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A note on the higher order strain and stress tensors within deformation gradient elasticity theories: Physical interpretations and comparisons
International Journal of Solids and Structures 90 (2016) 116–121
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A note on the higher order strain and stress tensors within deformation gradient elasticity theories: Physical interpretations and comparisons Castrenze Polizzotto∗ Università di Palermo, Dipartimento di Ingegneria Civile Ambientale Aerospaziale e dei Materiali, Viale delle Scienze, Ed. 8, Palermo 90128, Italy
a r t i c l e
i n f o
Article history: Received 19 December 2015 Revised 31 March 2016 Available online 9 April 2016 Keywords: Gradient elasticity Higher order strain and stress tensors Double and triple stresses
a b s t r a c t Higher order strain and stress tensors encompassed within gradient elasticity theories are discussed with a particular concern to the physical meaning of double and triple stresses. A single rule is shown to hold for the physical interpretation of the indices of a higher order stress tensor both within distortion gradient and strain gradient theories, whereas the analogous Mindlin’s rule holds only within distortion gradient theories. Double and triple stresses are discussed separately with the aid of simple illustrative examples. A corrigendum to a previous paper by the author (IJSS 50 (2013) 3749–3765) is also presented.
1. Premise Gradient theories of solid mechanics (elasticity, plasticity, damage mechanics and the like) are nowadays accepted by the research community as effective analytical tools to address a large variety of structural problems in which the effects of the micro-structural inhomogeneities cannot be disregarded. As it emerges from the related extensive literature (see e.g. the review papers by Askes and Aifantis, 2011; Javili et al., 2013; and the book by Gurtin et al., 2010), these gradient theories involve complex concepts of strain and stress states which need the use of tensors of order higher than two, to which researchers are likely not well acquainted. In particular within elasticity, two different types of deformation gradient theories have emerged, of which one is featured by a strain energy function depending on the strain, ε ij , along with the gradient(s) of the (compatible) distortion, hij := ∂ i uj , (Mindlin, 1965; Mindlin and Eshel, 1968; Wu, 1992); the other is featured by a strain energy function depending on the strain, ε ij , and on its gradient(s) (dell’Isola et al., 2009; Exadaktylos and Vardulakis, 20 01; Lazar et al., 20 06; Polizzotto, 2013). These gradient elasticity theories were distinguished by Mindlin and Eshel (1968) as Form I and Form II theories, respectively, but more frequently they are both referred to as “strain gradient elasticity” theories within the literature. For more clarity, the above theories are here distinguished with the names of distortion gradient elasticity (DGE)
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theory the former, strain gradient elasticity (SGE) theory the latter.1 These theories lead to boundary-value problems featured by a same displacement partial differential equation system (Mindlin, 1965; Mindlin and Eshel, 1968), however the inherent notions of double and triple stresses exhibit conceptual and qualitative differences which are in general not sufficiently clarified. Recently, the symmetry features associated to the mentioned Form I and Form II were addressed by Gusev and Lurie (2015) in a variational formulation for a simplified isotropic material model of gradient elasticity endowed with only four constants (including the Lamé ones). Explanations on the subject of higher order tensors as representations of double (or dipole) and triple (or quadrupole) stresses are available in the literature (see e.g. Gronwald and Held, 1993; Jaunzemis, 1967; Lazar and Maugin, 2005; Love, 1927; Mindlin, 1964). More recently, the present author addressed the subject in question with a particular concern to the extra indices of the higher order stress tensors, which represent the lever arm(s) of the double and triple stresses, see Appendix B in Polizzotto (2013). The purpose of the present note is to provide a further contribution aimed at improving the physical interpretation of the higher order strain and stress tensors, making a clear distinction between DGE and SGE theories. A second gradient theory will be considered, which will give us the opportunity to revise the content of Appendix B in Polizzotto (2013). The question of which model
1 In the literature, the wording “second displacement gradient” is often used in place of the nonstandard “(first) distortion gradient”. The two parallel concepts of “distortion” and “strain” are here believed to be more suitable to distinguish the above twin theories from each other, since these theories ultimately are each a particular form of second displacement gradient theory.
C. Polizzotto / International Journal of Solids and Structures 90 (2016) 116–121
among DGE and SGE may be better than the other is not discussed in the present work. For presentation clarity, a non-standard selfexplaining nomenclature is introduced. 2. Introductory concepts and basic analytical relations In the following, a continuum referred to Cartesian orthogonal axes, say xi , (i = 1, 2, 3 ), is considered along with the standard index notation. 2.1. Distortion gradient elasticity (DGE) theory The DGE theory is centered on a (Helmholtz) strain energy function as, say, ψd = ψd (εi j , ∂i h jk , ∂i ∂ j hkl ), where hij := ∂ i uj is the (compatible) distortion tensor and εi j = h(i j ) := 12 (∂i u j + ∂ j ui )
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2) (1968). Here we only observe that, since τi(jk1 ) and τi(jkl , like the 1) 2) work-conjugate variables ηi(jk and ηi(jkl , are not symmetric with re-
spect to their own last index pair, then the total stress Sij in (3) is not symmetric. For possible choices of the function ψ d in terms of invariants 1) 2) of the tensors εi j , ηi(jk , ηi(jkl we refer to Mindlin (1965) and Mindlin and Eshel (1968). 2.2. Strain gradient elasticity (SGE) theory The SGE theory is centered on a (Helmholtz) strain energy function as ψs = ψs (εi j , ∂i ε jk , ∂i ∂ j εkl ). The basic deformation states of the material element are described by ε ij (uniform strain within 1) the element), εi(jk := ∂i ε jk (strain ε jk linearly varying in the xi di-
1) is the standard strain tensor. The tensor variables ε ij , ηi(jk := ∂i h jk
2) rection), and εi(jkl := ∂i ∂ j ε jk (strain ε jk bi-linearly varying in the xi
volume element, namely: ε ij , a uniform strain within the whole
1) 2) The strain gradient variables εi(jk and εi(jkl are symmetric with
2) and ηi(jkl := ∂i ∂ j hkl describe basic deformation states of the generic
and xj directions).
