The linear analysis of the Rayleigh-Taylor instability in metal material is extended from the perfect plastic constitutive model to the Johnson-Cook and Steinberg-Guinan constitutive model, and from the constant loading to a time-dependent loading. The analysis is applied to two Rayleigh-Taylor instability experiments in aluminum and vanadium with peak pressures of 20 GPa and 90 GPa, and strain rates of 6 × 10 6 s −1 and 3 × 10 7 s −1 respectively. When the time-dependent loading and the Steinberg-Guinan constitutive model are used in the linear analysis, the analytic results are in close agreement with experiments quantitatively, which indicates that the method in this paper is applicable to the Rayleigh-Taylor instability in aluminum and vanadium metal materials under high pressure and high strain rate. From these linear analyses, we find that the constitutive models and the loading process are of crucial importance in the linear analysis of the Rayleigh-Taylor instability in metal material, and a better understanding of the Rayleigh-Taylor instability in metals is gained. These results will serve as important references for evolving high-pressure, high-strain-rate experiments and numerical simulations.
The perturbed interface in a metal material will undergo growth and mixing induced by interfacial instability under strong shock, high acceleration, and shear driving. There are numerous research works [
The RTI in metals has been studied by many authors, since the work of Miles in 1968 [
In this paper, we derive the equation of the perturbation amplitude based on the energy balance. There are three improvements over the classical linear analysis of the RTI in metals presented in this work. First, because the constant pressure driving used in previous works differs from the actual experimental driving process and is not applicable for a quantitative analysis of experiments, we introduce time-dependent pressure driving into the linear analysis of the RTI in metals. Second, we perform a linear analysis of the RTI in metals based on the Johnson-Cook (JC) and Steinberg-Guinan (SG) constitutive models. However, in the previous published papers [
For the linear analysis of interface instability of a finite thickness plate (
u x = ξ ˙ e − k y sin k x , u y = ξ ˙ e − k y cos k x . (1)
where ξ ( t ) is the perturbation amplitude on the interface (y = 0), k = 2 π / λ is the wave number.
The average potential and kinetic energies within a wavelength in the accelerated reference frame that moves with the plate are, respectively
〈 V 〉 = − 1 4 ρ g ( 1 − e − 2 k h ) ξ 2 ( t ) (2)
〈 T 〉 = 1 4 ρ k ( 1 − e − 2 k h ) ξ ˙ 2 ( t ) (3)
where ρ is the material density, g = P / ρ h is the acceleration of the plate, P is the driving pressure, h is the thickness of the plate. In a typical planar Rayleigh-Taylor experiment, the driving pressure is generated by a flowing plasma atmosphere [
Based on the energy balance method of Mises [
∂ ∂ t ( 〈 T 〉 + 〈 V 〉 ) + ∫ 0 h 〈 W ˙ 〉 d y = 0 (4)
where W ˙ = S i j D i j is the stress power, S i j is the deviatoric stress tensor, D i j is the deformation rate tensor.
The aforementioned analysis shows that it is very important to compute the stress power in the linear analysis. For the incompressible continuum media, the deformation rate tensor is
D i j = 1 2 ( ∂ u i ∂ x j + ∂ u j ∂ x i ) = k ξ ˙ e − k y [ cos k x − sin k x − sin k x − cos k x ] (5)
Based on the elastic constitutive relation, deviatoric stress tensor follows
S ˙ i j = 2 G D i j (6)
Under the initial condition ξ ( t = 0 ) = ξ 0 , the deviatoric stress tensor is
S i j = 2 G k ( ξ − ξ 0 ) e − k y [ cos k x − sin k x − sin k x − cos k x ] (7)
So, the stress power in the elastic range is
W ˙ = S i j D i j = 4 G k 2 ( ξ − ξ 0 ) ξ ˙ e − 2 k y (8)
While the material deformation is beyond the elastic range, the stress state can be calculated with radial return algorithm [
s * = 3 S i j S i j / 2 = 2 3 G k ( ξ − ξ 0 ) e − k y (9)
Here, we take the equivalent stress at y = 0 as the effective yield stress σ eff , namely
σ eff = s * ( y = 0 ) = 2 3 G k ( ξ − ξ 0 ) (10)
If σ eff < σ ¯ , the deformation is purely elastic,
s i j = S i j (11)
If σ eff ≥ σ ¯ , the deformation is elastic-plastic, the deviatoric stress tensor s i j is
s i j = σ ¯ / σ eff S i j (12)
σ ¯ is the material yield stress.
