5.5 The uniformly moving special case

For the case in which the dislocation moves uniformly with constant speed v, the past history function takes the form l(t) = vt. In that case, it is possible to find an analytical solution for elastodynamic fields of dislocations, either by the equations given in Table 2.2 or by directly solving the governing equations using η(x) = x · d where d = 1/v is the slowness of the dislocation.

In the latter case, one would obtain that the transformed potentials can be written as:

$\text{Ψ}\left(\lambda ,z,s\right)=-\frac{\text{Δ}u\left(b^2-2{\lambda }^2\right)}{s^2{b}^2\beta }\left[{\displaystyle {\int }_0^{\infty }{\text{e}}^{-s\left(d+\lambda \right)\xi }\text{d}\xi }\right]{\text{e}}^{-s\beta z}$

  (2.71)

$\text{Φ}\left(\lambda ,z,s\right)=-\frac{2\text{Δ}u\lambda }{s^2{b}^2}\left[{\displaystyle {\int }_0^{\infty }{\text{e}}^{-s\left(d+\lambda \right)\xi }\text{d}\xi }\right]{\text{e}}^{-s\alpha z}$

  (2.72)

These expressions can be directly integrated to obtain:

$\text{Ψ}\left(\lambda ,z,s\right)=-\frac{\text{Δ}u\left(b^2-2{\lambda }^2\right)}{s^2{b}^2\beta }\frac{1}{s\left(d+\lambda \right)}{\text{e}}^{-s\beta z}$

  (2.73)

$\text{Φ}\left(\lambda ,z,s\right)=-\frac{2\text{Δ}u\lambda }{s^2{b}^2}\frac{1}{s\left(d+\lambda \right)}{\text{e}}^{-s\alpha z}$

  (2.74)

Following the same procedure as described before, on would obtain the elastodynamic fields of a uniformly moving dislocation that begun its motion at t = 0. The resulting stress field components are shown in Table 2.3. It is worth noticing that, as with the general mobile contributions, the solutions presented in Table 2.3 must be added to the injection contributions.

6.1 The integration limits

If the elastodynamic fields of dislocations crucially depend on the past positions of the dislocation and the integration is performed over the spatial positions of the dislocation line, one must wonder whether the improper integration limits given in Table 2.2 entail a violation of causality; the ξ → ∞ upper integration limit seems to require knowledge of the dislocation line's position beyond its current one.

This is not the case. Consider the following integral extracted from Table 2.2:

$I\equiv I\left(x,z,t\right)=\frac{2\text{Δ}u}{\pi b^2}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}a\right)\frac{\tilde{t}z\left[{\tilde{t}}^2\left(z^2-3{\tilde{x}}^2\right)+a^2\left(2{\tilde{x}}^4+{\tilde{x}}^2{z}^2-z^4\right)\right]}{T_a{\tilde{r}}^6}}\text{d}\xi$

  (2.83)

where $\tilde{t}=t-\eta \left(\xi \right)$, $\tilde{x}=x-\xi$, $T_a=\sqrt{{\tilde{t}}^2-a^2{r}^2}$ with ${\tilde{r}}^2={\tilde{x}}^2+z^2$.

Notice that the integrand is multiplied by a Heaviside function, $H\left(\tilde{t}-a\tilde{r}\right)$. This function cancels the integrand for those values of ξ that, for a given spatial point (x, z) and instant in time t, the elastodynamic perturbations cannot have reached. Therefore, here $\tilde{t}-\tilde{r}a$ plays the role that retarded times play in electrodynamics. It follows that causality is not broken because there is no point and instant for which $H\left(\tilde{t}-a\tilde{r}\right)$ does not cancel before the current time.