1) volume element; ηi(jk , a distortion hjk linearly varying in the xi di-
respect to their own last index pair, the second one also with respect to the first index pair (i, j), therefore they possess, respectively, 3 × 6 = 18 and 6 × 6 = 36 independent components 5 1) (Lazar et al., 2006). The tensor εi(jk is not invariant with respect to
2) rection; ηi(jkl , a distortion hkl varying bi-linearly in the xi and xj
directions.2 1) 2) The distortion gradient variables ηi(jk and ηi(jkl possess the fol-
1) lowing symmetry features: ηi(jk = ∂i ∂ j uk is symmetric with respect
to the first index pair (i, j), hence it contains 3×6 = 18 independent 2) components, whereas ηi(jkl = ∂i ∂ j ∂k ul is symmetric with respect to
the (first) index triple (i, j, k), hence it has 3×10 = 30 independent 1) 2) components3 (Mindlin, 1965). Both ηi(jk and ηi(jkl are not symmetric with respect to their own last index pair. The stress state corresponding to a given deformation state is determined by the partial derivatives of ψ d , that is,
σi j =
∂ψd ∂ψd ∂ψd 2) , τi(jk1) = , τi(jkl = 2) ∂εi j ∂ηi(jk1) ∂ηi(jkl
(1)
which are the constitutive equations for the standard stresses σ ij ,
2) the double stresses τi(jk1 ) ,4 and the triple stresses τi(jkl , all of which
interchanges of the first index i with anyone within the last index pair (except, obviously, in the case i = j and/or i = k); the same can 2) be stated for εi(jkl , in the sense that none of the first two indices can be interchanged with anyone within the last index pair. The stress state corresponding to a given set of strain-like variables is obtained as
σi j =
∂ψs ∂ψs ∂ψs 2) , σi(jk1) = , σi(jkl = (1 ) 2) ∂εi j ∂εi jk ∂εi(jkl
which are the constitutive equations for the standard stress σ ij , the
2) double stress σi(jk1 ) , and the triple stress σi(jkl , all of which possess
the same symmetry features as the related work-conjugate strainlike variables. Indeed, the same nomenclature is used in the literature for both stress sets (1) and (4). The stress power Ws = ψ˙ s proves to be expressed as
possess symmetry features like the corresponding work-conjugate strain-like variables. The stress power Wd = ψ˙ d proves to be expressed as
1) 2 ) (2 ) Ws = σi j ε˙ i j + σi(jk1) ε˙ i(jk + σi(jkl ε˙ i jkl
1) 2 ) (2 ) Wd = σi j ε˙ i j + τi(jk1) η˙ i(jk + τi(jkl η˙ i jkl
(2 ) Ti j := σi j − ∂ p σ pi(1j) + ∂ p ∂q σ pqi j
(2)
The stresses (1), together with the so-called total stress defined as (2 ) Si j := σi j − ∂ p τ pi(1j) + ∂ p ∂q τ pqi j
(3)
are all required to satisfy the field and boundary equilibrium equations, for which we refer to Mindlin (1965) and Mindlin and Eshel 2 For an (m + n )th order tensor, A, equal to the mth order gradient of a nth order tensor B, a rule often adopted in the literature (e.g. Exadaktylos and Vardulakis, 2001; Fleck and Hutchinson, 1997; Mindlin and Eshel, 1968) is followed here, whereby the first m ≥ 1 indices denote the inherent gradient co-ordinates, that is, Ai1 ...im j1 ... jn = ∂i1 ...∂im B j1 ... jn , but many researchers (e.g. dell’Isola et al., 2009; Lazar et al., 2006) prefer to locate the mentioned indices in the last positions within the index string of A, such as to read A j1 ... jn i1 ...im = ∂i1 ...∂im B j1 ... jn . 2) 3 is equal to the product of the The number of independent components of ηi(jkl numbers of the analogous components of a fully symmetric third order tensor (10) 2) can be recognized to be of and of a vector (3). The independent components ηi(jkl the type: 3 × 3 = 9 components of the type (iiil), 6 × 3 = 18 of the type (ijjl), 1 × 3 = 3 of the type (ijkl), in which i = j = k, (no sum on repeated indices). 4 A symbol with a superscript (1) to denote a “double” stress may appear inappropriate. The superscript (1) on such a symbol just means that the double stress is associated to a “first” strain-like variable and possesses “one” lever arm. The same holds for the superscript (2) used for a triple stress, the latter being related to a “second” strain-like gradient and endowed with “two” lever arms.
(4)
(5)
The stresses (4) together with the related total stress defined as (6)
must satisfy the field and boundary equilibrium equations, for which we make reference to Polizzotto (2013). We note that, since 2) in (6) the variables σi(jk1 ) and σi(jkl are symmetric with respect to their own last index pair, then the total stress Tij of (6) is symmetric. The latter property makes the strain-gradient based approach to gradient elasticity preferable to the analogous distortiongradient based one. A simple example of Helmholtz strain energy function for SGE can be expressed in the form
ψs =
1 Ci jkl εi j εkl + (1 )2 ∂ p εi j ∂ p εkl + (2 )4 ∂ p ∂q εi j ∂ p ∂q εkl 2
(7)
where 1 and 2 are internal length scale parameters and Cijkl is the usual moduli tensor of isotropic elasticity, that is, denoting by λ, μ the Lamé constants,
Ci jkl = λ δi j δkl + μ(δik δ jl + δil δ jk )
(8)
2) 5 The tensor εi(jkl = ∂i ∂ j εkl , with ε kl being an arbitrary (symmetric) strain tensor, has 6 × 6 = 36 independent components, as stated above. If instead ε kl is compat2) = 12 (∂i ∂ j ∂k ul + ∂i ∂ j ∂l uk ) has (30 + 30 )/2 = ible, i.e. εkl = u(k,l ) , then the tensor εi(jkl
2) 30 independent components, just like ηi(jkl . Indeed, the compatibility conditions
2) 2) make εi(jkl have the same number (30) of independent components as ηi(jkl .
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From (7) we can obtain the classical Hooke law, that is,
σi j = Ci jkl εkl = λ ε pp δi j + 2μ εi j
(9)
along with the higher order constitutive equations as
σi(jk1) = (1 )2 ∂i σ jk , σi(jkl2) = (2 )4 ∂i ∂ j σkl
(10)
The latter constitutive law (featured by two length scale constants) extends to triple stresses and second strain gradients the analogous law for double stresses and first strain gradients advanced by Aifantis and co-workers (Aifantis, 1992; Altan and Aifantis, 1992; Ru and Aifantis, 1993), with which it identifies on taking 2 = 0. The extended law was advanced by Polizzotto (2003) and used by Lazar et al. (2006) for dislocation analysis and other applications. For brevity, we shall refer to the latter constitutive law as the “extended Alfantis law” in the following.
Fig. 1. Schematics of a cubic volume element of unit edges under constant distor(1 ) (1 ) = η211 = k amounting to either a distortion h21 varying linearly in tion gradient η121 the x1 direction, or a dilatation h11 varying linearly in the x2 direction: (a) deformed pattern and double stresses; (b) equivalence of the double stresses.
3. Distortion gradient variables and related higher order stresses In this section, the physical interpretation of the higher order 2) stress variables τi(jk1 ) and τi(jkl , work-conjugate of the distortion gra1) 2) dient variables ηi(jk and ηi(jkl , respectively, are discussed.
