Driving forces on dislocations: finite element analysis in the context of the non-singular dislocation theory

The configurational forces of dislocations are derived in the following using the basic ideas of Eshelby [10]. Consider a dislocated body that occupies a region V with the boundary S. Assuming the small strain approximation and quasi-static problem, the dislocated solid is governed by the mechanical equilibriumwhere \(()_{,j} = {\partial } / {\partial x_j}\), \(\sigma _{ij}\) is the stress tensor, and the body force is ignored. For the general anisotropic case, the constitutive relation for the stress \(\sigma _{ij}\) and elastic strain \(\varepsilon _{ij}^e\) is given aswhere \(c_{ijkl}\) is the stiffness tensor, \(\varepsilon _{kl}^e\) is the elastic strain, \(\varepsilon _{kl}^D\) is the eigenstrain of dislocations, and the total strain \(\varepsilon _{kl}\) is defined as the symmetric part of the displacement gradient \(u_{k,l}\) asEither displacement or traction boundary condition can be consideredwhere \({\bar{u}}_i\) is the prescribed displacement on the boundary, \(n_j\) is the normal vector of the boundary, and \(t_i\) is the applied traction on the boundary.

The associated eigenstrain \(\varepsilon _{ij}^D\) results from the plastic slip generated during the glide of the dislocation line. The eigenstrain tensor is defined as the symmetric form of a dyadic product of the Burgers vector \(b_i\) and the normal vector \(n_i\) of the slip plane Dwhere \(\delta ({\mathbf {x}}-{\mathbf {D}})\) is the one-dimensional Dirac delta function in the normal direction of the slip plane \({\mathbf {D}}\) [27]. Index-notation and Einstein’s summation convention for repeated indices are used here and in the following. Eq. (5) ensures that the eigenstrain is located merely on the slip plane, i.e., when \({\mathbf {x}} \in {\mathbf {D}}\). Consider the definition of the one-dimensional Dirac delta function [27]where \({\mathbf {x}}^D\) is the location vector of the points on the slip plane D. Substituting above equation into Eq. (5), the eigenstrain can be rewritten in terms of convolution aswhere \(\epsilon _{ij} = \left( b_i n_j + b_j n_i \right) /2\), and the symbol \(*\) means the convolution over the slip plane.

In the eigenstrain theory, by introducing the concept of dislocation density, the Burgers vector can be defined as [9]where C is the Burgers circuit, S is the Burgers surface bounded by C. \(\beta _{ij}^D = b_i n_j * \delta ({\mathbf {x}})\) is the plastic distortion due to the dislocation, \(\beta _{ij} = u_{i,j}-\beta _{ij}^D\) is the elastic distortion, and \(u_{i,j}\) is the total distortion. \(\alpha _{ij}\) is the dislocation density tensor, according to the Green theorem, it is defined aswhere \(\epsilon _{jkl}\) is the permutation tensor.

The strain energy density for the dislocated body can be rewritten aswhere the symmetry of the stress tensor is used in the second equation. In the following, a Volterra dislocation with constant Burgers vector is considered. The gradient of the strain energy is given asBy multiplying Eq. (9) with the permutation tensor \(\epsilon _{kjl}\), one can reformulate the definition of the dislocation density tensor asThen, consider the mechanical equilibrium, the gradient of the strain energy is reformulated asRearranging above equation yields the configurational force balance in the well-known local formwhere the generalized Eshelby stress tensor \(\Sigma _{kj}\) takes the following formand the generalized configurational body forces \(g_k\) is given asThe driving force per unit length on the dislocation can then be calculated by the surface integration of the divergence of the Eshelby stress tensor or the configurational body force. As an example, consider a straight dislocation line with dislocation density of \(\alpha _{il} = b_i \delta (x_1-x_1^C) \delta (x_2-x_2^C) {\hat{\xi }}_l\), one can derive the well-known Peach–Koehler formula [5]where \(x_i^C\) is the location of the dislocation line, \({\hat{\xi }}_l\) is the sense vector of the dislocation line, and \(\sigma _{ij}^0 = \left. \sigma _{ij} \right| _{x_i=x_i^C}\) is the stress field at dislocation core due to external loading. For more details regarding the physical interpretation of the Eshelby stress tensor and Peach–Koehler force, one can refer to Ref. [28, 29].

