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Abstract
We investigate the T(3)-gauge theory of static dislocations in continuous solids. We use the most general linear constitutive relations in terms of 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 ℓ1, ℓ2, ℓ3 and ℓ4, which are given explicitly. 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 ℓ1 = ℓ4. For the problem of an edge dislocation, only two characteristic lengths ℓ2 and ℓ3 arise with one of them being the same ℓ2 = ℓ1 as for the screw dislocation. Thus, this theory possesses only two independent lengths for generalized plane strain. If the two lengths ℓ2 and ℓ3 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.
Acknowledgements
The authors have been supported by an Emmy–Noether grant of the Deutsche Forschungsgemeinschaft (Grant No. La1974/1-2). One of us (M.L.) is very grateful to Friedrich W. Hehl for very useful and stimulating discussions about the pseudomoment stress tensor and the translational gauge theory in general.
Notes
Notes
1. We are using the notations and
.
2. We want to mention that Nye Citation35 originally derived Equation (Equation2.25) if the elastic strain is zero (see also the discussion in Li et al. Citation40). Here we have used the differential geometrical definition (Equation2.24) of the contortion in order to derive (Equation2.25). For non-zero elastic strain Equation (Equation2.25) was derived by deWit Citation38.
3. If γ = 0, we obtain c
2 = −c
1. If c
2 = 0 like in the Edelen choice, we obtain c
1 = 0, ,
and
. Thus, only the classical solution of a screw dislocation is allowed in the Edelen choice with symmetric force stresses. Edelen Citation17 used some ad hoc assumptions like
to avoid this problem. But in the classical theory it must be:
. Also, the approach of Citation16 possesses such deficiencies.
4. This relation is not fulfilled in the Einstein choice and c 1 − c 2 + c 3 = 0. In the Edelen choice c 3 = 0, c 2 = 0, we obtain from (Equation4.14) c 1 = 0, ℓ2 = 0 and ℓ3 = 0. Thus, only f = f 0 and Ψ = Ψ0 are allowed with the Edelen choice if the plane strain condition is fulfilled. Due to these reasons, Edelen Citation52, Kadić and Edelen Citation11 and Malyshev Citation18 used a stress function ansatz not fulfilling the plane strain condition.