3.1. First distortion gradient variables and related double stresses 1) As mentioned before, the variable ηi(jk = ∂i h jk denotes a defor-
mation state of a volume element determined by a (compatible) distortion h jk = ∂ j uk varying linearly along the xi direction. The 1) first index of ηi(jk is associated to the direction of the distortion
gradient, while the other two indices (j, k) are associated to the distortion component; more precisely, k → displacement component and j → displacement derivative. Correspondingly, the double stress τi(jk1 ) must be interpreted as a double force of direction xk , distributed over a plane element of normal xj and having a lever arm of direction xi . Indeed, a rule holds whereby the direction xi , indicating the distortion gradient di1) rection in ηi(jk , is associated to the lever arm direction in the related work-conjugate double stress τi(jk1 ) . The value of τi(jk1 ) can be thought
of as the (finite) limit of the product (ci σ jk ), where σ jk is a standard stress component of value tending to infinite, whereas ci is the length of a straight segment of direction xi of value tending to zero, that is,
τi(jk1) = lim (ci σ jk ) for ci → 0, σ jk → ∞
(11)
while the axis directions remain fixed. Mindlin (1964, 1965) interpreted a double stress tensor τi(jk1 ) as a double force of direction xk , distributed over a plane element of normal xi and having a lever arm in the direction xj . From this interpretation a particular rule—alternative to the one given above—comes up whereby the roles of the first two indices of τi(jk1 ) are interchanged, that is, the lever arm direction is indicated by the second index (j), the normal to the loaded plane is indicated by the first index (i). The latter rule (here referred as Mindlin’s rule) holds good within DGE (where double stresses are symmetric in the first index pair), but not within SGE (where the mentioned symmetry of double stresses does not occur). Mindlin seems not to have noticed this fact anywhere. Mindlin’s rule has gained some popularity (see e.g. Exadaktylos and Vardulakis, 2001; Lazar and Maugin, 2005) and, unfortunately, it has been applied extensively within both DGE and SGE by most researchers, including the present author (Polizzotto, 2013). Let us note that the double stresses acting on the boundary of a volume element may be graphically rendered in three alternative
ways. Namely, one way consists of an arrow representing the double force, which originates from a point of the surface element on which this force is applied and is equipped with one transversal stroke to allude to the existence of one lever arm, the direction of which is specified by the first index of the double stress therein involved (but if necessary the latter index may be specified aside the stroke). The second way consists in a discrete graphical representation which takes advantage from the fact that, as previously discussed in (Polizzotto, 2013), the double stresses (and the triple ones as well) are the external actions transmitted by the surrounding material, which are applied at the points of a thin boundary layer circumventing the volume element. At the micro-scale, this layer can be conceived formed up by a few particle networks parallel to the free surface of the cubic element, every pair of particles having a distance apart of a few atomic spaces, say c. The latter representation of the boundary external actions may be justified by means of discrete lattice models and of the nearest after to the nearest particle scheme often used to simulate the long distance action transmission mechanism (Toupin and Gazis, 1963). A third way to represent a double force, often used in the literature (Mindlin, 1965), consists in reproducing the dipole concept by means of two opposite forces applied at two points as the extremes of the inherent lever arm. For an illustrative example, let us consider a cubic volume element of unit edges, subjected to a first distortion gradient, say, (1 ) (1 ) η121 = η211 = ∂1 ∂2 u1 = k, where k is a constant scalar, whereas all other deformation modes are null, Fig. 1(a) and (b). The deformed pattern of the element can be considered either as the result of a distortion h21 = u1,2 = kx1 (linearly varying with x1 ), or as the result of a dilatation h11 = u1,1 = kx2 (linearly varying with x2 ). The deformed configuration of the element is determined by the displacements u1 = kx1 x2 , u2 = u3 = 0. The double stresses (1 ) (1 ) (1 ) (1 ) work-conjugate to η121 and η211 are, respectively, τ121 and τ211 , as schematically shown in Fig. 1(a). The latter double stresses constitute measures of a single homogeneous stress state of the volume element. Indeed, a sequence of double arrows representing (1 ) τ121 may equally be interpreted as an analogous sequence of dou(1 ) ble arrows representing τ211 as shown in Fig. 1(b). In this figure (1 ) (1 ) the stress τ211 equals the stress τ121 by symmetry. According to
(1 ) Mindlin’s rule, the first index of τ121 indicates the normal to the loaded plane, the second index indicates the lever arm direction; this is the reverse of what is stated by our own rule, but however no contradiction does arise here due to the symmetry in the first index pair. In Fig. 2 another example is reported, but the unit cubic volume (1 ) element finds itself under a uniform distortion gradient, η123 =
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Fig. 2. Schematics of a cubic volume element of unit edges under a constant distor(1 ) (1 ) (1 ) (1 ) = η213 = k and related work-conjugate double stresses τ123 = τ213 . tion gradient η123
(1 ) η213 = ∂1 ∂2 u3 = k, all other deformation modes being null. The
distortion h23 = u3,2 = kx1 varies linearly in the x1 direction and the distortion h13 = u3,1 = kx2 varies linearly in the x2 direction, hence the displacements u1 = u2 = 0 and u3 = kx1 x2 are deter(1 ) (1 ) mined. The work-conjugate double stresses τ123 = τ213 are also reported using the same graphical rules advanced before.
Fig. 3. Schematics of a cubic volume element of unit edged under a uniform second (2 ) (2 ) = η2211 = k amounting to either a distortion h21 varying distortion gradient η1221 linearly in the x1 and x2 directions, or a dilatation h11 varying quadratically in the (2 ) (2 ) = τ2211 are also reported. x2 direction. The triple stresses τ1221
3.2. Second distortion gradient variables and related triple stresses 2) A second distortion gradient variable, say ηi(jkl = ∂i ∂ j hkl denotes,
as stated before, a deformation state of a volume element determined by a distortion hkl = ∂k ul varying bi-linearly in the direc2) tions xi and xj . The related work-conjugate triple stress, say τi(jkl , is endowed with two lever arms, the directions of which are specified by the index pair (i, j) related to the gradient directions of the inherent distortion hkl . Therefore, in analogy to the double stress 2) τi(jk1) , the triple stress τi(jkl can be interpreted as a quadrupole force of direction xl , distributed over a plane of normal xk and having lever arms in the directions xi and xj . Also, similarly to τi(jk1 ) , the triple 2) stress τi(jkl can be thought of as the limit of the product (ci cj σ kl ),
where σ kl is a standard stress the value of which tends to infinite, whereas ci and cj are the lengths of two line segments of directions xi and xj , respectively, both of which tend to zero, that is (2 )
τi jkl = lim (ci c j σkl ) for ci , c j → 0 and σkl → ∞
(12)
Fig. 4. Schematics of a cubic volume element of unit edges under uniform second (2 ) (2 ) (2 ) (2 ) = η2223 = k and related triple stresses τ2123 = τ2213 . distortion gradient η2123
1 2 2 kx1 (x2 ) , (2 ) forces τ2123 =
determined by the displacements u1 = u2 = 0, u3 = hence h23 = ∂2 u3 = kx1 x2 . The related quadrupole (2 ) τ2213 are also graphically illustrated.
