We investigate the T (3)-gauge theory of static dislocations in continuous solids. We use the most general linear constitutive relations bilinear in the elastic distortion tensor and dislocation density tensor for the force and pseudomoment stresses of an isotropic solid. The constitutive relations contain six material parameters. In this theory both the force and pseudomoment stresses are asymmetric. The theory possesses four characteristic lengths ℓ 1 , ℓ 2 , ℓ 3 and ℓ 4 which are given explicitely. We first derive the three-dimensional Green tensor of the master equation for the force stresses in the translational gauge theory of dislocations. We then investigate the situation of generalized plane strain (anti-plane strain and plane strain). Using the stress function method, we find modified stress functions for screw and edge dislocations. The solution of the screw dislocation is given in terms of one independent length ℓ 1 = ℓ 4 . For the problem of an edge dislocation, only two characteristic lengths ℓ 2 and ℓ 3 arise with one of them being the same ℓ 2 = ℓ 1 as for the screw dislocation. Thus, this theory possesses only two independent lengths for generalized plane strain. If the two lengths ℓ 2 and ℓ 3 of an edge dislocation are equal, we obtain an edge dislocation which is the gauge theoretical version of a modified Volterra edge dislocation. In the case of symmetric stresses we recover well known results obtained earlier.

We derive conservation and balance laws for the translational gauge theory of dislocations by applying Noether's theorem. We present an improved translational gauge theory of dislocations including the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under the continuous group of transformations is studied. Through Lie's-infinitesimal invariance criterion we obtain conserved translational and rotational currents for the total Lagrangian made up of an elastic and dislocation part. We calculate the broken scaling current. Looking only on one part of the whole system, the conservation laws are changed into balance laws. Because of the lack of translational, rotational and dilatation invariance for each part, a configurational force, moment and power appears. The corresponding J, L and M integrals are obtained. Only isotropic and homogeneous materials are considered and we restrict ourselves to a linear theory. We choose constitutive laws for the most general linear form of material isotropy. Also we give the conservation and balance laws corresponding to the gauge symmetry and the addition of solutions. From the addition of solutions we derive a reciprocity theorem for the gauge theory of dislocations. Also, we derive the conservation laws for stress-free states of dislocations.Comment: 28 page

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The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether's theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity.

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