Effect of Volume Changes on Hot Rolling Deformation Behavior of Non-oriented Electrical Steel

Abstract

In order to quantify the effect of volume changes due to temperature drop and phase transformation on hot rolling deformation behavior of non-oriented electrical steel and make up for the deficiency of traditional austenitic rolling model, mathematical models derived from experiments are established, and programmed to subroutines to be incorporated into the coupled temperature-displacement strip model. The calculated results of temperature field, phase field and stress field are then transferred into the rolls-strip coupling model to participate in rolling process. The results indicate that the temperature deviation along strip width direction can lead to an obvious transverse transformation difference. When only the volume change due to temperature drop is considered, the strip shows completely elastic “tensile stress in the edge, compressive stress in the middle”, and the central thickness and quadratic crown of strip are increased slightly. When the volume changes due to temperature drop and phase transformation are both considered, the unexpected “secondary plastic deformation” is produced and the reverse distribution form of internal stress is presented, which significantly decrease the thickness and quadratic crown of strip. However, due to the average effect of tensile stress and compressive stress, both volume changes contribute little to the total roll force.

1. Introduction

Non-oriented electrical steel is a kind of high value-added functional material, which is widely used to manufacture the core of electric motor and generator. The performance of the core and the efficiency of the whole product are directly influenced by the quality of thickness and shape of rolled strip.^1,2) Because the phase transformation temperature of non-oriented electrical steel is about 100°C higher than that of carbon steel,³⁾ the temperature drop between finishing mill stands is accompanied by the phase transformation and the multiphase rolling is inevitable as shown in Fig. 1. Along rolling direction, the upstream stands locate in the austenite region, the middlestream stands drop into dual-phase region, and the downstream stands reach the ferrite region. Meanwhile, along strip width direction, the obvious temperature deviation causes the non-uniform transverse distribution of phase structure. It is worth noticed that there are generally volume changes of strip due to thermal expansion and phase transformation when phase transformation occurred in the continuous-cooling condition between middlestream stands, which will produce an additional strain and thus change the distribution of internal stress of strip. The variation of stress distribution will then affect the metal flow level in next roll gap. It can be found that for the dual-phase rolling of non-oriented electrical steel, in addition to the mixed effect of phase structures, the effect of volume changes on hot rolling deformation behavior should deserve much attention to make up for the deficiency of the exiting rolling models.

Fig. 1.

Multiphase rolling for non-oriented electrical steel.

The researches on the effect of volume changes due to temperature drop and phase transformation on internal stress of strip are mainly focus on the laminar cooling field. Z. Q. Zhou et al.⁴⁾ and X. D. Wang et al.⁵⁾ studied the residual stress of hot-rolled strip on the run-out table due to the difference of thermal expansion caused by non-uniform temperature drop along strip width. On this basis, X. D. Wang et al.⁶⁾ and W. Q. Sun et al.⁷⁾ additionally considered the volume change due to phase transformation and obtained more accurate residual stress during laminar cooling. However, it can be found that the comparative analyses on the tendency and quantity of the effects of temperature drop and phase transformation on the stress distribution are insufficient, which is unable to reveal the distinction of both effects. And for hot strip finishing rolling, the variation of stress distribution between mill stands will directly change the transverse thickness deviation in next roll gap. The researches are rarely reported in this area, which restricts the further improvement of the accuracy of gage and shape control. Therefore, it is necessary to make clear the stress evolution law and the effect of additional stress on roll gap profile in dual-phase region.

In the present study, the transformation kinetics model, the thermal expansion model, the phase transformation expansion model and the deformation resistance model are firstly established by experiments. And then the mathematical models are programmed to subroutine codes that are incorporated into coupled temperature-displacement strip model on ABAQUS software platform to calculate the transverse redistribution of phase structure and internal stress due to uneven temperature drop and asynchronous phase transformation. Eventually, the calculated results of temperature field, phase field and stress field are transferred into rolls-strip coupling model to analyze the effect of volume changes on hot rolling deformation behavior.

