8.4.1 The regimes of motion of a dislocation

The definition of the dissipative force requires a description of the physics of dislocation motion in some depth. First, it is necessary to recognize that dislocation motion is composed principally of glide and climb. Glide refers to the conservative motion of the dislocation within its own slip plane, and climb to the nonconservative motion of the dislocation perpendicular to the slip plane (Hirth & Lothe, 1991). Climb is a diffusion-assisted process (Argon, 2008), usually deemed too slow to be present in any high strain rate situation; hence, it will be omitted from further discussion. The reader is referred to Hirth and Lothe (1991) and Balluffi, Allen, and Carter (2005) for further discussions on dislocation climb. Glide, in turn, only requires mechanical stresses to occur (Hirth & Lothe, 1991); the process can be best understood in Fig. 2.42 for an edge dislocation in a simple cubic lattice.

As can be seen in Fig. 2.42, glide involves breaking and rebuilding interatomic bonds. At the core of the dislocation, it is an inherently atomistic process. It is conceivable that as the dislocation's core advances, breaking and rebuilding bonds acts as a dissipative process (Nabarro, 1967), where multiple lattice vibrational modes are excited, resulting in a net radiation of energy outward from the core. Relating this energy loss to the applied external fields that drive the dislocation motion—i.e., the Peach–Koehler force over the dislocation—ought to, in principle, allow the definition of a mobility law. However, the exact details of these processes are far from simple.

It is commonly observed, both experimentally (Johnston & Gilman, 1959; Nix & Menezes, 1971) and in MD simulations (Bitzek & Gumbsch, 2004, 2005; Koizumi et al., 2002; Olmsted et al., 2005; Tsuzuki et al., 2008), that dislocation glide is severely overdamped (Gilman, 1969) and directly proportional to the dislocation's velocity in a manner similar to a viscous drag force in fluid motion (Gilman, 1969; Hirth, 1996; Hirth & Lothe, 1991; Hull & Bacon, 2011). This was in fact first established by Leibfried (1950), who proposed the drag force of the form

$f_{\text{drag}}=d·v_{\text{glide}}$

  (2.130)

that can be found in most text books on the subject (vid. Argon, 2008; Bulatov & Cai, 2006; Hirth & Lothe, 1991; Hull & Bacon, 2011; Kubin, 2013; Nabarro, 1967; Reed-Hill & Abbaschian, 1994).

The most ubiquitous of mobility laws found in dislocation dynamics, both DDP and 3D DD, equates this viscous drag force f_drag (Eq. 2.130) to the glissile component of the Peach–Koehler force because, as has been said above, the dislocation moves so as to minimize the elastic energy of the system following the Peach–Koehler force, and that the energy reduction is dissipated by the dislocation in its motion through this viscous drag process. For the motion of straight edge dislocations of interest to D3P, the glissile component of the Peach–Koehler force is¹⁵ merely f_glide = τ · B with τ the resolved shear stress over the slip plane and B the magnitude of the Burgers vector, so the balance renders a mobility law of the form

$v_{\text{glide}}=\frac{\tau B}{d}$

  (2.131)

However, the validity of the mobility law defined in Eq. (2.131) is far from universal and depends on, among other things, the levels of applied stress. Figure 2.43 summarizes the regimes of motion of a dislocation. As can be appreciated there, there are at least three distinct regimes for the motion of a dislocation, each of which leads to different mobility laws.

8.4.2 Drag-controlled regime

The regime of dislocation velocities for which the viscous drag law (Eq. (2.131)) is applicable is called the “drag-controlled regime.” In the drag-controlled regime, dislocation motion is dominated primarily by phonon drag, i.e., by the interaction between the dislocation and the thermal vibrations in the lattice. Phonon drag is mainly mediated by phonon scattering (Gilman, 1969; Granato, 1973; Hirth & Lothe, 1991; Hull & Bacon, 2011; Nabarro, 1967), which seems to involve mainly two processes (vid. Granato, 1973). On one hand, the dislocation itself will strain the lattice around its core, thereby breaking its symmetry. As the dislocation moves through the lattice, this strained region will be met by the incoming thermal phonons from elsewhere in the lattice that will be refracted as a result. On the other hand, phonons may be absorbed by the dislocation itself, triggering vibrations of its core that, again, result in new phonons radiated outward. The result is that dislocations in the drag-controlled regime move with linear viscous drag laws such as that shown in Eq. (2.131).

8.4.3 Thermal activation regime

The lower limit of the drag-controlled regime marks the onset of dislocation motion. In this regime, dislocation motion occurs in the presence of low applied stresses, so low that the motion of dislocations is dominated by those mechanisms that offer direct resistance to the motion. A dislocation can encounter resistance to its motion from two kinds of sources: the crystalline lattice itself and obstacles. The intrinsic resistance of the lattice to the motion of dislocations refers to the energy cost associated with breaking and rebuilding interatomic bonds as the dislocation moves; this effectively manifests itself in the periodic¹⁶ Peierls barrier (Granato, 1973; Regazzoni et al., 1987). Obstacles refer to lattice imperfections that may include impurities, interstitials and vacancies, other dislocations, etc. These obstacles interact with the dislocation hindering its motion; unlike the Peierls barrier however, obstacles do not offer a periodic resistance. In either case, both the intrinsic lattice resistance and obstacles can be thought of as barriers of stress that the dislocation must overcome in order to move. Focusing on the intrinsic lattice resistance alone, the magnitude of the barrier, usually called the Peierls stress or intrinsic lattice resistance, sets up a mechanical threshold of applied stress below which dislocation motion is not, in principle, possible. This threshold can still be overcome, however, through thermally assisted processes (vid. Argon, 2008; Kocks, Argon, & Ashby, 1975; Regazzoni et al., 1987), whereby thermal energy helps the dislocation overcome the barrier. For this reason, the lower regime is sometimes called the “thermal activation regime” or “activation of motion regime.”

The thermal activation regime is often associated not only with the onset of dislocation motion, but of plasticity as a whole. This is true in a sense, but it may suggest that before moving into the drag-controlled regime, dislocations must overcome the thermal activation regime—i.e., that the onset of all dislocation motion is always thermally activated. This might be true at low strain rates and low levels of stress, and in those situations, it would be consistent with the empirical observation that the yield point tends to decrease with increasing temperature. However, as has been pointed out in the introduction, the yield point of most metals experiences a sudden upturn (vid. 2.2) that has been associated with a fundamental change in the dislocation's own motion regime, going from thermally activated to drag controlled Regazzoni et al. (1987). Undoubtedly, a dislocation in a material shock loaded to 20 GPa with a strain rate of 10¹⁰ s⁻¹ will hardly have time, if any, to go through the thermally activated regime. Hence, there are situations that are particularly relevant to D3P, where the thermally activated regime is not relevant.

