (DMM) criterion [63]

^{−1}. Zheng et al. [65] studied the high-temperature formability of high-strength Mg alloys through tensile experiments. The forming diagram of the Mg alloy was designed based on DMM, and it was found that, with the decrease of the strain rate and the increase of deformation temperature, the fracture instability patterns exhibited by the tensile specimens vary from quasi-cleavage to ductile fractures. For aluminum alloys, the tensile specimens’ fracture process is from brittle to ductile [66]. Liu et al. [67] researched the formability of cast steel by forming a graph method of DMM; it was found that, with the increase of temperature and the decrease of strain rate, the degree of dynamic recrystallization increases, and deformation instability is prone to occur.

- (5)
- Gegel’s and Alexander’s instability criterion [68]

- (6)
- Metallurgical instability criterion [69]

## 3. Deformation Instability Induced by Characteristics of Material

#### 3.1. Deformation Instability in Superplastics of Materials

- (1)
- Load instability criterion are as follows:

- (2)
- The geometric instability criterion is as follows:

#### 3.2. Deformation Instability in Hot Forming Process

^{−1}), flow instability occurs in both thermal and warm deformation regions. At low strain rates (0.01 s

^{−1}), flow instability occurs only in lower temperature ranges (700~900 °C). The reason for this is that low strain rates can cause microcracks or micro-void nucleation, leading to instability, while high strain rates can result in the formation of adiabatic shear bands within high temperature regions, ultimately causing shear instability within the materials.

^{−1}. The TC4 alloy is unstable in a range of 800~850 °C and 0.01~1.0 s

^{−1}. Macroscopic fracture morphology is shown in Figure 8b; it was found that, with temperature increases, the fractures have a characteristic that transitions from a quasi-cleavage plane to a ductile fracture. This is because the temperature increase leads to the homogenization of internal tissues and stress relaxation [102]. Sun et al. [103] analyzed the deformation instability characteristics of a CrMnFeNi high-entropy alloy cast and forged under high temperature tensile conditions from the perspective of physical mechanisms, as show in Figure 8c. It was found that the fracture instability of as-cast parts shifted from ductile fracture to brittle fracture as the parts reached moderate temperatures. At room temperature for the forged parts, the physical mechanism of instability was attributed to slip and twinning, and dynamic instability occurred when the temperature reached 950 °C, causing immediate softening after yielding.

## 4. Deformation Instability Induced by the Structural Geometry of Materials

#### 4.1. Deformation Instability of Sheet Metal

#### 4.2. Deformation Instability of Tubes

#### 4.3. Deformation Instability of Beams

## 5. Analytical Methods of Deformation Instability

#### 5.1. Theory Analysis

#### 5.2. FE Simulation and Experiment

## 6. Engineering Applications of Deformation Instability

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Schematic diagram of Storen and Rice model [48]: (

**a**) plastic flow theory with ideal yield surface; (

**b**) plastic flow theory with singular point.

**Figure 5.**Superplastic deformation behavior of the Mg alloy: (

**a**–

**f**) are the real stress–strain curves at different temperatures and different deformation rates; (

**g**) is the superplastic behavior of the actual tensile specimen [78].

**Figure 7.**Hot processing diagrams of different materials during compression: (

**a**) the gray area is the deformation instability area of the Ti-6Al-4V titanium alloy [94]; (

**b**) the dark blue area is the deformation instability area of the dual-phase Mg-9Li-3Al alloy [96]; (

**c**) ★ indicates the deformation instability point of stabilized micro-alloyed steel [98].

**Figure 8.**Deformation instability characteristics of different materials under hot tensile conditions: (

**a**) hot processing diagrams of MgNdZnZr alloy and TC4 alloy; (

**b**) fracture morphology of MgNdZnZr alloy and TC4 alloy at different temperatures [65,101]; (

**c**) engineering stress–strain curves of CrMnFeNi high-entropy alloys, with as-cast and forged samples [103].

**Figure 10.**Wrinkling behavior of sheet metal stamping: (

**a**) sheet metal stamping of a high curvature large flange [121,122]; (

**b**) wrinkling of GPa-grade steel, comparison analysis of representative wrinkle area (wrinkling area b, c and d) with the simulation results [123]; (