2. Foundation of Mathematical Models

In order to quantify the phase transformation process and the additional stress due to thermal expansion and phase transformation, as well as accurately obtain the deformation resistance of each phase region, the transformation kinetics model, the thermal expansion model, the phase transformation expansion model and the deformation resistance models are established by thermal dilatometric tests and hot compression experiments, which provide the basic mathematical models for the following ABAQUS secondary development.

2.1. Transformation Kinetics Model

The continuous-cooling transformation kinetics model can be realized by a combination of the isothermal transformation kinetics model and the Additivity Principle.⁸⁾ Firstly, the Ar3s temperature and Ar3f temperature of non-oriented electrical steel were respectively determined as 965°C and 940°C by continuous-cooling transformation test CCT. The test specimens were cut from continuous casting slab, of which chemical composition is listed in Table 1. Secondly, according to this critical transformation temperature range, four values of 945°C, 950°C, 955°C and 960°C were selected as the isothermal transformation temperature for TTT tests. The Avrami equation was employed to characterize the isothermal transformation kinetics as shown by Eq. (1). By fitting the measured Dilatometer-Time data, the expressions of the reaction rate k and the Avrami exponent n in Eq. (1) were derived as Eqs. (2) and (3). Finally, in combination with the Additivity Principle, the transformed fraction in the continuous-cooling transformation process at any time t_s+1 was expressed by Eqs. (4) and (5).   

$$X\left(t\right)=1-exp\left(-k{t}^n\right)$$(1)

  

$$k\left(T\right)=exp\left(-0.103T+96.989\right)$$(2)

  

$$n=1.6$$(3)

  

$$X\left(t_{s+1}\right)=1-exp[-k_s{\left(t_s^v+t_{s+1}-t_s\right)}^{n_s}]$$(4)

  

$$t_s^v=\sqrt[n_s]{-\frac{ln\left(1-X\left(t_s\right)\right)}{k_s}}$$(5)

Table 1. Chemical composition of non-oriented electrical steel tested (mass/%).

C	Si	Mn	P	S
0.0031	0.77	0.25	0.02	0.005

2.2. Thermal Expansion Model

As is known, thermal expansion is caused by temperature change. The thermal strain corresponding to the unit temperature change can be represented by thermal expansion coefficient α. The amount of thermal strain can be calculated by the multiplication of thermal expansion coefficient and temperature change, as shown by Eq. (6). The value of thermal expansion coefficient can be determined by dilatometric tests and its temperature dependence is shown as Fig. 2.   

$$\mathrm{\Delta }{\epsilon }_{\text{T}}\text{=}\alpha \cdot \mathrm{\Delta }T$$(6)

Fig. 2.

The temperature dependence of thermal expansion coefficient α.

2.3. Phase Transformation Expansion Model

As is known, in the process of transformation from austenite to ferrite, the volume of metal is increased and an additional strain is produced. It can be seen from Fig. 3 that the maximum expansion ΔL_max associated with full phase transformation at four different transformation temperatures are almost the same. The calculated average value of maximum expansion $\mathrm{\Delta }{L}_{max}^{\text{avg}}$ is 19.825 μm and the corresponding maximum strain can be derived as follow:   

$${\epsilon }_{\text{trans}}^{max}\text{=}\frac{\mathrm{\Delta }{L}_{max}^{avg}}{L}=1.983\times {10}^{-3}$$(7)

where L is the original length of the cylindrical specimen, of which value is 10 mm. It can be found that the strain produced by full phase transformation is equivalent to the thermal expansion caused by about 150°C temperature rise, and thus the effective role of phase transformation expansion should be considered.

Fig. 3.

Dilatometer-Time curves of TTT.