Furthermore, in the integral above one can write:

$I=\frac{2\text{Δ}u}{\pi b^2}{\displaystyle {\int }_{{\xi }_0}^{{\xi }_t}\frac{\tilde{t}z\left[{\tilde{t}}^2\left(z^2-3{\tilde{x}}^2\right)+a^2\left(2{\tilde{x}}^4+{\tilde{x}}^2{z}^2-z^4\right)\right]}{T_a{\tilde{r}}^6}}\text{d}\xi$

  (2.84)

where ξ₀ and ξ_t are such that, for each instant in time t, they cancel the retarded time. That is, ξ₀, ξ_t are such that

$t-\eta \left({\xi }_{\left(0,t\right)}\right)-a\sqrt{{\left(t-\eta \left({\xi }_{\left(0,t\right)}\right)\right)}^2-a^2\left({\left(x-{\xi }_{\left(0,t\right)}\right)}^2+z^2\right)}=0$

  (2.85)

It immediately follows that both ξ₀ and ξ_t are functions of both t and (x, z). That is, if the values of I at a certain point (x, z, t) are sought, then the integration limits will change accordingly so as to cancel the retarded time—cf. Eq. (2.85). It is clear that ξ₀ = 0 for all subsonic motion. As for ξ_t, it marks the spot the earliest radiated elastic perturbation might have reached at time t.

6.1.1 The past history function and the mobility law

Due to the explicit dependence on a past history function η(ξ) that is not generally known, a general analytic expression of the mobile contributions in Table 2.2 cannot be achieved. There are nonetheless special cases, such as that of the uniformly moving dislocation worked out in Section 5.5.

The past history function may not be known a priori. However, it is fundamentally related to the physics of the motion of a dislocation in that, to all effects, it stores its outcome. Consider a dislocation that, at a given time t₀, has its line at position x = ξ₀; after a time step of magnitude Δt, the current time will be t₁ = t₀ + Δt, and the position of the dislocation line will have been updated to some ξ₁ = ξ₀ + Δξ₁ in the past history function.

Mathematically, ξ and t are related in no other way than through the past history function t = η(ξ) itself. Physically, however, the dislocation line at time t₀ and position ξ₀ will be subjected to a series of external stimuli which, irrespective of their origin, will cause the dislocation to move in such a way that at time t₁ = t₀ + Δt the dislocation line is found at ξ₁ = ξ₀ + Δξ₁. The response of the dislocation to those stimuli (typically, external stress fields) is given by the dislocation's own mobility law, the forms and origin of which discussed in greater detail in Section 8.4.

Assume for instance a mobility law that requires that the dislocation moves with

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

where v_dislocation is the speed of the dislocation, d is some drag coefficient, τ an applied resolved shear stress, and B the magnitude of the Burgers vector of the dislocation. Assume that, due to numerical reasons, a D3P simulation can have a time step as small as Δt at most. Provided the value of $\frac{\tau B}{d}$ is known at instant t₀, the mobility law requires that, throughout the next time step, the dislocation's speed be ${\nu }_{\text{disloaction}}=\frac{\tau B}{d}$. In that case, the updated position of the dislocation can immediately be calculated as ξ₁ = ξ₀ + v_dislocation Δt.

This process of Euler-forward integration is employed in DDP (Van der Giessen & Needleman, 1995). D3P, as an extension of DDP, employs it as well (Gurrutxaga-Lerma et al., 2013). The integration scheme provides a natural way of building up (and storing) the past history function as a discrete sequence of pairs {(ξ_j, t_j)}, j = 0, 1, 2,… such that each pair can be obtained by applying the mobility law to the previous state.

This would lead to a discrete past history function. However, the expressions in Table 2.2 require a continuous η(ξ). In principle, one has no knowledge of the intermediate positions inside the intervals defined by the sequence of pairs (i.e., {(ξ_j, ξ_j+l), (t_j, t_j+1)}, j = 0, 1, 2,…), down to the exactitude of the DD method itself. In fact, it is easy to conceive, especially for small Δt, that within a time interval the dislocation the end position of the interval, ξ_j+1, and then back to the interval due to vibrations, etc. Nevertheless, here the following convention is adopted: the intermediate points within an integration interval {(ξ_j, ξ_j+1), (t_j, t_j+1)} satisfy the mobility law, with the same kinematic parameters (velocity, acceleration, etc.) as those the dislocation takes on the lower bound of the interval, (ξ_j, t_j).

