Theoretical analysis of the elastic Kelvin-Helmholtz instability in explosive weldings

Yuno Sun ,Jinning Gou ,Cheng Wng ,Qing Zhou ,Rui Liu *,Pengwn Chen ,Tonghui Yng , Xing Zho

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing,100081, China

b Beijing HIWING Scientific and Technological Information Institute, Beijing,100084, China

c School of Aerospace Engineering, Tsinghua University, Beijing,100084, China

Keywords:Explosive welding Hydrodynamic instabilities Elasticity

ABSTRACT By considering the joint effects of the Kelvin-Helmholtz(KH)and Rayleigh-Taylor(RT)instabilities,this paper presents an interpretation of the wavy patterns that occur in explosive welding.It is assumed that the elasticity of the material at the interface effectively determines the wavelength, because explosive welding is basically a solid-state welding process.To this end, an analytical model of elastic hydrodynamic instabilities is proposed, and the most unstable mode is selected in the solid phase.Similar approaches have been widely used to study the interfacial behavior of solid metals in high-energy-density physics.By comparing the experimental and theoretical results, it is concluded that thermal softening,which significantly reduces the shear modulus, is necessary and sufficient for successful welding.The thermal softening is verified by theoretical analysis of the increase in temperature due to the impacting and sliding of the flyer and base plates,and some experimental observations are qualitatively validated.In summary, the combined effect of the KH and RT instabilities in solids determines the wavy morphology, and our theoretical results are in good qualitative agreement with experimental and numerical observations.

1.Introduction

Explosive welding is a solid-state process in which explosive energy is used to create a high-velocity oblique collision between the flyer and base plates [1].This process can join a series of noncompatible materials, irrespective of the differences in their mechanical and chemical properties.In particular, explosive welding is applied to materials that cannot be joined by any other conventional means.To evaluate the welding quality, a striking feature is the emergence of a characteristic wavy morphology at the interface between the two work-pieces[2,3].The emergence of this wavy pattern, with its well-defined amplitude and wavelength, at temperatures below the melting point of the metal plates is generally taken as evidence of a successful weld [4-6].More generally,owing to the development of high-power devices such as high-power lasers [7-9], high-intensity heavy ion beams [10,11],and magnetically imploded cylindrical shells [12-15], analysis of the wavelengths and amplitudes of these perturbations has enabled the use of hydrodynamic instabilities in solids to determine the material properties under extreme conditions and to evaluate the experimental performance in high-energy-density physics [16,17].The underlying mechanisms that generate the wavy pattern at the interface due to these hydrodynamic instabilities are identical in explosive welding and in high-energydensity physics.The formation of the wavy of explosive welding is caused by the disturbance of the Helmholtz-Taylor combined instability and the properties of the materials [18].

Although hydrodynamic instabilities are frequently considered to occur in phenomena involving fluid matters[19,20],there are no fundamental difference between fluids and solids in terms of the continuum theory, apart from the constitutive properties that define them.Hydrodynamic instabilities in solids, such as the Rayleigh-Taylor (RT) instability driven by high pressure, is responsible for the generation of wavy morphologies.This was first experimentally observed by Barnes et al.,who used high explosives to accelerate solid metallic aluminum with a premachined corrugated surface to determine the mechanical properties of yield strength and shear modulus [21,22].Further theoretical and numerical analyses have demonstrated that both the initial perturbation amplitude and the perturbation wavelength determine the stability conditions of the solid plate[23-26].This fact constitutes a unique feature of the RT instability in accelerated solids.More recently,an exact asymptotic solution for the RT instability in solids was developed based on the normal mode method, verifying the experimental results in magnetically imploded cylindrical shells,where the most unstable mode in the elastic regime carries into the plastic regime [13].Based on the force balance method, Piriz et al.constructed the approximate model and discussed the evolution behavior of RT instability at the interface between elastic and plastic solids [10,11].In other words, the elasticity of the material determines the wavelength, while the plasticity controls the growth rate.