In this study, when austenite changes into ferrite between Ar3s temperature and Ar3f temperature, according to the lever law, the strain increment due to phase transformation is assumed to vary linearly with the temperature increment.⁹⁾ The strain increment can be estimated by using the linear approximation as follow:   

$$\mathrm{\Delta }{\epsilon }_{\text{trans}}\text{=}-\frac{{\epsilon }_{\text{trans}}^{\text{max}}}{A_{\text{r3s}}-A_{\text{r3f}}}\cdot \mathrm{\Delta }T$$(8)

2.4. Deformation Resistance Model

The distinct phase structures in austenite, ferrite and dual-phase regions will correspond to different micro hardening and softening mechanism. The method of introducing the dislocation density theory to construct the deformation resistance model is advisable as shown in Eq. (9).¹⁰⁾   

$$\sigma ={\left({\left(\alpha \mu b\right)}^{2}h\dot{\epsilon }/s\left(1-{\text{e}}^{-s\epsilon /\dot{\epsilon }}\right)+{\sigma }_0^2{\text{e}}^{-s\epsilon /\dot{\epsilon }}\right)}^{\frac{1}{2}}$$(9)

where α is the material constant, μ is the modulus of rigidity, and b is the Burgers vector, σ₀ is the yield stress, σ is the deformation resistance, h is the hardening coefficient representing growth rate of dislocation density, s is the softening coefficient representing decreasing rate of dislocation density.

The hot compression experiments were conducted in a wide temperature range from austenite to ferrite by Gleeble 3500. By the regression analysis of experimental data, the deformation resistance models for austenite, ferrite and dual-phase regions are shown from Eqs. (10), (11), (12), respectively.   

$$\{\begin{array}{l}{\sigma }_{\text{A}}={\left({\left({\alpha }_{\text{A}}\mu b\right)}^2{h}_{\text{A}}\dot{\epsilon }/s_{\text{A}}\left(1-{\text{e}}^{-s_{\text{A}}\epsilon /\dot{\epsilon }}\right)+{\left({\sigma }_0^{\text{A}}\right)}^2{\text{e}}^{-s_{\text{A}}\epsilon /\dot{\epsilon }}\right)}^{\frac{1}{2}}\\ {\alpha }_{\text{A}}=1.979\\ {\sigma }_0^{\text{A}}\left(T\right)=-0.098T+144.467\\ h_{\text{A}}\left(T\right)=-0.465T+600.685\\ s_{\text{A}}\left(T\right)=0.055T+8.778\end{array}$$(10)

  

$$\{\begin{array}{l}{\sigma }_{\text{F}}={\left({\left({\alpha }_{\text{F}}\mu b\right)}^2{h}_{\text{F}}\dot{\epsilon }/s_{\text{F}}\left(1-{\text{e}}^{-s_{\text{F}}\epsilon /\dot{\epsilon }}\right)+{\left({\sigma }_0^{\text{F}}\right)}^2{\text{e}}^{-s_{\text{F}}\epsilon /\dot{\epsilon }}\right)}^{\frac{1}{2}}\\ {\alpha }_{\text{F}}=1.993\\ {\sigma }_0^{\text{F}}\left(T\right)=-0.287T+306.487\\ h_{\text{F}}\left(T\right)=-1.072T+1\mathrm{\hspace{0.17em} }053.492\\ s_{\text{F}}\left(T\right)=0.036T+91.867\end{array}$$(11)

  

$$\sigma ={\sigma }_A\left(1-X\right)+{\sigma }_{F}X$$(12)

where X is the transformed fraction of ferrite, which can be calculated by the continuous-cooling transformation kinetics model established previously.

3. Thermal-Mechanical-Metallurgy Coupling Model

In order to accurately quantify the effect of phase transformation, the whole process can be decomposed into two parts, respectively between mill stands and in the next roll gap. The models for simulating these two processes are established on ABAQUS software platform, of which technical route is shown in Fig. 4. One is interstand model, which is used to calculate temperature, transformed fraction and additional stress due to temperature drop and phase transformation of strip between finishing mill stands. The other one is rolling model, which is used to quantify the effect of the additional stress on the hot rolling deformation. The distribution functions of temperature field and phase field and the result file of stress field, derived from the former model, are transferred into the latter model, so that the strip with initial thermal-mechanical-metallurgy states could be successfully involved in the next deformation.

Fig. 4.

Technical route of thermal-mechanical-metallurgy coupling model.

