Analytical Model for Dynamic Yield Strength of Metal

Strain rate effect of yield strength has been a hot topic for a long time in impact mechanics over decades, and it is important to explore the physical mechanism behind this phenomenon. In this study, a one-dimensional stress bar analytical model for the dynamic yield stress of metal materials under a sinusoidal stress wave pulse is presented based on the structural-temporal failure criterion, and the corresponding numerical results accord well with previous experimental data under the high strain rate. Moreover, the dynamic yield strength can be determined by the nondimensional parameters κ and χ as well as a material parameter α. Specifically, the first nondimensional parameter κ can be determined by the ratio between the loading amplitude and quasi-static yield strength. The second nondimensional parameter χ is calculated by the ratio between the loading period and the incubation time. 0herefore, the dynamic yield strength can be essentially determined by the quasi-static material parameters, incubation time and loading parameters. The so-called strain-rate effect on the yield strength should be treated as an interaction process parameter in a dynamic loading—material system and should not be considered as an intrinsic material property anymore. additionally, this study may help researchers to determine the parameters in the numerical models including the strain rate effect of the dynamic yield surface.


INTRODUCTION
The dynamic yield strength (or flow stress) for most metal materials and alloys can rise to a very high value at high strain rate (generally is greater than 10 3 s 1 ). .or decades, many papers [14] have been opened on such an issue for various metal materials. So far, there is still no recognized model to explain the relation between the dynamic yield strength and strain rate. In the well-known material constitutive models of metals, such as, JohnsonCook model [5,6] and Stein-bergCochranGuinan model [7], the relation between dynamic yield strength and strain rate is based on the experimental fitting method and empirical equations. They results in a classical guide ideology, which is the dynamic yield strength is considered as intrinsic material property, and it leads to dynamic yield strength only be measured instead of predicted. Such a guide ideology leads to the dynamic yield strength theory, where an instantaneous yielding begins when a certain stress condition is achieved. This criterion is directly extended from classical ultimate static yield stress criterion but excludes time effect to simplify problems. However, this classical criterion weakens the physical meaning for yielding and could further make more errors in the scenarios of high strain rate. Meanwhile, recently some researchers [8,9] have obtained the dynamic response of concrete prism by using the numerical methods based on some quasi-static material parameters and a quasi-static material model. In our previous works, we explicitly put forwarded that the dynamic strength of brittle materials could be calculated by means of the quasi-static material parameters and the boundary conditions, the corresponding physical mechanism were explained and applied in spalling study [1012]. Here, we try to extend this idea to metal materials.
In addition, the structural-temporal criterion is introduced to derivation in this study. The criterion was firstly proposed to explain dynamic strength of brittle materials by Petrov and Utkin [13], and further applied to calculate the dynamic fracture of brittle solids [14,15]. Petrov et al. developed this criterion to investigate other phenomena, such as dynamic plasticity, electrical breakdown, phase transformations and solid particle erosion, under high strain rate conditions [1618]. Moreover, this criterion was applied to explain the dynamic yielding and fracture of metals in the view of multiscale levels and study crack propagation [1924]. The temperature effect in dynamic yielding was also considered [2527]. .urthermore, Gruzdkov and Petrov emphasized that the incubation timebased criterion of dynamic yielding was meaningful to predict limiting loading parameters of plastic deformation and the instability of strain rate dependencies of dynamic yield limit [28]. In recent studies [29,30], Petrov et al. explored the microcosmic background hidden behind macrocosmic dynamic yielding, within which a function from dislocation energy was introduced to quantify the incubation time or relaxation time. The criterion was used to study the viscous-brittle transition effect and the wide range for dynamic yield strength under strain rate effect [31,32]. However, the value for strain rate was always considered as constant in these models, while experimental results show far more complex stress waveforms. Therefore, a more general loading boundary condition should be taken into consideration.
The current work presents an analytical model to investigate the strain-rate effect on the dynamic yield strength by means of applying a one-dimensional bar model with stress wave theory. Under the consideration of a sinusoidal loading boundary condition is taken into consideration, an explicit analytical expression for the dynamic yield strength of perfect plastic materials is derived and a numerical example is conducted to validate the analytical solution. .urthermore, the authors discuss the analytical solution and explore the parameters within it.

