Contents

1. Stress Upturn Phenomenon & Structure of Engineering Constitutive Model 

2.  A Stress-Upturn Constitutive Model

3. A Modified PTW model

1. Stress Upturn Phenomenon & Structure of Engineering Constitutive Model 

(1) stress-upturn phenomenon

Oxygen-Free-High-Conductivity (OFHC) copper is one of the representative materials that exhibits the flow stress upturn phenomenon. It is the phenomenon that the flow stress increases slowly at a low strain rate (the Arrhenius type increase), while beyond a certain rage of the strain rate (103 – 104 s-1), the flow stress turns upward very rapidly (Fig. 1). The phenomenon of flow stress upturn results from viscous drag on dislocations at high dislocation velocities. Examples of materials exhibiting the stress upturn phenomenon include iron [1,2], stainless steel [3], copper [4-7], aluminium [8], tantalum [9-10],  and zinc [1,11].

(2) Category of constitutive models

Physics- or Engineering-Based Model

Constitutive models can be categorized by two groups: physical models and phenomenological models (engineering models). In describing the relation between the flow stress and plastic strain, physical models considers associated mechanisms such as thermally activated dislocation movement. They are very useful to understand the mechanisms associated with the dynamic deformation. Of the many physical constitutive models developed, the SG model (Steinberg et al., 1980 [12]), the ZA model (Zerilli and Armstrong, 1987 [13]), and the MTS model (Follansbee and Cocks, 1988 [14]) have been the most widely utilized. Preston et al. [15] also have proposed a new model (PTW: Preston, Tonks, and Wallace) to cover wide ranges of pressure and strain rate. Briefly describing these models, the ZA model is designed considering the initial dislocation density and dislocation moving mechanism. The ZA models reasonably describe the yield stress and the strain hardening behavior of the materials deforming at a low and intermediate strain rates. The SG model focuses on high strain rate deformation and ignores the deformation at a low strain rate. The mechanical threshold stress (MTS) model was introduced with the idea that plastic deformation is controlled by the thermally activated interactions of dislocations with obstacles. In the PTW model, in addition to the thermally activated interactions that have dominant effect in a low strain rate regime, the dislocation drag mechanism is introduced to cover high strain rates over 107 s-1. Kim and Shin also have shown that the modification of the PTW model by replacing the strain hardening-related term with Voce hardening law successfully describes the flow stress and Taylor impact test results of tantalum in wide strain rate, strain, and temperature regimes [16-17].

In general, the more mechanisms that are reflected in the physical models, the more complicated the formulation and, thus, the greater the number of the parameters required to be determined by experiment. For instance, the so-called mechanical threshold (MTS) model requires as many as 24 material parameters. For the case of the PTW (Preston, Tonks, and Wallace [15]) model, which can be regarded as a semi-physical model, requires 12 material parameters to be fitted from experiment and 3 parameters to be determined from thermodynamic data etc.

Phenomenological constitutive models (engineering models) describe the observed apparent relation between the flow stress and plastic strain by using convenient mathematical functions, without considering the physical mechanisms associated with the dynamic deformation. Provided appropriate functions are selected, phenomenological models reasonably describe the observed deformation behavior of materials. Despite of the very informative nature of the physical models in understanding the mechanisms associated with the dynamic deformation of materials, the non-physics based phenomenological models are frequently used on a practical base for the simulation of the high-strain-rate events. Their powerful benefit arises from the simplicity for practical purposes. For the cases of the simulation of high-strain-rate phenomena via finite element analysis, in general, an extensive computation with a large number of elements and time steps is required. The simplicity of the constitutive model significantly lessen the computational burden; by reducing the computation time for each element and time step.  

Stress-Upturn or Non-Stress-Upturn Model

For the simulation of the high-strain-rate events, constitutive material models have to include not only the strain dependency of the flow stress (work hardening) but also its rate and temperature dependency. Among the constitutive material models describing the rate- and temperature- dependency, some models like the Johnson−Cook (JC) model [18] is incapable of describing the stress upturn phenomenon:

  σ = [A + Bεn] [1+C ln(ε̇/ε̇o)] [1−T*m ] 

where T* is [(T−Tref)/(Tm−Tref)]m,  σ is the flow stress,  ε is the strain, ε ̇ is the strain rate, ε ̇o is the reference strain rate, T is the temperature, Tref is the reference temperature,  Tm is the melting temperature, and  A,  B, n, C, and m  are the material parameters.  TheZerilli−Armstrong (ZA) [13] model is a semi-stress-upturn model. However, the rate- and temperature-dependent models like the MTS [14] and Preston−Tonks−Wallace (PTW) [15] models can describe the stress upturn phenomenon .