**c**) failure mode of a AA2024-T3 thin plate [124].

**Figure 12.**Modes of beam buckling: (

**a**) local buckling, global buckling and interactive buckling [155]; (

**b**) local buckling of a hollow microstructure beam [156]; (

**c**) buckling schematic diagram of a slender beam with irregular bilateral constraints; (

**d**) buckling modes of beams under linear and sinusoidal two-sided constraints [157].

**Figure 13.**Buckling instability of functionally gradient beams: (

**a**) numerical simulation, modal and experimental verification results of laminated functionally gradient materials (FGM) beams [160]; (

**b**) relationship between free vibration and buckling modes of functionally gradient composite beams (FG-CB) [161]; (

**c**) relationship of FG-CB load limit point number and loading position [162].

**Figure 15.**Research results of deformation instability by finite element simulation and experimental methods: (

**a**) flange forming of the spinning thin-walled plate [171]; (

**b**) stamping forming of a titanium alloy mechanical handle [184]; (

**c**) deep drawing of the AA1100 aluminum alloy [185]; (

**d**) 304 steel plate shear wrinkling test [187]; (

**e**) various buckling modes of tube compression instability [140].

**Figure 18.**Engineering applications of avoiding deformation instability: (

**a**) in-plane bending of plates [213]; (

**b**) hydroforming of sheet metal [216,218]; (

**c**) stamping of ultra-high strength steel [123,219]; (

**d**) stamping of high tensile strength steel [220]; (

**e**) stamping of AA2198 aluminum alloy [221].

Authors | Materials | Modified M–K Model | Representative Figures |
---|---|---|---|

Wang et al. [35,36] | 6061 Aluminum Alloy | ${\left({\displaystyle \sum {\beta}^{A}d{\epsilon}_{1}^{A}}\right)}^{{n}^{A}}\mathrm{exp}\left({\epsilon}_{1}^{B}-{\epsilon}_{1}^{A}\right){\phi}^{B}={f}_{0}{\left({\displaystyle \sum {\beta}^{B}d{\epsilon}_{1}^{B}}\right)}^{{n}^{B}}{\phi}^{A}$ where $\phi =\sigma /{\sigma}_{1}$, $\beta =d\epsilon /d{\epsilon}_{1}$, n is real-time strain hardening exponents, and A and B are zone-A and zone-B, respectively. | |

Yu et al. [37,38] | AA5182O Sheet | ${\psi}^{\ast}=\{\begin{array}{ll}arctan\sqrt{-\rho},& -0.5\le \rho \le 0\\ 0,& 0<\rho \le {r}_{{}_{0}}/{r}_{max}\\ \frac{\rho \cdot {r}_{max}-{r}_{{}_{0}}}{{r}_{max}-{r}_{{}_{0}}}{\theta}_{r-max},& {r}_{{}_{0}}/{r}_{max}<\rho \le 1\end{array}$ where ${\psi}^{\ast}$ is critical groove angle, $\rho $ is strain routes, ${r}_{{}_{0}}/{r}_{max}$ is degree of anisotropy, and ${\theta}_{r-max}$ is maximum r-value. | |

Hyuk et al. [39] | Ferritic Stainless Steel (FSS) Sheets | ${\sigma}_{XX}^{B}={\sigma}_{XX}^{A}\left(\frac{{t}^{A}}{{t}^{B}}\right)\left(\frac{{t}_{0}^{A}}{{t}_{0}^{A}-2R}\right)$ where t is thick, t _{0} is initial thick, R is surface roughness, and ${\sigma}_{XX}^{A}$ and ${\sigma}_{XX}^{B}$ are the stresses of REV-A and B, respectively. | |

Wang et al. [40,41,42,43] | Al-Mg-Li Alloy Sheet | ${f}_{0}={t}_{0}^{b}/{t}_{0}^{a}$ $f={f}_{0}\mathrm{exp}\left({\epsilon}_{3}^{b}-{\epsilon}_{3}^{a}\right)$ where f _{0} is imperfection coefficient and ${t}_{0}^{a}$ and ${t}_{0}^{b}$ are initial thick of a and b, respectively. | |

