c Allerton Press, Inc., 2014. ISSN 0025-6544, Mechanics of Solids, 2014, Vol. 49, No. 6, pp. 649–656. c A.E. Mayer, 2014, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2014, No. 6, pp. 58–67. Original Russian Text
Dynamic Shear and Tensile Strength of Iron: Continual and Atomistic Simulation A. E. Mayer* Chelyabinsk State University, ul. Br. Kashiribykh 129, Chelyabinsk, 454078 Russia Received July 20, 2014
Abstract—In this paper, continual and atomistic simulations are used to investigate the shear and spall strength of iron under high-rate strain conditions. The continual simulation is based on the use of models of dislocation plasticity and fracture due to formation and growth of microcracks and cavities; the molecular-dynamic simulation is based on the use of the LAMMPS software. The obtained results are analyzed in the light of experimental data for the high-speed impact and irradiation of iron films by ultrashort pulses of intense laser radiation. DOI: 10.3103/S0025654414060065 Keywords: high-rate strain, plastic flow, spall fracture, iron, molecular dynamics, continual theory of dislocations, fracture model.
1. INTRODUCTION The phenomena of high-rate deformation and fracture of metals brightly exhibit the specific characteristics of elementary processes of variations in the existing structure defects of the material and formation of new structure defects (dislocations, twins, microcracks, cavities); namely, the characteristic time of deformation becomes comparable with the time of elementary processes in defect ensembles as the strain rate increases. The experimental data on the dynamic shear and spall strength are of special importance in the study of these phenomena. The high-speed impact [1–4] and irradiation by ultrashort laser pulses [5, 6] are now actively used for dynamic loading of materials. The interferometric measurement of the velocity profile of the free surface of the loaded target is one of the most informative diagnostics of the process [2]. The study of loading of thin metal films [5–10] is of special interest, because the ultimately high strain rates are then attained, up to 0.1–1 ns−1 . Films of thickness up to several dozens of micrometers can be loaded by high-speed impacts [9, 10], and the thinner films are investigated by using ultrashort laser pulses [5–8]. For example, the reaction of iron films of submicron thicknesses to irradiation by powerful femtosecond laser pulses was studied in [6]. It was discovered that, under such conditions, the shear and tensile stresses in the material attain the values of the order of 8 GPa and 20 GPa, respectively, which is close to their theoretical ultimate values. Generalizing these data and the results of shock-wave tests for targets of greater thicknesses, one can construct experimental dependencies of the shear and spall strength of iron in a wide range of strain rates [11]. In the present paper, the experimental data obtained in [6, 11] are analyzed on the basis of continual and atomistic simulation. The continual simulation is based on the models of dislocation plasticity [12, 13] and fracture [13–15], and the molecular dynamic simulation is performed by using the LAMMPS software [16]. 2. CONTINUAL MODEL Consider the uniaxial deformation of a material along the x-axis. Such deformation conditions are usually ensured in experiments of high-speed impact of plates and intense irradiation. In the framework of the continuum approach, we determine the total deformation of the material as the sum of *
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contributions uik + Wik − wik [13]. Here uik is the tensor of macroscopic strain created by the motion of the matter. In the case under study, the only nonzero component of this tensor is uxx , ∂v duxx = , (2.1) dt ∂x where v is the speed of the matter motion along the x-axis. The plastic strain tensor wik is determined by the dislocation motion, and the strain tensor Wik related to fracture is determined by the characteristics of the ensemble of microdamages (microcracks or cavities). The expressions for the components of these tensors will be written out below. The set of basic equations contains the continuity equation ∂v dWkk dρ = −ρ + , (2.2) dt ∂x dt the equation of motion ρ
∂σ dv = , dt ∂x
(2.3)
and the equation of internal energy ∂v dWkk 3 ∂ ∂T dwxx dU = −P + + (1 − η) Sxx +Q+ κ , ρ dt ∂x dt 2 dt ∂x ∂x
(2.4)
