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On Nye’s Lattice ∗

Fabio Sozio1 and Arash Yavari†1,2

1School of Civil and Environmental , Georgia Institute of Technology, Atlanta, GA 30332, USA 2The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

May 4, 2021

Abstract We revisit Nye’s lattice curvature tensor in the light of Cartan’s moving frames. Nye’s definition of lattice curvature is based on the assumption that the dislocated body is stress-free, and therefore, it makes sense only for zero-stress (impotent) distributions. Motivated by the works of Bilby and others, Nye’s construction is extended to arbitrary dislocation distributions. We provide a material definition of the lattice curvature in the form of a triplet of vectors, that are obtained from the material covariant of the lattice frame along its integral . While the dislocation density tensor is related to the associated with the Weitzenböck , the lattice curvature is related to the . We also show that under Nye’s assumption, the material lattice curvature is the pullback of Nye’s curvature tensor via the relaxation map. Moreover, the lattice curvature tensor can be used to express the curvature of the material in the linearized approximation. Keywords: Plasticity; Defects; ; ; Torsion; Contorsion; Curvature.

1 Introduction their curvature. His study was carried out under the assump- tion of negligible elastic deformations: “when real crystals are distorted plastically they do in fact contain large-scale dis- The definitions of the dislocation density tensor and the lat- tributions of residual strains, which contribute to the lattice tice curvature tensor are both due to Nye [1]. Nye’s seminal curvature”. In the absence of external loads, elastic deforma- work was motivated by the observation that “when a single tions develop to restore compatibility of the total strain in crystal deforms by glide which is unevenly distributed over the crystal, inducing residual stresses. This difficulty can be the glide surfaces the lattice becomes curved”. The disloca- avoided by considering impotent plastic deformations, i.e., tion density tensor α is then defined as the operator that as- plastic deformations that, albeit incompatible in the sense signs to a unit vector l the Burgers vector B associated with that they are not associated with any configuration map, a circuit of unit area that is normal to l, i.e., Bi = αijlj, and still allow the crystal to relax locally into a stress-free con- it results from the distribution of dislocations in the lattice. figuration. In the language of modern dislocation theory, the Nye also showed that this tensor carries information on the absence of residual elastic strains is equivalent to the assump- change of the orientation of the lattice directions along the tion of plastic deformations inducing a Euclidean material coordinate dxj described by the infinitesimal axial vector dφi metric. This state is the same as Noll’s contorted aeolotropy via the relation dφi = κijdxj, where κ is a tensor that is re- [3], sometimes referred to as zero-stress or impotent disloca- 1 lated to the dislocation density tensor as κij = αji − 2 αkkδij. tion distributions [4,5,6]. This ensures the existence of a Nye called κ the curvature tensor, but since in the geometric local isometric embedding of the material manifold into the setting the expression “curvature tensor” usually implies the , so that the plastic deformation of the material Riemannian curvature associated with the material metric, can be relaxed into a stress-free configuration. The lattice we will be referring to κ as the lattice curvature tensor as curvature tensor has been studied by other researchers from in [2] (where the lattice curvature tensor is defined with the a more geometric perspective. Bilby et al. [7] and Bilby and opposite sign). Smith [8] reviewed Nye’s construction, and provided a mate- To prove the relation between α and κ, Nye looked at the rial version of the notion of lattice curvature, showing that it deformed configuration of lattice directions and planes and is related to the Ricci rotation coefficients. Steinmann [9] es- through the use of graphic techniques he was able to calculate

