Aifantis, E.C.

Lazar, M.

Maugin, G.A.

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International Journal of Solids and Structures 47 (2010) 738–739Contents lists available at ScienceDirect
International Journal of Solids and Structures
journal homepage: www.elsevier .com/locate / i jsolst rAddendum
Addendum to: ‘‘Dislocations in second strain gradient elasticity” [Int. J. Solids
Struct. 43 (2006) 1787–1817]
M. Lazar a,*, G.A. Maugin b, E.C. Aifantis c,d
a Emmy Noether Research Group, Department of Physics, Darmstadt University of Technology, Hochschulstr. 6, D-64289 Darmstadt, Germany
b Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, 4 Place Jussieu, Case 162, F-75252 Paris Cedex 05, France
c Laboratory of Mechanics and Materials, Polytechnic School, Aristotle University of Thessaloniki, P.O. Box 468, 54124 Thessaloniki, Greece
dCenter for the Mechanics of Material Instabilities and Manufacturing Processes, Michigan Technological University, Houghton, MI 49931, USAIn the present note we want to give some remarks on the gen-
eral case of anisotropic second strain gradient elasticity given by
Lazar et al. (2006) and especially on the symmetries of the material
tensors entering in the generalized stress–strain relationships. In
Lazar et al. (2006), the strain energy density W has the following
form
W ¼ 1
2
CijklEijEkl þ 12CijklmnEij;kElm;n þ
1
2
CijklmnpqEij;klEmn;pq
þ DijklmEijEkl;m þ DijklmnEijEkl;mn þ DijklmnpEij;kElm;np; ð1Þ
where the last three contributions are cross terms. From Eq. (1) we
obtain the following constitutive equations for the second (Cauchy-
like), third (double), fourth (triple) order stress tensors
rij ¼ CijklEkl þ DijklmEkl;m þ DijklmnEkl;mn;
sijk ¼ DlmijkElm þ CijklmnElm;n þ DijklmnpElm;np;
sijkl ¼ DmnijklEmn þ DmnpijklEmn;p þ CijklmnpqEmn;pq; ð2Þ
where rij ¼ @W=@Eij; sijk ¼ @W=@Eij;k and sijkl ¼ @W=@Eij;kl. Here
Cijkl;Cijklmn;Cijklmnpq;Dijklm;Dijklmn and Dijklmnp are the material tensors
with the constitutive coefﬁcients, which Lazar et al. (2006) as-
sumed, for simplicity, to possess additional symmetry properties
than those implied by their deﬁnitions (see also Agiasoﬁtou and La-
zar (2009)). In general, the material tensors in anisotropic second
strain gradient elasticity possess only the following symmetries:
CðijÞðklÞ  CðklÞðijÞ  CðijÞjðklÞ;
CðijÞkðlmÞn ¼ CðlmÞnðijÞk  CðijÞkjðlmÞn;
CðijÞðklÞðmnÞðpqÞ ¼ CðmnÞðpqÞðijÞðklÞ  CðijÞðklÞjðmnÞðpqÞ: ð3Þ
The material tensors DðijÞðklÞm;DðijÞðklÞðmnÞ and DðijÞkðlmÞðnpÞ do not possess
additional symmetries. The additional symmetries for CðijÞkðlmÞn and
CðijÞðklÞðmnÞðpqÞ given in (2.27) by Lazar et al. (2006) were obtained by
substituting Eq. (2) into the stress equilibrium condition
r

ij;j ¼ 0; r

ij ¼ rij  sijk;k þ sijkl;kl: ð4Þ
The terminology adopted by Lazar et al. (2006) for r