(3) Structure of an engineering (phenomenological) constitutive model 

Ways of Treating Hardening and Softening Phenomena

The simplest way of treating the phenomena of the strain hardening (f(ε)), rate hardening (g(ε̇)), and temperature softening (h(T)) are decoupling them in the mathematical expression:

σ = f(ε) g(ε̇) h(T)  

where σ is the equivalent plastic stress (flow stress in uniaxial stress state),  ε  is the equivalent plastic strain, ε̇  is the equivalent plastic strain rate, and T is the current temperature of the deforming specimen. Examples of this type are found in the Johnson-Cook model [18]. 

In the framework of the Zerilli−Armstrong (ZA) constitutive model [13], rate hardening (g(ε̇)) and temperature softening (h(T)) are coupled:

σ = f(ε) + g(ε̇, T)  

In the framework of the Khan−Huang−Li (KHL) constitutive model [19], the strain hardening and rate hardening are coupled:

σ = f(ε, ε̇) + g(T)  

From the viewpoint of deformation mechanisms of a solid, actually, in essence, all of the three phenomena, i.e., strain hardening, strain rate hardening, and thermal softening, may be coupled: 

σ = f(ε, ε̇, T) 

In what way and how much they are coupled (inter-dependent) are not yet currently clear and may be dependent on material types.  There are indeed numerous types of constitutive models, which were developed for capturing various aspects of complicated constitutive behaviours of versatile materials. The description capability of an engineering constitutive model can be judged after comparing the model prediction with experiment.

Strain-Hardening Laws

There are some commonly employed mathematical functions in  many constitutive models. For instance, for the description of the strain hardening phenomenon, followings have been used.

• Holomon's law [20]: f(ε) = Bεn

• Swift law [21]:  f(ε) = k(ε + εo)n

• Ludwik law [22]:  f(ε) = A + Bεn

• Voce law [23]:  f(ε) = A + B{1−exp(−Cε)} 

The Ludwik law was employed in the Johnson−Cook [18] and Zerilli−Armstrong [13] constitutive models.

Rate-Hardening Laws

The mathematical functions describing the phenomenon of rate hardening are as follows. 

• Allen−Rule−Jones law [24]: g(ε̇) = (ε̇)c 

• Cowper–Symonds law [25, 26]:  g(ε̇) = 1 + (ε̇/C)1/p

• Johnson−Cook law [18]:  g(ε̇) = 1 + C ln(ε̇/ε̇o)

• Tuazon et al's law [39]: 1 + C (ln(ε̇/ε̇o))p 

• Huh−Kang law [27]:  g(ε̇) = 1 + C1 ln(ε̇/ε̇o) + C2 (ln(ε̇/ε̇o))2 

• The  law presented in the study of Shin and Kim [28]: g(ε̇) = D ln(ε̇/ε̇o) + exp(E⋅ε̇/ε̇o)

Thermal-Softening Laws

The mathematical functions describing the phenomenon of rate hardening are as follows. 

• Anonymous law: h(T) = (T/Tm)p, where Tm is the melting temperature

• Johnson−Cook law [18]: h(T) = 1 − T*m where T* = [(T−Tref)/(Tm−Tref)]m

• The  law presented in the study of Shin and Kim [28]: h(T) = (1 − T*)m where T* = [(T−Tref)/(Tm−Tref)]m

The  last-introduced law is different from the JC thermal-softening law. They yield drastically different results, as will be shown later at this site. The Johnson−Cook thermal-softening law was employed in the Khan−Huang−Li (KHL) constitutive model [19].

2. A Stress-Upturn Constitutive Model

For materials exhibiting the stress upturn phenomenon, use of the stress-upturn constitutive model is desirable. In this section, an executive summary of an engineering constitutive model developed by Shin and Kim (Journal of Engineering Materials and Technology (ASME) 2010; 132: 021009) is provided, which model would be one of the simplest stress-upturn models with only six parameters. This model is called the 'current model' here.

(1) Formulation of the Current Model

In the current model [28], all of the phenomena of the strain hardening (f(ε)), rate hardening (g(ε̇)), and temperature softening (h(T)) were treated to be decoupled  and the law of each phenomenon was selected with reasonable backgrounds. The employed law for each phenomenon was as follows.