The classical solution of dislocations discussed above has singularity at the core center. However, the field singularity at the dislocation core has no sound physical interpretation and leads to inconvenience in the numerical simulations. One general idea is to spread the eigenstrain distribution on the slip plane over the surrounding region (regularization of dislocation slip) [2]. The regularization of the eigenstrain distribution is crucial to the accuracy of the numerical solution of dislocations. Cai et al. [22] proposed a non-singular continuum representation of the Burgers vector, which ensures that outside the pre-defined dislocation core the derived stress fields agree well with the classic solutions.

Though the results in this subsection are not limited to it, we demonstrate the configurational force theory in the context of the non-singular dislocation theory. For an arbitrary point on the slip plane, spreading the Burgers vector around the point in the three-dimensional space aswhere \(f_i({\mathbf {x}})\) is the Burgers vector density function. The convolution of the spreading function \({\tilde{w}}({\mathbf {x}})\) with itself defines the second spreading functionwhere \(r=||{\mathbf {x}}||\). The second spreading function \(w({\mathbf {x}})\) is used to modify the distance function \(R=\sqrt{x_1^2+x_2^2+x_3^2}\) to \(R_h=\sqrt{R^2+h^2}\). Then the dislocation core width parameter h is introduced in the formulation such that the modified distance function \(R_h\) ensures the resulting non-singular solution closely resemble the singular solution of the classical theory outside the dislocation core. The analytical form of \({\tilde{w}}({\mathbf {x}})\) is unknown, and Cai et al. [22] presented the following approximate expressionwhere \(h_1 = {0.9038} h\), \(h_2 = {0.5451} h\), and \(m={0.6575}\) are constants.

In the framework of the non-singular continuum theory of dislocations, after spreading the Burgers vector around the slip, the Dirac delta function shown in the last subsection is replaced now by the spreading function \({\tilde{w}}(||{\mathbf {x}}-{\mathbf {x}}^D||,h)\) for the convolutionComparison of Eq. (7) and Eq. (21) implies that the eigenstrain takes the similar form in both the singular and the non-singular theory. In the meantime, the first two equations in Eq. (8) are satisfied with the distribution of Burgers vector Eq. (18) in the non-singular dislocation theory. Therefore, following Eq. (7) to Eq. (16), the configurational force balance for the non-singular continuum theory can be derived by simply replacing \(\delta ({\mathbf {x}}-{\mathbf {x}}^D)\) with \({\tilde{w}}(||{\mathbf {x}}-{\mathbf {x}}^D||,h)\). The resulting configurational force balance takes the same form as Eq. (14), and the non-singular Eshelby stress tensor in the context of the non-singular continuum theory of dislocations is given aswhere \(\beta _{ik}^{ns} = u_{i,k} - b_i n_k * {\tilde{w}}({\mathbf {x}})\). From Eq. (22) it can be seen that the gradient of the distribution function does not show explicitly in the final form of the non-singular Eshelby stress tensor. Eq. (18) guarantees the magnitude of the regularized Burgers vector in the non-singular theory is the same as that of the singular theory, and Eq. (21) guarantees that the eigenstrain is obtained through the convolution over the slip plane similar to Eq. (7). Hence, the derivation of the non-singular Eshelby stress tensor (22) is mathematically consistent with the derivation of the singular Eshelby stress tensor (15) in the context of the non-singular continuum theory of dislocations developed by Cai et al. [22].

