1 Introduction

The process of microstructural evolution of materials through Severe Plastic Deformation (SPD) has been used in recent decades as a method to enhance the physical and mechanical properties of metals [1]. According to this process, after SPD operation, mechanical strength increases owing to changes in the microstructure of the metal. As yet, various methods have been developed to create the SPD, including the Equal Channel Angular Pressing (ECAP) or Extrusion (ECAE) [1, 2], High-pressure torsion (HPT) [3], Accumulative Roll Bonding (ARB) [4], Repetitive Corrugation and Straightening (RCS) [5], and Other new innovative methods[6,7,8,9,10]. Among the mentioned methods, the process of ECAP has been highly regarded by researchers due to the possibility of industrial-scale production, process repeatability, preservation of material dimensions before and after operations, as well as processing on various alloys and composites [11,12,13]. Because of the application of severe shear strain during the deformation of the sample, the structure of the material becomes finer so that the mechanical properties of the metal improve by increasing ultimate stress and fatigue strength [14, 15]. Therefore, it is necessary to investigate the effect of stresses applied during the process and the damage to the sample to identify regions being more sensitive to crack growth due to further damage. Researchers[16,17,18,19] have used the Cockcroft-Latham [20]criterion to investigate damage under the ECAE process. Figueiredo et al. [21] have declared that the maximum damage occurs in the upper surfaces of the materials with a perfect plastic behavior. In this regard, Ghaziani et al. [17], after calculating damage value in the TECAP process, introduced that the damage coefficient increases by decreasing the friction coefficient. Luri et al. [22] stated that by taking into account the smaller external radii and the larger internal radii of the die, the damage value will be reduced. Therefore, the possibility of cracking would be reduced after passing several times. Ebrahimi and his colleagues [16], in their study on aluminum alloys, expressed that the internal angle effect on strain and damage is greater than the friction effect. Moreover, a crack in the vicinity of the central region of the specimen could be created. During the passing of metals through the intersection of two channels in the ECAE process, severe shear stresses are applied on the metal, which has a significant effect on the stress state within the sample. However, Luri [22] stated that hydrostatic stresses during the ECAE prevents cracking due to shear strains in the metal, and on the other hand, the Cockcroft damage criteria do not consider the hydrostatic stresses on the specimen in damage prediction. Besides, this model has not considered the effect of variables such as stress triaxiality or Lode angle coefficient, which play an important role in predicting damage for complex processes. After experimental observations [23,24,25,26] and theoretical examinations, the researchers revealed that the stress triaxiality is not the only effective parameter in the ductile fracture strain, and the Lode angle parameter is considered by other researchers [27,28,29,30], also has a significant effect on the fracture strain. Accordingly, Xue [31] presented a phenomenological damage model that simultaneously consists of the effects of hydrostatic stresses and the Lode angle parameter. This model is dependent on the third invariant of the deviatoric stress. Gohari Rad et al. [32] investigated the damage accumulation on aluminum alloys before and after the shock loading process according to the Xue and Hosford–Coulomb damage criteria. Also, Gohari Rad and Zajkani [33] presented a stress state-based coupled plasticity – ductile damage model for aluminum alloys considering the influence of high-rate impulsive preload. Moreover, Bai and Wierzbicki [34] have modified the Mohr–Coulomb model (MMC) in such a way that their relationships depend on the stress triaxiality and the Lode angle. This model has been used for different loading paths in ductile damage fracture prediction. Recently, Li et al. [35]have used this method to investigate fracture during deforming processes that occur in several stages. Furthermore, Xiao et al. [36] declared the MMC damage criteria for measuring the ballistic resistance of metal sheets against projectiles.

According to our exploration of the literature, there is no available research work done to investigate the damage criterion for steel during the ECAE process that includes the effect of stress triaxiality along with the Lode angle. Therefore, the purpose of the present study is to evaluate damage model coefficients which will be obtained by a hybrid approach based on experimental and finite element simulation data optimized by the Genetic Algorithm (GA). In this research, the damage is applied to steel during the ECAE plastic deformation process based on MMC and Johnson–Cook (JC) criteria will be investigated. Accordingly, first of all, the strength coefficients of low-carbon steel were obtained based on the tensile test. Then, by experiments that are driven for various stress—states on the steel samples and using a hybrid method, the coefficients of the damage model are calculated for the steel material based on JC and MMC criteria. Subsequently, using the VUMAT written in ABAQUS finite element software, the damage values during the ECAE process are calculated based on MMC and JC criteria and compared with each other. There is no crack observed in the steel samples after the ECAE process, which is in good agreement with the results of the damage modeling performed.