8.4.4 Relativistic regime

The upper limit of the drag-controlled regime is usually a velocity of the order of tens or a few hundreds of meters per second. Surpassing this limit is unusual in most plasticity applications. In fact, DDP simulations typically cap the speed of dislocations at ≈ 20 m/s (cf. Cleveringa, Van der Giessen, & Needleman, 1997). The regime of motion beyond the drag-controlled regime is often called the “relativistic regime.” This is because the speeds reached in this regime are usually a significant fraction of the transverse speed of sound, and therefore, the relativistic effects discussed in Section 3.2 are expected to be present. Typically, in this regime the speed of dislocations saturates toward the transverse speed of sound (Gilman, 1969; Johnston & Gilman, 1959; Marian & Caro, 2006; Meyers, 1994; Olmsted et al., 2005).

As shown in Section 3.1, in the “relativistic regime” the elastic energy of dislocations tends to increase with the speed of dislocations, diverging at the transverse speed of sound. This is of great significance, as it shows that even the mobility laws commonly employed in dislocation dynamics (such as Eq. (2.131)) are in fact quasi-static. The reason is simple: as it moves, the self-energy of the dislocation itself must change. This change is negligible for the low speeds encountered in the drag-controlled regime, but it becomes significant at higher speeds—i.e., in the relativistic regime. If the speed of the dislocation is going to increase, the elastic energy of the dislocation will increase; the energy input due to the external fields (the Peach–Koehler force), that in quasi-static mobility laws only had to balance out the drag dissipation, will now also have to be spent on increasing the dislocation's elastic energy as well. To wit, quasi-static mobility laws do not account for changes in the elastic energy of the dislocation itself, which is expected to vary in dynamic cases.

Thus, the dynamic mobility law has to be modified to include one way or another the contribution of the dislocation's self-energy. This leads to several questions:

• Is the Peach–Koehler force itself, derived for time-independent fields, affected in dynamic situations?

• How does the increase of the dislocation's elastic self-energy affect the mobility law?

• Is the viscous drag force the main dissipative mechanism at play in the relativistic regime?

These questions will be addressed in the following sections.

8.4.5 The exactitude of the peach–koehler force

As originally derived by Peach and Koehler (1950), the Peach–Koehler force given in Eq. (2.129) accounts for variations in the external elastostatic energy. However, D3P uses a fully elastodynamic formulation, so one must establish its exactitude when time is an explicit elastic field variable. The considerations presented here were pioneered by Lothe (1961) and Stroh (1962), and rigorously formalized by Mura (1982). It is shown that the Peach–Koehler force as given by Eq. (2.129) is to be exact in a dynamic framework.

In order to show this, consider a moving dislocation loop in an infinite domain D. The associated Lagrangian functional shall be:

$\mathfrak{L}={\displaystyle {\int }_D\left(\frac{1}{2}\rho {\stackrel{.}{u}}_i{\stackrel{.}{u}}_i-\frac{1}{2}{\sigma }_{\mathit{ij}}{\epsilon }_{\mathit{ij}}\right)\text{d}D}$

  (2.132)

where $T=\frac{1}{2}{\displaystyle {\int }_D\rho {\stackrel{.}{u}}_i{\stackrel{.}{u}}_i\text{d}D}$ is the kinetic energy density and $V=\frac{1}{2}{\displaystyle {\int }_D{\sigma }_{\mathit{ij}}{\epsilon }_{\mathit{ij}}\text{d}D}$ the potential (elastic) energy density. Thus, $\mathfrak{L}=T-V$ is the Lagrangian density. Notice that the Lagrangian proper would be

$L={\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t}$

  (2.133)

Elastodynamics require that σ_ijε_ij = σ_iju_i,j, so

${\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t={\displaystyle {\int }_{t_0}^{t_1}\text{d}t{\displaystyle {\int }_D\left(\frac{1}{2}\rho {\stackrel{.}{u}}_i{\stackrel{.}{u}}_i-\frac{1}{2}{\sigma }_{\mathit{ij}}{u}_{i,j}\right)\text{d}D}}}$

  (2.134)

This expression is defined everywhere in the domain except on the slip surface S, where a discontinuity arises. The boundary of S is the dislocation line L, which shall be assumed to move with a velocity field ${\stackrel{.}{\xi }}_i$, so that ξ_i is taken to be the displacement of the line. Over the dislocation line, the dislocation's discontinuity can be expressed through the following boundary condition:

$u_i^{+}\left(\mathbf{x},t\right)-u_i^{-}\left(\mathbf{x},t\right)=b_i$

  (2.135)

The conditions that ξ must fulfill for the Lagrangian functional L to be minimized must be found. For that, in a dynamic situation, a virtual displacement δξ over the dislocation line is imagined that produces a change δS over the slipped region. In order to find out how this change is bounded by the principle of least action, the variation of the functional is considered:

$\delta \left\{{\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t}\right\}={\displaystyle {\int }_{t_0}^{t_1}\text{d}t{\displaystyle {\int }_D\left(\rho {\stackrel{.}{u}}_i\delta {\stackrel{.}{u}}_i-{\sigma }_{\mathit{ij}}\delta u_{i,j}\right)\text{d}D}}$

  (2.136)

Operating:

$\delta {\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t}={\displaystyle {\int }_{t_0}^{t_1}\text{d}t{\displaystyle {\int }_D\left(\rho {\stackrel{.}{u}}_i\frac{\partial }{\partial t}\left(\delta u_i\right)-{\sigma }_{\mathit{ij}}\frac{\partial }{\partial x_j}\left(\delta u_i\right)\right)\text{d}D}}$

  (2.137)

Applying Gauss's theorem, one must bear in mind that D is not simply connected, so that S + δS must be excluded by defining an infinitely closed enveloping surface made of S⁺ + δS⁺ and S⁻ + δS⁻. Excluding that surface, Gauss's theorem leads to

${\displaystyle {\int }_D{\sigma }_{\mathit{ij}}\frac{\partial }{\partial x_j}\left(\delta u_i\right)\text{d}D}={\displaystyle {\int }_{S+\delta S}{\sigma }_{\mathit{ij}}{n}_j\left[\delta u_i\right]}\text{d}S-{\displaystyle {\int }_D{\sigma }_{\mathit{ij},j}\delta u_i\text{d}D}={\displaystyle {\int }_{\delta S}{\sigma }_{\mathit{ij}}{n}_j{b}_i\delta S}-{\displaystyle {\int }_D{\sigma }_{\mathit{ij},j}\delta u_i\text{d}D}$

  (2.138)

where the apparent sign reversal is caused by the need of excluding the inner side of S + δS rather than the outer one. Here [u_i] is the difference of u_i being evaluated at the upper and lower surfaces of the slip plane (the integral is divided into ${\int }_{S^{+}+S^{-}}+{\int }_{\delta S^{+}}+{\int }_{\delta S^{-}}\equiv {\int }_{S+\delta S}$. Hence, [u_i] = b_i on S. After the virtual displacement δξ_i, its variation [δu_i] = δ[u_i] is therefore zero on S and b_i on the new virtual slipped region δS.