That is, if the motion at instant t_j obeys a mobility law of the form⁷

$v_{\text{dislocation}}={\frac{\tau B}{d}|}_{t=t_j}$

then all the pairs in the ((ξ_j, ξ_j+1), (t_j, t_j+1)) interval satisfy that very same mobility law, with the values of the kinematic variables (speed, acceleration, etc.) being those at t = t_j.

In this convention, η(ξ) has a form such that the position of the dislocation and its temporal derivatives (velocity, acceleration, …) satisfy the mobility law. For instance, for steps of constant speed, the past history function becomes a sequence of linear segments as depicted in Fig. 2.15. Each segment would correspond to the constant velocity of dislocations in the interval that delimits it. The relative noise appearing in elastodynamic solutions such as that shown later in Fig. 2.35B is in fact affected by the smoothness of η(x). Smoother results are obtained with approximations to η(ξ) that are smooth⁸ at the end of each time step, even if this means that intermediate points within the intervals do not satisfy the mobility law. There are several ways the latter can be done. For instance, one could interpolate past positions using cubic splines instead of segments, ensuring the smoothness at the end points of the past history intervals. Alternatively, smaller time steps are seen to decrease the noise in the solution. Because D3P is typically used to simulate fast phenomena such as shock loading, where the representative time can be of the order of a few picoseconds up to a few nanoseconds, the time step used is usually small enough to ensure the quality of the mobile contributions.

6.2 Numerical integration schemes

The need for a numerical integration scheme for the mobile contributions is justified by the lack of a priori knowledge of η(x). Even if it were known, the integrals to solve are usually elliptic integrals of the first, second, and third kind, which are difficult to solve analytically.

For numerical integration purposes, it is most convenient to subdivide the global integration interval, (0, ξ_t), into the intervals of the past history function, i.e., {(ξ_j, ξ_j+1), (t_j, t_j+1)}, j = 0, 1, 2,…, that is,

${\displaystyle {\int }_0^{{\xi }_t}=}{\displaystyle {\int }_0^{{\xi }_1}+{\displaystyle {\int }_{{\xi }_1}^{{\xi }_2}+\cdots +{\displaystyle {\int }_{{\xi }_j}^{{\xi }_{j+1}}+\cdots +{\displaystyle {\int }_{{\xi }_{t-1}}^{{\xi }_t}}}}}$

  (2.86)

where each (ξ_j, ξ_j+1) is one of the D3P integration intervals.

Subsequently, the numerical integration can be performed over each of those subintegrals

$I_j={\displaystyle {\int }_{{\xi }_j}^{{\xi }_{j+1}}}$

where for any integral in Table 2.2 one would have

$I={\displaystyle {\int }_0^{{\xi }_t}={\displaystyle \sum _{j=0}^{T-1}{\displaystyle {\int }_{{\xi }_j}^{{\xi }_{j+1}}}}}$

  (2.87)

where T corresponds with ξ_t.

The numerical method for the solution of each of the integrals is a matter of choice; in this work an adaptive Gauss–Kronrod (P. Davis & Rabinowitz, 1984) quadrature has been used. In general, adaptive methods seem to be the most cost effective.

However, numerical integration schemes cannot be directly used over the integrals in Table 2.2 without further care. This is due to the presence of several singularities in the integrand. These singularities require the specific treatment that is described in the following section.

6.3 Integration of the stress fields in the mobile contributions

As can be observed in Table 2.2, the stress component fields in the mobile contributions are expressed as the temporal derivative of integral expressions. There are several ways to proceed. Perhaps the most immediate one would be to numerically solve the integrals and then perform a numerical differentiation upon them. This is, however, computationally expensive and, as with all numerical derivatives of numerical data, inaccurate.

In principle, the use of algorithmic differentiation (AD) would allow for the stress field components to be obtained in parallel to the displacement field components, using forward AD schemes (Griewank & Walther, 2008) to obtain the spatial derivatives of displacement. The advantages of such a method as opposed to traditional numerical differentiation schemes such as finite differences are significant: for one thing, any discretization error is prevented, and the differential is expected to be as accurate as the underlying numerical method applied (Griewank & Walther, 2008)—in this case, that of the numerical integration scheme. Nonetheless, forward AD amounts to the analytic differentiation of the integrand of the primitives, i.e., it amounts to interchanging the order of differentiation and integration in the primitives.