2 Stress State Characterization

The normal effective stress is given as follows [37]:

$$\overline{\sigma } = \sigma_{eq} = \frac{1}{\sqrt 2 } \cdot \sqrt {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{1} - \sigma_{3} } \right)^{2} }$$
(1)

where \({\sigma }_{1}\),\({\sigma }_{2}\) and \({\sigma }_{3}\) are the principal stress components. The principal stresses for homogeneous materials in the Haigh-Westergaard space are defined based on the three principal stress components \({\sigma }_{3}.{\sigma }_{2} \cdot {\sigma }_{1}\) [37]. Hence the triple variables of the principal stress are determined as follows:

$${\varvec{I}}_{1} = \sigma_{ii} { }; \quad {\varvec{I}}_{2} = \frac{1}{2}\sigma_{ij} \sigma_{ji} { };\quad {\varvec{I}}_{3} = \frac{1}{3}\sigma_{ij} \sigma_{jk} \sigma_{ki}$$
(2)

The invariants of stress deviator tensor (\(s_{ij}\)) and the invariant stress tensor values (\({\varvec{J}}_{{\text{i}}}\)) are shown as:

$$s_{ij} = \sigma_{ij} - \frac{1}{3}\sigma_{kk} \delta_{ij}$$
(3)
$${\varvec{J}}_{1} = s_{ii} = 0{ };\quad {\varvec{J}}_{2} = \frac{1}{2}s_{ij} s_{ji} ;\quad {\varvec{J}}_{3} = \frac{1}{3}s_{ij} s_{jk} s_{ki} = s_{1} s_{2} s_{3}$$
(4)

The triaxiality stress parameter (\({\upeta }\)) is considered as the ratio of mean stress (hydrostatic stress) to equivalent stress [38]. This parameter has more applicable to predicting the soft ductile damage. The triaxiality stress can be expressed as the ratio of the first invariant stress tensor to the second invariant of stress deviator tensor.

$${\upeta } = \frac{{\sigma_{m} }}{{\sigma_{eq} }} = { }\frac{{{\varvec{I}}_{1} }}{{3\sqrt {3{\varvec{J}}_{2} } }}$$
(5)

the values of \({\sigma }_{m}\) and \({\sigma }_{eq}\), which represent the mean stress and equivalent stress, respectively, calculate as follows:

$${\sigma }_{m}=\frac{1}{3}{{\varvec{I}}}_{1}$$
(6)
$${\sigma }_{eq}=\sqrt{3{{\varvec{J}}}_{2}}$$
(7)

The second important parameter in the consideration of damage is the Lode angle \(\theta\). This value is defined as a ratio of the third and second invariant of stress deviator tensor as follows [39]:

$$\xi =\mathrm{cos}(3\theta ) \cdot \theta =\frac{1}{3}\mathrm{arccos}\left(\frac{3\sqrt{3}{{\varvec{J}}}_{3}}{2{{\varvec{J}}}_{2}^{3/2}}\right)$$
(8)

3 Material Model

3.1 Johnson—Cock (JC) criteria

The constitutive models are developed based on phenomenological and micro-mechanics approaches, based on their scale [40]. One of the phenomenologically models is Johnson–Cook(JC) model [41], which was expressed based on a combination of stress–strain, strain rate, and temperature conditions. This constitutive model is most widely used to study the plastic behavior of metals [42]. Moreover, this model determines damage value and the possibility of failure on it. Due to its simplicity, this approach has been increasingly considered between researchers also is used especially in industrial applications

$$\sigma =\left[A+B{\varepsilon }_{f}^{N}\right]\left[1+C\mathit{ln}\left(\frac{\dot{{\varepsilon }_{P}}}{\dot{{\varepsilon }_{0}}}\right)\right]\left[1-{\left(\frac{T-{T}_{0}}{{T}_{m}-{T}_{0}}\right)}^{q}\right]$$
(9)

In the above relation, \(A\), \(B\), \(C\), \(N\), and \(q\), are the five parameters of the model and, \(T\), \({T}_{0}\) , \({T}_{m}\) are the temperature at the time of testing, the reference room temperature, and metal melting temperature, respectively, and also \(\dot{{\varepsilon }_{0}}\) is the reference strain rate. In the Johnson–Cook model, the effective strain of failure based on the function of stress triaxiality and plastic strain rate and temperature rate is expressed as follows:

$${\varepsilon }_{f}(\eta \cdot {\varepsilon }_{P}\cdot T)=\left[{J}_{1}+{J}_{2}{e}^{-{J}_{3}\eta }\right]\left[1+{J}_{4}\mathrm{ln}\left(\frac{\dot{{\varepsilon }_{P}}}{\dot{{\varepsilon }_{0}}}\right)\right]\left[1+{J}_{5}\frac{T-{T}_{0}}{{T}_{m}-{T}_{0}}\right]$$
(10)

where \({J}_{1}\)، \({J}_{2}\) ،\({J}_{3}\) ،\({J}_{4}\) and \({J}_{5}\) are five constants of material determined by experiments.