∗To appear in Mechanics Research Communications. †Corresponding author, e-mail: [email protected] tablished a relation between the contorsion tensor and Nye’s This means that the lattice frame {eα} is orthonormal with lattice curvature. respect to G, i.e., hheα, eβiiG = δαβ. In other words, the In this paper we revisit Nye’s lattice curvature tensor in lattice frame represents an internal observer that is unaware the light of modern differential geometry, and particularly, of the plastic slip occurring in the solid [12, 13]. Cartan’s moving frames [10, 11], and teleparallelism. The The presence of dislocations in solids is associated with material nature of Nye’s tensor will be shown without assum- the anholonomicity of the lattice frame. A frame {eβ} is ing the absence of residual stresses. We do this by using a no- holonomic if there exist local coordinates {Y β} such that ∂ tion of lattice curvature due to Bilby and Smith [8], which is eβ = ∂Y β . This is equivalent to the vanishing of the Lie purely material and relies on the Riemannian structure inher- [eα, eβ] for all α, β [14, 15, 16, 17]. Holonomicity ited by the metric defined on the material manifold, without of the lattice frame can also be expressed in terms of its the need of a Euclidean material metric. More specifically, coframe as ϑβ = dY β, which is equivalent to requiring the the lattice curvature is defined starting from the material lattice forms to be closed.1 As a matter of fact one has of the lattice along the dϑγ (e , e ) = −hϑγ , [e , e ]i . (2.2) frame itself, a quantity that is independent of any mapping α β α β of the material manifold into the ambient space. In our ap- The presence of distributed dislocations can be detected by proach, using Cartan’s moving frames, the lattice curvature calculating the circulation of the lattice coframe along a is represented by a triplet of vectors. We show that starting closed γ, viz. from this more general definition of curvature, the material Z Z β β β variant of Nye’s tensor is the object that encodes it. As a B [γ] = ϕ∗ϑ = ϑ . (2.3) matter of fact, while the material dislocation density tensor ϕ(γ) γ is related to the torsion tensor associated with the Weitzen- The scalars Bβ[γ] represent the Burgers vector associated böck connection, the material lattice curvature tensor can be with γ. If γ is the only component of the boundary of a obtained from the contorsion tensor in a similar way. Carry- Σ, i.e., γ = ∂Σ, invoking Stokes’ theorem one can ing information on both the Weitzenböck and the Levi-Civita write (2.3) as connections, the lattice curvature tensor can be used to ex- Z Bβ[γ] = i∗dϑβ , (2.4) press the Riemann curvature of the material manifold in the Σ linearized approximation. where i :Σ ,→ B is the inclusion map. This paper is organized as follows. In §2 we introduce the notion of lattice frame in a dislocated solid and define all Remark 2.1. The dislocation-free case implies the existence β β β the geometric quantities associated with it. In §3 we review of local coordinates {Y } such that ϑ = dY , and hence, α β the dislocation density tensor. In §4 we introduce a mate- from (2.1) one obtains G = δαβ dY ⊗dY . This means that rial definition of lattice curvature, show its relation with the there exists a map ϕ : B → S whose Cartesian coordinate β ∗ contorsion tensor and, in the linear approximation, with the representation is {Y } and such that G = ϕ g locally, i.e., . a local isometric embedding. Such a map can be seen as a local relaxation for the body. We define a Weitzenböck connection ∇ˆ on B that par- 2 The dislocated lattice allelizes the lattice frame {eα}. As a derivative operator, it acts on a tensor as the ordinary derivative of the compo- We work in the framework of and con- nents of the tensor with respect to the lattice frame, whence sider smooth embeddings ϕ : B → S representing spatial the vanishing of the Weitzenböck derivative of the material configurations of a material body B in the ambient space S. metric G. Its torsion can be calculated by using Cartan’s The ambient space is endowed with a Euclidean metric g, formalism [14,6] and expressing the first structural equation expressing the standard scalar product in the ambient space. in terms of the lattice frame, viz. Crystalline solids carry additional information on the order β β β γ with which particles are arranged. In a geometric continuum T = dϑ + ω γ ∧ ϑ , (2.5) theory this information is encoded in a moving frame {e } β β β where ω γ are the connection 1-forms and T are the tor- on B [6], that we call lattice frame. Alternatively, one can sion 2-forms. As the Weitzenböck connection parallelizes use the associated lattice coframe {ϑβ}, i.e., a field of three β β β β ϑ , one sets ω γ = 0 to obtain T = dϑ , and hence T has 1-forms such that hϑβ, e i = δβ. The material metric G rep- β γ γ the expression T = eβ ⊗ dϑ . The first Bianchi identity is resenting the natural distances in the lattice is then defined obtained by differentiating the first structural equation and as reads dT β = 0, as ddϑβ = 0. By construction, ∇ˆ has zero α β G = δαβ ϑ ⊗ ϑ . (2.1) curvature [19].

