A Remark on stress of a spatially uniform dislocation density field
aa r X i v : . [ phy s i c s . c l a ss - ph ] M a r A REMARK ON STRESS OF A SPATIALLY UNIFORM DISLOCATIONDENSITY FIELD
SIRAN LI Introduction -dimensional: it works with the subgroup SO (2) ⊕ h Id i ⊂ O (3) . The objective of this note isto extend Acharya’s result in [1] to the O (3) , subject to one additional structural condition andless regularity assumptions.1.2. Nomenclature.
Throughout Ω ⊂ R is a simply-connected bounded domain with outwardunit normal vectorfield n . The group of × orthogonal matrices is denoted by O (3) ; i.e. , M ∈ O (3) if and only if M ⊤ = M − . The special orthogonal group SO (2) consists of thematrices in O (2) with determinant . The matrix field F : Ω → gl (3; R ) designates the elasticdistortion, and W := F − whenever F is invertible. T : gl (3; R ) → O (3) denotes a generallynonlinear, frame-indifferent stress response function, where gl (3; R ) is the space of × matrices.The composition T ( F ) is the symmetric Cauchy stress field applied to the configuration of body Ω . The constant matrix α ∈ gl (3; R ) denotes the dislocation density distribution specified on Ω .For a matrix field M = { M ij } ≤ i,j ≤ m : Ω → gl (3; R ) , its curl and divergence are understoodin the row-wise sense. In local coordinates it means the following: for each i, j, k, ℓ ∈ { , , } , curl M is the -tensor field î curl M ó ij := ∇ k M iℓ − ∇ ℓ M ik where ( k, ℓ, j ) is an even permutation of (1 , , , and div M is the vectorfield î div M ó i = X j ∇ j M ij . Moreover, recall the
Leray projector is the L -orthogonal projection P : L ( R ; R ) → L ( R ; R ) that sends a vectorfield in R onto its divergence-free part. On R it can be definedvia Fourier transform: ” P v ( ξ ) := Ç Id − ξ ⊗ ξ | ξ | å ˆ v ( ξ ) . The Leray projector plays an important rôle in the mathematical analysis of incompressibleNavier–Stokes equations; cf. e.g.
Constantin–Foias [3] and Temam [7]. For a matrix field M , Date : March 25, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Nonlinear elasticity; Stress; Dislocation; Uniform dislocation density; Load; Elastic Body;Non-existence. ( M ) is again understood in the row-wise sense. We denote by Q := Id − P the complementary projection of P .1.3. Differential Equations.
In the above setting, the governing equations for the internalstress field in the body subject to the Cauchy stress field T ( F ) was derived by Willis in [10]. Seealso Eq. (3) in [1]: curl W = − α in Ω , div Ä T ( F ) ä = 0 in Ω , T ( F ) · n = 0 on ∂ Ω . (1)Here α is a prescribed constant matrix. This PDE system is considered under the following Assumption 1.1. T ( F ) = if and only if F takes values in O (3) . Acharya proved in [1] the following result:
Theorem 1.2.
Let Ω , W , T , F , and n be as in Section 1.2 above. Let α be any nonzero constantmatrix. Then, under Assumption 1.1, there does not exist θ ∈ C (Ω; R ) such that W = R θ is asolution for Eq. (1) ; here R θ := cos θ − sin θ θ cos θ
00 0 1 . (2)The proof in [1] follows from concrete computations: with the ansatz (2), Eq. (1) reducesto a system of algebraic equations for sin θ and cos θ only, which is not soluble unless α ≡ .The goal of this note is to extend Acharya’s Theorem 1.2 in order to include more generalform of W and assuming lower regularity requirements. At the moment we are not able togeneralise to all of O (3) -valued W ; an additional structural condition is needed — Assumption 1.3. Q ( W ) is O (3) -valued ( Q is the complement of Leray projector in Section 1.2). Mechanics.
In the terminologies of continuum mechanics, Theorem 1.2 means that in the nonlinear regime, there is no C -stress-free spatially uniform dislocation density field, unlesssuch uniform dislocation density is everywhere vanishing.Various dislocation distributions producing no stress have been observed in the limit ofcontinuum elastic descriptions ( cf. Mura [6], Head–Howison–Ockendon–Tighe [4], Yavari–Goriely[11], etc.). This is the background for our work. In this note, we aim to further the investigationby Acharya [1] in the nonlinear regime.2.
Main Result
Theorem 2.1.
Let Ω , W , α , T , F , and n be as in Section 1.2. Under Assumptions 1.1 and 1.3,Eq. (1) has no solution W in C (Ω; O (3)) unless the uniform dislocation density field α ≡ . Theorem 2.1 agrees with the linear case. The following arguments are essentially takenfrom Section 3 in [1]. When U := F − Id is uniformly small, set C := D T ( I ) . The matrix eld U is known as the elastic distortion, and the rank- tensor field C is known as the elasticmodulus. Then the linearised system for Eq. (1) is curl U = − α in Ω , div Ä CU ä = 0 in Ω , CU · n = 0 on ∂ Ω . (3)By Kirchhoff’s uniqueness theorem for linear elastostatics, the symmetric part ǫ := U + U ⊤ must be zero. Thus, if U is O (3) -valued, then Eq. (3) is not soluble except when α ≡ . Thatis, α ≡ is a necessary (in fact, not sufficient in general) condition for the solubility of Eq. (3).Also note that W = R θ in Theorem 1.2 satisfies Assumption 1.3: direct computation inpolar coordinates shows that div R θ ≡ ; hence Q W ≡ W ≡ R θ , which is O (3) -valued.3. Proof
Proof of Theorem 2.1.