3.2 Modified Mohr–Coulomb(MMC) Model

Bai and Wierzbicki [34] have modified the MM model in such a way the relationships were depended on the stress triaxiality and the Lode angle parameters. This modified model has been widely used for the prediction of failure problems in soil mechanics and brittle materials [43,44,45]. It is found that the MMC model is able to adequately estimate the onset of fracture for materials such as TRIP-assisted steel sheets and 6260-T6 anisotropic aluminum alloy [46, 47]. The MMC model is developed based on the maximum shear stress of the fracture criterion [34]. Accordingly, the flow stress will be obtained as

$$\sigma ={C}_{2}{[\sqrt{\frac{1+{C}_{1}^{2}}{3}}\mathrm{cos}\left(\frac{\pi }{6}-\theta \right)+{C}_{1}\left(\eta +\frac{1}{3} \mathrm{sin}\left(\frac{\pi }{6}-\theta )\right)\right]}^{-1}$$
(11)

The relationships and extraction method related to the stress triaxiality η and the Lode angle θ are described in the reference [48]. Assuming the J2 plasticity relation and using the Ludwik strain-hardening law, the Eq. (11) can be rewritten as Eq. (12) to express a failure asymmetric evolution:

$${\varepsilon }_{f}={\left(\left({C}_{2}{\left[\sqrt{\frac{1+{C}_{1}^{2}}{3}}\mathrm{cos}\left(\frac{\pi }{6}-\theta \right)+{C}_{1}\left(\eta +\frac{1}{3}\mathrm{sin}\left(\frac{\pi }{6}-\theta \right)\right)\right]}^{-1}-a)\right/b\right)}^{1/n}$$
(12)

The two parameters \(C_{1}\) and \(C_{2}\) are the material constants determined employing the experiments. The researchers [34] showed that by increasing \(C_{1}\), the failure strain gets dependent on the stress triaxiality and it will be increased when \(C_{2}\) grows. Besides when \(C_{1} = 0\), the MMC relations will change to the maximum shear stress criterion.

4 Damage Evolution

Murakami and his colleague [49] have described the damage based on the effective cross-section ratio of the material. Bonora [50], Xue [31], and Bai [51] have stated the damage accumulation in nonlinear form of plastic strain(\(\varepsilon_{p}\)). Therefore, damage change(\(dD\)) is defined as follows

$$dD=g({\varepsilon }_{f}\cdot {\varepsilon }_{p}\cdot n)d{\varepsilon }_{p}$$
(13)

and the accumulation of damage under proportional loading is determined as

$$D={\int }_{0}^{{\varepsilon }_{f}}g\left({\varepsilon }_{f}\cdot {\varepsilon }_{p}\cdot n\right)d{\varepsilon }_{p}$$
(14)

In the particular case of \(g\) accordance to Xue [31]model, it can be written as:

$$g\left({\varepsilon }_{f}\cdot {\varepsilon }_{p}\cdot n\right)d{\varepsilon }_{p}=\frac{n}{{\varepsilon }_{f}^{n}}{\varepsilon }_{p}^{\left(n-1\right)}d{\varepsilon }_{p}$$
(15)

where \({ }n\) is the damage power, \(g\) is a weighting function and the failure strain \(\varepsilon_{f}\) is a function of the stress triaxiality and the Lode angle parameter. If any load has not been applied to the material or \(\varepsilon_{p} = 0\), the D value is zero and when \(\varepsilon_{p} = \varepsilon_{f}\), then \(D = 1\).

5 Samples Preparation and Determination of Steel Coefficients

To achieve a homogeneous structure, the cylindrical samples of low carbon steel with 100 mm length were annealed at 1173 K (900° C) for 2 h flowed by cooling in a furnace to ambient temperature. The chemical composition of used samples was stated in Table 1. The typical main microstructural features of annealed samples are revealed micrographs shown in Fig. 1. As can be observed, the microstructure of annealed sample is mainly composed of equiaxed polygonal grains of ferrite (Fig. 1a) with the sizes in the range of 18–110 µm. The pearlite with lamellar structure (Fig. 1b) is distributed along the ferrite grain boundaries and occupies less than 10% of the total area.