1A k-form α on B is closed if dα = 0, and is exact if there exists a k − 1-form χ such that α = dχ. An exact k-form is necessarily closed, while the converse holds only when the k-th de Rham group is trivial [18]. Since closedness can be seen as the local version of exactness, holonomicity becomes quite clear: the existence of local coordinates {Y β } such that ϑβ = dY β is guaranteed whenever the lattice forms are closed.

2 We denote with ∇ the Levi-Civita connection associ- object can be used to relate vectors and 2-forms through ated with G, having zero torsion by construction and non- the raised Hodge operator ?]. Given a 2-form β, the vec- ] vanishing curvature R. The contorsion tensor K is defined as tor ? β is defined as β = ι?]βµ (ι is the ] C the difference between the Weitzenböck and the Levi-Civita operator), which in components reads βAB = µABC (? β) . ] A 1 ABC connections, i.e., The inverse relation is written as (? β) = 2 µ βBC , with ABC − 1 ABC 2 µ = G 2  . A allows one to define K(X, Y ) = ∇ˆ X Y − ∇X Y , (2.6) the of a vector field V as (Div V ) µ = LV µ = dι µ.3 When µ is induced by a metric G one also has for all vectors X, Y . In components with respect to a coor- V Div V = tr ∇V , where ∇ is the Levi-Civita connection as- dinate chart {XA} on B, one has sociated with G. , raised Hodge operator 1 and divergence operator are related as KA = T A + T A + T A  , (2.7) BC 2 BC B C C B dβ = Div(?]β) µ , (3.1) where indices are lowered and raised using G. for any 2-form β. Remark 2.2. It is straightforward to prove that when tor- Geometric definitions of the dislocation density tensor us- sion T of the Weitzenböck connection vanishes (dislocation- ing the notion of holonomicity and torsion are due to Kondo free case of Remark 2.1), the Levi-Civita connection has [21], Bilby et al. [7], and Kröner [22]. We define the disloca- zero curvature (absence of residual stresses). The converse tion density as the triplet of vectors {αβ} given by does not hold: there exist distributions of dislocations (i.e., T 6= 0) associated with a lattice frame inducing a Euclidean αβ = ?]dϑβ , (3.2) material metric G, i.e., such that R = 0. In this case, the existence of a local isometric embedding is still guaranteed or αβ = ?]T β in Cartan’s formalism, where ?] is the by the Test Case theorem [14], and therefore the body is raised Hodge operator associated to G. Note that since allowed to locally relax. In other words, the plastic slips dT β = ddϑβ = 0, from (3.1) one necessarily has Div αβ = 0. are {ϑβ}-incompatible but G-compatible. These are called Thus, Div αβ = 0 can also be seen as a consequence of zero-stress or impotent dislocations by Mura [5], or contorted the first Bianchi identity for the Weitzenböck connection. aeolotropy by Noll [3]. As was mentioned earlier, the study The tensorial variant of the dislocation density is defined as β by Nye [1] is carried out under this assumption. It should the tensor α = eβ ⊗ α of type (2, 0), or equivalently, as also be noted that in this case the lattice frame can be ob- α = ?]T , where the raised Hodge operator acts on the lower tained through a rotation field superimposed to a defect-free indices. Note that denoting the extension of the divergence lattice frame. Hence, the stress-free state that Nye works operator to double contravariant with Div (acting on with cannot be achieved by pure plastic rotations, as was the second index), one has claimed in [20]. This can be shown by considering the ex- β ample of plastic bending of a slab presented by Nye, which Div α = (Div α ) eβ + ∇αβ eβ = ∇αβ eβ , (3.3) is a process that requires a change in length of the material fibers, and not just a change in their orientation. However, which in general does not vanish.4 This can also be written AB A CB by reparametrizing the material manifold via the relaxation as ∇Bα = −K BC α . It should be noted that by lin- map ϕ, i.e., working with ϕ(B) as reference configuration, earizing around a defect-free lattice coframe (see §4), where β one would be able to express the lattice structure through a both α and ∇eβ vanish, one obtains field of rotations of the Cartesian frame. β β Div δα = Div(δα ) eβ = δ(Div α ) eβ , (3.4) 3 The dislocation density tensor and hence, one recovers the classic identity Div δα = 0. The dislocation density tensor can be used to express the The volume form µ associated with G is called the mate- Burgers vector associated with a closed curve γ = ∂Σ. De- 1 rial volume form and has components µABC = G 2 ABC , noting with N the unit normal on Σ, and with ν the area where G = det G and  is the symbol. This 2-form induced by G on Σ, both induced by G, one can