2) while the axis directions are all taken fixed. τi(jkl has as many in-
4. Strain gradient variables and related higher order stresses
As a consequence of the symmetry of the index triple (i, j, k) at2) tached to τi(jkl , the role of the related indices can be interchanged.
In this section the physical interpretation of the higher or2) der stress variables σi(jk1 ) and σi(jkl , work-conjugate of the strain
2) dependent components as ηi(jkl .
This implies that either i, or j, may be interpreted as the normal to the loaded plane and k as a lever arm direction, which constitutes Mindlin’s rule extended to triple stresses. The graphical representation of the triple stresses is similar to the one used for a double stress, either by an arrow, but equipped with two transversal strokes to denote the presence of two lever arms, or by a discrete geometric sketch whereby the quadrupole force is specified within the element boundary layer. An illustrative example similar to that of Fig. 1 is reported in Fig. 3, where the unit cubic volume element is subjected to (2 ) (2 ) a second distortion gradient η1221 = η2211 = ∂1 ∂2 ∂2 u1 = k. The resulting deformation pattern is determined by the displacements u1 = 12 kx1 (x2 )2 , u2 = u3 = 0. The deformation of the element may be considered either as the result of a distortion h21 = kx1 x2 , or as a consequence of a dilatation h11 = 12 k(x2 )2 . The work-conjugate (2 ) (2 ) triple stresses, τ1221 = τ2211 are also reported. In Fig. 4 an example similar to that of Fig. 2 is reported, but the cubic volume element is here subjected to a second distortion (2 ) (2 ) gradient η2123 = η2213 = ∂1 ∂2 ∂2 u3 = k. The deformation pattern is
1) 2) gradient variables εi(jk and εi(jkl , respectively, is addressed. In
the present context, though not strictly necessary, the double and triple stresses may be evaluated by means of the extended Aifantis constitutive law (10) starting from the (known) ordinary stresses σ ij . This will be done in the following applications to make more evident the inapplicability of Mindlin’s rule in SGE. 4.1. First strain gradient variables and related double stresses 1) As stated previously, the strain gradient variable εi(jk = ∂i ε jk de-
notes a deformation state of the volume element determined by a strain ε jk varying linearly along the xi direction. The first in1) dex attached to εi(jk is still elected to designate the gradient di-
rection, that is, the direction in which the strain variation does occur, whereas the other two indices specify the strain component. As asserted in Polizzotto (2013), the double stress σi(jk1 ) , work-
1) conjugate to εi(jk , has to be interpreted in exactly the same way
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C. Polizzotto / International Journal of Solids and Structures 90 (2016) 116–121
(1 ) Fig. 5. Cubic volume element of unit edges under a constant strain gradient ε121 =
(1 ) (1 ) (1 ) ε112 = k. The deformation pattern and of the associated double stresses σ121 = σ112 are schematically reported.
(1 ) Fig. 6. Cubic volume element of unit edges under uniform strain gradient ε123 =
as τi(jk1 ) , that is, as a double force of direction xk , distributed over
(1 ) (1 ) ε132 = k. The deformation pattern and the work-conjugate double stresses σ123 = (1 ) (1 ) (1 ) are schematically reported along with the double stresses σ213 and σ312 . σ132
a plane surface element of normal xj and having a lever arm in the direction xi . Namely, the same rule as in DGE holds to specify the indices indicating the lever arm direction and the normal to the loaded plane. Due to the lack of symmetry in both the index pair (i, j) and (i, k), Mindlin’s rule cannot be applied to strain-gradient based double stresses σi(jk1 ) . The latter circumstance
4.2. Second strain gradient variables and related triple stresses
seems to be unknown to most literature; in particular it is in contrast to Polizzotto (2013), Appendix B. In the latter paper, the author, though guided by the (correct) idea that double (and triple) stress tensors should be interpreted in the same way for both distortion-gradient based and strain-gradient based approaches, arrived at the wrong conclusion that Mindlin’s rule may be applied in both cases. Indeed, this is not correct, except in particular cases. Therefore, the author suggests the following erratum to the paper (Polizzotto, 2013). Namely, the last four paragraphs of Appendix B of the latter paper have to be ignored by potential readers due to some misleading assertions therein reported; also, the last paragraph of Section 4.1 of the same paper, where the latter considerations were anticipated, has equally to be ignored. The mathematical representation as a limit given by (11) for τi(jk1 ) holds good also for σi(jk1 ) .
For comparison purposes, let us discuss the same examples of the previous section, but replacing the distortion gradients with the strain gradients as the pertinent deformation sources. In Fig. 5 a unit cubic volume element is considered under the (1 ) (1 ) action of a constant strain gradient ε121 = ε112 = k, all other deformation modes being zero. Then, the deformed configuration of the element is determined by the displacements u1 = kx1 x2 , u2 = 12 k(x1 )2 , u3 = 0, hence ε12 = kx1 , ε11 = kx2 , ε13 = ε23 = 0 and ε22 = ε33 = 0. The stress state can be computed as σ11 = [(1 − ν )/((1 + ν )(1 − 2ν ))]Ekx2 , σ22 = σ33 = [ν /(1 − ν )]σ11 , σ12 = (1 ) (1 ) 2μkx1 , σ13 = σ23 = 0. The double stresses σ121 = σ112 can be (1 ) computed using the extended Aifantis law (10), namely, σ121 =
(1 ) (1 ) (1 )2 ∂1 σ21 = 2(1 )2 μk, but σ211 = (1 )2 ∂2 σ11 = σ121 . In Fig. 5 the (1 ) (1 ) double stresses σ121 and σ112 are reported.
In Fig. 6 the unit cubic element is subjected to the strain (1 ) (1 ) gradient ε123 = ε132 = k. The deformed configuration of the element is determined by the displacements u1 = 0, u2 = kx1 x3 , u3 = kx1 x2 , such that ε23 = kx1 , whereas ε11 = ε22 = ε33 = 0, ε12 = kx3 /2, ε13 = kx2 /2. By the Hooke law, the stresses are computed as: σ11 = σ22 = σ33 = 0, σ23 = 2μkx1 , σ12 = μkx3 , σ13 = μkx2 . The double stresses can also be computed by the extended (1 ) (1 ) (1 ) Aifantis law, that is: σ123 = σ132 = (1 )2 ∂1 σ23 = 2(1 )2 μk, σ312 =
( 1
)2 ∂
3 σ12
= ( 1
)2 μk,
(1 )
σ213 = (1
)2 ∂
2 σ13
= ( 1
) 2 μk .
The latter two
(1 ) (1 ) double stresses happen to be equal to each other, but σ123 = σ213 (1 ) (1 ) and σ132 = σ312 .