3.1. Interstand Model

3.1.1. Initial State and Boundary Condition

Taking an actual production data as an example, based on the two-dimensional alternating difference temperature model,^11,12) the magnitude and distribution of temperature field of strip at the exit of each finishing mill stand was calculated. In order to most unlimitedly reflect the effect brought by asynchronous phase transformation, the temperature distribution of strip at F3 mill stand exit is selected as the initial temperature field, of which edge section will transform earlier than the center section as shown in Fig. 5(a). (In this study, the slightly uneven temperature along strip thickness direction is ignored and only the obvious temperature difference along strip width direction is considered.) The emissivity and convection coefficient of strip to air between mill stands, referred to the research results of Colas,¹³⁾ are equivalently treated as an equivalent heat flux, of which magnitude and distribution are shown in Fig. 5(b).

Fig. 5.

Initial temperature distribution and the cooling condition. (a) Transverse temperature distribution of strip at F3 mill stand exit. (b) Transverse distribution of equivalent heat flux.

Due to the high temperature and low strain rate in upstream stands, the effect of recovery and recrystallization are significant, which largely contribute to the elimination of internal stress of strip.¹⁴⁾ Thus it is assumed that the internal stress of strip at F3 mill stand exit is zero.

3.1.2. Coupled Temperature-Displacement Strip Model

The width and thickness of strip are respectively 1250 mm and 10.66 mm. According to symmetry, the 1/4 strip model is established. It has been found in the preliminary numerical experiments that if the strip size is set too short, the internal stress will be partially or completely released due to the free extension of end metal, which could be called “stress loss” phenomenon. In order to ensure the internal stress generated by thermal expansion and phase transformation expansion, especially the longitudinal component of stress that is closely related to the shape of strip, can be almost completely retained in the strip without loss, the strip length is determined as 5500 mm according to the real distance between finishing mill stands. The strip is divided into four layers along thickness direction and the mesh size along length direction and width direction are respectively 5 mm and 4 mm. The analysis type of “coupled temperature-displacement, implicit” is adopted. In order to accurately obtain the stress of each node in multiple coupled fields, the element type “3D, 8 nodes, hexahedron, linear, full integration, coupled temperature-displacement, C3D8T” is selected. The initial temperature field is set up in the initial step. The cooling boundary conditions are applied to all nodes in the form of load function. The uniform tension of 10 MPa are applied to the both ends of strip to simulate the tensile stress provided by looper between F3 and F4 mill stands.

3.1.3. Subroutine Description

The subroutines incorporated into the coupled temperature-displacement strip model are described as follows:

(1) SDVINI: initialization subroutine, which is used to initialize the proportion fraction of austenite and ferrite.

(2) UMAT: elastoplastic constitutive subroutine, is used to describe the stress-strain relation of material and only suitable for implicit analysis. The numerical solution strategy is shown in the Appendix.

(3) HARDSUB: deformation resistance subroutine, which is programmed according to the deformation resistance model established previously, is used to calculate the deformation resistance under different temperatures and phase fractions and only suitable for implicit analysis.

(4) KINETICS: transformation kinetics subroutine, which is programmed according to the front transformation kinetics model, is used to calculate the transformed fraction at any time under a given cooling condition.

(5) UEXPAN: thermal expansion subroutine, which is programmed according to the front thermal expansion model, is used to calculate the thermal strain.

(6) TRANSEXPAN: phase transformation expansion subroutine, which is programmed according to the front phase transformation expansion model, is used to calculate the additional strain due to phase transformation.

3.2. Rolling Model

3.2.1. Model Simplification

The period of strip in the roll gap is very short and the amount of phase transformation is relatively small. Furthermore, the temperature drop due to heat transfer between strip and rolls will be partially or fully compensated by the temperature rise due to plastic deformation of strip. Therefore, the variations of temperature and microstructure in the roll gap are not considered in this study.

3.2.2. Rolls-Strip Coupling Model

According to symmetry, the 1/4 rolls-strip coupling model is established, as shown in Fig. 6. The key size and mechanical parameters of rolling mill investigated is shown in Table 2. The analysis type is “dynamic, explicit”. Rollers are elastic and strip is elastic-plastic. In order to avoid the errors of interpolation caused by element mismatch, the number of integration points of strip in explicit rolling model should be consistent with that in implicit interstand model. Furthermore, the shear locking problem of full integrated element in the large elastic-plastic deformation also should be prevented. Based on the above reasons, the element type “3D, 8 nodes, hexahedron, linear, incompatible modes, C3D8I” is finally adopted. The geometry and mesh size of strip is same as the interstand strip model. The contact surfaces of backup roll and work roll, work roll and strip are specially refined to improve the accuracy of simulation results.