• Strain hardening law: f(ε) = A + B{1−exp(−Cε)} (Voce hardening law)

• Rate  hardening law: g(ε̇) = D ln(ε̇/ε̇o) + exp(E⋅ε̇/ε̇o)

• Temperature softening law: h(T) = (1 − T*)m where T* = [(T−Tref)/(Tm−Tref)]m

The formulation of the current model is

  σ = [A+B{1−exp(−Cε)}] [D ln(ε̇/ε̇o)+exp(E⋅ε̇/ε̇o)] [1−(T−Tref)/(Tm−Tref)]m

where  σ is the flow stress,  ε is the strain, ε ̇ is the strain rate, ε ̇o is the reference strain rate, T is the temperature, Tref is the reference temperature,  Tm is the melting temperature, and  A ,  B , C , D  , E  , and m  are the material parameters. 

(2) Characteristics of the Strain Hardening Law

The Ludwik and Voce strain hardening laws are compared first in Fig. 2. As can be seen in Fig. 2(a), the Ludwik hardening law has a limitation in describing the case when strain hardening is saturated. There exists unrealistic initial rise of the flow stress at a small strain to describe the saturation of the strain hardening by using small n values, e.g., n=0.1. However, Voce hardening law (Fig. 2(b)) is able to describe the saturation of the strain hardening as well as the case when the strain hardening continues to progress with strain. Thus, the Voce hardening formulation has been selected in the current model [28].

(2) Characteristics of the Rate Hardening Law

The rate hardening law includes additional term of the exponential of strain rate compared with the JC rate-hardening law, which is indispensable to account for the steep rise (upturn) of the flow stress at high strain rate. The strain rate hardening term in the current work will also be capable of describing the flow stress of a material showing only linear rise of the flow stress with logarithmic strain rate by assigning zero to the constant E. In such a case, the second term is approximated to , i.e., the JC type strain-rate hardening behavior. Fig. 3 compares the strain-hardening term of JC model (Fig. 3(a)) and the current model’s counterpart for the case when D=0.05 and 0.2. As illustrated in the Figs. 3(b) and 3(c), the D value describes the slope in the region where the flow stress increases linearly with the logarithmic strain rate, and E controls the strain rate value where the upturn of the flow stress takes place.

(3) Characteristics of the Thermal Softening Law

The current thermal-softening law is different from the JC thermal-softening law in that the exponent m is now applied to the total bracket. In the development of a new model, how drastically a formulation has been changed is not the essential part, but what is important is how the new model functions. Formulation of the thermal softening term in the current work indeed more properly describes the thermal softening behavior. As can be seen in Fig. 3(b), the increase of the flow stress at temperatures T < Tref is now properly modeled. In other words, there is no temperature limitation for the description of thermal behavior in the current model unlike the JC model. Further, the unrealistic abrupt decrease of the flow stress when temperature is slightly higher than the reference temperature (which is seen in the JC model in Fig. 4(a)) is now disappeared in the current model (Fig. 4(b)). The saturation of the thermal softening behavior can now be modeled properly by using higher values of m, e.g., m=6. 

(4) Description Capability of the Current Model

In all of the diagrams presented below, experimental data are illustrated as data points while the results of model predictions are  plotted as solid curves. 

Follansbee's Copper Data

The initial linear rise of the flow stress in logarithmic strain rate scale followed by the upturn at high strain rate above about 103 s-1, like the case of copper [4] shown in Fig. 5(a), would be one of the most challenging phenomena that is to be described by a single model. Note that the JC model will only predict a linear rise of the flow stress in logarithmic strain rate scale. However, as seen in Fig. 5(a), the current model [28] successfully describes the flow stress behavior of copper, which shows the initial linear rise followed by an upturn at strain rate over about 103 s-1. This finding indicates that, if the flow stress data at higher strain rates like the cases in shock regime are provided, the description capability of the current model is suitably extended to the higher strain rate regime.

It is noted that the PTW model can also describe the upturn of the flow stress of copper more or less similarly to the current model [28]. In the PTW model, the complicated error function and power law were employed to account for the upturn of the flow stress in logarithmic scale. Then, the result from the error function and the extrapolated flow stress from shock regime (strain rates of above about 109 s-1) were compared, from which the higher flow stress value was selected at a given strain rate. Although the power-law and error function may work well in the frame work of the PTW model, the simple formulation of the current strain-hardening term (exponential function) in the framework of the current model is also shown to work very well. The current model is fairly simple with only six material parameters to lessen the burdens of the parameter determination as well as computation. 