It has been already explained that a second strain gradient vari2) able, say εi(jkl = ∂i ∂ j εkl , indicates a deformation state of the volume element determined by a strain ε kl varying bi-linearly in the directions xi and xj (but quadratically if i = j). The fourth order tensor
2) εi(jkl is symmetric with respect to the first index pair (i, j) indicat-
ing the gradient directions, as well as with respect to the second index pair (k, l) indicating the strain component. 2) 2) The triple stress tensor, say σi(jkl , work-conjugate to εi(jkl , pos2) sesses the same symmetry features of εi(jkl , hence in particular
it also contains 36 independent components (but only 30 on imposing the compatibility conditions, i.e. εkl = u(k,l ) ) . As stated in Polizzotto (2013), the interpretation of the indices of a triple stress 2) tensor σi(jkl is the same as for the analogous distortion-gradient 2) 2) based triple stress tensor τi(jkl ; that is, σi(jkl can be interpreted as a
quadrupole force of direction xl , distributed over a plane element of normal xk and having two lever arms in the directions xi and xj , respectively. The mathematical representation as a limit like that 2) 2) given by (12) for τi(jkl is valid also for σi(jkl . Like for the strain-gradient based double stresses σi(jk1 ) , the
2) Mindlin rule does not hold for the triple stresses σi(jkl , since the
latter stresses are not invariant for interchanges of anyone of the first two indices (i, j) (associated to the lever arms) with anyone of 2) 2) the last index pair, therefore it is σi(jkl = σk(jil = σik(2jl) . The same examples considered in Section 3.2 are addressed again, but this time the cubic volume element is subjected to a specified second strain gradient. In Fig. 7, a constant strain (2 ) gradient ε1221 = ∂1 ∂2 ε21 = k is applied to the volume element. The displacements are given by u1 = 12 kx1 (x2 )2 , u2 = 12 k(x1 )2 x2 , u3 = 0, such that ε12 = kx1 x2 , ε23 = ε13 = 0, ε11 = 12 k(x2 )2 , ε22 = 1 2 2 k (x1 ) , ε33 = 0. By the Hooke law, we can obtain: σ12 = 2με12 = 2μkx1 x2 , σ13 = σ23 = 0, σ11 = 12 Ek[m1 (x2 )2 + m2 (x1 )2 ), σ22 = 12 Ek[m1 (x1 )2 + m2 (x2 )2 ), where m1 := (1 − ν )/((1 + ν )(1 − 2ν )) and m2 := ν /((1 + ν )(1 − 2ν )). By the extended Aifantis law, (2 ) (2 ) the triple stress σ1221 work-conjugate to ε1221 is computed as (2 ) (2 ) (2 ) σ1221 = (2 )4 ∂1 ∂2 σ12 = 2(2 )4 μk. The triple stresses σ1221 = σ1212
are reported in Fig. 7 using the already explained graphical rules. The other example is in Fig. 8, where the cubic element (2 ) is subjected to the uniform second strain gradient ε1223 = ∂1 ∂2 ε23 = k. The deformed configuration of the element is determined by the displacements u1 = 0, u2 = kx1 x2 x3 , u3 =
C. Polizzotto / International Journal of Solids and Structures 90 (2016) 116–121
121
discussed with a particular focus on their physical significance. A distinction has been made between distortion gradient elasticity (DGE) in which the gradient effects are carried in by the gradients of the (compatible) distortion tensor hi j = ∂i u j , and the strain gradient elasticity (SGE) in which instead the gradients of the strain tensor εi j = u( j,i ) are used for the same purpose. This distinction substantiates in a different behavior of the related sets of double and triple stresses. A same rule holds to specify, among the indices of a double (or triple) stress, which one(s) indicate the lever arm direction(s), which one the normal to loaded plane, whereas the alternative Mindlin’s rule holds good only within the DGE approach. Double and triple stresses have been separately discussed with the aid of simple illustrative examples. The presented results improve and in part amend previous results given in Polizzotto (2013). In the author’s belief, this note provides a useful contribution for a better understanding of higher order strain and stress tensors within the mechanics of generalized continua. (2 ) Fig. 7. Unit cubic volume element under uniform second strain gradient ε1221 = (2 )
∂1 ∂2 ε21 = k. Schematics of the deformation pattern and of the related triple (2 ) (2 ) = σ1212 . stresses σ1221
(2 ) = Fig. 8. Unit cubic element subjected to a constant second strain gradient ε2123
(2 ) ∂2 ∂1 ε23 = k. Schematics of the deformation pattern and the work-conjugate triple
(2 ) (2 ) stresses σ2123 = σ2132 .
1 1 1 2 2 2 kx1 (x2 ) , such that ε23 = kx1 x2 , ε13 = 2 k (x2 ) , ε12 = 2 kx2 x3 , ε11 = ε33 = 0 and ε22 = kx1 x3 . By the Hooke law we obtain: σ23 = 2μkx1 x2 , σ13 = μk(x2 )2 , σ12 = μkx2 x3 , σ22 = m1 Ekx1 x3 , σ11 = σ33 = m2 Ekx1 x3 , where m1 and m2 are the same scalar quan-
tities used within the example of Fig. 7. Then, by the ex(2 ) (2 ) tended Aifantis law, we can obtain: σ1223 = σ1232 = (2 )4 ∂1 ∂2 σ23 =
(2 ) (2 ) (2 ) (2 ) 2(2 )4 μk, σ2213 = σ2231 = (2 )4 ∂2 ∂2 σ13 = (2 )4 μk, σ2312 = σ2321 =
(2 ) (2 ) (2 )4 ∂2 ∂3 σ12 = (2 )4 μk. The double stresses σ1223 = σ1232 are re-
ported in Fig. 8. 5. Conclusion
In the present short note, the higher order strain and stress tensors encompassed within gradient theories of elasticity are
References Aifantis, E.C., 1992. On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 3, 1279–1299. Altan, S., Aifantis, E.C., 1992. On the structure of the mode III crack-tip in elasticity. Scr. Metall. Mater. 26, 319–324. Askes, H., Aifantis, E.C., 2011. Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990. dell’Isola, F., Sciarre, G., . Vidoli, S., 2009. Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A 2009 (465), 2177–2196. Exadaktylos, G.E., Vardulakis, J., 2001. Microstructure in linear elasticity and scale effects: a reconsideration of basic crack mechanics and rock fracture mechanics. Techtonophysics 335, 81–109. Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361. Gronwald, F., Held, W., 1993. Stress and hyperstress as fundamental concepts in continuum mechanics and relativistic field theories. In: Ferrarese, G. (Ed.), Advances in Modern Dynamics. Pitagora Editrice, Bologna, pp. 1–32. Gurtin, M.E., Fried, E., Anand, L., 2010. The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge, New York. Gusev, A.A., Lurie, S.A., 2015. Symmetry conditions in strain gradient elasticity. Math. Mech. Solids I-9 doi:10.1177/1081 2865 15606960. Jaunzemis, W., 1967. Continuum Mechanics. The McMillan Co., New York. Javili, A., dell’Isola, F., Steinmann, P., 2013. Geometrically nonlinear higher-order elasticity with energetic boundaries. J. Mech. Phys. Solids 61, 2381–2401. Lazar, M., Maugin, G.M., 2005. Nonsingular stress and strain fields of dislocations and dislocations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184. Lazar, M., Maugin, G.M., Aifantis, E.C., 2006. Dislocations in the second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817. Love, A.E.H., 1927. The Mathematical Theory of Elasticity, fourth edition Dover Publications, New York. Mindlin, R.D., 1964. Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78. Mindlin, R.D., 1965. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438. Mindlin, R.D., Eshel, N.N., 1968. On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 28, 845–858. Polizzotto, C., 2003. Gradient elasticity and nonstandard boundary conditions. Int. J. Solids Struct. 40, 7399–7423. Polizzotto, C., 2013. A second strain gradient elasticity theory with second velocity gradient inertia. Part I: constitutive equations and quasi-static behavior. Int. J. Solids Struct. 50, 3749–3765. Ru, C., Aifantis, E.C., 1993. A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68. Toupin, R., Gazis, D.C., 1963. Surface effects and initial stress in continuum and lattice models of elastic crystals. In: Willis, R.F. (Ed.), Lattice Dynamics. Pergamon Press, Osford, pp. 597–605. Wu, C.H., 1992. Cohesive elasticity and surface phenomena. Q. Appl. Math. 1, 73–103.