where Wkk denotes the sum of diagonal components of the corresponding tensor, σ = −P + Sxx is the stress tensor component along the x-axis, P is the pressure, Sxx is a component of the stress deviator tensor, ρ is the matter density, and T is temperature. The specific internal energy U does not include the elastic distortion energy determined by shear stresses and the energy of defects (dislocations). Thus, U is the internal energy of the material with the same density and temperature as the considered body but without shear stresses and structure defects. The quantities U , ρ, P , and T are related by the broadrange equation of state [17]. The heat release due to plastic deformation and removal of structure defects is taken into account by the second and third terms on the right-hand side in (2.4). Here η ≈ 0.1 is the plastic deformation energy fraction spent to form new defects and Q is the specific power of energy release spent to remove the defects (annihilation of dislocations). Equation (2.4) takes into account the fact that Syy = Szz − Sxx /2 and wyy = wzz = −wxx /2 by symmetry. To analyze the processes in thin films, it is also necessary to take the thermal conduction into account in (2.4); here κ is the thermal conductivity coefficient. The stress deviator is calculated by Hooke’s law Wkk 2uxx − wxx + Wxx − , (2.5) Sxx = 2G 3 3 where G is the shear modulus. The plastic strain tensor is determined on the basis of the dislocation plasticity model [12, 13]. The plastic strain rate can be calculated from the generalized Orovan relation [18], where it is generally necessary to sum over all possible systems of dislocation glide [12]. For simplicity, we consider the case of monocrystalline iron (BCC lattice) loaded in the crystallographic direction [100]. In this case, the Orovan relation can be simplified as b dwxx = √ VD ρD , dt 6
(2.6)
¨ where b is the Burgers vector, VD is the velocity of dislocation motion with respect to matter, and ρD is the scalar density of mobile dislocations. The equation for the dislocation velocity [13] contains quasirelativistic corrections, which take into account the fact that the dislocation velocity VD cannot exceed the transverse speed of sound ct = G/ρ, b VD 2 3/2 dVD 3 = Sxx − Y sign(Sxx ) 1 − − BVD , (2.7) m0 dt 2 2 ct MECHANICS OF SOLIDS
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where m0 = ρb2 is the dislocation mass at rest per unit length, Y is the static yield point, and B is the phonon friction coefficient. The dislocation motion and the plastic deformation begin if |Sxx | > Y 2/3. The kinetic equations for mobile and immobilized dislocations, which reflect the balance between the process of dislocation formation, immobilization, and annihilation, can be written as ρD dρ dρD = QD − QI − QDa + , (2.8) dt ρ dt ρI dρ dρI = QI − QIa + , (2.9) dt ρ dt where QD is the rate of formation of new mobile dislocations in plastic deformation, QI is the rate of dislocation immobilization, and QDa and QIa are the rates of annihilation of mobile and immobile dislocations. In [13], the following expressions were suggested for these quantities: η dwxx 3 √ Sxx , QI = VI (ρD − ρ0 ) ρI , QD = εD 2 dt (2.10) QDa = ka b|VD |ρD (2ρD + ρI ), QIa = ka b|VD |ρD ρI , where εD ≈ 8eV/b is the energy of formation of unit length of the dislocation line, ka is the annihilation coefficient, and ρ0 and VI are the model parameters. The energy release due to the dislocation annihilation can be expressed as Q = εD (QDa + QIa ).
(2.11)
The tensor Wik of strain caused by fracture is determined in terms of the volume fraction of cylindrical microcracks [13, 14] or spherical pores [15]. In the case under study, the stresses favor the development of cracks perpendicular to the x-axis. For thin cracks, the only nonzero component of the tensor Wik is [13] 1 dα dWxx =− , (2.12) dt 1 − α dt where α is the volume fraction of microcracks in the considered volume element. Similarly, for spherical cavities, we can obtain 1 dWkk 1 1 dα dWxx = =− . (2.13) dt 3 dt 3 1 − α dt The volume fraction α is calculated through the concentration and geometric parameters of the damage. A variation in the radius R of the existing cracks is described by the approximate equation [13] 2 dR 2 3 σ2 σ 2d R R R , (2.14) + ) + 6R = −4(γ + γ ρ G dt2 2 dt G where γ is the surface tension and γ is the irreversible surface energy related to the plastic deformation near the crack as follows: ρbρD ct R2 dR . (2.15) 2 dt For the tensile stress σ > 0, the cracks can grow with a radius greater than the critical one R > Rcr = (2/3)Gγ/σ 2 . For spherical cavities, we can write out an equation whose is similar in structure [15], 2 dR 2 3 2d R + R (2.16) = −2(γ + γ ) + Rσ, ρ R dt2 2 dt γ =
where γ stands for irreversible losses due to plastic deformation [15], dR . γ = 2πY R sign dt The critical radius of the cavity is Rcr = 2γ/(σ − 2πY ). MECHANICS OF SOLIDS
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Fig. 1.