The δS is the variation of the slipped region S, which arises from a virtual displacement δξ_i and thus must fulfill the same condition as in the static case:

$n_j\delta S={\epsilon }_{\mathit{jlh}}\delta {\xi }_l{\nu }_h\text{d}l$

  (2.139)

Furthermore, it is found that

${\displaystyle {\int }_{t_0}^{t_1}\rho {\stackrel{.}{u}}_i\frac{\partial }{\partial t}\left(\delta u_i\right)\text{d}t\text{d}D}=-{\displaystyle {\int }_{\delta S}\rho {\stackrel{.}{u}}_i{b}_i{\stackrel{.}{\xi }}_j{n}_j\delta S}-{\displaystyle {\int }_{t_0}^{t_1}\rho {\ddot{u}}_i\delta u_i\text{d}t\text{d}D}$

  (2.140)

Grouping everything into the functional

$\delta {\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t}=-{\displaystyle {\int }_{t_0}^{t_1}\text{d}t{\displaystyle {\oint }_L\left(\rho {\stackrel{.}{u}}_i{\stackrel{.}{\xi }}_j+{\sigma }_{ij}\right)b_i{\epsilon }_{\mathit{jlh}}\delta {\xi }_l{\nu }_h\text{d}l}}$

  (2.141)

The term $\rho {\stackrel{.}{u}}_i{\stackrel{.}{\xi }}_j{b}_i{\epsilon }_{\mathit{jlh}}{\nu }_h$ is called the Lorentz force for its analogy with the electrodynamic Lorentz force. A detailed analysis of the Lorentz force was presented by Lund (1998). It is of great mathematical interest, but of little physical significance because, as its electrodynamic counterpart, it does no work. Indeed, ${\stackrel{.}{\xi }}_j\text{d}t=\delta {\xi }_j$, which in the Lorentz force term leads to having ε_jhlδξ_jδξ_l = 0 necessarily. Hence, as stated by Lund (1998): “the additional “Lorentzian” force that is (…) orthogonal to the dislocation velocity, [so] it does not do any work”; in order to establish its relevance, it would be necessary to explain what the Lorentz force would physically entail. Otherwise, the variation of the functional leads to

$\delta {\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t}=-{\displaystyle {\int }_{t_0}^{t_1}\text{d}t{\displaystyle {\oint }_L{\sigma }_{\mathit{ij}}{b}_i{\epsilon }_{\mathit{jlh}}\delta {\xi }_l{\nu }_h\text{d}l}}$

  (2.142)

It could be imagined that the dislocation is subjected to a force rather than to an external stress field. Looking at the expression above, this would lead, as in the static case, to the Peach–Koehler force: the dynamic effects account for in ${\stackrel{.}{\xi }}_i$ via the Lorentz force do not contribute to the energy balance and, therefore, the Peach–Koehler force is valid in the dynamic case.

Indeed,

$\delta {\displaystyle {\int }_{t_0}^{t_1}\mathfrak{L}\text{d}t}=-{\displaystyle {\int }_{t_0}^{t_1}\text{d}t{\displaystyle {\oint }_L{f}_l\delta {\xi }_l\text{d}l}}$

  (2.143)

would be the variation in the action due to a force f_l, and by comparing with the expression above

$f_l={\epsilon }_{\mathit{jlh}}{\sigma }_{ij}{b}_i{\nu }_h$

  (2.144)

as in the static case, q.e.d.

8.4.6 Inertial forces

The Peach–Koehler force concerns forces over dislocations; these forces exert work during the dislocation's motion. In turn, the elastic energy of moving dislocations increases as the velocity of dislocations increases (and vice versa). Thus, in a dynamic situation part of the energy input due to the Peach–Koehler force must be spent in increasing the dislocation's self-energy. This is usually called the “inertial” effect (Hirth, Zbib, & Lothe, 1998), because as with inertia, it opposes changes in the motion of a dislocation, and because the effect can be translated into an equivalent thermodynamic force of the form of a Newtonian inertial force, directly proportional to a dislocation “mass” and the dislocation's acceleration.

The inertial force can be estimated with the following considerations, similar to those found in Hirth et al. (1998). Consider a straight edge dislocation in an infinite plane. Let v be its velocity. The Hamiltonian of the system can be written as

$H=T+V$

  (2.145)

where T is the kinetic energy of the dislocation, and V the potential (elastic) energy of the dislocation.

Of interest here is to derive from the Hamiltonian an equivalent configurational force. Consider a straight edge dislocation moving uniformly with speed v along the X-axis. Let x be the canonical coordinate along that very same direction. Define p to be the linear quasi-momentum of the dislocation. Then Hamilton's equations require that

$\frac{\text{d}p}{\text{d}t}=-\frac{\partial H}{\partial x}$

  (2.146)

$\frac{\text{d}x}{\text{d}t}=\frac{\partial H}{\partial p}$

  (2.147)

It is noticed that the force can be defined as

$F=\frac{\text{d}p}{\text{d}t}$

  (2.148)

Furthermore, $\frac{\text{d}x}{\text{d}t}=\nu$ where v is the dislocation's speed. In the context of moving dislocations, this force will be a configurational or self-force.

For the uniformly preexisting edge dislocation, Weertman (1981) found the expressions of the kinetic and elastic energy of a uniformly moving straight edge dislocation, given by:

$T=\frac{E_0}{2}\frac{1}{M_{\text{t}}^2}\left[4{\gamma }_l+\frac{4}{{\gamma }_l}+{\gamma }_t^3-5{\gamma }_t-\frac{5}{{\gamma }_t}+\frac{1}{{\gamma }_t^3}\right]$

  (2.149)

$V=\frac{E_0}{2}\frac{1}{M_{\text{t}}^2}\left[12{\gamma }_l+\frac{4}{{\gamma }_l}-{\gamma }_t^3-9{\gamma }_t-\frac{7}{{\gamma }_t}+\frac{1}{{\gamma }_t^3}\right]$

  (2.150)

where ${\gamma }_t=\sqrt{1-M_{\text{t}}^2}$, ${\gamma }_l=\sqrt{1-M_{\text{l}}^2}$, and $E_0=\frac{\mu B^2}{4\pi }\text{ln}\left(\frac{R}{r_{\text{c}}}\right)$ is the dislocation's energy at rest, where R and r_c are the inner and outer cutoffs of the core.