Table 1 Percentage of elements used in steel rods
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Fig. 1
figure 1

Typical in scanning electron microscopy micrographs of the microstructure of annealed sample represented a the polygonal grains of ferrite and b the lamellar structure of pearlite

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5.1 Extraction of Material Parameters

To obtain the mechanical properties of the steel, the tensile specimens were prepared according to the ASTM-E8M standard (Fig. 2. RB). Tensile testing was carried out at a constant strain rate of \(10^{ - 3} s^{ - 1}\) in room temperature by applying an Instron testing machine SANTAM -STM-250. The dog bone tensile test result of the sample that has near to 10 mm elongation are shown in Fig. 3. Swift [52] and Ludwik [53] have proposed the following equations to express the material stress–strain relationship.

$$\sigma =A*{\left(B+{\varepsilon }^{p}\right)}^{n} \mathrm{swift Model}$$
(16)
$$\sigma = A + B{{\varepsilon^{P}}^{n}} {\text{Ludwik Model}}$$
(17)
Fig. 2
figure 2

Tensile specimens of a smooth round bar (RB) and notch round bar(NRB) with notch radii equal to b 8 mm, c 5 mm, d 2 mm and e compression specimen

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Fig. 3
figure 3

Comparison between experimental and FE load–displacement diagrams for tensile tests

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In the above relations, \(\sigma\) and \(\varepsilon^{p}\) are the flow stress and equivalent plastic strain respectively, and \(A, B and n\) are the constant coefficients of the material. These coefficients were determined from the true stress–strain curve of samples subjected to tensile testing. Table 2 shows the values of the coefficients of annealed steel for two strain hardening laws.

Table 2 Strain hardening coefficients
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Each predicting damage model should be dependent on the material's stress states. Further, failure occurs in the vast domain of stress states, so it is not possible to test all stress states for calibration of damage parameters. Therefore, samples having different shapes have been tested to investigate the behavior of the material due to different stress states. These shapes of tests envelope different stress states for damage parameters on complex loadings (Fig. 2). All specimens were tested in the laboratory with a strain rate \(\dot{\varepsilon } = 0.001 s^{ - 1}\). All specimens’ cross-sectional diameter used for the tensile test was 5 mm. The above coefficients (Table 2) have been used as input data in the finite element software (ABAQUS 6-14), to compare output data of FE with experimental tests.

Figures 3 and 4 show the comparison between the experimental and the FE load–displacement curves obtained from the Swift model for the compression and the tensile tests. The obtained values indicate the good accuracy of modeling and experimental tests.

Fig. 4
figure 4

Comparison between compression tests and FE load–displacement diagrams

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To study a wide range of stress—state conditions in steel, it has been subjected to a torsion test. Therefore, the values of the stress triaxiality and the Lode angle parameter are equal to zero. According to Fig. 5, the specimen failure torsion is 25.49 rad, so the fracture strain value has been calculated \(\varepsilon^{f} = 1.15\).

Fig. 5
figure 5

The original torque related to the angle during the torsion test

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6 Simulation and Evaluation of Mesh Sensitivity

To investigate the stress states during the loading paths, the finite element simulation results have been used to determine the values of the stress triaxiality and the Lode angle parameter during the process. Therefore, the steel specimen has been simulated with C3D8R elements (8-node linear brick with reduced integration) under strain rate \(\dot{\varepsilon } = 0.001 s^{ - 1}\). Considering the symmetry of the mechanical system, only 1/8 of the cylindrical specimens are modeled (Fig. 6). This led to saving simulation time and allowed the using of the finer mesh at the critical geometric regions (Fig. 6b). Furthermore, the influence of mesh size on numerical results is studied for notch round bar with notch radii equal to 2 mm (NRB-R2 sample). It is found that increasing the mesh number up to 35,100 meshes could enhance the accuracy of the of results. Beyond this point, the effect of mesh number becomes somewhat less (the estimated results changes by less than 0.06% when using 75,352 meshes instead). Therefore, 35,100 elements are selected through the one eighth of the specimens dimensions for all computations.