Samanta's  Copper Data

The combined effect of strain-rate hardening and thermal softening has been checked. Fig. 6(a) compares the current model with the experimental data for copper (from Samanta [29]). As demonstrated by the figure, the current model describes the data set of Samanta’s copper fairly well, while the PTW model (Fig. 6(b)) underestimated or over estimated the same data set [29]. The parameter set used for the description of the Follansbee’s copper (Fig. 5) is different from the Samanta’s copper (Fig. 6), because each of the parameter set was determined from the respective experimental data set.

Copper at Various Rates and Temperatures [30]

Figs. 7-10 compare the flow stress description capability of the MTS, PTW, JC, and the current models with 14 experimental data sets available elsewhere [13-15]. The same experimental data sets are used for each model. The strain rate and temperature of each data set are: 1: 0.066 s-1 and 1173 K; 2: 960 s-1 and  1173 K; 3: 1800 s-1 and 1023 K; 4: 2300 s-1 and 873 K; 5: 0.1 s-1 and 296 K; 6: 8000 s-1 and 296 K; 7: 4000 s-1 and  1096 K; 8: 4000 s-1 and  896 K; 9: 4000 s-1 and 696 K; 10: 4000 s-1 and  496 K; 11: 4000 s-1 and 77 K; 12: 6.93×105 s-1 and 571 K; 13: 6.93×105 s-1 and 768 K; 14: 6.93×105 s-1 and 964 K. 

The data set No. 1 through 4 (compression test data) are adapted from Samanta [29] and 5 through 11 (tension test data) are from Nemat-Nasser [31]. The data set No. 1~11 were also used in Banerjee [32]. The high strain rate data set No. 12 through 14 (shear test data), obtained at high strain rate of 6.93×105 s-1 and varying temperatures, are adapted from Frutschy and Clifton [33], which were converted to equivalent plastic strain and equivalent plastic stress in the current work.

In each figure, the frames are divided into (a), (b), and (c) to avoid complication of the experimental data and prediction results. The data set shown in figure frames of (a) and (b) are determined at low and intermediate strain rates, while the data set in figure frame (c) are at high strain rate (6.93×105 s-1 [33]). 

As seen in Figs. 7 and 8, the most complicated and time-consuming MTS and PTW models show a reasonable agreement with experiment at low and intermediate strain rates (frames (a) and (b) in each figure). At high strain rate, however, these models certainly show limitations in predicting the experiment (frame (c) in each figure).

Fig. 9 compares the JC model prediction (solid lines) and experiment (dot data). This simple phenomenological model is inferior to the MTS and PTW models at low and intermediate strain rates (frames (a) and (b) in each figure). At high strain rate, the limitation of the JC model is even augmented (Fig. 9(c).

On the other hand, the current model is capable of reasonably predicting the experiment at low and intermediate strain rates (Figs. 10(a) and 10(b)), like the complicated MTS and PTW models.  Furthermore, at high strain rate (Fig. 10(c)), the prediction capability of the current model is the best among the comparing models.

The performance of the current model for copper at various rates and temperatures is notable considering its simplicity and resultant mild computational burden: the current model is composed of simple exponential and logarithmic functions unlike the MTS (which contains complicated algorithms and requires looping routine) or PTW (which contains complicated error functions composed of the series expansion).

4340 Steel

Figs. 11(a)-(c) also check the combined effect of strain-rate hardening and thermal softening by using the experimental data of AISI 4340 steel (adapted from Lee and Yeh [34]). As can be seen in the figures, the current model reasonably describes the experiment of these materials. Such description capability for various metals with different behaviors of strain hardening, strain-rate hardening, and thermal softening, is remarkable considering the simplicity of the current model. It is also noted that the data set for AISI 4340 steel (Fig. 6) was also described well by the MTS model [35], which requires as many as 24 material parameters. 

Tungsten Heavy Alloy (WHA)

Figs. 12(a)-(c) additionally check the combined effect of strain-rate hardening and thermal softening by using the experimental data of tungsten heavy alloy (adapted from Lee et al. [36]). As can be seen in the figures, the current model reasonably describes the experiment of these materials. 

Beryllium

Also, the description capability of the model for the strain-rate hardening associated with the strain hardening has been checked by using the experimental data of beryllium with respect to strain at 573 K and varying strain rate (Fig. 13), adapted from Montoya [37]. In Fig. 13, the relation between shear stress and the equivalent plastic strain is presented. As the current model is the phenomenological (empirical) model, it can fit the relation between shear stress and equivalent plastic strain as well. In the caption of Fig. 13, material parameters to predict the equivalent plastic stress (flow stress, ) with respect to the equivalent plastic strain are also given. As can be seen in Fig. 13 the current model describes the flow stress of beryllium fairly well with respect to strain at varying strain rate. When the PTW model was applied for the data sets of beryllium shown in Fig. 13, it notably deviated from the data [37].