It follows that, in this framework, T = T(v) and V = V(v), so accordingly H = H(v) alone. Hence,

$\frac{\partial H}{\partial x}=0$

  (2.151)

so from Eq. (2.146) it is found that the quasi-momentum is invariant in time $\frac{\text{d}p}{\text{d}t}=0$, and from Eq. (2.151) that the Hamiltonian (here, the total energy) of the system is symmetric with respect to any given displacement. To wit, as in the quasi-static case, the uniformly moving dislocation does not radiate energy, and the fields are always the same as it moves.

However, ignore this last point for the moment. Invoking the chain rule

$v=\frac{\text{d}x}{\text{d}t}=\frac{\partial H}{\partial p}=\frac{\partial H}{\partial t}\frac{\text{d}t}{\text{d}p}=\frac{1}{F}\frac{\partial H}{\partial t}$

  (2.152)

so that the force

$F=\frac{1}{v}\frac{\partial H}{\partial t}=\frac{1}{v}\frac{\partial H}{\partial v}\frac{\partial v}{\partial t}$

  (2.153)

Equation (2.153) takes the form of an inertial force provided that the effective “mass” m of the dislocation is defined as

$m=\left\{\forall \left\{F,v,t\right\}\exists m|F=\frac{\text{d}v}{\text{d}t}\right\}$

  (2.154)

From Eq. (2.153), it is found that

$m=\frac{1}{v}\frac{\partial H}{\partial v}$

  (2.155)

Since H = T + V, substituting Eqs. (2.149) and (2.150) in Eq. (2.155), one obtains that the uniformly moving edge dislocation's effective mass:

$m=\frac{E_0}{2}\frac{1}{c_{\text{t}}^2{M}_{\text{t}}^4}\left[-8{\gamma }_l-\frac{20}{{\gamma }_l}+\frac{4}{{\gamma }_l^3}+7{\gamma }_t+\frac{25}{{\gamma }_t}-\frac{11}{{\gamma }_t^3}+\frac{3}{{\gamma }_t^5}\right]$

  (2.156)

This expression is entirely comparable to that obtained by Hirth et al. (1998), who used the Lagrangian formalism instead.

Thus, there seems to exist a force associated with the elastic and kinetic fields of dislocations, given in the form of an inertia force:

$F=m\frac{\partial v}{\partial t}$

  (2.157)

This force exists solely because the dislocation has a velocity. As with the Peach–Koehler force, it is important to recognize that it is not a mechanical force, but a thermodynamic virtual force.

At this point, it is natural to be perplexed by the inherent paradox that an uniformly moving dislocation—i.e., a dislocation where v = constant—could have an inertia. From Eq. (2.157), it seems obvious that it does not, for $\frac{\partial \nu }{\partial t}=0$ when the dislocation is moving uniformly. Using a far more sophisticated approach, Ni and Markenscoff (2008) reached the same conclusion for a uniformly moving dislocation.

This casts doubt on the dislocation's effective mass as defined by Eq. (2.156). As mentioned above, from Eqs. (2.146), (2.149), and (2.150), it follows that for a uniformly moving dislocation, the quasi-momentum is invariant in time. In Eq. (2.152), however, one must assume that $\frac{\text{d}t}{\text{d}p}$ exists. However, if p is not a function of t, then its inverse function t = t(p) does not exist,¹⁷ and hence $\frac{\text{d}t}{\text{d}p}$ is illegitimate.¹⁸ It follows that in that case v cannot be written as

$v=\frac{1}{F}\frac{\partial H}{\partial t}$

which suggests that the effective mass should not be defined in the case of a uniformly moving dislocation. It is important to note that the derivation above is essentially correct apart from the above mentioned contradiction.

This raise the question, what could the effective mass obtained in Eq. (2.156) possibly mean? First of all, it must be pointed out that, mathematically, it remains legitimate to define a function m such that it fulfills Eq. (2.155) even if, as said, there is no inertial force as such. This is because H = H(v) and therefore $\frac{\partial H}{\partial \nu }$ exists. Consider Eq. (2.155), from which the effective mass has been derived. There, the mass is expressed as a measure of the change in the total energy (the Hamiltonian) of the dislocation as the dislocation's speed is varied. In the case of the uniformly moving dislocation, the dislocation cannot change its speed by construction; however, one can compare two dislocations moving at different speeds. Their associated energies are different; hence, it is possible to study how the energy varies for dislocations moving uniformly with different speeds. Then m as defined in Eq. (2.156) is a measure of that radiation.

It can then be argued that, as a rough approximation, the inertial force defined through Eq. (2.156) is a measure of the additional energy that is required for the dislocation to increase its steady-state speed v. This would entail that the dislocation transitions from a uniform speed to another, different, uniform speed. This clearly goes against the hypothesis employed here. However, since the dynamic fields of dislocations reach their steady-state values in a short amount of time, it can be argued that it serves as a measure of the energy required to accelerate the dislocation, and hence as an approximation to the true inertia.

A complete treatment of the inertia of a dislocation requires the expressions of T and V for the nonuniform motion of a dislocation. Unfortunately, as in the derivation of the elastodynamic fields of dislocations provided in Section 5, obtaining the inertial force of a nonuniformly moving dislocation is far from simple. Ni and Markenscoff (2008) have recently achieved an expression for the mass of the nonuniformly moving straight screw dislocation that is of much greater complexity than the one presented here. Using their method, a derivation for the mass of the nonuniformly moving edge dislocation can be achieved as well (Ni & Markenscoff, 2008).

Either way, assuming one has a valid expression (or an approximation) of an inertial force, then the mobility law that balances inertial and dissipative effects with the Peach–Koehler force would take the form

$f_{\text{pk}}=m\frac{\partial v}{\partial t}+f_{\text{drag}}$

  (2.158)

This expression would account for the inertial effect, the dissipative mechanisms, and the action of the external fields over the moving dislocation, provided that adequate expressions for m and f_drag were used. From the discussion above, one can assume that for D3P, where dislocations are expected to move nonuniformly, the mass m should take a form akin to that provided in Ni and Markenscoff (2008), and f_drag = d · v, with d a drag coefficient.

8.4.7 Other considerations

Several questions remain open. Fundamentally, it has been questioned (Gilman, 1969; Nabarro, 1967) whether the drag mechanism at play is always solely of the phonon scattering mechanism assumed to occur at low speeds (Granato, 1973; Leibfried, 1950). In that sense, several additional dissipation mechanisms have been proposed, especially at higher speeds; these include anharmonic effects of the lattice (Brailsford, 1972), electronic effects (Brailsford, 1969; Huffman & Louat, 1967; Nabarro, 1967), quantum tunneling (Coffey, 1986, 1994), thermoelastic effects (Eshelby, 1949a; Nabarro, 1967; Zener, 1940), etc. Nabarro (1967), Gilman (1969), Granato (1973), Hirth and Lothe (1991), and Meyers (1994) offer detailed accounts of many of these dissipative mechanisms.