Using these results, the stress power integration is obtained,
∫ 0 h W ˙ d y = { 2 G k ( ξ − ξ 0 ) ξ ˙ ( 1 − e − 2 k h ) , σ eff < σ ¯ σ ¯ / σ eff 2 G k ( ξ − ξ 0 ) ξ ˙ ( 1 − e − 2 k h ) , σ eff ≥ σ ¯ (13)
Taking the average kinetic and potential energies and the stress power integration into the energy balance Equation (4), and assuming that ξ ˙ ≠ 0 , the following evolution equation of the perturbation amplitude of the RTI is obtained:
ξ ¨ − g k ξ = { − 4 G k 2 ( ξ − ξ 0 ) / ρ , σ eff < σ ¯ , − σ ¯ / σ eff 4 G k 2 ( ξ − ξ 0 ) / ρ , σ eff ≥ σ ¯ , (14)
where ξ ( t ) is the perturbation amplitude on the interface, ξ 0 is the initial perturbation amplitude, k = 2 π / λ is the wave number, ρ is the material density, G is the material shear modulus, g = P / ρ h is the plate acceleration, P is the driving pressure, and h is the plate thickness. When the perfect plastic constitutive model is used, the analytic solution is obtained from (14). Here, the JC and SG constitutive models are applied. The JC constitutive model [
σ JC = [ A + B ( ε p ) n ] [ 1 + C ln ε ˙ ε ˙ 0 ] [ 1 − ( T − T r T m − T r ) m ] , (15)
where σ JC is the yield stress of the JC model, A is the initial yield stress at a reference temperature, B and n are the material strain hardening parameters, ε p is the plastic strain, C and ε ˙ 0 are the strain-rate hardening parameter and the defect density characteristic parameter, respectively, m is the heat softening parameter, T r and T m are the reference temperature and the melting temperature, respectively, and commonly T r = 300 K . Because density and pressure effects are not considered in the JC model, the model is only applicable to the low-pressure region. Moreover, the yield stress is simply linearly related to the logarithmic strain rate in the JC model, so the model cannot characterize the transition from a dislocation slide to a dislocation drag mechanism of distortion, restricting the applicable validity of the model to the condition in which strain rate < 104 s−1. The pressure, temperature, and strain-rate terms are added into the elastic-plastic constitutive equation of the SG model [
G ( P , T ) = G 0 [ 1 + 1 G 0 ( ∂ G ∂ P ) 0 η − 1 / 3 P + 1 G 0 ( ∂ G ∂ T ) 0 ( T − 300 ) ] ,
σ SG = Y 0 ( 1 + B ε ) n [ 1 + A η − 1 / 3 P − α ( T − 300 ) ] ,
respectively, where Y 0 is the initial yield strength, G 0 is the initial shear modulus, ( ∂ G / ∂ P ) 0 and ( ∂ G / ∂ T ) 0 are the derivatives of the shear modulus with respect to pressure and temperature at the initial condition, respectively, A and a correspond to ( ∂ G / ∂ P ) 0 / G 0 and ( ∂ G / ∂ T ) 0 / G 0 , respectively, B and n are strain hardening parameters, and η = ρ / ρ 0 is the compression ratio. The SG model is independent of strain rate formally, but it limits the extent of the strain rate; namely, the strain rate must be >105 s−1. The reason for this limitation is that, in the SG model, one assumes that the intenerating effect induced by the temperature rise under high-velocity impact counteracts the hardening effect of the strain rate, resulting in a stain-rate-independent constitutive equation. The SG model is the most widely applied high-pressure constitutive model at present, which is capable of characterizing the yield stresses of many metals at high pressures.
