13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Si,u j = Si,u j =−1 (i+1,j)− (i,j) (i+1,j)− (i,j) (1) where ρ is the electrical resistivity of material, and A (=wb) is the cross-sectional area of wire. Here, as an example, Si,l j, Si,l j are current and current density for mesh segment si,l j ; (i,j) and (i,j), (i,j) are electrical potential and coordinates of node (i, j). It should be noted that as shown in Fig. 1, the origin of the coordinate system is set at the bottom left corner of the mesh, i.e., node (0, 0). The x-axis points right and the y-axis points up, both of which are taken to be along the wire. Moreover, at any mesh node of (i, j), according to Kirchhoff's current law, we have internal + external =0 (2) Here, internal(= Si,l j − Si,r j+ Si,d j − Si,u j) means the sum of the current flowing into the node (i, j) from different adjacent nodes, and Iexternal represents the external input/output current where the external output current takes the minus value. By considering Eqs. (1) and (2) for all the nodes, the current density at any mesh segment (i.e., Si,l j, Si,r j, Si,d j, Si,u j), and the electrical potential at any mesh node can be obtained. On the other hand, for any mesh segment, the heat energy, flowing in any mesh segment between two adjacent nodes, can be calculated using Fourier’s law of heat conduction as Si,l j = Si,l j =− (i,j)− (i,j−1) (i,j)− (i,j−1) Si,r j = Si,r j =− (i,j+1)− (i,j) (i,j+1)− (i,j) SI,d j = SI,d j =− (i,j)− (i−1,j) (i,j)− (i−1,j) Si,u j = Si,u j =− (i+1,j)− (i,j) (i+1,j)− (i,j) (3) where λ is the thermal conductivity of material. Here, as an instance, Si,l j, Si,l j are heat energy and heat flux for mesh segment si,l j; T(i,j) is the temperature at node (i, j). Moreover, at any mesh node, according to the law of conservation of heat energy, we have internal+ external =0 (4) Here, internal(= Si,l j − Si,r j + Si,d j − Si,u j) means the sum of the heat energy flowing into the node (i, j) from different adjacent nodes, and Qexternal represents the external input/output heat energy where the external output heat energy takes the minus value. It should be noted that when a metallic nanowire is subjected to a steady direct current flow, Joule heating occurs, which, in turn causes increase in temperature of the wire. For simplicity, it is assumed that there is no heat transfer from the surface of the wire to the ambient, and the time-dependence of temperature can be neglected. Then, the heat conduction in the above four mesh segments can be governed by the following one-dimensional Poisson’s equations [11]: 2 S i,l j 2 + S i,l j 2 =0 2 S i,r j 2 + Si,r j 2 =0 2 S i,d j 2 + Si,d j 2 =0 2 S i,u j 2 + Si,u j 2 =0 (5) where Si,l j as an example is the temperature of mesh segment si,l j.
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