Li et al. [44] | Aluminum Alloy | $e=\{\begin{array}{ll}{\sigma}_{1}^{A}{t}^{A}-{\sigma}_{1}^{B}{t}^{B}& \mathrm{Classical}\text{}\mathrm{\text{M-K}}\text{}\mathrm{model}\\ {\left({\epsilon}_{i}^{A}+d{\epsilon}_{i}^{A}\right)}^{n}{t}_{0}^{A}\mathrm{exp}\left[-\left({\epsilon}_{i}^{A}+d{\epsilon}_{i}^{A}\right)\right]& \\ -{\left({\epsilon}_{i}^{B}+d{\epsilon}_{i}^{B}\right)}^{n}{t}_{0}^{B}\mathrm{exp}\left[-\left({\epsilon}_{i}^{B}+d{\epsilon}_{i}^{B}\right)\right]& \mathrm{Modified}\text{}\mathrm{\text{M-K}}\text{}\mathrm{model}\end{array}$ | |

Hu et al. [45,46] | AA5754 Aluminm Alloy | $\begin{array}{ll}{\sigma}_{nn}& =\left({\sigma}_{1}-{\sigma}_{3}\right){\mathrm{cos}}^{2}\theta +\left({\sigma}_{2}-{\sigma}_{3}\right){\mathrm{sin}}^{2}\theta -2{\sigma}_{12}\mathrm{sin}\theta \mathrm{cos}\theta \\ & =\left({\mathrm{cos}}^{2}\theta +\alpha {\mathrm{sin}}^{2}\theta -2\delta \mathrm{sin}\theta \mathrm{cos}\theta \right)\left({\sigma}_{1}-{\sigma}_{3}\right)\end{array}$ $f={f}_{0}\mathrm{exp}\left\{\left[{\epsilon}_{1}^{a}+{\epsilon}_{2}^{a}+\Delta {\epsilon}_{1}^{a}\left(1+{\beta}_{a}\right)\right]-\left[{\epsilon}_{1}^{b}+{\epsilon}_{2}^{b}+\Delta {\epsilon}_{1}^{b}\left(1+{\beta}_{b}\right)\right]\right\}$ where $\alpha =\left({\sigma}_{2}-{\sigma}_{3}\right)/\left({\sigma}_{1}-{\sigma}_{3}\right)$; $\delta ={\sigma}_{12}/\left({\sigma}_{1}-{\sigma}_{3}\right)$. | |

He et al. [47] | AA6061-F Tube | ${f}_{0\theta}=\frac{{\overline{t}}_{B0}}{{\overline{t}}_{A0}}=\frac{{t}_{\mathrm{min}}}{{R}_{1}-\sqrt{\left({R}_{2}^{2}-{\mathrm{sin}}^{2}\theta \cdot \Delta {d}^{2}\right)}-\Delta d\cdot \mathrm{cos}\theta}$ ${f}_{0\theta}=\frac{{\overline{t}}_{B0}}{{\overline{t}}_{A0}}=\frac{{t}_{\mathrm{min}}}{{R}_{1}-{R}_{2}-\Delta d\cdot \mathrm{cos}\theta}=\frac{\overline{t}-\Delta d}{\overline{t}-\Delta d\cdot \mathrm{cos}\theta}$ where t _{min} is minimum thickness on the tube, $\theta $ is angle, ${\overline{t}}_{A0}$ and ${\overline{t}}_{B0}$ are the thicknesses of zone-A and zone-B, Δd is the eccentric distance of extrusion mandrel, and R_{1} and R_{2} are the radius of outer and inner profiles of the tube, respectively. |

**Table 2.**A review of recent research on the superplastic deformation instability of alloy materials.

Author | Year | Material | Main Points |
---|---|---|---|

Demirel et al. [80] | 2023 | Ti6Al4V | For high-temperature superplastic formation of Ti alloys, the main causes of deformation instability are grain boundary slip (GBS) and creep mechanisms. |

Bobruk et al. [81] | 2023 | 2021Al | For ultrafine grained (UFG) Al alloys, according to the analysis of strain rate sensitivity, they showed stable superplastic behavior at the test temperature of 240~270 °C. |

Myshlyaev et al. [82] | 2023 | Al-Mg-Li | The important role of intra-grain slip during superplastic flow was demonstrated through experimental analysis of strain hardening, the formation of typical deformation textures, and the increase of dislocation density within grains. Superplastic materials exhibited pronounced porosity near the instability point. |