As argued by Markenscoff and Clifton (1981), the interchange of the order of differentiation and integration in this case is not legitimate. It would give rise to terms of the order of T⁻³_a and T⁻³_b that are, in principle, nonintegrable singularities. AD or a numerical differentiation scheme over the integrals in Table 2.2 would amount to such an interchange, so further considerations are required.

The way to proceed here was devised by Markenscoff and Clifton (1981). Consider for instance a troublesome term from the transverse wave component of σ_xz:

${{\sigma }_{\mathit{xz}}|}_b=\frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)}\frac{b^4{\left({\tilde{x}}^4-z^4\right)}^2+T_b^2\left(8{\tilde{t}}^2{\tilde{x}}^2{z}^2-{\tilde{r}}^4{\tilde{t}}^2\right)}{r^8{T}_b}\text{d}\xi$

  (2.88)

Two terms can be identified. The second one does not produce a T⁻³_b singularity, so the order of integration and differentiation can be interchanged directly:

$\begin{array}{ll}{i}_1\hfill & =\frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)\frac{T_b\left(8{\tilde{t}}^2{\tilde{x}}^2{z}^2-{\tilde{r}}^4{\tilde{t}}^2\right)}{{\tilde{r}}^8}\text{d}\xi }\hfill \\ \hfill & ={\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)\frac{\left(8{\tilde{x}}^2{z}^2-{\tilde{r}}^4\right)}{{\tilde{r}}^8}\frac{\partial }{\partial t}\left[T_b{\tilde{t}}^2\right]\text{d}\xi =}\hfill \\ \hfill & ={\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)\frac{\tilde{t}\left(8{\tilde{x}}^2{z}^2-{\tilde{r}}^4\right)\left(3{\tilde{t}}^2-2b^2{\tilde{r}}^2\right)}{{\tilde{r}}^8{T}_b}\text{d}\xi }\hfill \end{array}$

In turn, the first term gives rise to the Tb⁻³ singularity mentioned above. It can be integrated by parts (Markenscoff & Clifton, 1981) so that the resulting integral allows the interchange as well:

$\begin{array}{ll}{i}_2\hfill & =\frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)\frac{b^4{\left({\tilde{x}}^4-z^4\right)}^2}{{\tilde{r}}^8{T}_b}\text{d}\xi }=\frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)\frac{b^4{\left({\tilde{x}}^2-z^2\right)}^2}{{\tilde{r}}^4\left(b^2\tilde{x}-{\eta }^{\prime }\left(\xi \right)\tilde{t}\right)}\text{d}{T}_b=}\hfill \\ \hfill & =\frac{\partial }{\partial t}\left[\frac{b^4{\left(x^2-z^2\right)}^2{T}_{b_0}}{r^4\left(b^{2}x-{\eta }^{\prime }\left(0\right)t\right)}\right]H\left(t-rb\right)-{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)\frac{\partial }{\partial t}\left[T_b\frac{\partial }{\partial \xi }\left[\frac{b^4{\left({\tilde{x}}^2-z^2\right)}^2}{{\tilde{r}}^4\left(b^2\tilde{x}-{\eta }^{\prime }\left(\xi \right)\tilde{t}\right)}\right]\right]\text{d}\xi =}\hfill \\ \hfill & =\frac{b^6{\left(x^2-z^2\right)}^2\left[tx-r^2{\eta }^{\prime }\left(0\right)\right]}{r^4{T}_{b_0}{\left(b^{2}x-t{\eta }^{\prime }\left(0\right)\right)}^2}H\left(t-rb\right)-{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}b\right)F\left(\tilde{x},z,\tilde{t}\right)\text{d}\xi }\hfill \end{array}$