Fig. 6.

Rolls-strip coupling model.

Table 2. Size and mechanical parameters of rolling mill investigated.

Type of mill	4-high
Strip width/mm	1250
Entry strip thickness/mm	10.66
Exit strip thickness/mm	6.48
Body diameter of WR/BUR/mm	730/1372
Body length of WR/BUR/mm	2000/1800
Bending Force/kN	−300

The distribution functions of temperature field and phase field derived from the interstand model are applied to the strip in the rolling model by subroutine TRANSFER. Meanwhile, the result file of stress field calculated by interstand model is applied to the strip in the rolling model by setting the initial stress in the predefined field. Through the above operations, the state initialization of strip before rolling is completed.

3.2.3. Subroutine Description

The subroutines incorporated into the rolling model are described as follows:

(1) VUMAT: elastoplastic constitutive subroutine, is similar as UMAT subroutine and only suitable for explicit analysis. The numerical solution strategy is shown in the Appendix.

(2) TRANSFER: state variables transfer subroutine, is used to apply the distribution functions of temperature field and phase field to strip in forms of state variables at the first incremental step.

(3) VHARDSUB: deformation resistance subroutine, is similar as HARDSUB subroutine and only suitable for explicit analysis.

4. Results and Discussion

In order to accurately quantify the effect of volume changes due to temperature drop and phase transformation on the internal stress of strip between mill stands, and make clear the effect of non-uniform internal stress produced by uneven temperature drop and asynchronous phase transformation on hot rolling deformation, three typical simulation conditions are designed, as shown in Table 3.

Table 3. Typical simulation conditions.

Simplified symbol	Description
A+F	Only considering variation of phase structure
A+F+Thermal	Considering variation of phase structure and thermal expansion due to temperature drop
A+F+Thermal+Trans	Considering variation of phase structure and the volume changes due to temperature drop and phase transformation

4.1. Distribution of Temperature Field and Phase Field of Strip between Stands

Figure 7(a) shows the comparison of transverse distribution of temperature at exit of F3 mill stand and entrance of F4 mill stand. It can be seen that the temperature of whole strip decreases due to air cooling between mill stands. Because the contact area between the strip edge and air is larger than the middle section, thus the temperature drop in the edge is larger and the maximum transverse temperature deviation increases from 29.90°C to 40.17°C between F3 and F4 mill stands.

Fig. 7.

Transverse distribution of temperature field and phase field of strip between mill stands. (a) Comparison of temperature field. (b) Distribution of phase field.

Figure 7(b) shows the transverse distribution of phase field at entrance of F4 mill stand. Because the initial temperature of strip edge is lower, the phase transformation will occur in the edge with a priority. And the closer to the edge, the greater the temperature drop, the longer the period of transformation, and the lager the transformed fraction. Eventually, the transformed fraction of ferrite in the edge has reached 97.63%, which is close to the pure ferrite. The temperature in the middle of strip maintains above the critical transformation temperature, so the transformed fraction is zero and the phase type remains austenite. Obviously, at the entrance of F4 mill stand, the strip shows a significant transverse difference of phase structure.

4.2. Effect of Volume Changes on the Stress Field of Strip between Stands

Figure 8 shows the effect of volume changes on the stress distribution of strip between mill stands. When the volume changes due to temperature drop and phase transformation are not considered, the strip only suffers the uniform tension from looper and the internal stress keeps 10 MPa unchanged. When the volume change due to temperature drop is only considered, because the temperature drop of strip edge is larger than the middle section, according to the principle of “heat expansion and cold contraction”, the contraction of fibers in the strip edge are larger than in the middle, which leads to non-uniform deformation of fibers along strip width direction. As a result, the strip shows “tensile stress in the edge, compressive stress in the middle” (due to the superposition effect of looper tension, the value of stress in the edge is positive, but it is compressive stress relative to the middle section), and the maximum absolute value of stress is not more than 30 MPa. When the volume changes due to temperature drop and phase transformation are both considered, because the expansion induced by phase transformation is much greater than the contraction caused by temperature drop in the strip edge, the length of fibers in the edge is large than that in the middle, which ultimately leads to an reverse distribution form of “compressive stress in the edge, tensile stress in the middle” and the maximum absolute value of stress is close to 50 MPa.