Mochugovskiy et al. [83] | 2023 | Al-Mg-Si-Cu | When the strain rate was low, the residual cavitation after superplastic forming was relatively large; the impurity particles inside the grains also caused the surrounding cavities to increase, which would easily lead to superplastic deformation instability. |

Authors | Year | Material | The Conditions of Deformation Instability | |
---|---|---|---|---|

Temperature/°C | Strain Rate/s^{−1} | |||

Shabani et al. [104] | 2023 | FeCrCuMnNi | 750~850 | 0.1, 0.01, 0.001 |

Singh et al. [105] | 2023 | EN30B Steel | 1000~1150 | 0.1~0.8 |

Jeong et al. [106] | 2023 | AlSi4340 Steel | 1000~1100 | 0.1, 0.2, 0.9, 1.0 |

Azizi et al. [107] | 2023 | AlSiAA4032 | 427~527 | 0.01~0.1 |

Yang et al. [108] | 2023 | Al4.6Mg0.2Sr | 300~400, 400~450 | 0.018~1, 0.018~0.1 |

Lin et al. [109] | 2022 | Ti47.5Al2.5V1.0Cr0.2Zr | 1050~1140, 1180~1200 | 0.006~1 |

Yang et al. [110] | 2022 | 215AlLi | 390~520 | 0.1~10 |

Qiao et al. [111] | 2022 | Fe2.5Ni2.5CrAl | 1020~1100 | 0.01~1 |

Ghosh et al. [112] | 2022 | Ti14Cr | 850~950 | 0.01 |

Yi et al. [113] | 2022 | Al0.5Mg0.4Si0.1Cu | 350~500 | 0.316~10 |

Authors | Materials | Types of Forming | Conditions of Instability |
---|---|---|---|

Yu et al. [143] | ST12 | Push bending | Gap between punch and U/O die, and excessive stock at the end of elbow causing wrinkling. |

Tao et al. [144] | 5A02 Al Alloy | Push bending | Due to the tangential tensile stress concentration at the front end of the tube, the smaller the relative bending radius, the easier it is to have instability. |

Xiao et al. [145] | 5A02 Al Alloy | Push bending | The stress distribution on the compression side is greater than the tension side, indicating that inner side of the tube is more prone to instability. |

Österreicher et al. [146] | AA2024 | Three-roll-push bending | Only solution-annealed material leads to a wrinkle-free bend. |

Cheng et al. [147] | AA6061-T6 | Free bending | When t_{0} < 0.8 mm, plastic instability and wrinkling occurred in the inner flange, and the smaller the wall thickness, the more obvious the wrinkling. |

Wang, Hu and Cheng et al. [148,149,150] | Stainless Steel SS304 | Free bending | The smaller the distance between the center point of the bending die and the front end of guide, the easier it is for the tube to wrinkle. |

Yang et al. [151] | SS304 | Free bending | The inner side of the rectangular tube is subjected to uneven compressive stress, which makes the material flow unevenly, resulting in increased wall thickness on the inner side of the tube, and wrinkled instability. |

Authors | Equation | Explanation | |
---|---|---|---|

Chawla et al. [164] | ${\left(\frac{{\sigma}_{lb}}{{S}_{lb}}\right)}^{2}+{\left(\frac{{\sigma}_{tc}}{{S}_{tc}}\right)}^{2}+{\left(\frac{{\tau}_{xz}}{{S}_{xz}}\right)}^{2}\le 1$ | (31) | ${\sigma}_{lb}$$,{\sigma}_{tc}$$\text{}\mathrm{and}\text{}{\tau}_{xz}$ are longitudinal compressive bending stress, transverse compressive stress and shear stress, respectively; ${S}_{lb}$, ${S}_{tc}$ and ${S}_{xz}$ are bending compressive strength, transverse compressive strength and shear strength, respectively. |