Uranium

The description capability of the current model for the thermal softening phenomenon associated with the strain hardening was checked by using the experimental flow stress data of uranium with respect to strain at the strain rate of 3500 s-1 and varying temperature, adapted from Armstrong and Wright [38]. In Fig. 9, it is seen that the current model describes the experiment fairly well. When the PTW model was applied for the uranium data shown in Fig. 14, the description was significantly erroneous for the data set at 373 K and more deviated than the current model at 573 K [38]. 

(5) Conclusion

The description capability of the current model for considered metallic materials with different behaviors of strain hardening, strain-rate hardening, and thermal softening, is notable considering the simplicity of the current model. 

(6) References

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[28] Shin H and Kim JB. A Phenomenological Constitutive Equation to Describe Various Flow Stress Behaviors of Materials in Wide Strain Rate and Temperature Regimes. J Eng Mater Technol 2010; 132: 021009. 

[29] Samanta SK. Dynamic Deformation of Aluminium and Copper at Elevated Temperatures. J Mech Phys Solids 1971; 19: 117–122.

[30] Shin H and Kim JB.  Description Capability of a Simple Phenomenological Model For Flow Stress of Copper in an Extended Strain Rate Regime. Proc 4th Int’l Conf Design Analysis Protective Structures, Jeju, Korea (Paper No. T8-10, pp.1–6) June 2012; 19–22.

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[33] Frutschy KJ and Clifton RJ. High-Temperature Pressure-Shear Plate Impact Experiments on OFHC Copper. J Mech Phys Solids 1998; 46: 1723–1743.

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3. A Modified PTW Model

This section provides an executive summary of a constitutive model developed by Kim and Shin (Comparison of Plasticity Models for Tantalum and a Modification of the PTW Model for Wide Ranges of Strain, Strain Rate, and Temperature,  International Journal of Impact Engineering 2009; 36(5): 746–753).

(1) Introduction

With the merits of high density and high ductility, tantalum has been widely used as a liner material for explosively formed projectiles (EFPs). As tantalum is highly ductile, it can form a long aerostable projectile (EFP) out of a dish-shaped liner. Tantalum EFPs are efficient in penetrating a target due to the high velocity (1500-2000 m/s) and high density of tantalum (16.65 g/cm3). Fig.1 shows a schematic illustration of the EFP formation process and the final morphology of penetration into the target. For both the formation and penetration processes, strain, strain rate, and temperature change significantly. In order to ensure an accurate numerical simulation, therefore, a constitutive material model has to reflect the effects of strain, strain rate, and temperature in a reasonably wide range. 

Thus far, indeed, numerous constitutive models have been proposed to describe plastic deformation.  Some of them include, namely, the Steiberg-Guinan model (SG) [1],  the Zerilli-Armstrong model (ZA) [2], the Johnson-Cook model (JC) [3], and the mechanical threshold stress model (MTS) [4], Voyiadjis and Abed (VA) model [5], and Preston-Tonks-Wallace (PTW) model [6]. Briefly introducing the proposed models, the SG model focused on a high-strain-rate deformation by ignoring deformation at low strain rates. The ZA model was designed to account for initial dislocation density and dislocation moving mechanism. The JC and ZA models are known to describe the flow stress and strain hardening behavior of material deforming at a low strain rate and at temperature near room temperature. So-called mechanical threshold stress (MTS) model was introduced with the idea that plastic deformation is controlled by the thermally activated interactions of dislocations with obstacles [4]. The VA model aimed to improve the ZA model by modifying the evolution of mobile dislocation density, and it reportedly predict the flow stress better at high strain rates. In the PTW model, in addition to the thermally activated dislocation interactions that have dominant effect in a low strain rate regime, the dislocation drag mechanism was introduced to cover strain rates over 107 s-1. 

There have been also many works regarding the experimental determination of the material parameters for the constitutive models. Briefly addressing some of them, Chen and Gray III [7] determined the parameters of the constitutive relations of Ta and Ta-W alloys for the JC, ZA, and the MTS models, and also investigated the fitting characteristics of the measured flow stress in the range of strain from 0 to 0.8, strain rate from 0.001 s-1 to 5000 s-1, and temperature from 25oC to 1000oC. In the case of Maudlin et al. [8-9], the MTS model parameters for DoD tantalum were determined by comparing the simulated anisotropic deformation of a tantalum rod with a piece-wise yield surface that was determined by an experimentally derived orientation distribution function.  In order to describe the constitutive behavior using the MTS model, however, as many as seven parameters and three functions had to be decided.