Fig. 8.

Effect of volume change due to temperature drop and phase transformation on stress field of strip between mill stands.

Furthermore, it can be seen from Fig. 9 that different from the common elastic deformation induced by thermal expansion, there is unexpected plastic deformation occurred in the strip edge when the volume change due to phase transformation is considered. It means the “secondary plastic deformation” of strip can be produced by phase transformation between mill stands, which has not been taken into account seriously by previous researchers.

Fig. 9.

Comparison of distribution of equivalent plastic strain.

4.3. Effect of Volume Changes on Hot Rolling Deformation

Figure 10(a) shows the effect of volume changes on the total roll force. The prediction error of total roll force can be controlled within 4% by dual-phase rolling model proposed. The volume changes due to temperature drop and phase transformation have little effect on the total roll force, which attributes to the average effect of tensile stress and compressive stress.

Fig. 10.

Effect of volume changes on hot rolling deformation. (a) Effect on total roll force. (b) Effect on roll gap profile.

Figure 10(b) shows the effect of volume changes on the roll gap profile. When the volume change due to temperature drop is only considered, due to “tensile stress in the edge, compressive stress in the middle”, the edge metal is more likely to convert the thickness reduction into longitudinal plastic flow, and the deformation of middle metal becomes difficult due to triaxial compressive stress, which makes the central thickness and quadratic crown increase by 0.01 mm and 24 μm, respectively. When the volume changes due to temperature drop and phase transformation are both considered, due to “compressive stress in the edge, tensile stress in the middle”, the edge drop decreases and the middle metal deforms easily, which makes the roll gap profile smooth and reduces the central thickness and quadratic crown to 6.48 mm and 85 μm, respectively. It can be found that owing to the reverse stress distribution forms caused by temperature drop and phase transformation, the effects of both volume changes on the roll gap profile are opposite.

5. Conclusion

(1) The transformation kinetics model, the thermal expansion model, the phase transformation expansion model and the deformation resistance model based on dislocation density theory for non-oriented electrical steel are established based on experiments, which lay the theoretical foundation for the quantification of phase transformation process and effects of phase transformation.

(2) The volume changes due to temperature drop and phase transformation have different effects on the distribution form and amplitude of internal stress of strip. When the volume change due to temperature drop is only considered, the strip shows “tensile stress in the edge, compressive stress in the middle” and the maximum absolute value of stress is not more than 30 MPa. When the volume changes due to temperature drop and phase transformation are both considered, the strip shows the reverse distribution form of “compressive stress in the edge, tensile stress in the middle” and the maximum absolute value of stress is close to 50 MPa.

(3) Different from the common elastic deformation induced by temperature drop, there is unexpected plastic deformation occurred in the strip edge when the volume change due to phase transformation is considered, which means the volume change due to phase transformation is much greater than the volume change due to temperature drop, and the “secondary plastic deformation” of strip can be produced by phase transformation between finishing mill stands.

(4) The both volume changes have little effect on the total roll force, which attributes to the average effect of tensile stress and compressive stress.

(5) The effects of both volume changes on the roll gap profile are opposite. When the volume change due to temperature drop is only considered, due to “tensile stress in the edge, compressive stress in the middle”, the central thickness and quadratic crown of strip are increased by 0.01 mm and 24 μm. When the volume changes due to temperature drop and phase transformation are both considered, due to “compressive stress in the edge, tensile stress in the middle”, the roll gap profile becomes smooth and the central thickness and quadratic crown are reduced to 6.48 mm and 85 μm, respectively.

Acknowledgement

The authors would like to thank the National Natural Science Foundation of China (No. 51674028 and No. 51404021) for the support to this research.