THEORETICAL MODEL, SOLUTION AND ANALYSIS
In this paper, the dynamic yield strength for metal materials just before yielding is studied. In order to simplify the deduction, the bar is treated as perfectly plastic, thus the material can be considered as an elastic material before equivalent stress achieves yield surface. Accordingly, a one-dimensional stress linear elastic bar model is presented. Under the dynamic loading acting on the left end of the bar, a one-dimensional stress plane longitudinal wave begins instantaneously at t = 0 to propagate along the positive x axis direction (.ig. 1), which can be described by where u is the particle displacement, ρ is the plane longitudinal wave velocity of the bar, and E and ρ are Youngs modulus and the mass density, respectively.
.or convenience, the bar is assumed to be initially stress-free and at rest. The sinusoidal stress waveform is taken as the boundary loading (.ig. 2), which can be written as: where 0 σ and T are the amplitude and the half cycle of the loading, respectively, and 1 t is the time variable used on the boundary x = 0. Moreover, we take 1 2, t T ≤ which implies that the yield of materials during unloading is not considered.
.ig. 1. One dimensional strength plane longitudinal wave in the bar (color online). (3), the displacement field in the bar can be obtained as Thus the strain and strain rate fields in the bar can be deduced as It means there is no particle motion before stress wave arrives b ( ), t x c < and only the particle motion after wave arriving By assuming that the yield is independent on the location, we can adopt time-axis shifting y b t x c ξ = − to make the time when the stress wave arrives at the yielding location y x x = at new initial time ξ = 0, the strain, stress and the strain rate fields described by Eqs. (5)(7) can be modified as The structural-temporal criterion [17,28] for a metal material can be written as where Y is the quasi-static yield strength, α is a constant exponent for certain perfectly plastic material, and τ is the incubation time for this metal resist deformation from elastic deformation to plastic deformation. It should be noted here that the structural-temporal criterion is different from the one adopted for brittle materials, where a new parameter α is introduced and more details is presented in the discussion.
In order to obtain the dynamic yield strength, some important deductions are given as follows. .irstly, the left side of the inequity (11) can be rewritten by combining Eq. (9) as y y According to Gruzdkov and Petrov [28], the constant α changes from 12 to 30. Assume α is integer, and then the integral term can be derived by using Eulers identity and then binomial theorem, Taylor series as following: By adopting Eulers identity again, (2 ) i k x e −α can be rewritten as (2 ) cos (2  ) sin (2  ) .
By combining Eqs. (13) and (14), the expression for sin x α can be expressed as following: According to the equity for the real parts in Eq. (15), we have: is odd, sin ( 1) ( 1) cos(2 ) , 2 is even.
By combining Eqs. (12) and (16), we have the following equations as follows: When α is odd, then the left side of the structuraltemporal criterion Eq. (11) can be rewritten as When α is even, then the left side of the structuraltemporal criterion Eq. (11) can be rewritten as y y 2 0 0 ( , ) d Thus the left side of Eq. (12) can be written as following 1 ( 1) , 2 2 when is even.
Assume the material yields at y , t t = and then the dynamic yield strength can be written as following by Eq. (9): In order to get the analytical solution for dynamic yield strength, y t can be eliminated by combining Eqs. (19) and (20). Then we can obtain the implicit function for dynamic yield strength y σ as following: Taking the following two nondimensional parameters Then the implicit function for dynamic yield strength y σ can be rewritten as  (23) .urther, the dynamic yield strength d σ can be expressed as following by solving Eq. (23) above: d ( , , ), f σ = κ χ α (24) which indicates that the relation between dynamic yield strength d σ and the strain rate ε can be deduced without introducing any dynamic parameters.
In addition, note that the strain rate is which means that strain rate can be completely determined by external loading parameters and material constants.
.inally, the relation between dynamic loading carrying capacity and strain rate can be deduced by eliminating y t by Eqs. (23) to (25) as d d ( , , , ). f σ = ε κ χ α (26) Therefore, we can also obtain the relation between dynamic yield strength and strain rate without introducing any dynamic parameters.