Whether or not any of these proposed mechanisms play a significant role is not always easy to ascertain. In many cases, the mechanisms at play are out of the reach of molecular dynamics simulations and would require a full quantum-mechanical treatment that, in some cases, goes beyond the current capabilities of any of the variants density functional theory or GW methods.

Nevertheless, using molecular dynamics simulations alongside an ad hoc mobility law similar to the one given in Eq. (2.158), Bitzek and Gumbsch (2004, 2005) were able to estimate the value of dislocation mass and the drag coefficient. It might seem that this kind of study merely shows the consistency reached between MD models, that allow only for phonon-based dissipation and long-range elastic fields,¹⁹ and mobility laws that solely consider precisely the terms that MD simulations can capture: a viscous phonon drag term and an inertial term. However, Bitzek and Gumbsch (2004, 2005) fundamentally clarify the form of the dissipative forces when inertial effects are present. Other MD simulations, such as those by Wang, He, and Wang (2010), also show that phonon viscosity is a major dissipative mechanism at high speeds. Thus, these models serve to clarify the effect of lattice-based dissipative mechanisms, showing the relative importance of phonon scattering and other mechanisms such as thermoelastic and anharmonic effects that can, in principle, be captured by MD models.

Relevant to D3P is an observation regarding the effect of inertia, that the predicted acceleration times are usually very short compared to the rise time of a shock front (Gillis & Kratochvil, 1970). Consider the equation of motion of a dislocation with an inertia term

$m\frac{\text{d}v}{\text{d}t}+\text{d}v=b·\tau$

  (2.159)

where m is the dislocation “mass.”

Assume m is constant. Then Eq. (2.159) can be solved directly by separation of variables as

${\displaystyle \int \frac{m\text{d}v}{b\tau -\text{d}v}}={\displaystyle \int \text{d}t}$

  (2.160)

whereupon

$-\frac{m}{d}\text{ln}\left(b\tau -\text{d}v\right)=t$

  (2.161)

Hence,

$v\left(t\right)=\frac{b\tau }{d}\left[1-{\text{e}}^{-\frac{\mathit{td}}{m}}\right]$

  (2.162)

Take typical values of the parameters involved: B ≈ 10⁻⁴ Pas, τ ≈ 1 GPa, b ≈ 2.5 × 10⁻¹⁰ m, m ∝ ρb² ≈ 10⁻¹⁶. If the steady-state speed is around v ≈ 2000 m/s, the time a dislocation would take to acquire that speed from v = 0 m/s would be around t_accel ≈ 1 ps.

However, dislocation inertia is typically velocity dependent. For simplicity, consider the velocity dependency to be of the approximate form (Weertman, 1981)

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c_{\text{t}}^2}}}$

  (2.163)

with m₀ ≡ ρb².

The combination of this form of the dislocation effective “mass” with Eq. (2.159) results in a nonlinear differential equation. On first approach, the acceleration times can be studied by considering the form of the acceleration of the dislocation, which can be deduced from Eq. (2.159):

$\frac{\text{d}v}{\text{d}t}=\frac{1}{m}\left(b\tau -\text{d}v\right)$

  (2.164)

Consider d = 5 × 10⁻⁴ Pa s, τ = 5.8 GPa, b ≈ 2.5 × 10⁻¹⁰ m, m ∝ ρb² ≈ 10⁻¹⁶. The resulting acceleration time is plotted in Fig. 2.44; exceedingly large values of acceleration can already be appreciated there. This kind of curve can also be found in an analogous analysis by Meyers (1994), that reaches the same conclusions.

This curve takes the drag coefficient to be constant; one can complicate matters further by considering a nonlinear drag coefficient such as Taylor's (1969),

$d=\frac{d_0}{1-v^2/c_{\text{t}}^2}$

  (2.165)

and combining it with the mass given in Eq. (2.163) and the mobility law given by Eq. (2.159):

$\frac{m_0}{\sqrt{1-\frac{v^2}{c_{\text{t}}^2}}}\frac{\text{d}v}{\text{d}t}+\frac{d_0}{1-v^2/c_{\text{t}}^2}v=B\tau$

  (2.166)

The resulting acceleration curve for d₀ = 5 × 10⁻⁴ Pa s is also shown in Fig. 2.44, where the additional effect of a larger drag as the dislocation's speed increases manifests itself in lower values of the acceleration, even if its order of magnitude is similar.

Equation (2.166) should be treated with some scepticism. Originally, J. Taylor (1969) introduced the saturating drag coefficient given in Eq. (2.165) as a phenomenological expression able to reproduce the steady-state (i.e., nonaccelerating) mobilities observed experimentally by Johnston and Gilman (1959). Specifically, Taylor wanted to address the observed saturation of dislocation velocities about the transverse speed of sound. This “saturation” is generally associated with inertia effects—i.e., with the associated increase of the dislocation's self-energy as its speed approaches the speed of sound—and not with a relativistic increase of the magnitude of the dissipative forces themselves. Therefore, the combination of inertial with Eq. (2.165) has a questionable physical meaning, as it entails that not only inertial effects are present, but that the dissipative mechanisms themselves evolve in a relativistic way. Nevertheless, Eq. (2.166) can be considered as an extreme case of dislocation mobilities, useful for sensitivity analyses such as those presented here.

From the curves in Fig. 2.44, it can already be appreciated that the order of magnitude of the acceleration of the dislocation (10¹⁵ m/s²) is high in either case. In the figure, it is also apparent that Eq. (2.159) leads to different terminal speeds depending on the definition of the mass and drag coefficient. The terminal speed, i.e., the steady-state speed, is reached when the viscous drag force equates the applied stress:

$v_{\text{terminal}}=\left\{\forall \left\{m,d,v\right\};m\frac{\text{d}v}{\text{d}t}+\text{d}v=b·\tau \exists v_{\text{terminal}}\subset v|d·v_{\text{terminal}}=b·\tau \right\}$

  (2.167)

With the parameters given above, for the linear drag case this entails a terminal speed of 2958 m/s (vid. Fig. 2.44); with the saturating drag law, the terminal speed is lower, but can also be determined analytically by solving Eq. (2.166) for the condition given in Eq. (2.167):

$v_{\text{terminal}}=\frac{\sqrt{4b^2{c}_{\text{t}}^2{\tau }^2+c_{\text{t}}^4{d}_0^2}-c_{\text{t}}^2{d}_0}{2b\tau }$

  (2.168)

This renders a terminal speed of 1842.38 m/s (vid. Fig. 2.44).