In the conventional finite element simulation, the strain is calculated on integration points rather than on the nodes. The distribution of the eigenstrain on integration points is crucial to the accuracy of the numerical solution of dislocations. In the following, we consider the eigenstrain distribution calculated by using the concept of distributed Burgers vector [22], as introduced in Eq. (18). We demonstrate the calculation by the basic examples of the straight and circular dislocation line. For these dislocations, the normal vector will be a constant in local coordinate system. Therefore, the distribution of eigenstrain in Eq. (21) can be written aswhereThe function \(W({\mathbf {x}},h)\) defines the distribution of eigenstrain around the slip plane in the non-singular continuum theory of dislocations. In order to reduce the computational cost, the integration is truncated at a distance \(r_c\) such that the distribution function becomesandwhere \(H(r_c-r)\) is the Heaviside step function. A small \(r_c\) leads to under estimation of the eigenstrain. \(r_c=2h\) is used in this work and it leads to less than 5% error compared to the analytical solution (\(r_c \rightarrow {} \infty \)) [8].

The two-dimensional setup of the numerical integration of \({\tilde{w}}(r,h)\) is shown in Fig. 1a. The dislocation is located at \({\mathbf {x}}=\{0,0,0\}^T\) and the dashed line represents the slip plane. It can be applied to the straight edge dislocation or straight screw dislocation with the dislocation line lying in the \(x_3\) direction. The distribution of the Burgers vector along the \(x_3\) direction remains constant. Therefore, the range of the integration domain for \(x_3^D\) is from \(-r_c\) to \(r_c\). Then the surface integration of w(r, h) can be represented aswhere L represents the integration path along the \(x_1\) direction on the slip plane, and \(H(r_c-r)\) is removed from the integration by changing the integration domain to be always inside \(-r_c\) to \(r_c\) around the integration point \({\mathbf {x}}\). The analytical solution of the integration along the \(x_3^D\)-axis can be determined aswhere \(X = \left( x_1-x_1^D \right) ^2+x_2^2 +h^2 \). The integration along the \(x_1^D\) direction is done numerically, and the trapezoidal numerical integration is used in this work. In other words,where N is the number of integration points (\(N={20}\) is used in this work), and \(X_i\) is the X calculated at the ith integration point. For \({\bar{x}}_1 \in [0,r_c]\), the slip plane is assumed to be extended outside the sample such that the integration domain is \(x_1^D \in [x_1-r_c,x_1+r_c]\). For \({\bar{x}}_1 \in [r_c,(n+1)r_c]\), the integration domain is \(x_1^D \in [x_1-r_c,x_1+r_c]\). For \(x_1 \in [-r_c,r_c]\), the integration domain is \(x_1^D \in [x_1-r_c,0]\). For the rest part of the sample, \(W_1({\mathbf {x}},h)=0\). It should be noted that \(W({\mathbf {x}},h)\) is only a function of the coordinate \(x_i\), therefore, it only has to be calculated once at the initial step.

Figure 1b shows the numerical results of \(W({\mathbf {x}},h)\), which also represents the distribution of the eigenstrain. The eigenstrain is constant along the slip plane, except for the region on the right side of the dislocation core. For illustration, only a circular dislocation loop will be considered in this work. The distribution function \(W({\mathbf {x}},h)\) for a circular loop is axis-symmetric. It means that the analytical solution Eq. (28) can also be obtained in the circumferential direction, and the numerical integration along the radius direction is the same with a straight dislocation line. Therefore, the distribution function \(W({\mathbf {x}},h)\) of the straight edge dislocation can also be used along the radius direction for the circular dislocation loop. For the more general application of the distribution of eigenstrain, readers can refer to Ref. [8].