2 In general, the Hodge operator assigns to a k-form β the (n − k)-form ?β such that for any G-orthonormal frame {Xα} one has ] ] (?β)(XI1 , ..., XIk ) = β(XIk+1 , ..., XIn ). The raised Hodge operator is defined by raising all the indices of the , i.e., ? β = (?β) . The result is an alternating contravariant tensor. 3 The second equality is a consequence of Cartan’s formula LV µ = dιV µ + ιV dµ, and dµ = 0, where L is the operator. 4This might seem to disagree with what was obtained by Yavari and Goriely [6], i.e., that the dislocation density tensor is divergence-free. In that work, however, Cartan’s exterior covariant derivative was used to define a divergence operator for tensors of type (2, 0) that only operates on the second index. Therefore, this extended divergence operation is equivalent to taking the of triplets of vectors and assembling them together. As a matter of fact, it is straightforward to prove that if one denotes with Divg the divergence operator defined by Yavari and Goriely [6], β then Divgα = (Div α ) eβ . Thus, our result agrees with what was obtained by Yavari and Goriely [6].

3 rewrite (2.3) as manifold. Using the coefficients Λµν one defines the 2-form µ ν Z Λ = Λµν ϑ ⊗ ϑ . The axial vector associated with Λ is β β ] B [γ] = hhα , NiiG ν , (3.5) defined through the raised Hodge operator as W = ? Λ. Σ Remark 4.1. We emphasize that the coefficients Λµν do not meaning that each Bβ[γ] is given by the flux of the corre- β transform tensorially with the frame that is used to define sponding vector α across Σ. them. As a matter of fact, if one considers a different frame ω fβ, related to eβ as eβ = A βfω, then one can easily see 4 Nye’s lattice curvature tensor that ρ σ ρ σ The incompatibility of the lattice structure can be described hh∇teµ, eν iiG = A µA ν hh∇tfρ, fσiiG + ∇tA µ δρσA ν . by Nye’s lattice curvature tensor as well. This object is (4.5) equivalent to the dislocation density tensor, but instead of If one considers the Frenet frame {fβ} = {t, n, b} associated being associated with the circulation of the lattice coframe, with a curve γ, where n and b are respectively the normal it represents the way the lattice frame changes along its own and binormal unit vectors, the coefficients hh∇tfµ, fν iiG have integral curves. We start by providing three equivalent def- the following representation: initions of the lattice curvature tensor in the material man-  0 κ 0 ifold. We will also discuss their geometric interpretations. h i hh∇tfµ, fν iiG = −κ 0 τ , (4.6) First, we introduce the lattice curvature as a triplet of vec- 0 −τ 0 tors {κβ} defined by where κ and τ are, respectively, the curvature and torsion of 1 γ κβ = αβ − hϑ , αγ i eβ , (4.1) γ (here the should not be confused with 2 the torsion of a connection). The axial vector associated to η β where αβ = δβηα