Abstract
The displacement and stress function fields of straight dislocations and lines forces are derived based on three-dimensional anisotropic incompatible elasticity. Using the two-dimensional anisotropic Green tensor of generalized plane strain, a Burgers-like formula for straight dislocations and body forces is derived and its relation to the solution of the displacement and stress function fields in the integral formalism is given. Moreover, the stress functions of a point force are calculated and the relation to the potential of a Dirac string is pointed out.
Highlights
Anisotropic elasticity is an important theory for deformed bodies, which can be used for three-dimensional and two-dimensional problems
We show that the integral formalism is nothing but a result of the inverse Fourier transform of the Green tensor and the F-tensor
If we substitute Equation (57) into Equation (55), we obtain the six-dimensional displacement-stress function vector of a straight dislocation with Burgers vector b and a line force with strength F located at the position (0, 0) known in the integral formalism: u(r, ω )
Summary
Anisotropic elasticity is an important theory for deformed bodies, which can be used for three-dimensional and two-dimensional problems. In two-dimensional (2D) anisotropic elasticity, the displacement and stress function fields of straight dislocations and straight line forces were derived by Stroh [4,5]. Infinitely long straight dislocation lines with Burgers vector b and body forces of strength F, and the corresponding field quantities, displacement u and stress function vector Φ, are treated by the so-called “six-dimensional integral theory”, developed by Barnett and Lothe [9]. In 3D, there seems to be no particular advantage in putting together the dislocation density tensor α and the body force vector f , and displacement vector u and stress function tensor Φ, into aggregates. We show how the integral form of the displacement field and stress functions of straight dislocations and line forces can be derived directly from the 3D framework given by Lazar and Kirchner [15]. The φ-integration is the remnant of the inverse Fourier transform in polar coordinates in anisotropic elasticity, a fact which is blurred in the original formulation of the integral formalism
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