The acceleration curves serve to highlight that, in general, the acceleration times of dislocations must be extremely short. In the case shown in Fig. 2.44, an average acceleration of 4 × 10¹⁵ m/s² entails an acceleration time to the 3000 m/s terminal speed of about 7.5 × 10⁻¹³ s, which is of the same order as the values calculated above for a constant mass. Similar calculations made by Meyers (1994) render even smaller acceleration times.

These estimations can be improved further by solving Eq. (2.159) numerically for the two cases shown above. Numerically, for the case without saturating drag, the acceleration time is estimated at 2 ps, and at 0.5 ps for the case with saturating drag. Similar calculations can be performed for different values of drag coefficient and applied stresses, all of which render large accelerations and, concurrently, small acceleration times. This would support the adequacy of neglecting inertial effects entirely—i.e., to define mobility laws assuming the dislocation reaches its speed instantaneously.

This is further backed by the work done by Beltz, Davis, and Malén (1968) and Gillis and Kratochvil (1970), who, using even more sophisticated models of inertia than those presented here concluded that the acceleration time of dislocations was so much smaller than the rise time of a shock front that it could be neglected. Furthermore, the works of Beltz et al. and Gillis and Kratochvil pointed out that, because of the short acceleration times, the use of an inertial term leads to dislocation mobilities that can be equally reproduced using an adequately characterized dislocation drag coefficient. This latter point was acknowledged by Zbib and Diaz de la Rubia (2002) whom, nonetheless, favored inertial laws.

It is worth devoting a few lines to elaborate this last point. It has been seen that, because the acceleration of dislocations is large, the corresponding acceleration times of dislocations are short. This entails that a dislocation will reach its steady-state speed almost instantaneously. However, this does not convey much about the terminal speed of the dislocation itself. In the relativistic regime, as shown in Fig. 2.43, a saturation of speed with increasing applied stress is expected. Physically, this is effectively explained through inertial effects—the increase in the dislocation's self-energy as the dislocation's speed increases. However, because the resulting acceleration times are small, Beltz et al. (1968) and Gillis, Kratochvil (1970) point out that, rather than solving a nonlinear differential equation such as Eq. (2.166), one can approximate the expected saturation behavior by considering alternative (and instantaneous) mobility laws that address the saturation instead.

These alternative mobility laws commonly favor modification of the drag coefficient. The main requirement then becomes obtaining a mobility law that, as in Fig. 2.43, saturates in the vicinity of the transverse speed of sound. Along those lines, one can define a mobility law using the saturating drag coefficient presented above (Eq. 2.165) as originally proposed by J. Taylor (1969)

$\frac{d_0}{1-\frac{v^2}{c_{\text{t}}^2}}·v=B\tau$

  (2.169)

where d₀ is the asymptotic viscosity coefficient at low velocities. The velocity law is then given by Eq. (2.168).

An alternative expression is that of the power law

$v={\left(\frac{\tau }{{\tau }_0}\right)}^m$

  (2.170)

This equation, originally a phenomenological law (vid. Gilman, 1969), can be physically justified for the thermal activation regime (vid. Kocks et al., 1975).

The power law can be used to reflect grosso modo empirical results in a mobility law (Meyers, 1994). The exponent m is the slope of the $\log \overline{\nu }-\log \tau$ curve, which is generally seen to vary with the regime of motion of the dislocation. Hence, a mobility law could be constructed by modifying the values of m for each regime of motion. If the dislocation is moving in the thermally activated regime, m_I > 1. For the drag-controlled regime, m_II ≈ 1. For the relativistic regime, m_III < 1.

Thus, provided that the exponent m is modified accordingly, this would ensure the validity of Eq. (2.170). The values of m can vary sharply: Johnston and Gilman (1959) estimated m ≈ 15 − 20 for the thermally activated regime,²⁰ and should reduce to m = 1 for the drag-controlled regime, where conventional knowledge holds that the mobility should be linear τ ∝ v. However, in practice m ≈ 1 − 10 (Gilman, 1969, 2003). Notice that Eq. (2.170) is a rather unphysical approximation, as there is no upper limit on velocity (Gilman, 2003). However, it provides a good first approach toward estimating v if it is used carefully. Values of the m exponent for a number of materials are collected in Nix and Menezes (1971).

Further expressions can be derived from direct fits of molecular dynamics simulations of dislocation motion. For instance, from the data for aluminum by Olmsted et al. (2005), one can reach a fit of the form:

$v=\{\begin{array}{ll}2.066929885\times {10}^{-5}\left|\frac{f_{\text{pk}}}{b}\right|\hfill & v<1152.67\text{m}/\text{s}\hfill \\ 22961.54+1.00876\times {10}^{13}\frac{b^2}{f_{\text{pk}}^2}-1.00876\times {10}^{11}\left|\frac{b}{f_{\text{pk}}}\right|\hfill & v>1152.67\text{m}/\text{s}\hfill \end{array}$

  (2.171)

Similar fits can be obtained from experimental data (Nix & Menezes, 1971). However, experimental data are seldom available for very high speeds, so one has to rely most of the time on MD data fits.

The use of direct MD data fits might seem not as desirable as other options shown above. However, one must bear in mind that the equations such as Eq. (2.169), Eq. (2.170), or even the inertial law Eq. (2.159) are fits in their own right, and with their own shortcomings as well. Figure 2.45 shows the fit of Taylor's equation (2.169) to the data by Olmsted et al. (2005); it provides a good fit, but so does Eq. (2.171). Both get the saturation of the mobility law at the transverse speed of sound right, and there is no physical reason to believe that Taylor's fit is in any way a reflection of a physical process. The same can be argued about the power law equation (2.170) that would require calculating three m exponents as explained above, or the inertial law which, despite being more physically motivated, it would still need to be fitted to the MD data for the values of the drag coefficient.

8.4.8 The way forward

The biggest challenge when defining the mobility law in D3P is that it needs to be able to describe the motion of a nonuniformly moving high speed dislocation. Unfortunately, there is no complete theory of the motion of dislocations and, hence, there is no consensus as to its specific form. Most of the mobility laws presented here have a highly speculative nature, especially in the relativistic regime.

On one hand, the use of an inertial force seems physically motivated, but it still requires the fitting of the drag coefficient (vid. Bitzek & Gumbsch, 2004) and the effect of inertia itself seems to be small. On the other hand, MD or experimental fits seem to produce behaviors similar to those obtained through inertial laws, but are obviously mere fits. The main advantage of the latter is their smaller computational cost, as they require the evaluation of a polynomial, while the inertia-based laws usually require solving a nonlinear first order differential equation, often numerically. In the D3P simulations that will be presented in this chapter, numerical fit laws are used for that reason.