  (2.89)

with $T_{b_0}=\sqrt{t^2-b^2{r}^2}$, $r=\sqrt{x^2+z^2}$ and

$F\left(\tilde{x},z,\tilde{t}\right)=\frac{b^4\left({\tilde{x}}^2-z^2\right)}{{\tilde{r}}^6{T}_b{\text{Γ}}_b^3}\left\{\tilde{t}{\text{Γ}}_b\left[b^2\left({\tilde{x}}^4-8{\tilde{x}}^2{z}^2-z^4\right)+8\tilde{t}{z}^2\tilde{x}{\eta }^{\prime }-\left({\tilde{x}}^2-z^2\right){\tilde{r}}^2{{\eta }^{\prime }}^2+\tilde{t}\left({\tilde{x}}^2-z^2\right){\tilde{r}}^2{\eta }^{″}\right]+T_b^2\left[8\tilde{t}\tilde{x}{z}^2{{\eta }^{\prime }}^2-2\left({\tilde{x}}^2-z^2\right){\tilde{r}}^2{{\eta }^{\prime }}^3+b^2\tilde{x}\left({\tilde{x}}^2-z^2\right){\tilde{r}}^2{\eta }^{″}+{\eta }^{\prime }\left(2b^2\left({\tilde{x}}^4-4{\tilde{x}}^2{z}^2-z^4\right)+\tilde{t}\left({\tilde{x}}^2-z^2\right){\tilde{r}}^2\right){\eta }^{″}\right]\right\}$

  (2.90)

with ${\text{Γ}}_b=b^2\tilde{x}-{\eta }^{\prime }\left(\xi \right)\tilde{t}$.

Throughout the derivations of the elastodynamic fields above, it is assumed that the dislocation is quiescent before it begins its motion, that is, v_dislocation (t = 0⁺) = 0, whence η′(0⁺) = ∞. Therefore,

$\frac{\partial }{\partial t}\left[\frac{b^4{\left(x^2-z^2\right)}^2{T}_{b_0}}{r^4\left(b^{2}x-{\eta }^{\prime }\left(0\right)t\right)}\right]H\left(t-\mathit{rb}\right)$

  (2.91)

must vanish. A dislocation jumping from rest to some velocity would, on the other hand, have a η(0) ≠ ∞; this case is a mere mathematical subtlety bearing little physical significance.

The procedure presented above can be equally extended to the rest of terms and components of the elastic fields. The derivatives obtained in this way are collected in Table 2.4.

Integration along the dislocation's path: the z = 0 special case Besides the subtleties in the order of differentiation and integration, some of the integrals in Table 2.2 present an anomalous behavior when z = 0. This is particularly important because z = 0 corresponds with the dislocation's slip plane, i.e., the direction of motion.

For example, consider the longitudinal part of σ_xz in Table 2.2:

$I\left(x,z,t\right)=\frac{4\text{Δ}u}{b^2}\mu \frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\tilde{r}a\right)\frac{a^4{\tilde{x}}^2{z}^2{\tilde{r}}^4+T_a^2\left(8{\tilde{t}}^2{\tilde{x}}^2{z}^2-{\tilde{r}}^4{\tilde{t}}^2\right)}{{\tilde{r}}^8{T}_a}\text{d}\xi }$

  (2.96)

Make z = 0. The integral becomes

$I\left(x,0,t\right)=\frac{4\text{Δ}u}{b^2}\mu \frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }H\left(\tilde{t}-\left|\tilde{x}\right|a\right)\frac{-{\tilde{t}}^2{T}_a}{{\tilde{x}}^4}\text{d}\xi }$

  (2.97)

In this integral, at any time t the integrand is singular if x = ξ. This is not problematic as long as the singularity does not fall within the integration interval [ξ = 0, ξ = ξ_t]. However, for any point (x, z = 0) laying ahead of the injection site (i.e., x = 0) but behind the current position of the dislocation core (i.e., x = ξ_t), the denominator in Eq. (2.97) cancels (i.e., $\tilde{x}=x-\xi =0$) within the integration path. Hence, the integral diverges when one tries to compute current values of the elastic field in former positions of the core of the dislocation. Not giving this case a proper consideration results in large numerical instabilities that preclude a proper computation of the values of the fields obtained along the path of the dislocation.