The expression for the rate of formation of defects of both types has the same form [13–15] 2 /(3k T )] − exp(−γ/Δγ) exp[−2πγRcr dn B = f n0 (1 − α), 2 /(3k T )][1 − exp(−γ/Δγ)] dt [1 − 2πΔγRcr B
(2.18)
where n is the damage concentration, n0 is the matter atom concentration, f is the frequency factor, and kB is the Boltzmann constant. The mean-square deviation Δγ characterizes the material imperfection, i.e., the presence of regions with decreased threshold of damage nucleation. The above-written system of equations was solved numerically by using the method [19] for integration equations of continuum mechanics. The action of pressure pulse of amplitude 10 and 30 GPa and duration 10 ns on the iron surface was modeled. The elastic precursor attenuation with the distance into the interior of the target was analyzed. The simulation results compared with the experimental data are shown in Fig. 1: σ is the precursor amplitude (GPa), and x is the distance away from the loaded surface (mm). Here markers 1 correspond to the results of experiments with laser irradiation of thin films [6], markers 2 correspond to the results obtained in [20], markers 3 corrspond to the results of experiments with high-speed impact [11], and curves 4 and 5 illustrate the computational results for the pressure pulses of 30 GPa and 10 GPa, respectively. In the computations, just as in experiments, the elastic precursor rapidly decays in the surface layer, i.e., at the depths up to 300 μm counted from the loaded surface; the further decay becomes slower. In the surface layer, the unstable initial profile of compression wave transforms into a stable shock wave. Computations show that the initial amplitude of the elastic precursor at the moment of its formation is approximately equal to the amplitude of the successive plastic shock wave [13]. As a result, the precursor decay follows different trajectories for different pressures on the surface (Fig. 1). Therefore, for small thicknesses of the film, the amplitude of the fixed elastic precursor must significantly depend on the intensity and duration of the action creating the compression pulse. The large values of shear stresses in the precursor at the initial stages of its evolution can be explained by the fact that the amount of dislocations existing in the material cannot ensure the plastic strain rate wxx corresponding to the macroscopic (external) strain rate uxx . 3. MOLECULAR-DYNAMIC SIMULATION OF UNIAXIAL DEFORMATION The processes at the atomistic level were studied by molecular-dynamic simulation of the uniaxial deformation of iron by using the LAMMPS software [16]. A specimen of monocrystalline (BCC) iron MECHANICS OF SOLIDS
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Fig. 2.
of initially cubic shape was deformed at a constant strain rate dε/dt in the direction [100] (along the x-axis). Periodic boundary conditions were posed on all boundaries of the computational region. The model statement of the problem was considered at a constant temperature of 300 K of the matter (one computation was performed at a temperature of 10 K) controlled by a thermostat. The particle interaction was determined by the potential of the immersed atom [21], and the main part of computations was performed using the potentials [22, 23]. The extension (dε/dt > 0) and compression (dε/dt < 0) with subsequent extension were considered. Computations with various numbers of atoms were carried out to determine the required size of the modeled volume of the matter. At a strain rate of 50 ns−1 , the calculated spall strength varies within 3% as the system volume increases from 54 to 250 hundreds of atoms. A qualitative criterion for the sufficiency of the system dimension is that many cavities should formed within the modeled volume as fracture occurs. Figure 2 illustrates the calculated dependence of the pressure P (GPa) on time t (ps) in the case of extension with strain rate 5 ns−1 (curve 1 ), 10 ns−1 (curve 2 ), and 50 ns−1 (curve 3 ). At the initial stage, the deformation is elastic and the crystalline order in the atomic arrangement is preserved. As the tensile stresses attain a value of 11–12 GPa (depending on the strain rate), the mechanism of homogeneous formation of dislocations and plastic relaxation of shear stresses starts, which is accompanied by a decrease in the total stresses. The corresponding failure of the crystalline structure is illustrated in Fig. 3, where only the atoms near which the structure differs from the regular structure are shown. Such atoms are concentrated in the dislocation gliding planes. The slope of the curve P (t) decreases after the beginning of the plastic deformation. No dislocation formation was observed in the compression of matter, at least prior to the pressures of 70 GPa (the shear stress is 15 GPa). Apparently, the energy threshold of homogeneous nucleation in compression is somewhat greater compared with the case of extension or pure shear [24]. The defect formation is related to the appearance of “empty” space, which is unfavorable for the compressed state. Further extension leads to formation of nearly spherical cavities (see Fig. 4). The formation of cavities leads to a decrease in tensile stresses until values of the order of 1 GPa (see Fig. 2). The subsequent deformation of matter is reduced to the growth of cavities. The tensile stress maximum in absolute value attained in the material is the spall strength σsp . MECHANICS OF SOLIDS
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Fig. 3.