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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
A note on the higher order strain and stress tensors within deformation gradient elasticity theories: Physical interpretations and comparisons Castrenze Polizzotto∗ Università di Palermo, Dipartimento di Ingegneria Civile Ambientale Aerospaziale e dei Materiali, Viale delle Scienze, Ed. 8, Palermo 90128, Italy
a r t i c l e
i n f o
Article history: Received 19 December 2015 Revised 31 March 2016 Available online 9 April 2016 Keywords: Gradient elasticity Higher order strain and stress tensors Double and triple stresses
a b s t r a c t Higher order strain and stress tensors encompassed within gradient elasticity theories are discussed with a particular concern to the physical meaning of double and triple stresses. A single rule is shown to hold for the physical interpretation of the indices of a higher order stress tensor both within distortion gradient and strain gradient theories, whereas the analogous Mindlin’s rule holds only within distortion gradient theories. Double and triple stresses are discussed separately with the aid of simple illustrative examples. A corrigendum to a previous paper by the author (IJSS 50 (2013) 3749–3765) is also presented.
1. Premise Gradient theories of solid mechanics (elasticity, plasticity, damage mechanics and the like) are nowadays accepted by the research community as effective analytical tools to address a large variety of structural problems in which the effects of the micro-structural inhomogeneities cannot be disregarded. As it emerges from the related extensive literature (see e.g. the review papers by Askes and Aifantis, 2011; Javili et al., 2013; and the book by Gurtin et al., 2010), these gradient theories involve complex concepts of strain and stress states which need the use of tensors of order higher than two, to which researchers are likely not well acquainted. In particular within elasticity, two different types of deformation gradient theories have emerged, of which one is featured by a strain energy function depending on the strain, ε ij , along with the gradient(s) of the (compatible) distortion, hij := ∂ i uj , (Mindlin, 1965; Mindlin and Eshel, 1968; Wu, 1992); the other is featured by a strain energy function depending on the strain, ε ij , and on its gradient(s) (dell’Isola et al., 2009; Exadaktylos and Vardulakis, 20 01; Lazar et al., 20 06; Polizzotto, 2013). These gradient elasticity theories were distinguished by Mindlin and Eshel (1968) as Form I and Form II theories, respectively, but more frequently they are both referred to as “strain gradient elasticity” theories within the literature. For more clarity, the above theories are here distinguished with the names of distortion gradient elasticity (DGE)
∗
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© 2016 Elsevier Ltd. All rights reserved.
theory the former, strain gradient elasticity (SGE) theory the latter.1 These theories lead to boundary-value problems featured by a same displacement partial differential equation system (Mindlin, 1965; Mindlin and Eshel, 1968), however the inherent notions of double and triple stresses exhibit conceptual and qualitative differences which are in general not sufficiently clarified. Recently, the symmetry features associated to the mentioned Form I and Form II were addressed by Gusev and Lurie (2015) in a variational formulation for a simplified isotropic material model of gradient elasticity endowed with only four constants (including the Lamé ones). Explanations on the subject of higher order tensors as representations of double (or dipole) and triple (or quadrupole) stresses are available in the literature (see e.g. Gronwald and Held, 1993; Jaunzemis, 1967; Lazar and Maugin, 2005; Love, 1927; Mindlin, 1964). More recently, the present author addressed the subject in question with a particular concern to the extra indices of the higher order stress tensors, which represent the lever arm(s) of the double and triple stresses, see Appendix B in Polizzotto (2013). The purpose of the present note is to provide a further contribution aimed at improving the physical interpretation of the higher order strain and stress tensors, making a clear distinction between DGE and SGE theories. A second gradient theory will be considered, which will give us the opportunity to revise the content of Appendix B in Polizzotto (2013). The question of which model
1 In the literature, the wording “second displacement gradient” is often used in place of the nonstandard “(first) distortion gradient”. The two parallel concepts of “distortion” and “strain” are here believed to be more suitable to distinguish the above twin theories from each other, since these theories ultimately are each a particular form of second displacement gradient theory.
C. Polizzotto / International Journal of Solids and Structures 90 (2016) 116–121
among DGE and SGE may be better than the other is not discussed in the present work. For presentation clarity, a non-standard selfexplaining nomenclature is introduced. 2. Introductory concepts and basic analytical relations In the following, a continuum referred to Cartesian orthogonal axes, say xi , (i = 1, 2, 3 ), is considered along with the standard index notation. 2.1. Distortion gradient elasticity (DGE) theory The DGE theory is centered on a (Helmholtz) strain energy function as, say, ψd = ψd (εi j , ∂i h jk , ∂i ∂ j hkl ), where hij := ∂ i uj is the (compatible) distortion tensor and εi j = h(i j ) := 12 (∂i u j + ∂ j ui )
117
2) (1968). Here we only observe that, since τi(jk1 ) and τi(jkl , like the 1) 2) work-conjugate variables ηi(jk and ηi(jkl , are not symmetric with re-
spect to their own last index pair, then the total stress Sij in (3) is not symmetric. For possible choices of the function ψ d in terms of invariants 1) 2) of the tensors εi j , ηi(jk , ηi(jkl we refer to Mindlin (1965) and Mindlin and Eshel (1968). 2.2. Strain gradient elasticity (SGE) theory The SGE theory is centered on a (Helmholtz) strain energy function as ψs = ψs (εi j , ∂i ε jk , ∂i ∂ j εkl ). The basic deformation states of the material element are described by ε ij (uniform strain within 1) the element), εi(jk := ∂i ε jk (strain ε jk linearly varying in the xi di-
1) is the standard strain tensor. The tensor variables ε ij , ηi(jk := ∂i h jk
2) rection), and εi(jkl := ∂i ∂ j ε jk (strain ε jk bi-linearly varying in the xi
volume element, namely: ε ij , a uniform strain within the whole
1) 2) The strain gradient variables εi(jk and εi(jkl are symmetric with
2) and ηi(jkl := ∂i ∂ j hkl describe basic deformation states of the generic
and xj directions).