ij was ‘total
stress’ whereas the terminology ‘Cauchy stress’ was adopted for0020-7683/$ - see front matter  2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijsolstr.2009.11.005
DOI of original article: 10.1016/j.ijsolstr.2005.07.005.
* Corresponding author.
E-mail address: lazar@mechanik.tu-darmstadt.de (M. Lazar).the stress rij with the dimension of force=ðlengthÞ2, while the ter-
minology ‘double stress’ and ‘triple stress’ was adopted for the
hyperstress sijk which has the dimension of force=length and sijkl
which has the dimension of force. Alternatively, the self-equilibrat-
ing stress r

ij could be identiﬁed with the Cauchy stress (e.g. Aifantis
(2010) and references quoted therein) and then, Eq. (4.b) would
play the role of a ‘constitutive equation’ for r

ij.
In order to obtain the additional symmetries for DðijÞðklÞm and
DðijÞkðlmÞðnpÞ given in (2.27) by Lazar et al. (2006), we assumed that
the crossing terms of the tensors of odd rank cancel each other
out in Eq. (4). Moreover, the symmetry for the material tensor
DðijÞðklÞðmnÞ which is of even rank was assumed for simplicity. Of
course, from the group theoretical point of view, the material ten-
sors possess only the symmetries given above, i.e. those listed in
Eq. (3).
If we substitute the constitutive relations given by Eq. (2) into
the equilibrium condition given by Eq. (4), we ﬁnd for the general
anisotropic case the result:
CðijÞjðklÞEkl;j  CðijÞkjðlmÞnElm;jkn þ CðijÞðklÞjðmnÞðpqÞEmn;jklpq
þ ðDðijÞðklÞm  DðklÞðijÞmÞEkl;jm þ ðDðijÞðklÞðmnÞ þ DðklÞðijÞðmnÞÞEkl;jmn
 ðDðijÞkðlmÞðnpÞ  DðlmÞkðijÞðnpÞÞElm;jknp ¼ 0: ð5Þ
In addition, the stress tensor r

ij, for the general case of anisotropic
second strain gradient elasticity, reads
r

ij ¼ CðijÞjðklÞEkl  CðijÞkjðlmÞnElm;kn þ CðijÞðklÞjðmnÞðpqÞEmn;klpq
þ ðDðijÞðklÞm  DðklÞðijÞmÞEkl;m þ ðDðijÞðklÞðmnÞ þ DðklÞðijÞðmnÞÞEkl;mn
 ðDðijÞkðlmÞðnpÞ  DðlmÞkðijÞðnpÞÞElm;knp: ð6Þ
Lazar et al. (2006) considered, in Eqs. (2.23)–(2.27), the case of
anisotropic second strain gradient elasticity. Afterwards they spe-
cialized to the isotropic case. All the solutions derived by Lazar
et al. (2006) are valid for the isotropic case.References
Lazar, M., Maugin, G.A., Aifantis, E.C., 2006. Dislocations in second strain gradient
elasticity. Int. J. Solids Struct. 43, 1787–1817.
Agiasoﬁtou, E., Lazar, M., 2009. Anisotropic elasticity of grade three: conservation
and balance laws. In: Steinmann, P. (Ed.), IUTAM Symposium on Progress in the
M. Lazar et al. / International Journal of Solids and Structures 47 (2010) 738–739 739Theory and Numerics of Conﬁgurational Mechanics, IUTAM Bookseries, vol. 17.
Springer, pp. 179–191.
Aifantis, E.C., 2010. A personal view on current generalized theories of elasticity and
plastic ﬂow. In: Maugin, G.A., Metrikine, A.V. (Eds.), Advances in Mechanics andMathematics (Proc. of the Euromech 510 ‘‘Mechanics of generalized continua: A
hundred years after the Cosserats”, Paris, May 13–16, 2009). Springer, NY, pp.
181–192. Chapter 20.


Addendum to: “Dislocations in second strain gradient elasticity” [Int. J. Solids Struct. 43 (2006) 1787–1817] 

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Addendum to: “Dislocations in second strain gradient elasticity” [Int. J. Solids Struct. 43 (2006) 1787–1817]

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