Fig. 6
figure 6

Schematic illustration of a geometric symmetry in notch round bar and b simulated 1/8 sample

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7 Calculation of Friction Coefficients in Compression Test

The friction has a significant effect during the compression test. So, to the calibration of the models coefficients with the experimental compression test results, it is necessary to evaluate the effects of friction on dimensions of pressed sample. Here, according to Cao et al. [44], in the first step, the geometric dimensions of the sample are experimentally measured after the compression test. The geometric dimensions of the sample define as Rt, upper surface radius (largest radius of the sample after compression) and Ru, the maximum radius of the specimen in the barrel part (maximum radial distance in the barrel position from the center of the specimen) as shown as points 1 and 2 in Fig. 7, respectively. The measured values were equal to \(R_{T} =\) 5.42 mm and \(R_{u}\) = 6.1 mm.In the next step, the friction is defined as \(\tau = {\text{min}}\left( {m\frac{{\sigma_{eq} }}{\sqrt 3 } \cdot \mu \sigma_{n} } \right)\), where, \(\sigma_{eq}\),\({ }\sigma_{n}\), μ and m are respectively the equivalent stress, normal contact stress, the Coulomb coefficient friction and friction factor. The friction conditions are determined so that the dimensions of the simulation sample are similar to the actual dimensions measured. According to measured values of \(R_{T}\) and \(R_{u}\), the friction coefficient values are considered as μ = 0.15 and m = 0.3, in the FE simulation.

Fig. 7
figure 7

Schematic illustration of a Rt and Ru in front view of specimen, b whole model, c a 1/2 simulated sample after the compression test

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8 Identification of Damage Parameters

To determine the damage coefficients in the MMC criterion (Eq. 12), first of all, values of the fracture strains are obtained from experimental tests, and the values of the stress triaxiality and the Lode angle parameter in the loading path are determined from the simulation. In the hybrid method, the Genetic Algorithm (GA) program is written in MATLAB software for determining the optimum parameters of steel damage. For this purpose, the cost function is defined to minimize the difference between the failure strain obtained based on the phenomenological modeling and the experimental failure strain. The cost function of this program is defined as follows.

$$\mathrm{Min Error} \left(\eta \cdot \xi \cdot {\varvec{D}}\right)=\mathrm{Min}\left\{\sum_{i}^{pop}\sum_{j}^{n}\frac{{[{\varepsilon }_{f\cdot j}(\eta \cdot \xi )-{\varepsilon }_{f\cdot j}(\eta \cdot \xi \cdot {D}_{i})]}^{2}}{{({\varepsilon }_{f\cdot j}(\eta \cdot \xi ))}^{2}}\right\}$$
(18)

To validate the damage parameters, the fracture strain values of the NRB_R5 were not applied in the calibration of the GA program, they were used to validate the obtained parameters. Overall procedure of present hybrid modeling has been illustrated by a general flowchart in Fig. 8. Table 3 shows the nonlinear coefficients of the MMC in Eq. (12) as well as, the coefficients of the JC in Eq. (19), which is obtained by assuming that JC model (Eq. 10) is independent from the strain rate and temperature.

Fig. 8
figure 8

General flowchart for procedure of the present hybrid modeling

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Table 3 Damage parameters of annealed steel
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$${\varepsilon }_{f}(\eta )=\left[{J}_{1}+{J}_{2}{e}^{-{J}_{3}\eta }\right]$$
(19)

9 Predicted Fracture Strain Locus Based on MMC Criterion

The plastic fracture strain of steel can be predicted in a wide range of stress triaxiality and Lode angle parameters by using parameters in Table 3. By substituting the values from Table 3 in Eq. (12), Fig. 9 is obtained, which indicates the plastic strain locus predicting fracture based on the MMC criterion for annealed steel. The black curve in the loci of fracture strain indicates plane stress conditions.

Fig. 9.
figure 9

3D symmetric fracture locus for annealed steel using MMC criterion

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10 Plane Stress with Plastic Fracture Strain

The MMC and JC damage criteria are used for various ranges of the plane—stress problems. Figure 10 illustrates the fracture plastic strain curve for this condition. Here, the values of MMC and JC are mutually different in shear stress (\(\eta = 0\)) and biaxial tensile stress (\(\eta = 0.6\)) regions. On the other hand, triaxiality stress of predicted plastic fracture strain values in uniaxial tension (\(\eta = 0.33\)) and \(\eta < - 0.2\) regions are very close to each other.

Fig. 10
figure 10

Plastic fracture strain in-plane stress state for MMC and JC criteria

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11 Damage Accumulation

Figures 11a–d show the comparison between the predicted results by MMC and JC models and the experimental load–displacement curves obtained from the tensile tests on the unnotched and notched round bars. The solid triangular point represents the phenomenon of fracture during the tensile test. Remarkably, the damage accumulation increases instantly after the necking region in RB specimen (Fig. 11a). The damage evolution of MMC and JC methods are so similar to each other in RB and NRB specimens.