1) volume element; ηi(jk , a distortion hjk linearly varying in the xi di-
respect to their own last index pair, the second one also with respect to the first index pair (i, j), therefore they possess, respectively, 3 × 6 = 18 and 6 × 6 = 36 independent components 5 1) (Lazar et al., 2006). The tensor εi(jk is not invariant with respect to
2) rection; ηi(jkl , a distortion hkl varying bi-linearly in the xi and xj
directions.2 1) 2) The distortion gradient variables ηi(jk and ηi(jkl possess the fol-
1) lowing symmetry features: ηi(jk = ∂i ∂ j uk is symmetric with respect
to the first index pair (i, j), hence it contains 3×6 = 18 independent 2) components, whereas ηi(jkl = ∂i ∂ j ∂k ul is symmetric with respect to
the (first) index triple (i, j, k), hence it has 3×10 = 30 independent 1) 2) components3 (Mindlin, 1965). Both ηi(jk and ηi(jkl are not symmetric with respect to their own last index pair. The stress state corresponding to a given deformation state is determined by the partial derivatives of ψ d , that is,
σi j =
∂ψd ∂ψd ∂ψd 2) , τi(jk1) = , τi(jkl = 2) ∂εi j ∂ηi(jk1) ∂ηi(jkl
(1)
which are the constitutive equations for the standard stresses σ ij ,
2) the double stresses τi(jk1 ) ,4 and the triple stresses τi(jkl , all of which
interchanges of the first index i with anyone within the last index pair (except, obviously, in the case i = j and/or i = k); the same can 2) be stated for εi(jkl , in the sense that none of the first two indices can be interchanged with anyone within the last index pair. The stress state corresponding to a given set of strain-like variables is obtained as
σi j =
∂ψs ∂ψs ∂ψs 2) , σi(jk1) = , σi(jkl = (1 ) 2) ∂εi j ∂εi jk ∂εi(jkl
which are the constitutive equations for the standard stress σ ij , the
2) double stress σi(jk1 ) , and the triple stress σi(jkl , all of which possess
the same symmetry features as the related work-conjugate strainlike variables. Indeed, the same nomenclature is used in the literature for both stress sets (1) and (4). The stress power Ws = ψ˙ s proves to be expressed as
possess symmetry features like the corresponding work-conjugate strain-like variables. The stress power Wd = ψ˙ d proves to be expressed as
1) 2 ) (2 ) Ws = σi j ε˙ i j + σi(jk1) ε˙ i(jk + σi(jkl ε˙ i jkl
1) 2 ) (2 ) Wd = σi j ε˙ i j + τi(jk1) η˙ i(jk + τi(jkl η˙ i jkl
(2 ) Ti j := σi j − ∂ p σ pi(1j) + ∂ p ∂q σ pqi j
(2)
The stresses (1), together with the so-called total stress defined as (2 ) Si j := σi j − ∂ p τ pi(1j) + ∂ p ∂q τ pqi j
(3)
are all required to satisfy the field and boundary equilibrium equations, for which we refer to Mindlin (1965) and Mindlin and Eshel 2 For an (m + n )th order tensor, A, equal to the mth order gradient of a nth order tensor B, a rule often adopted in the literature (e.g. Exadaktylos and Vardulakis, 2001; Fleck and Hutchinson, 1997; Mindlin and Eshel, 1968) is followed here, whereby the first m ≥ 1 indices denote the inherent gradient co-ordinates, that is, Ai1 ...im j1 ... jn = ∂i1 ...∂im B j1 ... jn , but many researchers (e.g. dell’Isola et al., 2009; Lazar et al., 2006) prefer to locate the mentioned indices in the last positions within the index string of A, such as to read A j1 ... jn i1 ...im = ∂i1 ...∂im B j1 ... jn . 2) 3 is equal to the product of the The number of independent components of ηi(jkl numbers of the analogous components of a fully symmetric third order tensor (10) 2) can be recognized to be of and of a vector (3). The independent components ηi(jkl the type: 3 × 3 = 9 components of the type (iiil), 6 × 3 = 18 of the type (ijjl), 1 × 3 = 3 of the type (ijkl), in which i = j = k, (no sum on repeated indices). 4 A symbol with a superscript (1) to denote a “double” stress may appear inappropriate. The superscript (1) on such a symbol just means that the double stress is associated to a “first” strain-like variable and possesses “one” lever arm. The same holds for the superscript (2) used for a triple stress, the latter being related to a “second” strain-like gradient and endowed with “two” lever arms.
(4)
(5)
The stresses (4) together with the related total stress defined as (6)
must satisfy the field and boundary equilibrium equations, for which we make reference to Polizzotto (2013). We note that, since 2) in (6) the variables σi(jk1 ) and σi(jkl are symmetric with respect to their own last index pair, then the total stress Tij of (6) is symmetric. The latter property makes the strain-gradient based approach to gradient elasticity preferable to the analogous distortiongradient based one. A simple example of Helmholtz strain energy function for SGE can be expressed in the form
ψs =
1 Ci jkl εi j εkl + (1 )2 ∂ p εi j ∂ p εkl + (2 )4 ∂ p ∂q εi j ∂ p ∂q εkl 2
(7)
where 1 and 2 are internal length scale parameters and Cijkl is the usual moduli tensor of isotropic elasticity, that is, denoting by λ, μ the Lamé constants,
Ci jkl = λ δi j δkl + μ(δik δ jl + δil δ jk )
(8)
2) 5 The tensor εi(jkl = ∂i ∂ j εkl , with ε kl being an arbitrary (symmetric) strain tensor, has 6 × 6 = 36 independent components, as stated above. If instead ε kl is compat2) = 12 (∂i ∂ j ∂k ul + ∂i ∂ j ∂l uk ) has (30 + 30 )/2 = ible, i.e. εkl = u(k,l ) , then the tensor εi(jkl
2) 30 independent components, just like ηi(jkl . Indeed, the compatibility conditions
2) 2) make εi(jkl have the same number (30) of independent components as ηi(jkl .