Comparison of existing models is important for an accurate numerical simulation of the plastic deformation phenomenon [10]. Of many constitutive models, the current work has compared the prediction capability of four well-known plastic constitutive models, Johnson-Cook (JC), Zerilli-Armstrong (ZA), Voyiadjis and Abed (VA), and Preston-Tonks-Wallace (PTW), for the case of the plastic deformation of tantalum. Through the comparison of each model’s prediction with existing experimental data elsewhere, it will be shown that while none of above models is appropriate in wide ranges of strain, strain rate, and temperature, the PTW model has good fitting capability for high strain rate and high temperature, provided the strain hardening term is modified properly. The modification process as well as the capability of the modified PTW for tantalum in wide-range of strain, strain rate, and temperature, will be presented.

(2) REVIEW OF FOUR CONSTITUTIVE MODELS 

The effects of strain, strain rate, and temperature on the flow stress are not independent of each other: there exists an interaction effect. Thus, accurate determination of the flow stress as a function of the above mentioned three parameters is not an easy task. Also, the minimum number of fitting coefficients is preferred, because in most cases the coefficients are determined from experiment. The four constitutive models compared in the current work are reviewed hereinafter.

JC Model [3]

The JC model describes the plastic flow stress by the relation,

where,  is the equivalent plastic strain,  is the equivalent plastic strain rate, is the reference strain rate which is usually set to 1 s-1, and T* is the homologous temperature defined as . T is the absolute temperature and A, B, n, C, and m are material constants. In this model, strain hardening, strain rate hardening, and thermal softening are taken into account as a multiplication form. The expression in the first set of parenthesis gives the flow stress as a function of equivalent plastic strain for unit strain rate and =0. The expressions in the second and third sets of parenthesis represent the effects of the equivalent plastic strain rate and temperature, respectively. The JC model has some shortcomings. First, the linear relation of the flow stress is predicted with temperature while such behavior is often not the case in practice, especially at high temperature. The flow stress at high temperature does not decrease linearly as the temperature increases. Second, the model predicts a linear increase of flow stress with log strain rate. However, some materials, such as tantalum, are known to show an abrupt increase in flow stress at a certain strain rate [5]. Finally, Eq. (1) gives a negative flow stress when the plastic strain rate is nearly zero.

ZA model for BCC materials [2] 

The ZA model has different forms of constitutive relation for body-centered-cubic (BCC) materials and face-centered-cubic (FCC) materials. The constitutive equation for  BCC materials such as tantalum is as follows:

where, C0, C1, C3, C4, C5, and n are material constants. The first term C0 is related to Hall-Petch

relation , where d is the grain size of the material. In this model, it is presumed that the work hardening is independent of temperature and strain rate. This means that the thermally activated movement of a dislocation is independent of the plastic strain. The saturation of strain rate hardening, however, is related to the temperature softening. The power-law stress-strain relationship in Eq. (2) exhibits a continual work hardening without saturation of flow stress at a large strain.

VA model [5] 

To improve the prediction capability of flow stress at high strain rates and temperatures Voyiadjis and Abed [5] modified the ZA model as follows:

where, , , , Ya, B, p, q, and n are material constants. The last two terms are athermal components of flow stress and they are the same forms as the ZA model in Eq. (2). The first term is thermal flow stress and it is related to the strain rate and temperature. The first term was modified from the ZA model and derived by using the concept of thermal activation energy as well as the dislocation interaction mechanism. The mobile dislocation density evolution was also taken into account. By modifying the thermal component of flow stress, the prediction capability at high strain rates and temperatures is reportedly improved [5].

PTW model [6] 

Thermal activation mechanism of dislocation has a significant influence on the deformation by weak shocks of strain rate up to 105 s-1. The strain rate in explosively driven deformations or in high-velocity impacts is sometimes much higher than 105 s-1, and thus the plastic constitutive model based on only the thermal activation mechanism can result in a significant error. In order to model the material behavior accurately at a strain rate up to 1012 s-1, Preston et al. [6] proposed a plastic constitutive model considering nonlinear dislocation drag effects that are predominant in a strong shock regime. The model is given by

where,  is a normalized flow stress (=  where is the shear stress and G is the shear modulus) ,  and  and  are the normalized work hardening saturation stress and normalized  yield stress, respectively. The variables, p, q , and s0 are dimensionless material constants.  and  are defined as,

where the material constants s0 and s¥  are the values that takes at zero temperature and very high temperature, respectively.  y0 and y¥ have analogous interpretations. k  and g are dimensionless material constants. Scaled temperature is defined by where T is temperature and Tm is melting temperature. The parameter  is defined by 