Fig. 11
figure 11figure 11

Force–displacement diagrams in comparison with damage evolution based on JC and MMC criteria for specimens a RB, b NRB-R8, c NRB-R5 and d NRB-R2

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In the uncoupled methods, the damage is defined as the ratio of plastic strain(\(\varepsilon_{p}\)) to failure strain(\(\varepsilon_{f} )\) [31] and assumes that failure occurs when the value of damage verify the condition D = Dc = 1(\(\varepsilon_{p} = \varepsilon_{f}\)), where, Dc is a critical value of damage. Table 4 shows the ratio of the fracture strain values predicted by JC and MMC models to the amount of the maximum strain at failure in experimental tensile testing. By considering the effect of the Lode angle parameter in the MMC criterion, therefore, the predictive values ​​of damage of these criteria are slightly higher than the JC criteria. However, because the value of the Lode angle parameter in the RB and NRB samples is the same and equal to one, the strain of the predicted failure by both criteria are close to each other. Although, if the stress triaxiality was in the vicinity of 0 or 0.6 values, the effect of the Lode angle parameter makes the MMC criteria have a better prediction concerning damage accumulation.

Table 4 The ratio of the predicted to the experimental failure strain values
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In general, it can be concluded that, the MMC method for (\(\eta > - 0.2\)) more cautious predicts the probability of fracture in the steel and it predicts the possibility of failure presently before fracture and earlier than the JC criterion.

The above discussion is based on the comparison of the fracture strain values predicted by models JC and MMC(\(\varepsilon_{f - pr}\)) with the amount of the maximum strain at failure in tensile testing(\(\varepsilon_{f - Ex}\)). Now, it is assumed that the plastic strain corresponding to the appearance of a visible crack on the surface of sample (\(\varepsilon_{cr}\)) is a more suitable criterion for comparison. Obviously, the value of \(\varepsilon_{cr}\) is less than \(\varepsilon_{f - Ex}\), and therefore the value of \(\varepsilon_{f - pr} /\varepsilon_{cr}\) is greater than the \(\varepsilon_{f - pr} /\varepsilon_{f - Ex}\) value. This means that the damage values predicted by both models for samples NRB_5 and NRB_2 are closer to the actual (experiment) value of failure criterion when the strain corresponding to the appearance of crack is taken into account.

12 ECAE Process Simulation in Finite Element Method

In this study, ABAQUS 6.12 FE software is used to consider the mechanical behavior, plastic deformation, and damage accumulation evolution during the ECAE process. Elastoplastic behavior is considered based on the strain hardening and Ludwik relation obtained from the tensile test (Table 2). To increase the accuracy of the results, a total of 10,400 elements have been used to mesh the specimen. Mesh sensitivity was considered; therefore, element size was determined \(0.7{\text{ mm}}^{3}\) for a stable state. The C3D8R (eight-node linear brick with reduced integration) element type is considered. Besides, R3D4 (4-node 3-D bilinear rigid quadrilateral) element type is assumed for Punch and Die that defined discrete rigid and no deformation in ECAE Process. Hence, the steel specimen is simulated with a length of 10 cm and a diameter of 9.5 mm at the center of the vertical channel. Die internal and external angle is \(90^{^\circ }\) and \(30^{^\circ }\) (Fig. 12a), respectively and it was fixed in all degrees of freedom (Fig. 12b). Also, punch speed is 0.3 mm/s according to the experimental value. In the reality, to better usage of lubricant (MoS2) and preventing the adhesion of the sample and die, the die was made including tolerance of + 0.1 mm. Therefore, in the FEM this tolerance is taken into account. Many researchers [54,55,56,57,58,59] have used the Coulomb friction condition to simulate the contact constraint between the specimen and the die. Based on research by Balasundar et al. [60], the inhomogeneity index difference in the ECAE process is about 1% for friction values of about 0.02 based on the Columbus friction and shear friction. In this study, the friction coefficient between the specimen and the die is assumed to be μ = 0.024. It has been established [60,61,62,63,64] that during the initial stages of deformation, the load is increased gradually with increasing the ram displacement up to a peak value, due to frictional and the work hardening effects. Thereafter reaching to maximum value, it is continuously decreased due to the reduction of tri-axial stress state and domination mechanical softening processes. Then, it is followed by steady state or increasing with a small slope. Therefore, in this study, the deformation force was not measured and simulated throughout the all ECAP processing of the sample. It is limited to the initial 45 mm of the sample displacement. In Fig. 12c. the experimental results is compared with those that obtained by the FEM. It is seen that the FEM can predict the force with the rational accuracy.