Appendix

Numerical Solution Strategy for UMAT and VUMAT

In any interval [t_n, t_n+1], given the incremental strain Δε, the equivalent plastic strain ${\overline{\epsilon }}_n^p$ and the stress σ_n at time t_n, the values of ${\overline{\epsilon }}_{n+1}^p$, σ_n+1 at the end of the interval should be updated by the numerical solution as follows:

The first step is the evaluation of the elastic trial stress tensor:   

$${\sigma }^{trial}={\sigma }_n+2G\mathrm{\Delta }\epsilon +\lambda I\mathrm{\Delta }\epsilon :I$$(A1)

where G is the shear modulus, λ is the Lame constant.

The von Mises yield function is defined by   

$$f={\sigma }_e^{trial}-{\sigma }_{yield}=\sqrt{\frac{3}{2}{\sigma }^{trial}:{\sigma }^{trial}}-\sigma \left({\overline{\epsilon }}_n^p,X\right)$$(A2)

where $\sigma \left({\overline{\epsilon }}_n^p,X\right)$ is calculated by the deformation resistance model proposed.

Then the plastic admissibility is checked:

i) f < 0: Elastic deformation, the real state is equal to the trial state:   

$${\overline{\epsilon }}_{n+1}^p={\overline{\epsilon }}_n^p,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\sigma }_{n+1}={\sigma }^{trial}$$(A3)

ii) f ≥ 0: Plastic deformation, the following equation can be derived from the Hooke’s law and the plastic flow rule:   

$${\sigma }_e^{trial}-3G\mathrm{\Delta }{\overline{\epsilon }}^p=\sigma \left({\overline{\epsilon }}_{n+1}^p\right)=\sigma \left({\overline{\epsilon }}_n^p+\mathrm{\Delta }{\overline{\epsilon }}^p\right)$$(A4)

The Newton-Raphson algorithm is adopted to solve the Eq. (A4) to obtain the unknown $\mathrm{\Delta }{\overline{\epsilon }}^p$, and the consistent tangent modulus used in each iteration is as follow:   

$$\begin{array}{l}\frac{\partial \sigma }{\partial {\overline{\epsilon }}^p}=\frac{\partial {\sigma }_{\text{A}}}{\partial {\overline{\epsilon }}^p}\left(1-X\right)+\frac{\partial {\sigma }_{\text{F}}}{\partial {\overline{\epsilon }}^p}X\\ \mathrm{\hspace{1em} }=\frac{1-X}{2{\sigma }_{\text{A}}}{\text{e}}^{-s_{\text{A}}{\overline{\epsilon }}^p/\dot{\epsilon }}\left[{\left({\alpha }_{\text{A}}\mu b\right)}^2{h}_{\text{A}}-{\left({\sigma }_0^{\text{A}}\right)}^2{s}_{\text{A}}/\dot{\epsilon }\right]\\ \mathrm{\hspace{1em} }+\frac{X}{2{\sigma }_{\text{F}}}{\text{e}}^{-s_{\text{F}}{\overline{\epsilon }}^p/\dot{\epsilon }}\left[{\left({\alpha }_{\text{F}}\mu b\right)}^2{h}_{\text{F}}-{\left({\sigma }_0^{\text{F}}\right)}^2{s}_{\text{F}}/\dot{\epsilon }\right]\end{array}$$(A5)

Finally the state variables are updated:   

$${\overline{\epsilon }}_{n+1}^p={\overline{\epsilon }}_n^p+\mathrm{\Delta }{\overline{\epsilon }}^p$$(A6)

  

$$\mathrm{\Delta }{\epsilon }^p=\frac{3}{2}\mathrm{\Delta }{\overline{\epsilon }}^p\frac{{\sigma }^{{trial}^{\prime }}}{{\sigma }_e^{trial}}$$(A7)

  

$$\mathrm{\Delta }{\epsilon }^e=\mathrm{\Delta }\epsilon -\mathrm{\Delta }{\epsilon }^p$$(A8)

  

$${\sigma }_{n+1}={\sigma }_n+2G\mathrm{\Delta }{\epsilon }^e+\lambda I\mathrm{\Delta }{\epsilon }^e:I$$(A9)

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