Based on the finite element method, the weak form formulation is constructed for numerical implementation. Multiplying the mechanical equilibrium (1) with the test function \(\eta _i\) and integration over the domain VIntegrating by parts leads to the weak form formulation asAccording to the conventional linear finite element method, the test function and its gradient can be interpolated through the shape function \(N^I\) aswhere \(\eta _i^I\) are the nodal values of the test function. Evaluating Eq. (31) over each element yieldswhere \(V_e\) is the volume of an element.

Similarly, multiplying the configurational force balance (14) with the test function \(\eta _i\) and integrating over the domain V, the weak form formulation is obtained through integrating by parts asThe weak form formulation of mechanical equilibrium is used for calculation of the stress \(\sigma _{ij}\) and the displacement field \(u_i\) of the system. Then the numerical solution of the elastic fields is used in the weak form formulation of the configurational force balance to obtain the nodal configurational forces. Since the elastic fields are calculated based on the non-singular continuum theory of dislocations, according to Eq. (34), the nodal configurational forces acting on a node I are calculated aswhere the assembly operation \(\bigcup \) is performed over all \(n_{el}\) elements adjacent to node I. Then the driving forces on the dislocation can be calculated by summing up all nodal configurational forces around the dislocation core aswhere \(n_{nod}\) is the total number of nodes in the chosen integration volume.

The user element for dislocations is developed and the finite element simulations are performed using the open source software MOOSE [30].

We start with a straight edge dislocation with the Burgers vector \({\mathbf {b}}=[b_0\,0\,0]\) with \(b_0=1\), and the normal vector of the slip plane \({\mathbf {n}}=[0\,1\,0]\). The edge dislocation is placed in the center of the sample. The straight edge dislocation is modeled as a plane strain problem. To assure that there are enough integration points for calculating the distribution of eigenstrain, it is suggested that two element should be covered in \(r_c\) range. The minimum h used in this work is 1, thus, we choose the minimum element size of 0.5. The sample size is \(200\times 200\) and the mesh size is \(400\times 400\). In order to compare the numerical solution of a finite size with the analytical solution of a single dislocation in an infinite medium, one can either use a large-enough sample or apply asymptotically the traction \(t_i=\sigma _{ij}^a n_j\) on the surfaces based on the analytical stress solutions \(\sigma _{ij}^a\). Since the computation is not very expensive, we apply the traction free boundary condition on a sufficiently large sample to reduce traction free boundary effects in this work. We studied the influence of the sample size on the relative error of the numerical solution compared to the analytical solution. For a square sample with a width of 50, the relative error of \(\sigma _{11}\) at \(x_2 = \pm 10\) is around \({-39}{\%}\). When the width of the sample reaches 150, the relative error reduces to around \({-4}{\%}\). Therefore, the sample size used in this numerical example is large enough to get good approximation of the stress field of dislocations near the dislocation core. The stress distribution near the dislocation core are shown in Fig. 2(a-c). Outside the dislocation core \(x_i\in [-h,h]\), the numerical solution of the stress fields agrees well with both the singular analytical solution and Cai’s non-singular solution [22]. Inside the dislocation core region, the numerical solution lies between the singular analytical solution and Cai’s non-singular solution. The displacement \(u_1\) on the left surface are shown in Fig. 2d. The displacement jump near the slip plane is equal to the Burgers vector, which indicates the validity of the numerical solutions.

The sample is subsequently subjected to uniform stretch in the \(x_1\) direction, as shown in the left top corner of Fig. 3a. A uniform stress \(\sigma _0=1\) is applied on left and right surfaces, the top and bottom surfaces are traction free. The simulated elastic fields are then used to calculate the driving force on the dislocation following the scheme outlined in subsection 3.2. A convergence study is performed to estimate the required integration area to obtain accurate results. A square integration area with a width of s is chosen, as shown in the left top corner of Fig. 3a. Figure 4 shows the driving forces calculated using Eq. (36). The result shows that the driving force is convergent when the sample width is larger than 6.