The Kelvin-Helmholtz(KH)instability in solids arises be-tween two materials of different densities when the tangential velocity is discontinuous at the perturbed interface.This instability plays important roles in explosive welding for industrial applications and in determining material properties under extreme conditions[27,28].Specifically,in the process of explosive welding to join two distinct and incompatible metals,the KH instability can be used to interpret the wavy morphology that determines the quality of the welding products [3,29].The mechanical properties of the materials strongly suppress the growth rate of the KH instability, and this suppression could in turn induce certain properties under extreme conditions,as widely investigated for the RT instability in solids [27].In experiments related to high-energy-density physics,the KH instability often occurs in shear flow, resulting in the turbulent mixing of materials and affecting the energy release.Although the KH instability has been thoroughly investigated in fluids and magnetized plasmas, no comprehensive understanding yet exists for solids under extreme conditions, and its effects in combination with other phenomena remain unknown.Few literature quantitatively concentrates on the joint effects of the RT and KH instabilities on the underlying mechanism for the formation of wavy patterns in explosive welding.

By carefully examining a physical schematic of an explosive welding process, as shown in Fig.1, it appears that the combined effects of the RT and KH instabilities determine the wavy pattern at the interfaces.The RT instability arises when the ultrahigh pressure generated by the high explosive pushes the flyer plate to accelerate the base plate.The KH instability arises when the shear velocity and tangential velocity generated by the oblique explosive create wavy patterns along the interface.The mechanical interactions in explosive welding satisfy the mechanisms of the abovementioned instabilities.Therefore, this paper first investigates the wavy morphologies by analyzing the coupling effects of the RT and KH instabilities in solids.The elasticity, which determines the most unstable modes, is compared with the theoretical value, because thermal softening may significantly weaken the elastic shear modulus.

The remainder of this manuscript is organized as follows.Section 2 presents a brief description of the governing equations for studying the joint effects of the RT and KH instabilities in solids.Section 3 compares the theoretical results with experimental findings to qualitatively validate the accuracy of this approach.Section 4 summarizes the conclusions from this study and suggests some prospective applications of hydrodynamic instabilities inelastic-plastic materials.

2.The mathematical formulations

We herein derive the dispersion relation of a shear-driven KH instability by including the acceleration-driven RT instability of elastic solids based on potential flow theory.The experimental configurations are shown in Fig.1,in which the flyer plate is driven at high velocity by a high-explosive to impact the base plate at an oblique angle, joining the two materials together.The impact velocity Vpand the shear velocity Vcsatisfy

in which β means the oblique angle between the flyer and base plates.At the interface, the KH and RT instabilities in the solids together form wavy patterns, providing an indication of the welding performance.To investigate these patterns analytically, we focus on the occurrence of the most unstable mode and how the driving pressure,temperature,and shear velocity Vcmay affect the generation of the mode of interest.Another method of forming the wavy morphology is to build up an elastic-plastic model, because most of the material remains solid during the welding process.To deal with the most unstable mode, we assume that the interface behavior is dominated by the KH and RT instabilities in the solid state, and select the most unstable wavelength in the elastic regime, which corresponds to theoretical analysis [13] and experimental results[12].Briefly,this issue can be interpreted as material 1 moving parallel to material 2 to present the classical KH instability,while the RT instability occurs as a result of the acceleration g generated by the detonation pressure in the form of g =P0/(ρ1h1+ρ2h2), where P0denotes the detonation pressure exerted on the flyer plate, ρ1, ρ2are the densities of the flyer plate and the base plate,respectively,and h1,h2are the thickness of the flyer plate and base plate,respectively.Throughout this paper,we assume that the interface evolves according to ξ(t)=A0eσtin the linear regime,requiring ξ(t)k≪1, where k=2π/λ is the wavenumber of the perturbation, ξ(t) is the amplitude of the perturbation, A0is the initial amplitude, and σ is the growth rate.The most unstable wavelength is selected based on the local physical parameters in the linear regime.The thickness hiof the fluid layers satisfies khi≫1, so the two materials involved in the KH instability can be regarded as being semi-infinitely thick.