is a simple reindexing of the triplet {α }. the Frenet frame {t, n, b} is W = τt + κb. The lattice curvature can also be defined as a tensor κ of type (2, 0) as Next we let the curve γ be an integral curve for the lat- 1 tice vector eβ, i.e., we assume t = eβ. This means that κ = α − (Tr α) G] . (4.2) 2 G we are looking at infinitesimal rotations of the lattice frame AB AB 1 H AB along its own integral curves. Hence, we define the triplet of In components, κ = α − 2 αH G . It can also be αβ 2-forms {Hβ} as the analogue of (4.4), with components written as κ = δ eα ⊗κβ. Finally, a definition very similar to that of Nye [1] is the following tensor of type (1, 1): (Hβ)µν = hh∇eβ eµ, eν iiG = Γνβµ , (4.7) 1 k = α[1 − (Tr α) I , (4.3) where Γ are the Christoffel symbols of the first kind of 2 G νβµ G in the moving frame {eβ}. The (Hβ)µν ’s represent Ricci where [1 denotes the lowering of the first index, while I rotation coefficients [23] associated with the frame {eβ}. It A is the identity operator. In components, one has k B = should be noted that from the orthonormality of the lattice A 1 H A β αB − 2 αH δB. It can be written as k = ϑ ⊗ κβ, and vice frame one has versa κβ = keβ. The (1, 1) and (2, 0) variants are related as [1 A A Γµβν = −hh∇e eν , eµiiG k = κ , i.e., k B = κB . β ˆ Next we provide a geometric interpretation of Nye’s con- = −hh∇eβ eν − K(eβ, eν ), eµiiG struction. Instead of assuming the existence of a relaxed (4.8) = hhK(eβ, eν ), eµiiG configuration and expressing all the quantities with respect = K , to it, we carry out our analysis entirely in the material man- µβν ifold. Let us consider an arbitrary curve γ in the material where use was made of the definition of the contorsion ten- manifold B, with G-unit tangent vector t. Then, ∇t indicates sor (2.6) and the fact that the Weitzenböck derivative of the covariant derivative along γ corresponding to an arc-length lattice frame vectors vanishes. In other words, the contorsion parametrization, where ∇ is the Levi-Civita connection as- tensor can be used to express the Ricci rotation coefficients. sociated with G. Let us define the following symbols Hence, we define the symbols Hβνµ = (Hβ)µν = Kµβν . This allows us to define the tensor H as Λµν = hh∇teµ, eν iiG , (4.4) hhH(X, Y ), e ii = H (X, Y ) , (4.9) representing the way eµ changes along γ with respect to eν . β G β It is straightforward to check that since the lattice frame or {eβ} is orthonormal with respect to G, by construction, one hhH(X, Y ), ZiiG = hhK(Z, Y ), ZiiG . (4.10) has Λµν = −Λνµ. Therefore, the coefficients (4.4) repre- A A AF D sent the infinitesimal rotation that the lattice frame under- In components H BC = KB C = GBDG K FC . It goes while moving by ds along the curve γ in the material should be noted that the anti-simmetry in the lower indices