Throughout the proof we denote by W , W , W the row-vectorfields ofthe matrix field W . Also, let e α be the field of differential -forms dual to α , namely e α i = α i dx ∧ dx + α i dx ∧ dx + α i dx ∧ dx . Thus, by Hodge duality, the first equation in Eq. (1) becomes d W i = − e α i for each i ∈ { , , } , (4)which is an identity of -forms. Here and hereafter, we identify W i with a -form (not relabelled).Under Assumption 1.1 the second and the third equations in Eq. (1) are satisfied automat-ically. So it remains to solve for Eq. (4) in the space of O (3) -valued matrix fields.Recall that the divergence operator acting on differential -forms on Ω ⊂ R is nothingbut the codifferential d ∗ := ⋆d⋆ , where ⋆ is the Hodge star operator. Also, the Laplacian equals ∆ = dd ∗ + d ∗ d. (5)Let us split W into W i = d ∗ II i + dφ i + c i on Ω , (6)where II i is a field of differential -form, φ i is a scalarfield, and c i is a constant in R . This isdone by the Hodge decomposition theorem and that Ω is simply-connected; see, e.g. , Chapter 6in [9]. In local coordinates, Eq. (6) can be expressed as follows: W ij = X k =1 ∇ k II ikj + ∇ j φ i + c ij for each i, j ∈ { , , } . By standard elliptic regularity theory (see [8]), II i and φ i have C ,γ -regularity for any γ ∈ [0 , .Now we claim that n ∇ j φ i o ≤ i,j ≤ is equal to a constant O (3) -matrix . (7) ndeed, since the Leray projector maps onto the divergence-free part of W , we have Q W i = dφ i for φ i ∈ C ,γ (Ω) . By Assumption 1.3 we have X k =1 ∇ k φ i ∇ k φ j = δ ij , namely that φ is an isometric embedding from Ω ⊂ R into R . The classical rigidity theoremof Liouville [5] yields that φ i is an affine map globally on Ω (in fact, C -regularity of φ i sufficeshere). Thus the claim (7) follows.To conclude the proof, taking d ∗ to both sides of Eq. (6) and noting the claim (7), we get d ∗ W i = 0 . This together with Eqs. (4) and (5) implies that ∆ W i = 0 . (8)That is, W i is a harmonic -form for each i ∈ { , , } . Eq. (8) is understood in the sense ofdistributions; nevertheless, by Weyl’s lemma (see [8]) W i is automatically C ∞ . In view again ofthe Hodge theory (see Chapter 6 in [10]), it is represented by generators of the first cohomologygroup. But Ω is simply-connected, so there is no non-trivial such generator. Thus W i is constant.Therefore, we infer from Eq. (4) that α i equals zero. The proof is complete. (cid:3) Remarks
It would be interesting to consider the same problem for Ω being a -dimensional manifold,which falls into the framework of incompatible (non-Euclidean) elasticity.The mechanical problem considered in this paper may have deep underlying geometricalconnotations. In particular, it is related to constructions for coframes with prescribed (closed)differential. See Bryant–Clelland [2] for analyses via exterior differential systems. Acknowledgement . The author is deeply indebted to Amit Acharya for kind communicationsand insightful discussions. We also thank Janusz Ginster for pointing out a fallible argument inan earlier version of the draft.
References [1] A. Acharya, Stress of a spatially uniform dislocation density field,
J. Elasticity (2019), no. 2, 151–155[2] R. L. Bryant and J. N. Clelland, Flat metrics with a prescribed derived coframing,
SIGMA SymmetryIntegrability Geom. Methods Appl. (2020), Paper No. 004, 23 pp[3] P. Constantin and C. Foias, Navier–Stokes Equations , University of Chicago Press, 1988[4] A. K. Head, S. D. Howison, J. R. Ockendon, and S. P. Tighe, An equilibrium theory of dislocation continua,
SIAM Rev. (1993), 580–609[5] J. Liouville, Théorème sur l’équation dx + dy + dz = λdα + dβ + dγ , J. Math. Pures Appl. (1850).[6] T. Mura, Impotent dislocation walls,
Materials Science and Engineering: A (1989), 149–152[7] R. Temam,
NavierâĂŞStokes Equations: Theory and Numerical Analysis , AMS Chelsea Publishing, 2001[8] D. Gilbarg and N. S. Trudinger,
Elliptic partial differential equations of second order , Classics in Mathematics.Springer-Verlag, Berlin, 2001[9] F. W. Warner,
Foundations of differentiable manifolds and Lie groups . Corrected reprint of the 1971 edition.Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983[10] J. R. Willis, Second-order effects of dislocations in anisotropic crystals,
Intern. J. Engineering Sci. (1967),171–190
11] A. Yavari and A. Goriely, Riemann–Cartan geometry of nonlinear dislocation mechanics,
Arch. Ration. Mech.Anal. (2012), 59–118
Siran Li: Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston,Texas, 77251, USA.