where r is density and M is atomic mass. has the meaning of the time required for a transverse wave to cross an atom, thereby the term in Eqs. (6) and (7) is the dimension-less strain rate variable. Shear modulus G is taken to be a function of density and temperature: , where a is the G dependency on scaled temperature and G0 is the shear modulus at absolute temperature. Constants y0, y1, y2, y¥, s0, s1, s¥, k,  g , p, and b are fitted from flow stress experiment (e.g., split Hopkinson pressure bar test) at varying strain rate and temperature. The parameter q  is determined from the work hardening slope data (or fitted from flow stress experiment) and other material parameters such as G0, a, and  are usually obtained from other sources in the literature. 

The first terms in the braces of Eqs. (6) and (7) are associated with the thermally activated dislocation movement in a low-strain-rate regime, and the second terms are associated with the dislocation drag through phonon in a high-strain-rate regime. The strain hardening behavior is associated the second term of the right-hand side in Eq. (4).

(3) PREDICTION CAPABILITY OF MODELS 

Prediction capability of models

In order to investigate he prediction capability of the models, experimentally determined relation between flow stress and equivalent strain at various strain rates and temperatures have been compared with the predictions by the PTW, ZA1, ZA2, VA, and JC models. Fig. 2 compares the prediction capability of the models with the experimental data obtained at room temperature and two different strain rates, i.e., 0.1 and 1300 s-1, by Maudlin et al. [8]. Isothermal assumption was employed for the strain rate of 0.1 s-1, while adiabatic for 1300 s-1. At the fairly slow strain rate (0.1 s-1), only the PTW is reliable, while at 1300 s-1, the PTW, ZA1, and JC predict the experiment fairly well. 

Fig. 3 compares the models with the experiment performed at moderately high strain rates (2200-2900 s-1) and at elevated temperatures by the same authors as Fig. 2, i.e., Maudlin et al. [8]. Sets of data in Fig. 3 are listed in the order of temperature. At all of the elevated temperatures, again, the PTW, ZA1, and JC predict the experiment fairly well. The underestimation by ZA2 increases as temperature increases. The VA predict the flow stress very well at temperature of 474 K, however, it underestimates the flow stress as temperature increases to 873 and 1073 K. 

Fig. 4 shows the prediction capability of the models for moderately wide range of strain (0.0~0.6). The experimental data are adapted from Nemat-Nasser and Isaacs [13]. All of the models, except PTW, predict the flow stress very well at room temperature (Fig. 4(a)), while all of the models, except ZA2, predict well at elevated temperature of 798 K (Fig.4(b)).

The capability of the models in predicting the relation between flow stress and equivalent strain is now presented for a very high strain rate near 106 s-1 (9.24×105 s-1) in Fig. 5, by comparing with the experimental data[1] of Duprey and Clifton [12]. In contrast to the results at lower strain rates (Figs. 2-4), no models are successful in predicting the experiment. The ZA1 and JC models now give worst result. Although no models are successful in predicting the experiment at such high strain rate (~106 s-1), PTW, VA, and ZA2 describe the flow stress at an equivalent strain of about 0.2, although they underestimate thereafter. The underestimation is ascribed to the fact that the strain hardening is underestimated compared to the temperature softening by deformation heat.  Thus, the work hardening phenomenon needs to be properly reflected in the PTW, VA, and ZA models. 

Comparison of the models and their limitation

At a fairly low strain rate (10-1 s-1), only PTW is reliable (Fig. 2). At moderately high strain rates and elevated temperatures, PTW, ZA1 and JC predict the experiment fairly well (Figs.3-4). The VA model predicts the flow stress well at moderately high strain rate and temperatures below 500 K and underestimates the flow stress at temperature over 870 K (Fig.3). At a very high strain rate (~10-6 s-1), PTW, VA and ZA2 describe the flow stress at an equivalent strain of about 0.2, but underestimate thereafter. In overall, the prediction capability of PTW (by using the new material parameters [14]) is remarkable up to moderately high strain rates (~103 s-1) as compared to any other models, and it could also be good at about 106 s-1 as well, provided the work hardening is  properly reflected, as aforementioned. Therefore, it is desirable to limit the attempt of improving a model or material parameter set to the PTW model. 