4 of H coming from the orthonormality of {eβ} can be verified approach consists of defining the following spatial dislocation by using (2.7), viz. density tensor and lattice curvature tensor: 1 A 1 A A A β ] β γ H = T + T + T α¯ = ?gdϕ∗ϑ , κ¯β = α¯ β − hϕ∗ϑ , α¯ γ i ϕ∗eβ , (4.14) BC 2 B C BC CB 2 1 A A A (4.11) where ?] denotes the raised Hodge operator in the ambient = − TC B + T CB + TBC g 2 space induced by the metric g, and ϕ is a configuration map. A A = −KC B = −H CB . The quantities defined in (4.14) are the spatial analogues of (3.2) and (4.1), and describe the deformed lattice struc- The following proposition shows that the lattice curvature ture with respect to the metric g. Under Nye’s assumption, tensor of Nye can be obtained from the permutated contor- the configuration map ϕ is a local isometric embedding, and sion tensor. hence g = ϕ∗G (see Remark 2.2). This means that all the β Proposition 4.2. The lattice curvature tensor is the axial quantities involved in the definition of α¯ and κ¯β (metric, vector associated with the permutated contorsion tensor, i.e., covariant derivative, Hodge operator) are preserved by the κ = ?]H and κ = ?]H . tangent map T ϕ, and hence, they coincide with their ma- β β β β β β terial counterparts, i.e., α¯ = ϕ∗α and κ¯ = ϕ∗κ (note Proof. We use components, and invoke (2.7) and (4.11) to that d and ϕ∗ commute). write Next we look at the lattice curvature tensor in the lin- 1 earized approximation. In nonlinear compatibility HA = T A + T A + T A CD 2 C D CD DC can be expressed in terms of the strain field as the vanishing 1 of the curvature tensor associated with the pulled-back met- = µA α F + µ αAF + µ A α F  , 2 DF C CDF C F D ric C [24]. These conditions can be linearized to obtain the (4.12) compatibility equations in terms of the infinitesimal strain A AD where use was made of the relation T BC = µBCD α .  = 1 δC in the linearized setting [25]. It should be noted ] 2 Therefore, we can calculate ? H as that in three, curvature can be expressed by sev- eral equivalent tensors [26]. In particular, the linearization 1 1 µBCDHA = µBCDµA α F of the Einstein tensor allows one to obtain compatibility as 2 CD 4 DF C BCD AF curl ◦ curl  = 0 [24]. In a similar way, the compatibility of + µ µCDF α plastic deformations can be written in terms of the material BCD A F  + µ µC F αD metric G as the vanishing of the curvature R associated with 1 (4.13) G. As was mentioned in Remark 2.2, this does not ensure the = αAB − α H GAB + 2αAB 4 H absence of dislocations, but the lack of residual stresses. An + αAB − α H GAB incompatibility object for the plastic deformation is usually H defined by linearizing the curvature tensor [27]. AB 1 H AB = α − αH G , According to Kröner [2], the lattice curvature tensor can 2 be used to express the incompatibility content of the plastic strain. Therefore, we would like to recover Kröner’s result in which coincides with (4.2). The expression for κβ can be obtained by lowering the first index of κ and contracting it our geometric approach via the linearization of the Einstein tensor. We linearize around a defect-free lattice coframe, in- with eβ. ducing a Euclidean material metric and a Weitzenböck con- Proposition 4.2 implies that the lattice curvature tensor nection that coincides with the flat Levi-Civita connection. can be obtained from Hβ and H in the same way that the This means that the initial plastic deformation is compat- dislocation density tensor is obtained from T β and T . This ible, while we look at a small plastic deformation carrying also shows that the lattice curvature tensor represents rota- the entire dislocation content. Therefore, in this zeroth-order tions of the lattice frame. In particular, the geometric inter- structure, the tensors T , K, α, κ and R vanish. We start by pretation of the operator k is the following: given a vector linearizing the . As all the second- field Z on B, the vector kZ is the axial vector describing the order terms vanish, one can write [28] rotation of the lattice along integral curves of Z. Similarly, A δR BCD the vector κβ is the axial vector associated with the rotation A A of the lattice along integral curves of the lattice vector eβ. = δΓ DB|C − δΓ CB|D (4.15) ˆA A ˆA A Remark 4.3. Under the assumption of a stress-free crystal, = δΓ DB|C − δK DB|C − δΓ CB|D + δK CB|D A A Nye’s work was carried out entirely in the deformed (relaxed) = δK CB|D − δK DB|C , configuration. In geometric terms, Nye studied the deformed A A A lattice structure on ϕ(B) represented by {ϕ∗eβ} with respect where Γ BC = Γˆ BC − K DB, and a vertical bar denotes to the metric g. Although not explicitly stated by Nye, this the covariant derivative ∇. Recalling the definition of per-

5 muted contorsion tensor H and the lattice curvature tensor Acknowledgement κ, one writes This work was partially supported by NSF – Grant No. A A A δR BCD = δHC B|D − δHD B|C CMMI 1561578, 1939901, and ARO Grant No. W911NF- (4.16) A F F  18-1-0003. = µ BF δκC |D − δκD |C . Next, we linearize the relation References