For the benchmark of the numerically calculated driving force, the Peach–Koehler force is used [5]. It is known that the driving force on a dislocation can be evaluated from the following equationwhere \(\epsilon _{kij}\) is the permutation tensor, \({\hat{\xi }}_j\) is the dislocation sense vector, and \(\sigma _{il}^0\) is the applied loading.

For the uniformly stretched sample, according to Eq. (37), the driving force \(F_i\) on dislocation is in the negative \(x_2\) direction, as shown in Fig. 3a. The sample subjected to pure shear stress is also considered, and the loading is illustrated in the left top corner of Fig. 3b, where uniform shear stress \(\tau _0=1\) is applied on all surfaces. For the sample under pure shear loading, according to Eq. (37), the driving force \(F_i\) is in \(x_1\) direction, as shown in Fig. 3b.

In the inset on the right bottom corner of Figs. 3a and 3b, the details of the distributed nodal configurational forces are shown. The configurational force is in principle defined as the gradient of the potential energy. Hence, Fig. 3a implies the dramatic drop of the potential energy in the compressive and tensile regions. Nevertheless, the drop on the tensile side takes over that of the compressive side. Thus the driving forces on the dislocation core is towards the tensile side. In other words, under the uniform stretch, the dislocation tends to move downwards. Similar discussion can be made on the inset of Fig. 3b, and one can conclude that the dislocation tends to move towards right under the given shear loading.

An integration area of \(20 \times 20\) around the dislocation core is chosen to sum up all nodal configurational forces \(G_i\) to calculate the driving forces \(F_i\). According to the convergence study discussed above, this integration area should give an accurate result to compare with the Peach–Koehler force. In Table 1 the numerically calculated driving force and the analytically evaluated Peach–Koehler force are contrasted. It can be seen that the numerical results agree well with the analytical one. The differences in both loading cases, defined as \((F_i-F_i^{PK})/|{\mathbf {F}}^{PK}|\), are less than 1%. We compare also the results for the dislocation core size \(h=1\) and \(h=2\). The numerical result implies that the numerically determined driving forces are not sensitive to this parameter.

In the next we consider a straight screw dislocation with the Burgers vector \({\mathbf {b}}=[0\,0\,b_0]\) with \(b_0=1\), and the normal vector of the slip plane \({\mathbf {n}}=[0\,1\,0]\). The sample size is \(80 \times 80 \times 5\) and the mesh size is \(160 \times 160 \times 10\). The periodic boundary condition for the displacement is applied along the \(x_3\) direction (front and back surfaces).

The dislocation induced elastic fields are first simulated, assuming the traction free boundary condition. The stress fields are shown in Figs. 5(a–b). The displacement \(u_3\) at the left surface is shown in Fig. 5c. Similar to the edge dislocation, good agreement is observed between the numerical solutions and the singular and non-singular analytical solutions outside the dislocation core region. The distribution of nodal configurational forces without external loads is shown in Fig. 5d. The nodal configurational forces decrease to 0 very quickly within the distance of 2h to the dislocation core.

According to Eq. (37), the Peach–Koehler force is nonzero when \(\sigma _{13}\) and \(\sigma _{23}\) are nonzero for the screw dislocation. Therefore, the pure shear stress is applied as \(\tau _0 = \sigma _{i3} n_i, \, (i=1,2)\), where \(n_i\) is the normal vector of the corresponding surface. The shear stress is applied on the front and back surfaces, thus, the periodic boundary condition will not be applied for this simulation. In order to eliminate boundary effects, a thick sample along \(x_3\) direction is used. The sample size is \(50 \times 50 \times 50\) and mesh size is \(100 \times 100 \times 100\). The straight screw dislocation line is coincident with the \(x_3\)-axis in the center of the sample. The driving forces per unit length on the dislocation line are calculated by the total nodal configurational forces evaluated on the \(x_3=0\) plane averaged along the dislocation line length. The comparison between the numerical results and the Peach–Koehler force is shown in Table 2. The difference remains below 1%. It shows that the driving forces on the screw dislocation are also not sensitive to the dislocation core size h, similar to the edge dislocation case.