Markenscoff (1980) and Markenscoff, Clifton (1981) pointed out that a divergence in Eq. (2.96) within the integration path implies an overlooked burden in its original derivation. Take Eq. (2.96). This expression, as all the rest in Table 2.2, is derived from an integral in the reciprocal Laplace space which, prior to obtaining the Cagniard form of the integral, takes the form

$I=\frac{4\text{Δ}u}{b^2}\mu {\displaystyle {\int }_0^{\infty }\alpha {\lambda }^2{\text{e}}^{-s\eta \left(\xi \right)}{\text{e}}^{-s\left[\alpha z-\lambda \xi \right]}\text{d}\xi }$

  (2.98)

The first inverse Laplace transform (in the spatial variable) leads to

$\widehat{i}=\frac{s}{2\pi \text{i}}\frac{4\text{Δ}u}{b^2}\mu {\displaystyle {\int }_{-i\infty }^{i\infty }\left[{\displaystyle {\int }_0^{\infty }\alpha {\lambda }^2{\text{e}}^{-s\eta \left(\xi \right)}{\text{e}}^{-s\left[\alpha z-\lambda \xi \right]}\text{d}\xi }\right]{\text{e}}^{s\lambda x}\text{d}\lambda }$

  (2.99)

Once Eq. (2.99) is obtained, the derivation of the elastodynamic fields in Section 5 proceeds to interchange the order of integration from ξ to λ by invoking Fubini's theorem. This puts the integral in its Cagniard form. It is a well-known problem, however, that Fubini's theorem is pertinent only if both the integral in ξ and the integral in λ converge (A. Taylor, 1986). Equation (2.96) highlights that the latter is not true when x falls behind the dislocation line but ahead of the injection site.

As proposed by Markenscoff (1980), the integral in Eq. (2.99) can be regularized extracting the singular part of the integral (Abramowitz, 1954). The regularization relies on noting that Eq. (2.99) diverges when η(ξ) takes values in the path of ξ. However, the integrand maintains its behavior about x = ξ if η(ξ) is approximated with its first-order Taylor about x = ξ: η(x) + (x − ξ) η′(x). Hence, consider the following integral

$I_{\text{extra}}=\frac{4\text{Δ}u}{b^2}\mu {\displaystyle {\int }_0^{\infty }\alpha {\lambda }^2{\text{e}}^{-s\left[\left(\eta \left(x\right)-{\eta }^{\prime }\left(x\right)\left(x-\xi \right)\right)-\lambda \xi \right]}\text{d}\xi }$

  (2.100)

The first Laplace inversion of this integral becomes,

${\widehat{i}}_{\text{extra}}=\frac{s}{2\pi \text{i}}\frac{4\text{Δ}u}{b^2}\mu {\displaystyle {\int }_{-i\infty }^{i\infty }\left[{\displaystyle {\int }_0^{\infty }\alpha {\lambda }^2{\text{e}}^{-s\left[\eta \left(x\right)-{\eta }^{\prime }\left(x\right)\left(x-\xi \right)\right]}{\text{e}}^{s\lambda \xi }\text{d}\xi }\right]{\text{e}}^{-s\lambda x}\text{d}\lambda }$

  (2.101)

Clearly, the integral in Eq. (2.99) can be regularized by adding and subtracting the integral in Eq. (2.101) to it, because by subtracting the term in Eq. (2.101) from Eq. (2.99) the singularity at x = ξ is canceled at that point (i.e., η(ξ) would cancel with η(x)).

Notice that the additional term corresponds to that of a displaced uniformly moving dislocation with speed η′(x), the fields of which have been solved in Section 5.5 and can be found in Table 2.3.