First, we assume the materials are incompressible, which requires

where Uiis the perturbed velocity vector in the fluids and i = 1, 2 represents the materials separating the interface.Throughout this paper,we perform linear analysis of the hydrodynamic instabilities based on potential flow theory and ignore the vorticity, following several classical textbooks[19,30,31].The irrotational velocity field described by the potential functions can be obtained as follows:

where uiand wirepresent the velocity components parallel and perpendicular to the perturbed interface, respectively.As the flow is irrotational and incompressible, for a velocity potential φ satisfying the Laplace equation,it holds that

In each material, φican be written as

Fig.1.Schematic of explosive welding in metals.

where±k represents the decay modes,indicating that the velocity vanishes at infinity away from the perturbed interface, and A1, A2are coefficients related to the disturbance amplitude.Based on the expression for the potential function in Eq.(6), the perturbed velocity along the z direction can be obtained as follows:

In the linear regime and for an infinitesimal perturbation, the interface subjected to the KH instability is given by

Due to the existence of shear velocities,the kinematic boundary conditions at the perturbed interface can be expressed as

where P is the perturbed pressure at the interface, τijis the stress tensor depending on the constitutive property of the material,and g is the acceleration acting on the perturbed interface.

By substituting the expression for the normal velocity in Eq.(7)into Eq.(11)and integrating from the equilibrium state η=0 to the current state η=η(x,t) along the z direction, we obtain the pressure at the perturbed interface.

Linear analysis of hydrodynamic instabilities based on potential flow theory requires the continuity of stress tensors along the normal direction.Therefore, we have

Substituting Eqs.(6),(8)and(10)into Eq.(15),we then obtain an equation for the perturbation amplitude.

Recall that,for the KH instability in a solid,an analogy between elasticity and viscosity is given by μ ～G/(σ + ikU), because the interface deformation is determined by Eq.(9) and the uniform velocity affects the deformation of the interface.The equation for the perturbation amplitude of the KH instability between elastic solids can also be obtained by substituting Eq.(18)into Eq.(16),and the dispersion relation for the KH instability between elastic solids is found to be

Eq.(16) describes the interface behavior determined by the KH and RT instabilities depending on the material properties under extreme conditions.In many respects, the interface exhibits wavy patterns as a consequence of multiple hydrodynamic mechanisms.For example,the effect of a horizontal magnetic field,analogous to the surface tension effect, is widely used in magnetically accelerated flyer plates, in which the magnetic pressure, which can be higher than the order of magnitude of the target strength, significantly influences the instability growth when the plates retain their mechanical properties [14,32].In this paper, we focus on the mechanism that forms the wavy patterns at the interface in an attempt to understand the characteristic wavelength.The elasticity is essential for determining the most unstable mode,and once this mode is selected,it carries into the plastic regime to de-termine the amplitude of the wavy morphologies.

In other words, the wavelength selection is determined by the most unstable mode in the elastic regime,and this then determines its amplitude in the plastic regime,even in nonlinear cases[13].In an elastic solid, the constitutive relation reads

To discuss the effect of elasticity on the evolution of the interface, we first investigate a simple case in which we neglect the mechanical properties of the upper flow (G1=0) and assume that the lower material is stationary(u2=0).Therefore,the dispersion relation becomes

Using a similar mathematical approach, Eq.(20) can be rewritten as

where A = ρ1+ ρ2, Bi= kρ1u1, and CR= (ρ2- ρ1)gk- k2ρ1u21+2k2G2.

A perturbation with positive Atindicates that the KH and RT instabilities act on the disturbed interface together;in this case,the dimensionless velocities gradually increase and finally saturate as the wavenumber increases.Therefore,as long as the dimensionless velocity U of the upper fluid exceeds the corresponding saturation values, the interface will remain disturbed and there will be no cutoff wavenumber.This explains why the growth rates increase linearly when the values of U remain sufficiently high.

Interestingly, the cutoff wavenumber between elastic solids with positive values of Atsatisfies

which recovers the conclusion for the cutoff wavenumber of the RT instability in elastic solids when U = 0.Therefore, a perturbation with k0.Note that the cutoff wavenumber indicates that the stability region not only depends on the initial amplitude, but also on the wavenumber,as first proposed by Drucker[23]and then experimentally confirmed by Barnes et al.[22].This is a unique characteristic feature of hydrodynamic instabilities inelastic solids.

The most unstable mode could dominate the instability evolution and mainly affect the formation of the wavy patterns.The most unstable mode kmaxof interest can be obtained as follows:

By exploring the most unstable mode,we can fathom the wavy patterns and evaluate the performance of the explosive welding.