It has been attempted to account for both data sets from Maudlin et al. [8] and Nemat-Nasser and Isaacs [13] at moderately high strain rates (10-1~5´103 s-1) and the set achieved at a very high strain-rate (~106 s-1) by Duprey and Clifton [12]. It is very important to include such high strain-rate (~106 s-1) because the application of tantalum, e.g., EFP, imposes very wide range of strain rate, which is not limited to ~103 s-1. In order to reflect such a complicated demand, here we aims at the modification of the PTW model itself instead of finding a new material parameter set (Section 4). Then, its prediction capability has been checked in Section 5 by comparing with experimental data sets at both the very high strain-rate (~106 s-1: Duprey and Clifton [12]) and lower strain rates (10-1~5´103 s-1: Maudlin et al. [8] and Nemat-Nasser and Isaacs [13]).

(4) MODIFICATION AND CAPABILITY OF THE MODIFIED PTW 

In general, the strain-rate controlling mechanism at least up to 104 s-1 is the thermally activated interaction of dislocations with obstacles, usually other dislocations. At higher strain rates (109~1012s-1), the flow stress may be governed by the phonon-drag stress, which is the stress required to move dislocation through phonons, and not much hardening is associated with this process. In the intermediate regime (105 s-1~108 s-1), Preston et al. [6] adopted either the phonon-drag mechanism or thermally activated dislocation glide mechanism, whichever yields higher flow stress value. 

If only the phonon drag mechanism were operative at ~106 s-1 in the experiment by Duprey and Clifton [12], the flow stress would decrease with strain due to the thermal softening of material by the converted heat from the plastic work (Eq. (9)). However, in reality, no such thermal softening is observed (Fig. 5), but the flow stress increases slightly with strain. It means that there certainly exists a strain-hardening mechanism, i.e., the thermally activated dislocation glide, to compensate for the influence of the thermal softening. 

Thus, we aim at accounting for the observed strain hardening at ~106 s-1 [12] as well as at lower strain rates [8, 13] in terms of the mechanism of thermally activated dislocation glide. For this purpose, we first tried to extend application limit of Eq. (4) by finding new material parameter set to explain the strain hardening both at both 106 s-1 and lower rates, although Eq. (4) was applied only up to 104 s-1 to predict the strain hardening in Preston et al. [5]. The modification of the material parameters for such purpose was not successful in our separate analysis. 

Then, the strain hardening term in the original PTW model (Eq. (4)) has been modified next by employing the Voce equation,

where,  is the initial yield stress defined as 

and A, B, and C are material constants. A represents the maximum strain hardening amount for unit strain rate. B controls the saturation speed of the hardening. C represents the dependency of the maximum hardening amount on the strain rate.

Fig. 6 illustrates how the hardening behavior of tantalum is varied by the varying combination of the constants A, B, and C. As B increases, the hardening saturates rapidly. It is assumed that the maximum strain hardening is dependent on the strain rate. The constant C is obtained from the experimentally determined yield stress versus strain rate relation in log scale, as seen in Fig. 7, with the assumption that the effect of the strain rate on the maximum hardening is the same as the effect of the strain rate on the initial yield stress.  In Fig. 7, the experimental data were taken from Hoge and Mukherjee [16], Maudlin et al. [8], and Duprey and Clifton [12]. It is shown that the logarithmic value of the strain rate and the yield stress has a linear relation and the slope is presented as follows by the modification of Eq. (10).

From Eq. (12), the value of constant C is determined as 0.167. Constants A and B are determined by trials. The material constants for the modified PTW model are listed in Table 5.

Having modified PTW and determined the parameters for the modified PTW (the modified PTW model is referred to MPTW hereinafter), its capability of prediction has been checked by comparing with experiment as before. As seen in Figs. 8 through 11, the strain hardening of tantalum is well described in wide ranges of strain up to 1.5, a strain rate up to ~106 s-1, and temperature from room temperature to 1273 K.

(5) Conclusion

Four well-known constitutive models for plastic deformation of materials, i.e., Johnson-Cook (JC), Zerilli-Armstrong (ZA), Voyiadjis and Abed (VA), and Preston-Tonks-Wallace (PTW), have been compared with reference to existing deformation data of tantalum in wide ranges of strain, strain rate, and temperature. At a very low strain rate, e.g., 0.1 s-1, only PTW was reliable. At moderately high strain rates, e.g., ~103 s-1, the PTW, ZA and JC predict the experiment fairly well. At a very high strain rate, e.g., ~106 s-1, no models were successful, but PTW and VA describe the flow stress at an equivalent strain of about 0.2, followed by underestimation thereafter. Thus, strain hardening term of the PTW model was modified in the current work. The modified PTW demonstrated very good prediction for the deformation of tantalum in wide ranges of strain, strain rate, and temperature.

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