Unfortunately, this solution leads to even more lengthy formulas. In the example above, the solution would be:

$I\left(x,0,t\right)=\frac{4\text{Δ}u}{\pi b^2}\mu \frac{\partial }{\partial t}{\displaystyle {\int }_0^{\infty }\left[-H\left(\tilde{t}-\left|\tilde{x}\right|a\right)\frac{\tilde{t}{T}_a}{{\tilde{x}}^4}+H\left({\tilde{t}}^{\prime }-\left|\tilde{x}\right|a\right)\frac{{\tilde{t}}^{\prime }{T^{\prime }}_a}{{\tilde{x}}^4}\right]}\text{d}\xi +\frac{{{\tilde{t}}^{\prime }}^2{T^{\prime }}_a}{x^3\left(t-\eta \left(x\right)\right)}H\left(t-\eta \left(x\right)+x{\eta }^{\prime }\left(x\right)-\left|x\right|a\right)$

  (2.102)

$I\left(x,0,t\right)=\frac{4\text{Δ}u}{\pi b^2}\mu {\displaystyle {\int }_0^{\infty }\left[-H\left(\tilde{t}-\left|\tilde{x}\right|a\right)\frac{\tilde{t}\left(3{\tilde{t}}^2-2a^2{\tilde{x}}^2\right)}{{\tilde{x}}^4{T}_a}+H\left({\tilde{t}}^{\prime }-\left|\tilde{x}\right|a\right)\frac{{\tilde{t}}^{\prime }\left(3{{\tilde{t}}^{\prime }}^2-2a^2{\tilde{x}}^2\right)}{{\tilde{x}}^4{T^{\prime }}_a}\right]}\text{d}\xi +\frac{{{\tilde{t}}^{\prime }}^2{T^{\prime }}_a}{x^3\left(t-\eta \left(x\right)\right)}H\left(t-\eta \left(x\right)+x{\eta }^{\prime }\left(x\right)-\left|x\right|a\right)$

  (2.103)

where ${\tilde{t}}^{\prime }=t-\eta \left(x\right)+\tilde{x}{\eta }^{\prime }\left(x\right)$, ${T^{\prime }}_a=\sqrt{{\widetilde{t^{\prime }}}^2-a^2{\tilde{x}}^2}$.

Notice that I(x, 0, t) must be differentiated in time in order to obtain the transverse part of σ_xz(x, 0, t); this shall be done as explained above, integrating by parts where necessary.

The z = 0 case is relevant for the σ_xz and u_z components alone; σ_xx, σ_zz, as well as u_x vanish for z = 0.

6.4 Singularities at the injection front and behind the injection front

The elastic fields of dislocations are singular (i.e., they diverge) at the dislocation core. This is a well-known feature related to the way the core is modeled in elasticity (Hirth & Lothe, 1991). Several attempts to regularize the core analytically the core have been proposed, from the Peierls–Nabarro model (Hirth & Lothe, 1991; Nabarro, 1997; Pellegrini, 2010) to gradient elasticity theory (Lazar, 2013). Despite some recent attempts to perform computer simulations of dislocation dynamics with regularized cores (Pillon, Denoual, Madec, & Pellegrini, 2006; Po, Lazar, Seif, & Ghoniem, 2014), usually dislocations dynamics methods are primarily concerned with the long-range effects of the dislocations. For the study of long-range effects, the classical, core-diverging elastic fields are exact and, in many cases, easier to handle computationally.

However, the presence of an infinity in a computer simulation is neither possible nor desirable, and simplified regularizations have been proposed to deal with them, the most typical one being imposing cutoffs about the core. In 3D dislocation dynamics, where the Peach–Koehler forces are usually directly derived from the elastic energy of dislocations (Bulatov & Cai, 2006), the cutoff is exerted over the energy landscape; this requires the additional estimation of the core energy itself. However, in DDP simulations the Peach–Koehler forces are directly obtained from the analytical expressions of the elastic fields of the dislocations, so the core is regularized through a cutoff of the stress fields about the core itself (Van der Giessen & Needleman, 1995). In either DDP or 3D-DD, the regularization typically consists of a radial cutoff around the singular core that is a few Burgers vector wide; in DDP, the stress field within the cutoff is assumed to be constant and of a value equal to that at the radius of the cutoff.

It can be argued that the same can be done in D3P simulations, and the cutoff concept further extended to deal with an additional feature that was already commented by Markenscoff (1982): the stress field components at the injection wave fronts show singularities of order $1/\sqrt{t}$ as well.