3.Results and discussions

3.1.Reduction of elastic moduli

In the typical explosive welding case, the parameters have typical orders of magnitude of λ ～10-3m, ρ ～103kg/m3, g ～109m/s2, Vc～1000 m/s.Selecting the wavelength to be of order λ ～10-3m and substituting the relevant parameters into Eq.(23),we have

From Eq.(24), the shear modulus is of order ～108Pa, which corresponds with the numerical result reported by Mousavi et al.for the Cu/Cu collision with a sliding velocity of Vc=650 m/s[33].Generally, a remarkable reduction in the elastic shear modulus is necessary for successful explosive welding, and the temperature plays an important role in softening the metals.In the next subsection,we verify that the temperature increment due to impacting and sliding during the welding process is sufficient to weaken the metals’ shear modulus.

By using AUTODYN,the welding process between the base plate and fly plate for the Cu/Cu collision has been calculated.The size of the fly plate is 3 mm×40 mm and that of the copper substrate is 15 mm×40 mm.The sliding velocity Vpis 650 m/s and the oblique angle is 15°.And the other physical parameters are similar to those used in Ref.[31].The numerical result of wavy pattern for the Cu/Cu welding is shown in Fig.2.Through the numerical simulation,when the yield strength of Cu is reduced to 23 GPa,the wavelength of wavy patterns in explosive welding can be about 0.75 mm that well corresponds with our theoretical predictions.

3.2.Heat increments

The heat effect plays a critical role in the quality of explosive welding.The interface temperature rises in response to two factors:the impact between the two plates and the sliding at the interface.The flyer plate is driven by an industrial explosive to attain its initial velocity V, which follows the Gurney equation [34] and reads

where C and M represent the unit area masses of the explosive and metal,respectively,and 2E is the Gurney characteristic velocity for the industrial explosive.E can be estimated by[35].

where D is the detonation velocity of the explosive and γ=3 is the constant polytropic coefficient for a perfect gas.When the flyer plate impacts the base plate with velocity V, the full theory only considers the elastic state, and so the pressure P at the impacting point can be calculated by elasticity wave theory as

where (ρc)Tiand (ρc)Fedenote the acoustic impedance of the two plates,respectively.

In the case of a titanium flyer plate and a steel base plate, the pressure at the interface will be around 11 GPa under the condition of an industrial explosive.Further, according to the shock compression energy [36], the temperature increment can be estimated as around 411 K.

During explosive welding, high-speed sliding occurs at the interface,and the frictional heating induces a temperature rise.The frictional heat generation power Q is given by[37].

where μ denotes the frictional coefficient between the flyer plate and the steel base plate,P represents the pressure at the interface,and Vsis the sliding velocity.Generally,for most metal materials,μ ranges from 0.1 to 0.5.Vsis considered as the detonation velocity of the industrial explosive.Under energy conservation, the temperature can be estimated as

Fig.2.The wavy pattern for the Cu/Cu welding.

where Δt is the sliding action time,ρ is the material density,and CVis the material specificheat capacity.Again,for a titanium flyer plate and a steel base plate,the temperature ΔT would exceed 1000 K.

The temperature of the metal material subjected to the impact compression can be estimated by the Grüneisn equation of state,as shown in the following equation:

Combined with the effects of shock compression,the increment in temperature during the welding process is sufficient to soften the solid metal and enable successful welding.Consequently, the increment of temperature will lead to the decrease of the shear modulus of the metal material,as described in the following Fig.3 that describes the shear modulus(represented by the blue line)and the shock temperature (represented by the red line) of the impacted material as a function of the shock pressure.During the welding process, the impact pressure will cause the temperature increment of the impacted metal.The higher the impact pressure,the faster the shock temperature increases.When the impact pressure increases to 90 GPa, the temperature will increase by about 1500 K,and in this case the shear modulus of the metal may drop to zero.The heat increments will lead to weakening the metals' shear modulus, which is necessary to achieve successful welding.