DISCUSSION
In this section, the dynamic behavior for yield strength of metal materials will be further explained.
.irst, in order to validate the analytical solution of Eq. (23), the previous experimental data for cold-rolled plain carbon steel 1018 [3] is compared with our model. As shown in .ig. 3, the solution can agree with the experimental data well. Here the quasi-static materials parameters are listed as following: Y = 370 MPa, E = 200 GPa, τ = 0.1 µs, α = 12 and the loading parameter T = 1.44 µs. It should be mentioned that the incubation time τ for metals can be considered from several microsecond to 0.1 µs [34]. In our work, the incubation time τ is considered as the lower boundary. Meanwhile, the material response constant α can be approximately taken as 12 in this work, which is similar with the number used for wild steel [16]. By changing loading amplitude, the relation between dynamic yield strength and different strain rates can be obtained. It is shown in .ig. 3, the analytical solution can agree well with the experimental data in the range of 6800 s 1 < ε < 7800 s 1 , while some deviation still exists in other ranges, which can be explained in the following. It is because that the shape of the loading in the experiments was not easy to be controlled as a perfect sinusoidal wave, while the theory developed in this work assumes that the shape of the loading has to follow a perfect sinusoidal wave. To some extent, the theory result can still be acceptable and match with experimental data well. In addition, a similar theory has successfully been developed to predict the tensile strength of concrete only below strain rate ε = 150 s 1 in our previous work [10]. Current work has improved the theory applied for metal under a large strain rate range. The advantage of the theory is only a few of parameters can determine the dynamic strength of metal.
Second, the material parameter α can be discussed further. According to Campbell [2], the approximated relation between dynamic yield strength d σ and the material response constant α is where s T represents the static loading half cycle, and is assumed to be a great enough value (several seconds). .urther, the material response constant α can be obtained as s d log ( ) . log ( ) By combining with Eqs. (23) and (28), the semianalytical solution for dynamic yield strength d σ can be rewritten as d ( , ), f σ = κ χ (29) which means the dynamic yield strength of metal materials can be determined completely by the two nondimensional parameters κ and χ. According to the definition of the two parameters, it demonstrates that the dynamic yield strength of metal materials is just related to the quasi-static material parameter and the exterior loading boundary conditions. On the one hand, it stresses that the thoughts by adopting experiments to search the relation between dynamic yield strength and strain rate should be reconsidered. On the other hand, some methods to measure the incubation time should be developed. .urthermore, the term for dynamic yield strength is no longer suitable and thus dynamic yield capacity should be applied. To be mentioned, the only additional parameter to determine dynamic strength of metal materials is the parameter α, but it is probably deleted by using relation Eq. (29). However, only perfectly plastic materials are considered in this study, which means only linear effect of strain rate is considered while hardening and plastic wave effect are omitted. If the hardening is considered, a nonlinear effect is then introduced, which in turn results in more complex problem. Therefore, the parameter α could take effects in that case, which will be discussed in the future work.
It should be noted that the incubation time τ governs the material yield in this work, which has been studied in previous studies. .or example, Selyutina and Petrov [32] predicted more accurate yield stress curves than the JohnsonCook model for different metals under a wide range of strain rates based on the incubation time criterion. .rom this point of view, we believe that the incubation time can be considered as a static parameter. Additionally, the incubation time for metals is recently considered to be associated with the certain types of lattice defects or whiskers [29,30]. According to Refs. [29,30], the incubation time for metals connects with impurity atmospheres on dislocation lines and dislocation starvation for different cases. In detail, the incubation time or relaxation time is defined as where G, D E and b represent shear modulus, the total elastic energy of dislocation lines per unit volume and the Burgers vector of the dislocation, respectively, characterizes the scattering rate for the kinetic energy of dislocations, D L~10 13 s 1 is the frequency close to the Debye frequency. Meanwhile, as indicated in Ref. [32], the incubation time can be obtained from the static experiment instead of dynamic experiments. In addition, it should be emphasized that the yield strength is sensitive to temperature according to the previous study [27]. But in this study, in PHYSICAL MESOMECHANICS Vol. 22 No. 4 2019 order to simply the problem, the authors assume that the dynamic yielding occurs in a fixed temperature.

CONCLUSIONS
In summary, some key conclusions are presented as following.
The analytical solution for dynamic yield strength of metal can be derived by combing the one-dimensional bar model with stress wave theory.
The incubation time based analytical model can predict the relation between the dynamic yield strength and strain rate well within the strain rate 6000 10 000 s 1 , which is a wide and high strain rate range. It should be emphasized that in the previous study [10], we predicted the tensile strength of brittle materials in case of low strain rate. Current work extended the incubation time model well.
The dynamic yield strength of metal materials can be determined by two nondimensional parameters κ and χ, material parameter α and loading boundary parameters. Note that the nondimensional parameter κ is calculated by the maximum loading value and the quasi-static yield strength, while the nondimensional parameter χ is calculated by the period of the loading T and the incubation time τ.
Since the dynamic yield strength of metal materials can be completely determined completely by the quasistatic material parameters, incubation time and loading parameters, that is, the so-called strain-rate effect on the yield strength, should be treated as an interaction process parameter in a dynamic loading material system under a general stress wave. It strongly supports the conclusion previously inferred in [17,32] that the strain rate effect is not an intrinsic material property.