Fig. 12
figure 12

Representations of a Die and steel specimen, b die modeled in FE software and c comparison between the force applied by the punch experimentally and FE simulation

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13 VUMAT Validation

To investigate the damage according to the MMC and JC criteria, the subroutine is written in finite element software. To validate the subroutine, a comparison of equivalent stresses and equivalent strains calculated by ABAQUS software and a written subroutine has been performed. The ECAE process is considered with the same conditions for both methods. The amount of equivalent plastic strain (PEEQ) and Von-Mises stress are shown in Fig. 13. The difference between calculated values of the written subroutine and ABAQUS is less than 1% for both of them. This slight difference indicates the accuracy of the subroutine.

Fig. 13
figure 13

Von mises stress (a), equal plastic strain (PEEQ) (c, e) obtained by ABAQUS software conversely, Von misses stress (b), equal plastic strain (d, e) obtained by VUMAT

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As shown in Fig. 13e, f, the maximum equivalent plastic strain has occurred at the upper surface of the specimen and this value will be reduced by moving downward in the cross-section.

The section shown in Fig. 14 is a cross-section of the specimen at 3 cm from the beginning of it (Fig. 14a). In addition, the direction A to L for 12 nodes is assumed from the upper surface of the sample to the lower surface for considering of damage evolution (Fig. 14b).

Fig. 14
figure 14

The assumption cross-section (A to L) shows the values from the upper to the bottom surface

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As can be seen in Fig. 13c, d, the values of equivalent plastic strain to different parts of the body cross-section, unlike the strain obtained from the Iwahashi [65] relation, are not equal to only unit. Iwahashi [63] predicted the value of strain has occurred in billet base on die features (\(\phi = 90\), \(\psi = 30\)) is equal to 1. However, the FE simulation showed this value is higher than 1 (Fig. 12). Because the strain hardening of steel and friction between the die and specimen are very influential factors on the plastic strain. Besides, by moving away from the inner angle and approaching the lower level (from A to L (Fig. 14)), the amount of strain applied by the sample decreases by about 66%.

14 Distribution of Damage During the ECAE Process

14.1 Investigation of JC Damage Distribution

Luri et al. [22] declared that the Cockcroft-Latham [20] method would not be a suitable criterion for predicting damage, because it does not consider the hydrostatic stresses in the ECAE. Accordingly, the amount of damage caused during the ECAE process has been investigated using Eq. (18) JC [35]. Therefore, using the damage parameters introduced in Table 3 and the VUMAT code, the damage value is illustrated in Fig. 15. Besides, the influence of plastic deformation Zone (PDZ) (the region of intersection two channels) on damage accumulation is shown in Fig. 15.

Fig. 15
figure 15

The Johnson–Cook damage value distribution in annealed steel during the ECAE process

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In the head part of the sample, the maximum amount of damage is spread from the upper surfaces diagonally and parallel to the intersection angle of the two channels. Luri et al. [22] in research on different radii in internal and external angles introduced that the aluminum alloy sample has severe cracking in the initial parts. As can be seen, the maximum of the damage, excepted at the head of the sample, occurs in the center and slightly below it. Figueiredo et al. [21] stated that maximum accumulation of damage occurs in the center and slightly below of it in aluminum alloy with high strain hardening property but damage accumulation evolution is not the same for perfect plastic material. According to the JC criterion, the maximum damage value in the center part of the specimen is less than 0.7, which is less than the value of fracture damage D = 1. So, there is no fracture predicted to occur in the steel. During the examination of the cross section of the processed sample no cracks were observed, neither on the top surface nor in its center.

For considering the evolution and accumulation of damage in the specimen, the damage value has been calculated in each of the 12 nodes in the direction of A to L (Fig. 14). Therefore, node 1 corresponds to node A and node 12 corresponds to node L. Node 1 is the nearest region to the inner angle, so in ECAE process its direction changes rapidly from vertical to horizontal. Moreover, the damage value rises instantaneously (Fig. 16). On the other hand, for node 12 the damage accumulation occurs over a longer period. However, the final damage value in this node is less than in other nodes. Moreover, maximum damage occurs in node 7. The duration of this node is under the influence of shear strain at PDZ so longer than node number one, so the damage accumulation in this node is longer than node 1.

Fig. 16
figure 16

Damage accumulation trend according to JC criteria at three points identified from the direction of A (node 1) to L (node 12)

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The damage value in node 7 is about 10% higher than in node 1. Besides, the maximum damage in node 7 is about 64% more than the accumulated damage in node 12, which indicates a tremendous reduction in damage at the bottom of the specimen where contact with the die surface.