The variation of the metal shear modulus is dictated by its thermodynamic state during the impacting and sliding interactions, and the magnitude of this modulus can be estimated using the Steinberg-Cochran-Guinan constitutive equation [38].The elastic shear modulus is governed by

where G0refers to the reference state and G′p, G′Tare materialdependent parameters [38].For a titanium/steel collision, the temperature rise is sufficient to soften the materials as expected by Eq.(24).

3.3.Effect of flyer plate thickness

The correlation between the thickness of the flyer and base plates and the wavelength of the wavy patterns has been demonstrated through experiments and numerical simulations[33,39].Experiments performed by Al-Hassani et al.first showed that increasing the thickness of the flyer or base plate increases the wavelength of the deformation at the interface [39].To determine the correlation, Eq.(23) can be rewritten as

A similar correlation has been observed in numerical simulations[40],where Mousavi carried out simulations of 1.5-,3-,and 6-mm-thick copper flyer plates impacting a 15-mm-thick base plate at a velocity of 650 m/s and an angle of 15°[33].The thickness of the flyer and the corresponding wavelength given by our present model and found by Mousavi are presented in Table 1.Our theory yields a correlation between flyer thickness and wavelength that is within 8% of the numerical results provided that κ remains unchanged during the explosive welding process.

3.4.The effect of plasticity

It is well recognized that the elasticity determines the most unstable wavenumber while the plasticity dominates the evolution of the perturbation amplitude.Therefore, to well capture the interfacial behavior,one needs to explore the effect of plasticity on the KH instability, which, however, has not been fully studied yet.Fortunately,in high energy density physics,when the solid metals are accelerated by shock waves or by ultrahigh pressures, the interfaces are subjected to the RM or RT instabilities and the elasticplastic properties will strongly suppress the interfacial growth[10,11,41,42].More importantly,Dimonte et al.and Polavarapu et al.respectively confirmed that the RM and RT instabilities act as powerful tools to diagnose the material properties under extreme conditions [43,44].By solely considering the plasticity, the temporal evolution of the amplitude, which turns out to be

where the coefficients a, b, c, and d are defined as

In short, the previous studies about RT and RM in elasticplastic materials could allow us to examine their effects on KH instability due to those similarities.

Fig.3.The variation of the shear modulus and shock temperature with the impact pressure.

Table 1 Comparisons of the wavelength of the wavy patterns from Ref.[31]and the present theory with varying thickness of the flyer.

4.Conclusions

In this paper, we first proposed that the joint effects of the RT and KH instabilities are the underlying mechanism for the formation of wavy patterns at the interface in explosive welding.To verify this theory, the configurations of the conventional RT and KH instabilities driven by high explosives and high-power lasers were elaborated and compared, and it was found that the coupling effects of the RT and KH instabilities account for the generation of wavy patterns in explosive welding.In addition, because this process is basically solid-state welding, it was assumed that the welding occurs in the solid state and the most unstable mode was selected in the elastic regime, as widely used for the RT instability in metals driven by high explosives and magnetic liners.

A comparison with experimental results shows that our formulation gives a qualitatively good approximation and illustrates that a significant reduction in the elastic shear modulus is necessary to achieve successful welding.The increase in temperature as a result of impacting and sliding during the welding process creates a strong softening of the solid metals.Several theories on the variation of the temperature suggest that the temperature rise is sufficient to soften the metals, in agreement with our assumptions and Steinberg’s model.This simple model accurately recovered the experimental correla-tions between flyer plate thickness and wavelengths,allowing us to optimize the experimental results by focusing on the con-stitutive relations during the impact process.

In short, this paper has described how both the RT and KH instabilities can be incorporated to interpret the formations and structures of the wavy morphologies formed at the interface during explosive welding.Theoretical analysis showed that a remarkable reduction in the shear modulus is essential for successful welding,and the temperature rise in the welding process was found to be in agreement with the theoretical results.Additionally, our theory validates the correlations between wavelength and plate thickness obtained through numerical simulations.However, this theory is limited to investigating the effect of elasticity as the most unstable mode.To complete this theory,an elastic-plastic model is required to understand the wavelength selection and amplitude evolution during the explosive welding process.This will be presented in a future publication.

Data availability

The data that supports the findings of this study are avail-able within the article.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant Nos.12002037 and 12141201).