In this subsection a circular dislocation loop is simulated. The dislocation loop is located at the \(x_2=0\) plane, and the Burgers vector is \({\mathbf {b}}=[b_0\, 0\, 0]\) with \(b_0=1\). The radius of the dislocation loop is 25. It exists in a cylinder with a radius and a height of 100 each. The finite element model is shown in Fig. 6. The mesh around the dislocation loop is refined with a mesh size around 0.5. Figure 6a shows the eigenstrain distribution function \(W({\mathbf {x}},h)\) at the \(x_2=0\) plane.

At the two intersection points of the dislocation loop with the plane \(x_3 = 0\), it is of edge dislocation type, since the tangent of the dislocation line is perpendicular to the Burgers vector, whereas at the intersection points of the loop with the plane \(x_1=0\) it is of screw dislocation type, since the tangent of the dislocation loop is parallel to the Burgers vector. The corresponding stress distributions confirm these features, as shown in Figs. 6(c–d), respectively. The driving forces on the dislocation loop are compared with the Cai’s non-singular analytical model [22]. In the analytical model, the dislocation loop is approximated by 12 dislocation line segments connected by 12 nodes, as shown by the black line segments in Fig. 6a. The analytical driving forces on the 12 nodes are obtained by summing up the contributions from the Peach–Koehler force integrated over the segments. In the numerical model, the driving forces \(F_i\) are calculated by summing up the nodal configurational forces in the dashed line box around the dislocation core shown in Fig. 6c, including the volume for \(\theta \in [-\pi /12+i\pi /6,\pi /12+i\pi /6],\, (i=0,1,2,...,11)\) in the circumferential direction.

The distribution of the 12 numerical driving forces \(F_r\) along the radial direction is shown in Fig. 6a. The distributed nodal configurational forces \(G_i\) are shown in Figs. 6(c–d). The results show that the driving forces \(F_i\) are oriented towards the center of the dislocation loop, which means the dislocation loop tends to shrink without external loads. Fig. 6b shows the distribution of \(F_r\) as a function of \(\theta \) along the dislocation loop. The numerical results agree well with the analytical solutions, in both the cases of \(\textit{h}=1\) and \(\textit{h}=2\). The driving forces obtained with \(\textit{h}=1\) are larger than that in the case \(\textit{h}=2\). The reason is that the stress at dislocation core region is larger for \(\textit{h}=1\). As result, the driving forces due to this stress field on the adjacent points will be larger. Due to the same reason, the element length along the circumferential direction can affect the accuracy of the numerically predicted driving forces. In this work, the element size on the dislocation loop is around 1.6. Fig. 6b also shows that the driving forces \(F_r\) at the two intersection points of the screw dislocation type are larger than those at the two intersection points of the edge dislocation type.

The fourth numerical example is the interaction between a straight edge dislocation and a vacancy loop. The vacancy loop is made up of edge dislocations with the Burgers vector perpendicular to the loop plane. The setup of the model is illustrated in Fig. 7a. The Burgers vectors of the dislocation loop and the straight edge dislocation are both \({\mathbf {b}}=[b_0\,0\,0]\) with \(b_0=1\). The sample size is \(200 \times 200 \times 100\), and the mesh size around the dislocation line is 0.5. The loop is located on the \(x_1=0\) plane. The dislocation loop has a radius of 6. The distance from the center of the loop to the straight edge dislocation is \(H=12\). Fig. 7b shows the distribution function \(W({\mathbf {x}},h)\) for the vacancy loop and the straight edge dislocation.