AB 1 AMN BPQ Ein = − µ µ RMNPQ , (4.17) [1] J. Nye. Some geometrical relations in dislocated crys- 4 tals. Acta Metallurgica, 1(2):153–162, 1953. for the raised Einstein tensor,5 and recalling that all the [2] E. Kröner. Dislocations and continuum mechanics. Ap- defect-related zeroth-order quantities vanish, (4.15) can be plied Mechanics Reviews, 15(8):599–606, 1962. written as [3] W. Noll. Materially uniform simple bodies with inhomo- 1 δEinAB = − µAMN µBPQ µ δκ F − δκ F  geneities. Archive for Rational Mechanics and Analysis, 4 MNF P |Q Q |P 27(1):1–32, 1967. 1 = µBQP δκ A − δκ A  2 P |Q Q |P [4] T. Mura. Micromechanics of Defects in Solids. Springer BQP A = µ δκP |Q . Science & Business Media, 2013. (4.18) [5] T. Mura. Impotent dislocation walls. and Engineering: A, 113:149–152, 1989. AB BQP A One can also write (4.18) as δEin = µ δK P |Q. This coincides with what was obtained by Kröner [2]. [6] A. Yavari and A. Goriely. Riemann–Cartan geometry of nonlinear dislocation mechanics. Archive for Ratio- nal Mechanics and Analysis, 205(1):59–118, 2012. 5 Conclusions [7] B. A. Bilby, R. Bullough, and E. Smith. Continu- In this paper we revisited Nye’s lattice curvature tensor and ous distributions of dislocations: A new application of presented a modern perspective. While Nye’s construction the methods of non-. Proceedings is based on the assumption of no residual stresses, motivated of the Royal Society of London A, 231(1185):263–273, by the works of Bilby and others the notion of lattice curva- 1955. ture was extended to arbitrary dislocation distributions. In [8] B. A. Bilby and E. Smith. Continuous distributions of the framework of Cartan’s moving frames, the lattice curva- dislocations. ill. Proceedings of the Royal Society of Lon- ture is a representation of the rotation of the lattice moving don. Series A. Mathematical and Physical Sciences, 236 frame with respect to an affine connection. In particular, the (1207):481–505, 1956. lattice curvature is related to the Ricci rotation coefficients associated with the lattice frame and is therefore a purely [9] P. Steinmann. Views on multiplicative elastoplastic- material object. We started by expressing the material ver- ity and the continuum theory of dislocations. Inter- sion of Nye’s lattice curvature as a triplet of vectors, and national Journal of Engineering Science, 34(15):1717– showed that it can be obtained from the contorsion tensor 1735, 1996. via the raised Hodge operator. It was also shown that if [10] É. Cartan. La structure des groupes de transformations one works under Nye’s assumption of a Euclidean material continus et la théorie du trièdre mobile. Bulletin Sci- metric (i.e., zero-stress dislocations), which ensures the exis- ences mathématiques, 34:250–284, 1910. tence of a (local) stress-free reference configuration in which the material metric is preserved, the material and the spatial [11] S.-S. Chern. Moving frames. The Mathematical Her- definitions of κ coincide. As a matter of fact, since all the itage of Élie Cartan (Lyon, 1984), Astérique, 1985:67– quantities involved in its geometric definition—metric, co- 77, 1985. variant derivative, and Hodge operator—are preserved, the [12] E. Kröner. On in defect mechanics. In spatial lattice curvature is the push forward of the material Trends in Applications of Pure to Mechan- lattice curvature via the relaxation map. Moreover, we were ics, pages 281–294. Springer, 1986. able to show that in the linearized approximation the lattice curvature tensor can be used to express the Riemann curva- [13] M. Lazar. Dislocation theory as a 3-dimensional trans- ture of the material manifold. In particular, its curl is equal lation gauge theory. Annalen der Physik, 9(6):461–473, to the linearization of the Einstein tensor. 2000.

5This can be found in [26] with the opposite sign coming from a different definition of the .

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