Pozorski et al. [166] | $\begin{array}{l}{\sigma}_{w}=\sqrt[3]{\frac{3}{4}}\cdot \sqrt[3]{{E}_{C}{G}_{C}{E}_{F}}\cong 0.909\cdot \sqrt[3]{{E}_{C}{G}_{C}{E}_{F}}\\ {\sigma}_{w}=\frac{3}{2\cdot \sqrt[3]{6}}\sqrt[3]{{E}_{C}{G}_{C}{E}_{F}}\cong 0.825\cdot \sqrt[3]{{E}_{C}{G}_{C}{E}_{F}}\\ {\sigma}_{w}=\sqrt[3]{\frac{9}{2\left(1+{v}_{c}\right)\cdot {\left(3-{v}_{c}\right)}^{2}}}\cdot \sqrt[3]{{E}_{C}{G}_{C}{E}_{F}}=r\cdot \sqrt[3]{{E}_{C}{G}_{C}{E}_{F}}\end{array}$ | (32) | E_{C} and G_{C} are the modulus of elasticity and shear modulus of the isotropic core material; E_{F} is the modulus of elasticity of the isotropic facing material. |

He et al. [172] | ${\sigma}_{cr}=D\frac{{\gamma}^{2}\sqrt{2\beta +1}\left[{s}_{1}{\beta}^{2}{\delta}^{3}+{s}_{1}S{\delta}^{3}+2{s}_{2}\zeta {\beta}^{4}\left(\delta -1\right)/\left(\beta -1\right)\right]}{12\sqrt{k}\sqrt{S}{\beta}^{2}{\delta}^{3}\left[\beta -1-\mathrm{ln}\beta +\eta \left(\beta -1\right)\left(\delta -1\right)/\delta \right]}$ | (33) | D is the plastic modulus; k is a coefficient related to the flange width and Poisson’s ratio, k = 1.5; $\gamma ,\beta ,\zeta ,\eta $ are coefficients related to the geometrical parameters of the material; ${s}_{1}\text{}\mathrm{and}\text{}{s}_{2}$ are the coefficients representing the increase in the moment of inertia caused by the shift of the neutral surface after stiffening; $S=k/\left[4\left(2\beta +1\right)\right]$. |

Li et al. [173] | ${\sigma}_{\mathrm{max}}=\frac{{\left|M\right|}_{\mathrm{max}}}{{W}_{y}}=\frac{6{R}_{t}{F}_{G1}{\left|{Q}_{k}\left({\alpha}_{1},{\alpha}_{2},\phi ,{k}_{g}\right)\right|}_{\mathrm{max}}}{h{b}^{2}}\le {\sigma}_{s}$ | (34) | ${\left|M\right|}_{\mathrm{max}}$ is the maximum section bending moment; W_{y} is section modulus in bending; h and b are the width and height of the section, respectively; R_{t} is the radius; ${Q}_{k}\left({\alpha}_{1},{\alpha}_{2},\phi ,{k}_{g}\right)$ is section bending moment factor; ${F}_{G1}$ is guide forces. |

$I={\displaystyle {\int}_{{\epsilon}_{b}}^{{\epsilon}_{a}}Bd{\epsilon}_{eq}}={\displaystyle {\int}_{{\epsilon}_{b}}^{{\epsilon}_{a}}\left\{\left(\frac{d{\sigma}^{h}}{d{\epsilon}_{eq}}-\frac{d{\sigma}^{s}}{d{\epsilon}_{eq}}\right)-\left({\sigma}^{h}-{\sigma}^{s}\right)\right\}}d{\epsilon}_{eq}$ | (35) | ${\epsilon}_{eq}$ is equivalent strain; h and s are hard and soft floor, respectively; B < 0 indicates necking progression, larger absolute values. |

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**MDPI and ACS Style**

Wan, M.; Li, F.; Yao, K.; Song, G.; Fan, X. Theory, Method and Practice of Metal Deformation Instability: A Review. *Materials* **2023**, *16*, 2667.
https://doi.org/10.3390/ma16072667

**AMA Style**

Wan M, Li F, Yao K, Song G, Fan X. Theory, Method and Practice of Metal Deformation Instability: A Review. *Materials*. 2023; 16(7):2667.
https://doi.org/10.3390/ma16072667

**Chicago/Turabian Style**

Wan, Miaomiao, Fuguo Li, Kenan Yao, Guizeng Song, and Xiaoguang Fan. 2023. "Theory, Method and Practice of Metal Deformation Instability: A Review" *Materials* 16, no. 7: 2667.
https://doi.org/10.3390/ma16072667