Fig. 4.
The dependence of the spall strength σsp (GPa) on the strain rate dε/dt (s−1 ) is shown in Fig. 5. Curve 1 illustrates the computational results obtained by the above-described model of microcrack formation [13, 14] taken from [25]; the model parameters are γ = 1 J/m2 and Δγ = 0.036 J/m3 [25]. Markers 2 –7 correspond to the experimental results obtained in [11], where markers 2 illustrate the results in the case of laser irradiation of thin films [6]. Marker 11 corresponds to molecular-dynamic computations [26]. Markers 8 –10 present the molecular-dynamic computations performed in this paper: 8, with the potential used in [22]; 9, with the potential in [23]; and 10, with the potential in [23] but at a temperature of 10 K. The results of molecular-dynamic studies show that, in the originally homogeneous material, the dynamic sensitivity of the spall strength in the region of strain rates greater than 1 ns−1 is less than that MECHANICS OF SOLIDS
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Fig. 5.
observed in experiments at smaller strain rates. This agrees well with the conclusions in [14] about the change of the damage formation mechanism from heterogeneous to homogeneous. This transition is described by the fracture model considered in Section 2. CONCLUSION The common use of continual and atomistic simulation is a promising approach in the study of highrate deformation and fracture of metals. The initial amplitude of the elastic precursor is equal to the plastic shock wave amplitude. Therefore, for small thicknesses of the film (< 300 μm), the precursor amplitude significantly depends on the intensity and duration of the external action. Greater values of shear stresses in the precursor at the initial stage of evolution are stipulated by the fact that the amount of dislocations existing in the material is insufficient for ensuring the required plastic strain rate. The homogeneous nucleation of dislocations in compressed matter turns out to be suppressed and cannot ensure the relaxation of shear stresses either. As the rate of strain exceeds 0.1–1 ns−1 , the damage formation mechanism is changed from heterogeneous to homogeneous. ACKNOWLEDGMENTS The part of the studies related to molecular-dynamic computations of iron strength was supported by the Russian Science Foundation (project No. 14-11-00538). The study of the elastic precursor decay in thin films of iron was supported by the Ministry of Education and Science of the Russian Federation (project No. 3.1334.2014/K). REFERENCES 1. G. I. Kanel, S. V. Razorenov, K. Baumung, and J. Singer, “Dynamic Yield and Tensile Strength of Aluminum Single Crystals at Temperatures up to the Melting Point,” J. Appl. Phys. 90 (1), 136–143 (2001). 2. G. I. Kanel, V. E. Fortov, and S. V. Razorenov, “Shock Waves in Condensed-State Physics,” Uspekhi Fiz. Nauk 177 (8), 809–830 (2007) [Phys. Uspekhi (Engl. Transl.) 50 (8), 771–791 (2007)]. 3. G. V. Garkushin, O. N. Ignatova, G. I. Kanel, et al., “Submicrosecond Strength of Ultrafine-Grained Materials,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 155–163 (2010) [Mech. Solids (Engl. Transl.) 45 (4), 624–632 (2010)]. MECHANICS OF SOLIDS
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