8.5.1 The source strength: strain–rate dependence of the strength of a frank–read source

In DDP and D3P, the selection of the source strength is physically justified. As done by Frank and Read (1950) and later corrected by Foreman (1967), the Frank–Read source strength is given by

${\tau }_{\text{nuc}}={\beta }_{\text{nuc}}\frac{\mu b}{l_{\text{FR}}}$

  (2.173)

where l_FR is the segment's length, μ the shear modulus, b the magnitude of the Burgers vector, and β_nuc a material-dependent parameter said to be of the form (Foreman, 1967)

${\beta }_{\text{nuc}}=\frac{A}{2\pi }\left[\text{ln}\frac{l_{\text{FR}}}{r_0}+B\right]$

  (2.174)

where A and B are material constants of order unity, and r₀ is the core cutoff radius.

Thus, according to Eq. (2.173), the source strength is inversely proportional to the length of the pinned dislocation segment. Realistically, it is not possible to know the length of the Frank–Read source segments in a given sample and, hence, even 3D DD models must estimate it statistically. Traditionally, in DDP the source strength was assumed to follow a normal (gaussian) distribution of a given variance with respect to the mean (Van der Giessen & Needleman, 1995). This has always had the problem of allowing, especially in large samples with many sources, strengths much larger and smaller than the mean, to the point that they stood the chance of having a negative source strength.

Shishvan and Van der Giessen (2010) have recently argued that the length of the source segments must follow a log-normal distribution where the maximum source length l_FR^max is limited by the dimensions of the sample (for a rectangular sample of dimensions h × d, it would be $l_{\text{FR}}^{\text{max}}=\sqrt{h^2+d^2}$, the maximum length that can fit inside the sample), whereas the minimum source length l^min_FR must be such that the resulting source strength is still lower than the lattice resistance or a distance of a single Burgers vector, whichever is reached first.

Either way, if l_FR follows a log-normal distribution, then it is ln τ_nuc and not τ_nuc that follows a normal distribution, while τ_nuc is also log-normally distributed. Therefore, τ_nuc has associated minimum and maximum values defined by the maximum and minimum lengths, respectively. This prevents negative source strength values altogether. An additional offset value τ⁰_nuc was introduced by Shishvan and Van der Giessen (2010) to account for other effects such as image forces, obstacles in the nucleation path, and size effects, leading to a source strength of the form

${\tau }_{\text{nuc}}={\tau }_{\text{nuc}}^0+{\tau }_{\text{nuc}}^{\text{log-norm}}$

  (2.175)

where τ⁰_nuc is the offset, and τ^log-norm_nuc the value obtained from the log-normal distribution of source lengths.

8.5.2 Activation times

The source activation time is the time it takes for the Frank–Read source segment to reach the unstable position. The activation time is of foremost importance both for DDP quasi-static models, where it is linked to size effects and dislocation starvation processes (Balint et al., 2006; Deshpande, Needleman, & Van der Giessen, 2005), and to D3P, where any viable dislocation generation mechanism must be at least as fast as the rise time of the shock front.

Benzerga, Bréchet, Needleman, and Van der Giessen (2004) and Benzerga (2008) were able to calculate the activation time analytically for the quasi-static case, using the considerations that are reproduced here. Consider a dislocation segment of length l_FR as depicted in Fig. 2.47. A resolved shear stress τ is applied over it, as a result of which the segment begins to bow out. Define the distance h(x, t) as the distance between any one infinitesimal element of the bowing out loop and the unbowed position. Consider the force balance between the resolved shear stress, the line tension, and the drag on the central segment, x = 0:

$\tau ·b=\text{d}v\left(t\right)+\frac{\mu b^2}{R\left(t\right)}$

  (2.176)

where $\nu =\frac{\text{d}h\left(0;t\right)}{\text{d}t}$ is the segment velocity and R(t) the radius of curvature of the central segment. Call h(0; t) ≡ h(t) for brevity.

The radius of curvature R(t) can be related to h(t) as follows (Benzerga et al., 2004):

$R\left(t\right)=\frac{h\left(t\right)}{2}+\frac{l_{\text{FR}}^2}{8h\left(t\right)}$

  (2.177)

Substituting that into Eq. (2.176) leads to the following expression:

$\frac{b}{d}{\displaystyle {\int }_0^{t_{\text{nuc}}}\text{d}t}={\displaystyle {\int }_0^{l_{\text{FR}}/2}g\left(h\right)\text{d}h}$

  (2.178)

where

$g\left(h\right)=1+\frac{{\tau }_{\text{nuc}}}{\tau }\frac{h{l}_{\text{FR}}}{h^2-\left({\tau }_{\text{nuc}}/\tau \right)l_{\text{FR}}h+\left(l_{\text{FR}}^2/4\right)}$

  (2.179)

The nucleation time can therefore be expressed as

$t_{\text{nuc}}=\frac{1}{2}\frac{\text{d}{l}_{\text{FR}}}{{\tau }_{\text{nuc}}b}\mathcal{F}\left(\xi \right)$

  (2.180)

with ξ = τ/τ_nuc and

$\mathcal{F}\left(\xi \right)=1+\frac{2}{\xi }\left[\frac{1}{2}\text{ln}\left(2\frac{\xi -1}{\xi }\right)+\frac{1}{\sqrt{{\xi }^2-1}}\times \left(\text{arctan}\frac{1}{\sqrt{{\xi }^2-1}}+\text{arctan}\sqrt{\frac{\xi -1}{\xi +1}}\right)\right]$

  (2.181)

This derivation refers to circumferential loops. Benzerga (2008) tackled the case of elliptical loops with different Burgers vector characters. Although more accurate expressions can be obtained by doing that, the underlying physics remains unchanged and the differences between circumferential and elliptical loops are relatively minor. eor this reason and the inherent simplicity of the circular case, the case of circumferential loops alone will be considered here.

Dynamic case. The main underlying assumption in the derivation above is that the linear viscous drag mobility law can be applied in the force balance between the line tension, the applied stress, and the drag force itself. This has been done in Eq. (2.176), from which the rest of the derivation follows. This assumption is valid if the dislocation segment is expected to move at very low speeds (M_t = v/c_t ≈ 0.01–0.1); this is the case when the applied stress τ is expected to be of the same order of magnitude as the source strength itself. However, of relevance to D3P are situations such as shock loading, where the applied stress can easily reach several gigapascals in magnitude. This is usually thought to lead to dislocation speeds that are a significant fraction of the transverse speed of sound. As discussed in Section 8.4, if the dislocation segment reaches high enough speeds, the linear drag law is not applicable any longer. In that case, the force balance defined in Eq. (2.176) must be modified to capture dynamic effects on dislocation motion. Thus, the effect of large “dynamic” loads on the activation time of Frank–Read sources will be discussed next. This entails modifying the mobility law of dislocations to account for dynamic effects as those discussed in Section 8.4.