The Peach–Koehler forces on the loop due to the stress fields of the straight edge dislocation can be calculated as [1]where \(\sigma _{11}^{ns}\) is the non-singular stress field of the straight edge dislocation [22]. In Cai’s non-singular continuum theory of dislocations, the non-singular stress \(\sigma _{11}^{ns}\) is not Cauchy stress [26], but rather an effective stress obtained through the convolution of the \({\tilde{w}}({\mathbf {x}})\) with the Cauchy stress (stress field of an “source” dislocation). The influence of the convolution will only be limited inside the dislocation core region, because \( {\tilde{w}}({\mathbf {x}}) \rightarrow 0\) outside the core region. In finite element simulation, the stress field at an arbitrary point is resulting from superposition of the stress fields of “source” dislocations in linear elasticity. As a consequence, for a given dislocation core size h, the numerical stress fields will be different to \(\sigma _{ij}^{ns}\) inside the core region, as shown in the numerical results of the edge and screw dislocation above. The difference will further affect the determination of the interaction force between two very close dislocations in numerical simulations.

In the numerical simulation, the driving forces on the loop consists of two contributions. The first contribution is the driving forces due to the stress fields of the line segments of the loop, which can be calculated when the straight edge dislocation is not present, similar to the subsection 4.3. The results are shown in Fig. 7c. The second contribution is the driving forces due to the stress fields of the straight edge dislocation. This contribution can be compared with the analytical solution defined in the last equation (38). Since the driving forces on the loop due to the loop itself is always present, we simulated two problems: one with both the loop and the straight edge dislocation, and the other with only the vacancy loop. The difference between the driving forces on the loop of the two problems leads to the second contribution, which is comparable to the analytical solution (38). For numerical results, total nodal configurational forces on a point of the dislocation line are calculated by summing up all the nodal configurational forces on the cross section perpendicular to the dislocation line. Then the driving forces per unit length are calculated as the total nodal configurational forces divided by the element length in the circumferential direction.

When there is only the vacancy loop, Fig. 7c shows the nodal configurational forces are uniformly distributed along the dislocation loop in the circumferential direction. Figure 7d shows the nodal configurational forces on the straight edge dislocation increase slightly near the vacancy loop in the \(x_2\) direction, which indicates the attraction between the vacancy loop and the straight edge dislocation. The driving force on the vacancy loop due to stress field of the straight edge dislocation are shown in Fig. 8. Figure 8a shows that the attractive forces \(F_2\) at \(\theta =0\) are larger than the repulsive forces at \(\theta =\pi \). In the meantime, Fig. 8b shows that the driving forces \(F_3\) are positive in the radial direction. Therefore, the interaction force tends to expand the vacancy loop, as illustrated in Fig. 7a.

The comparisons between the analytical and the numerical solutions of driving forces are shown in Fig. 8. Minor differences between the analytical and the numerical solution can be found for the \(\theta \approx 0\) and \(\theta \approx \pi \), which is the bottom and the top of the loop, respectively. It may be due to the fact that for the bottom part, the vacancy loop is located very close to the dislocation core region and the stress fields are highly dependent on the dislocation core size h. At the top, the dislocation on the vacancy loop is far away from the straight edge dislocation; therefore, the stress fields of the straight edge dislocations are significantly influenced by the traction free boundary condition on a small sample. Increasing the sample size can reduce the difference between the numerical and the analytical results.

