# Influence of Different Strain Hardening Models on the Behavior of Materials in the Elastic–Plastic Regime under Cyclic Loading

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## Abstract

**:**

## 1. Introduction

## 2. An Analytical Description of Elastic–Plastic Material Behavior

- Initial area of plasticity, which specifies the interface between the elastic and plastic deformation area.
- Law of plastic transformation, which specifies the direction of plastic deformation velocity during plastic deformation (transformation).
- Hardening condition, which specifies the position, size and shape of the subsequent load area.

#### 2.1. Initial Area of Plasticity

#### 2.2. Law of Plastic Transformation

#### 2.3. Condition and Function of Hardening

#### 2.4. Isotropic Hardening

#### 2.5. Kinematic Hardening

## 3. Numerical Simulations of the Effects of Isotropic and Kinematic Hardening

- control method: controlled by force increments
- iterative method: Newton-Raphson method
- integration method: Newmark method.

#### 3.1. Numerical Simulation with Tensile Stress Followed by Unloading

#### 3.2. Numerical Simulation with Torsional Stress Followed by Unloading

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Detail of the collider assembly [12].

**Figure 11.**Tensile simulations. (

**a**) Meshing, boundary conditions and load for the element α. (

**b**) Meshing, boundary conditions and load for the element β. (

**c**) Time dependence of the loading force.

**Figure 12.**Field and extreme values of normal stresses. (

**a**) ${\sigma}_{\mathrm{yI}}$ for isotropic hardening. (

**b**) ${\sigma}_{\mathrm{yK}}$ for kinematic hardening.

**Figure 13.**Identification of concentration points of compressive stresses using the isotropic material hardening model.

**Figure 14.**Field and extreme values of strains. (

**a**) ${\epsilon}_{\mathrm{yI}}$ for isotropic hardening. (

**b**) ${\epsilon}_{\mathrm{yK}}$ for kinematic hardening

**Figure 15.**Graphical dependencies. (

**a**) of stress ${\sigma}_{\mathrm{yI}\mathrm{max}}$ on deformation ${\epsilon}_{\mathrm{yI}}$ for isotropic hardening; (

**b**) of stress ${\sigma}_{\mathrm{yK}\mathrm{max}}$ on deformation ${\epsilon}_{\mathrm{yK}}$ for kinematic hardening.

**Figure 16.**Torsional simulations. (

**a**) Meshing, boundary conditions and load. (

**b**) Time dependence of the loading torque.

**Figure 17.**Field and extreme values of equivalent stresses. (

**a**) ${\mathbf{\sigma}}_{\mathrm{VM}\hspace{0.33em}\mathit{I}}$ for isotropic hardening. (

**b**) ${\mathbf{\sigma}}_{\mathrm{VM}\mathit{K}}$ for kinematic hardening.

**Figure 18.**Field and extreme values of strains. (

**a**) ${\epsilon}_{\mathrm{eq}\hspace{0.33em}I}$ for isotropic hardening. (

**b**) ${\epsilon}_{\mathrm{eq}\hspace{0.33em}K}$ for kinematic hardening.

**Figure 19.**Graphical dependencies. (

**a**) Of stress ${\sigma}_{\mathrm{VM}\hspace{0.33em}I\hspace{0.33em}\mathrm{max}}$ on deformation ${\epsilon}_{\mathrm{eq}\hspace{0.33em}I}$ for isotropic hardening of the element α. (

**b**) Of stress ${\sigma}_{\mathrm{VM}\hspace{0.33em}K\hspace{0.33em}\mathrm{max}}$ on the deformation ${\epsilon}_{\mathrm{eq}\hspace{0.33em}K}$ for kinematic hardening of the element α. (

**c**) Of stress ${\sigma}_{\mathrm{VM}\hspace{0.33em}I\hspace{0.33em}\mathrm{max}}$ on the deformation ${\epsilon}_{\mathrm{eq}\hspace{0.33em}I}$ for isotropic hardening of the element β. (

**d**) Of stress ${\sigma}_{\mathrm{VM}\hspace{0.33em}K\hspace{0.33em}\mathrm{max}}$ on the deformation ${\epsilon}_{\mathrm{eq}\hspace{0.33em}K}$ for kinematic hardening of the element β.

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**MDPI and ACS Style**

Sivák, P.; Frankovský, P.; Delyová, I.; Bocko, J.; Kostka, J.; Schürger, B.
Influence of Different Strain Hardening Models on the Behavior of Materials in the Elastic–Plastic Regime under Cyclic Loading. *Materials* **2020**, *13*, 5323.
https://doi.org/10.3390/ma13235323

**AMA Style**

Sivák P, Frankovský P, Delyová I, Bocko J, Kostka J, Schürger B.
Influence of Different Strain Hardening Models on the Behavior of Materials in the Elastic–Plastic Regime under Cyclic Loading. *Materials*. 2020; 13(23):5323.
https://doi.org/10.3390/ma13235323

**Chicago/Turabian Style**

Sivák, Peter, Peter Frankovský, Ingrid Delyová, Jozef Bocko, Ján Kostka, and Barbara Schürger.
2020. "Influence of Different Strain Hardening Models on the Behavior of Materials in the Elastic–Plastic Regime under Cyclic Loading" *Materials* 13, no. 23: 5323.
https://doi.org/10.3390/ma13235323