The effect of ultrahigh pressures on the material strength was studied using an Al-6061-T6 plate with a pre-imposed sine perturbation driven by plasma ramp loading on the Omega laser, where the peak pressure was ~20 GPa and the average strain rate was ~ 6 × 10 6 s − 1 . During the course of the experiment, the loading rate is not so fast that the compression wave evolves into a strong shock, so the aluminum sample remains shock-free, with less of a temperature rise. In ref. [
In contrast to the constant pressure driving in previous work, here, a time-dependent pressure driving term P ( t ) = 7.5 × 10 − 3 ( t + 20 ) 4 e − 0.2 ( t + 20 ) , where P is in GPa and the unit of t is nanoseconds, is introduced to approximate the experimental driving, as shown in
A (GPa) | B (GPa) | C | Tm (K) | n | m |
---|---|---|---|---|---|
0.3243 | 0.11385 | 2.0 × 10 − 3 | 950.2 | 0.42 | 1.34 |
G0 (GPa) | ( ∂ G / ∂ P ) 0 | ( ∂ G / ∂ T ) 0 (GPa・K−1) | Y0 (GPa) | B | n | A (GPa−1) | α (kK−1) |
---|---|---|---|---|---|---|---|
27.6 | 1.7940 | −0.017 | 0.29 | 125.0 | 0.10 | 0.0652 | 0.616 |
region of pressure. In
The effects of three different constitutive models―namely, the JC, SG, and PP models―on the RTI in metals are further analyzed under time-dependent pressure, as shown in
growth factor with the three different constitutive models, which shows that the result with the SG model agrees with the experiment quantitatively but the results with the JC and PP models depart from the experiment. The aforementioned results indicate that the SG model is applicable for the high-pressure, high-strain-rate loading condition with a peak pressure of about 20 GPa and a strain rate of ~106 s−1 in the linear analysis. As shown from
The linear analysis method can be applied to Lorenz’s experiment with loading pressure and strain rate of the are ~20 GPa and 6 × 10 6 s − 1 . Whether or not the linear analysis is applicable to an experiment with a higher loading pressure and a higher strain rate is not clear. Hence, the linear analysis was performed on the RTI in vanadium, which was quasi-isentropically plasma-driven on the Omega laser by Park et al. [
Because the loading pressure and the strain rate in Park’s experiment are substantially beyond the applicable extent of the JC model, here only the SG model is applied in the linear analysis. Both constant and time-dependent pressure
G0 (GPa) | ( ∂ G / ∂ P ) 0 | ( ∂ G / ∂ T ) 0 (GPa・K−1) | Y0 (GPa) | B | n | A (GPa−1) | α (kK−1) |
---|---|---|---|---|---|---|---|
48.1 | 0.4906 | −0.0099 | 0.60 | 10.0 | 0.10 | 0.0102 | 0.206 |
drives are considered, and the time-dependent pressure is P ( t ) = 14.53 ( t + 10 ) 1.4 e − 0.14 ( t + 10 ) , where the unit of t is nanoseconds, as shown in
Based on the energy balance equation, the linear equation of RT interfacial instability for finite thickness plate metal is derived. The constitutive model of materials is extended to JC and SG constitutive models. Using the linear analysis method, we numerically solve the perturbation growth experiment of the RT instability on the Omega aluminum laser and the quasi-isentropic loading of aluminum and vanadium. We have obtained the growth evolution of metal aluminum and vanadium interface perturbation and compared it with the experimental results.
The main results of this paper are as follows: 1) Some nonlinear effects are introduced into the linear analysis by the imported time-dependent pressure and the JC and SG models, and the new method is more applicable to the nonlinear regime of the RTI in metals than the classical method with a constant pressure and an ideal constitutive model. 2) For the RTIs in aluminum and vanadium
quasi-isentropically plasma-driven with high pressures and high strain rates > 105 s−1, the computed results obtained by implementing the SG model in the linear analysis are quantitatively comparable to the experimental data; however, these extreme conditions of high pressures and high strain rates are beyond the applicable extent of the JC model. 3) At the condition of a peak pressure of ~100 GPa and a strain rate of 106 - 107 s−1, the linear analysis of the RTI in aluminum and vanadium is also applicable. From the linear analysis of the RTI in metal material, it is found that constitutive models and loading processes are both important, and a better cognitive understanding of the RTI in metals is gained. These results will serve as important references to evolving high-pressure, high-strain-rate experiments and numerical simulations.
This research was supported by the National Natural Science Foundation of China under Grants No. 51572208, and the Joint Fund for Equipment pre Research of the Ministry of Education of China No. 6141A02022209.
Bai, X.B., Wang, T., Zhu, Y.X. and Luo, G.Q. (2018) Expansion of Linear Analysis of Rayleigh-Taylor Interface Instability of Metal Materials. World Journal of Mechanics, 8, 94-106. https://doi.org/10.4236/wjm.2018.84008