The proposed numerical scheme of determining the distributed driving force on the dislocation line is very general and can be used for complex situations. In the last example we show that the driving forces can be used to calculate the critical passing stress and determine the movement tendency of each individual dislocations in a dislocation cluster. The passing stress due to interaction between dislocations is crucial to understand the yield stress and strain hardening behavior of crystals. The Taylor hardening formula is widely used to predict these behaviors, which is defined aswhere \(\tau _c\) is the critical passing stress for passing of the oppositely signed edge dislocations in an array, as shown in Fig. 9a. \(b_0=1\) is the magnitude of the Burgers vector \({\mathbf {b}}=[b_0\,0\,0]\). \(\rho =1/(2 a^2)\) is the density of dislocations, where \(a=5\) is the distance between adjacent dislocations in the \(x_2\) direction. It should be noticed that the value of a used in this numerical example is rather small and corresponds to high dislocation density. We choose the small a in order to achieve a computationally affordable but legitimate example, because increasing a implies larger simulation domain and thus higher computation cost. In fact, the constant \(\alpha \) in the general law of the critical passing stress (39) is independent from the specific choice of the dislocation density and thus also independent from the distance a. In the case of varying slip plane distances in a complex dislocation cluster, the dislocation density is then related to the mean values of a. The analytical solution of the constant \(\alpha \) for two oppositely signed edge dislocations (one dipole) is 0.08 for the current material. For a cluster of multiple dislocations or a dislocation cloud in general, this constant is unknown. For this complicated case, the proposed numerical scheme of calculating the driving force can be applied.

In this numerical simulation, the sample size is assumed to be \(50 \times 50\) and the mesh size is \(100 \times 100\). The dislocation core size is \(h=2\). The relative position between the oppositely signed dislocation arrays is described by d. The distance between the same signed dislocations (lie in the same line in the horizontal direction) remains 2a and we vary the value of d to achieve different interaction forces between oppositely signed dislocation arrays. When the sample is under pure shear load \(\tau _0\), the driving force on a dislocation can be calculated using the Peach–Koehler formula (37) as \(F_1^{PK} = \tau _0 b_0\). In order to move this dislocation, the driving force on the dislocation due to external load should be larger than that due to the stress fields of the surrounding dislocations. Then a constant \(\alpha _0\) can be calculated at different d aswhere the driving force \(F_1(d)\) is numerically calculated by summing up all nodal configurational forces in the integration area of \(6 \times 6\), as shown in Fig. 9c. The constant \(\alpha \) is determined by the maximum value of \(\alpha _0\). Fig. 9b shows that the maximum \(\alpha _0\) is 0.175 at \(d=2\). The driving force on the center dislocation in the \(x_1\) direction can be calculated as \(F_1 = \sigma _{12} b_0\), where \(\sigma _{12}\) is the stress at the core of the center dislocation. With the Burgers vector in the \(x_1\) direction, \(\sigma _{12}\) is symmetric with respect to the \(x_2\)-axis. Therefore, since one oppositely signed dislocation in a dipole contribute to \(\alpha _0 = 0.08\), the two oppositely signed dislocations close to the center dislocation (for a small d) already contribute to \(\alpha _0 = 0.16\). The numerical result \(\alpha _0 = 0.175\) shows that the other surrounding dislocations also contribute to larger passing stress. In reality, there are much more dislocation in a plastically deformed copper material than the simulated example, and the experimentally measured \(\alpha \) is around \({0.35} \pm {0.15}\) [34].

For the unloaded case \(\tau _0=0\), the distribution of nodal configurational forces in Fig. 9c shows that the dislocation array tends to shrink in the \(x_2\) direction while it expands along the \(x_1\) direction. In order to show the tendency of the movement of dislocations, the maximum critical shear stress is applied on the sample, which means \(\tau _0 = \left. \tau _c \right| _{\alpha ={0.5}}\). Fig. 9d shows that the external load largely changes the distribution of potential energy on the compressive and tensile site of the dislocation. The distribution of nodal configurational forces clearly shows that the oppositely signed edge dislocations tend to move towards each other. The total nodal configurational forces on the boundary should be equal to the total driving forces on the dislocation array in magnitude but in opposite direction. It shows that the total driving force on the array is along the \(x_1\) direction, because there are more dislocations with positive Burgers vector in the dislocation array.

The results above show that the driving forces can be used to determine the critical passing stress of dislocations. More importantly, the tendency of the movement of each individual dislocation can be also clearly indicated by the nodal configurational forces. The methodology can be in principle used to study any arbitrary dislocation arrays with irregular distributions.