14.2 Investigation of MMC Damage Distribution

Due to the complex stress conditions applied to the steel specimen during the ECAE process, a criterion that simultaneously uses the effects of the stress triaxiality and the Lode angle to predict fracture plastic strain could provide an acceptable prediction of damage accumulation. The MMC criteria have been calculated damage based on the stress triaxiality ​​and the loading angle (Eq. 12). Therefore, according to Table 4, this method has good accuracy in predicting the damage value. The damage values during the ECAE process are evaluated presented in Fig. 17. As can be seen, the amount of damage in the central area and slightly below it reaches the maximum value. Also, the damage value is shown in the upper region (Fig. 17d) is slightly lower than the damage value of the center. The damage is reduced to a minimum value in the region close to the bottom surface of the die. In other words, the MMC criterion predicts shear strain has a minimum effect on the bottom region of the specimen.

Fig. 17
figure 17

The MMC damage value distribution in annealed steel during the ECAE process

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Ebrahimi et al. [16] introduced that the possibility of occurring cracks in the middle area is greater than the outer areas of the specimen for the dies by an internal angle of 90. As shown in Table 4, the predictive damage value by the MMC criterion reaches one before the fracture occurs in steel. Therefore, this criterion is somewhat more cautious in predicting fracture damage rather than the JC criterion. According to Fig. 17, discernible that the maximum damage predicted by the MMC criterion in the middle points is 0.91, which is close to the value of 1, but the MMC criterion also does not predict the possibility of steel fracture during the ECAE process.

Figure 18 shows the damage values for the top, maximum, and bottom points, of the specimen cross-section according to Fig. 14 b. As can be seen, the damage accumulation in node 1 is faster than other areas and reaches the maximum value, while node 7 later than node 1 enters to channels intersection regions (PDZ), therefore, its rapid growth occurs later (Fig. 18) nevertheless, eventually reaches the maximum value compared to other nodes.

Fig. 18
figure 18

Damage accumulation trend according to MMC criteria at three points identified from the direction of A (node 1) to L (node 12)

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However, the damage value in node 12 grows simultaneously by other points, due to the maximum distance from the internal angle and the lower influence of shear stresses among the intersection of two channels, the damage growth occurs with a lower ratio by minimum value. Moreover, Figueroido et al. [21, 66] have stated that owing to the vicinity of node 12 to the lower surface of the specimen, the effects of shear stresses and strains are reduced and the effects of element rotation have a dominant role in the influence of damage accumulation. Therefore, the damage growth in node 12 is lower than in other regions of the cross-section. Moreover, the maximum damage value in node 7 is about 68% greater than the damage value in the end node 12, Also the difference between nodes 1 and 7 is about 24%. It indicates a low reduction in damage distribution from node 1 to node 7 and drastically reduced from node 7 to node 12. Figure 18 compares the damage values based on the JC and the MMC criteria together (Fig. 14b).

As shown in Fig. 19, the JC and MMC criteria have the same trend in predicting damage accumulation for steel in the ECAE process, except that the MMC criterion predicts the damage accumulation value in the vertical direction A to L more than the JC criterion. Somehow, the maximum damage value of the MMC criterion is 26% more than the maximum damage value of the JC criterion. Damage reduction approaches from the center of the specimen to the end of the sample for both criteria have the same trend. Researchers [67,68,69,70] declared the reduction of damage in metals causes the micro-hardness of the sample to be reduced. For this reason, they depicted lower micro-hardness values measured in lower regions of the specimen. According to Fig. 18 has been illustrated there is a slight difference between the damage value in the upper surface of the sample compared with the maximum damage region in the middle of it.

Fig. 19
figure 19

Comparison between MMC and JC damage distribution from A to L direction

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15 Conclusion

  • Due to the wide usage of low carbon steel in the industry, calculation, validation, and then presentation of the elastoplastic coefficients for modeling processes is so essential. Therefore, in this study, the material coefficients of annealed low carbon steel based on Ludwik and Swift models are presented with good accuracy.

  • In this study, a hybrid method is used to obtain MMC and JC damage parameters. Somehow, coefficients obtained by simultaneously employing the experimental results and calculating stress invariants which are simulated in the finite element method and now are presented for other future applications.

  • Prediction failure strain locus is presented by MMC criterion and by considering the damage accumulation in smooth and notch round bars, revealed that MMC criterion demonstrated a cautious prediction compare to JC criterion.

  • The MMC damage values are determined in low carbon steel during the ECAE process depends on the stress triaxiality and the Lode angle. The damage accumulation grows from the surface close to the inner angle of the die and reaches a maximum value in the middle region of the core. moreover, the MMC predicted damage value does not reach the fracture limit (D = 1) after one ECAE pass.

  • The amount of accumulated damage predicted in the steel alloy according to the JC method is lower than the damage is predicted by the MMC method. However, the trend of damage accumulations in both criteria is similar to each other. Besides, the least damage value occurs in the bottom of the specimen that is closer to the outer angle of the die.