As it has been discussed in Section 8.4, there is no unique way of defining the mobility of a dislocation segment at high speeds. On one hand, the use of saturating drag coefficients such as that proposed by J. Taylor (1969) has been discussed. That would involve employing the force balance defined in Eq. (2.176) with a drag coefficient d of the form

$d=\frac{d_0}{1-v^2/c_{\text{t}}^2}$

  (2.182)

When this coefficient is employed in Eq. (2.176), a nonlinear differential equation is reached instead:

$\tau ·b=\frac{d_0}{1-\frac{1}{c_{\text{t}}^2}{\left(\frac{\text{d}h}{\text{d}t}\right)}^2}\frac{\text{d}h}{\text{d}t}+\frac{\mu b^2}{\frac{h\left(t\right)}{2}+\frac{l_{\text{FR}}^2}{8h\left(t\right)}}$

  (2.183)

On the other hand, as it has also been discussed in Section 8.4, Eq. (2.176) could be modified by introducing an inertia term. In that case, the force balance would be modified to

$m\frac{{\text{d}}^{2}h\left(t\right)}{\text{d}{t}^2}+d\frac{\text{d}h\left(t\right)}{\text{d}t}+\frac{\mu b^2}{R\left(t\right)}=\tau ·b$

  (2.184)

where m is the dislocation mass, and d is the drag coefficient. Depending on the model employed, both m and d can be velocity dependent.

Both Eqs. (2.183) and (2.184) are nonlinear differential equations, so achieving an analytical solution is unlikely. Through careful manipulation of the equations, and depending on the inertial and drag model employed, numerical solutions show (Gurrutxaga-Lerma, Balint, Dini, Eakins, & Sutton, In preparation) that taking typical values like l_FR = 10⁻⁷ m (this corresponds to a source strength of about 70 MPa), τ = 1 GPa, d₀ = 10⁻⁴ Pa s, b = 2.85 Å, c_t = 3200 m/s, one finds values of t_nuc ≈ 20–100 ps. Caeteris paribus, the activation time is seen to increase with the source length, and decrease with the applied stress. The computation of these values is numerically expensive, so in D3P it is more cost-effective to first tabulate them, and then interpolate exact values falling between the tabulated ones.

These results suggest that the activation times of Frank–Read sources are, for high applied stresses, of the order of tens of picoseconds at their smalls. The analysis performed in Section 8.4 suggests that the acceleration times of dislocations are in general very small. If one drops the inertia term from Eq. (2.184) and integrates it to calculate t_nuc when h = l_FR/2, one gets

$t_{\text{nuc}}=\frac{l_{\text{FR}}}{2\tau b}d$

  (2.185)

Substituting in the usual values gives t_nuc = 20 ps. This compares very well with the predictions made by Benzerga et al. (2004) given in Eq. (2.180), that

$t_{\text{nuc}}=\frac{1}{2}\frac{\text{d}{l}_{\text{FR}}}{\tau b}\mathcal{F}\left(\xi \right)$

  (2.186)

The value of $\mathcal{F}\left(\xi \right)\approx 1$ when ξ 1, as it is in this case. As expected, the static prediction matches the one above.

These values also compare well with those predicted using a constant inertial mass (t_nuc = 2 × 10⁻¹¹ s); and with those obtained using Taylor's saturating coefficient, albeit the latter seems to produce even larger values as the applied stress drives the dislocation segment toward the transverse speed of sound.

In general, for the high strain rates associated with high loads in shock loading, these results would suggest that Frank–Read sources play a secondary role in relaxing shock fronts. If the rise time of the shock front—i.e., the inverse of the strain rate—is very low, Frank–Read sources hardly have enough time to be activated. For instance, if the minimum source activation time is 40 ps and the strain rate is 10¹⁰ s⁻¹ (rise time 100 ps), then the Frank–Read source will be activated only twice at the front. This reduces considerably the ability of Frank–Read sources to relax shock fronts at high strain rates.

8.6.1 Spatial distribution of frank–read sources

In DDP, Frank–Read sources represent out-of-plane pinned dislocation segments. In the same way, the Frank–Read source segment's length cannot be ascertained a priori, and it is distributed as a random variable, the spatial distribution of these segments cannot be deterministically specified. Hence, DDP distributes Frank–Read sources randomly throughout the sample. D3P proceeds exactly in the same manner as DDP, because there is no physical reason why preexisting Frank–Read sources ought to be distributed in a different way simply because they are going to be subjected to a dynamic loading such as a shock front.

Thus, the total number of Frank–Read sources to be randomly allocated is determined by defining the density of sources ρ_source as the number of sources per unit area. Typical values in DDP (and D3P) are about 100 sources/μ m². Once the source density is defined, the total number of sources, calculated as the product of ρ_source and the total area of the sample, is allocated in random positions of the slip systems as defined in Section 8.3. This usually entails randomly selecting, out of their numbers, the slip plane and then randomly positioning the source in it.

In D3P, Frank–Read sources are randomly distributed only once, at the beginning of the simulation, and their position remains throughout so as to simulate ever-pinned Frank–Read source segments.

8.7.1 Constitutive rules for homogeneous nucleation in D3P

In D3P, homogeneous nucleation is included, taking into account the considerations above, to provide an alternative, fast-operating dislocation generation mechanism. Homogeneous nucleation sources operate in a manner akin to how Frank–Read sources do in DDP and D3P. Thus, they will have a source strength, an “activation” time, an injection distance, and a spatial distribution that will resemble the specific physics of homogeneous nucleation.

Regarding the spatial distribution, homogeneous sources are randomly allocated every time step; any position in the slip plane was considered a potential nucleation site down to a spacing of 10b. On a first approach approximation, the homogeneous source strength is set to be almost the ideal shear strength (Tschopp & McDowell, 2007), estimated at τ_hom = 1/4π μ, with μ the shear modulus; temperature dependance and other effects can possibly lower this barrier (Aubry et al., 2011; Tschopp & McDowell, 2008), but 1/4π μ is an upper limit that should be relatively close to the actual value. It is important to highlight, however, that, unlike Frank–Read sources, the homogeneous source strength is not a statistical variable in D3P. Otherwise, as with Frank–Read sources, upon overcoming the source strength, a dipole is injected into the system. The injection distance is randomly allocated following a Poisson distribution²¹ peaking at ≈ 5b; this reflects the observation made in Gutkin and Ovidko (2008), Aubry et al. (2011) that homogeneous nucleation occurs through nanodisturbances in the crystalline lattice: new loops are generated through nonquantized nanodislocation loops where the Burgers vector of the loop grows in size from 0 to b, and the dislocation loop can either grow or collapse back depending on its size—this is reflected through the injection of randomly spaced dipoles. Thus, many dipoles collapse due to their mutually attracting forces being larger than the external driving forces. The generation time for homogeneous nucleation is set to be instantaneous with respect to the time step of the simulation (typically 10⁻¹² s).