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From (7) we can obtain the classical Hooke law, that is,
σi j = Ci jkl εkl = λ ε pp δi j + 2μ εi j
(9)
along with the higher order constitutive equations as
σi(jk1) = (1 )2 ∂i σ jk , σi(jkl2) = (2 )4 ∂i ∂ j σkl
(10)
The latter constitutive law (featured by two length scale constants) extends to triple stresses and second strain gradients the analogous law for double stresses and first strain gradients advanced by Aifantis and co-workers (Aifantis, 1992; Altan and Aifantis, 1992; Ru and Aifantis, 1993), with which it identifies on taking 2 = 0. The extended law was advanced by Polizzotto (2003) and used by Lazar et al. (2006) for dislocation analysis and other applications. For brevity, we shall refer to the latter constitutive law as the “extended Alfantis law” in the following.
Fig. 1. Schematics of a cubic volume element of unit edges under constant distor(1 ) (1 ) = η211 = k amounting to either a distortion h21 varying linearly in tion gradient η121 the x1 direction, or a dilatation h11 varying linearly in the x2 direction: (a) deformed pattern and double stresses; (b) equivalence of the double stresses.
3. Distortion gradient variables and related higher order stresses In this section, the physical interpretation of the higher order 2) stress variables τi(jk1 ) and τi(jkl , work-conjugate of the distortion gra1) 2) dient variables ηi(jk and ηi(jkl , respectively, are discussed.
3.1. First distortion gradient variables and related double stresses 1) As mentioned before, the variable ηi(jk = ∂i h jk denotes a defor-
mation state of a volume element determined by a (compatible) distortion h jk = ∂ j uk varying linearly along the xi direction. The 1) first index of ηi(jk is associated to the direction of the distortion
gradient, while the other two indices (j, k) are associated to the distortion component; more precisely, k → displacement component and j → displacement derivative. Correspondingly, the double stress τi(jk1 ) must be interpreted as a double force of direction xk , distributed over a plane element of normal xj and having a lever arm of direction xi . Indeed, a rule holds whereby the direction xi , indicating the distortion gradient di1) rection in ηi(jk , is associated to the lever arm direction in the related work-conjugate double stress τi(jk1 ) . The value of τi(jk1 ) can be thought
of as the (finite) limit of the product (ci σ jk ), where σ jk is a standard stress component of value tending to infinite, whereas ci is the length of a straight segment of direction xi of value tending to zero, that is,
τi(jk1) = lim (ci σ jk ) for ci → 0, σ jk → ∞
(11)
while the axis directions remain fixed. Mindlin (1964, 1965) interpreted a double stress tensor τi(jk1 ) as a double force of direction xk , distributed over a plane element of normal xi and having a lever arm in the direction xj . From this interpretation a particular rule—alternative to the one given above—comes up whereby the roles of the first two indices of τi(jk1 ) are interchanged, that is, the lever arm direction is indicated by the second index (j), the normal to the loaded plane is indicated by the first index (i). The latter rule (here referred as Mindlin’s rule) holds good within DGE (where double stresses are symmetric in the first index pair), but not within SGE (where the mentioned symmetry of double stresses does not occur). Mindlin seems not to have noticed this fact anywhere. Mindlin’s rule has gained some popularity (see e.g. Exadaktylos and Vardulakis, 2001; Lazar and Maugin, 2005) and, unfortunately, it has been applied extensively within both DGE and SGE by most researchers, including the present author (Polizzotto, 2013). Let us note that the double stresses acting on the boundary of a volume element may be graphically rendered in three alternative
ways. Namely, one way consists of an arrow representing the double force, which originates from a point of the surface element on which this force is applied and is equipped with one transversal stroke to allude to the existence of one lever arm, the direction of which is specified by the first index of the double stress therein involved (but if necessary the latter index may be specified aside the stroke). The second way consists in a discrete graphical representation which takes advantage from the fact that, as previously discussed in (Polizzotto, 2013), the double stresses (and the triple ones as well) are the external actions transmitted by the surrounding material, which are applied at the points of a thin boundary layer circumventing the volume element. At the micro-scale, this layer can be conceived formed up by a few particle networks parallel to the free surface of the cubic element, every pair of particles having a distance apart of a few atomic spaces, say c. The latter representation of the boundary external actions may be justified by means of discrete lattice models and of the nearest after to the nearest particle scheme often used to simulate the long distance action transmission mechanism (Toupin and Gazis, 1963). A third way to represent a double force, often used in the literature (Mindlin, 1965), consists in reproducing the dipole concept by means of two opposite forces applied at two points as the extremes of the inherent lever arm. For an illustrative example, let us consider a cubic volume element of unit edges, subjected to a first distortion gradient, say, (1 ) (1 ) η121 = η211 = ∂1 ∂2 u1 = k, where k is a constant scalar, whereas all other deformation modes are null, Fig. 1(a) and (b). The deformed pattern of the element can be considered either as the result of a distortion h21 = u1,2 = kx1 (linearly varying with x1 ), or as the result of a dilatation h11 = u1,1 = kx2 (linearly varying with x2 ). The deformed configuration of the element is determined by the displacements u1 = kx1 x2 , u2 = u3 = 0. The double stresses (1 ) (1 ) (1 ) (1 ) work-conjugate to η121 and η211 are, respectively, τ121 and τ211 , as schematically shown in Fig. 1(a). The latter double stresses constitute measures of a single homogeneous stress state of the volume element. Indeed, a sequence of double arrows representing (1 ) τ121 may equally be interpreted as an analogous sequence of dou(1 ) ble arrows representing τ211 as shown in Fig. 1(b). In this figure (1 ) (1 ) the stress τ211 equals the stress τ121 by symmetry. According to
(1 ) Mindlin’s rule, the first index of τ121 indicates the normal to the loaded plane, the second index indicates the lever arm direction; this is the reverse of what is stated by our own rule, but however no contradiction does arise here due to the symmetry in the first index pair. In Fig. 2 another example is reported, but the unit cubic volume (1 ) element finds itself under a uniform distortion gradient, η123 =
C. Polizzotto / International Journal of Solids and Structures 90 (2016) 116–121
119
Fig. 2. Schematics of a cubic volume element of unit edges under a constant distor(1 ) (1 ) (1 ) (1 ) = η213 = k and related work-conjugate double stresses τ123 = τ213 . tion gradient η123
(1 ) η213 = ∂1 ∂2 u3 = k, all other deformation modes being null. The
distortion h23 = u3,2 = kx1 varies linearly in the x1 direction and the distortion h13 = u3,1 = kx2 varies linearly in the x2 direction, hence the displacements u1 = u2 = 0 and u3 = kx1 x2 are deter(1 ) (1 ) mined. The work-conjugate double stresses τ123 = τ213 are also reported using the same graphical rules advanced before.
Fig. 3. Schematics of a cubic volume element of unit edged under a uniform second (2 ) (2 ) = η2211 = k amounting to either a distortion h21 varying distortion gradient η1221 linearly in the x1 and x2 directions, or a dilatation h11 varying quadratically in the (2 ) (2 ) = τ2211 are also reported. x2 direction. The triple stresses τ1221
3.2. Second distortion gradient variables and related triple stresses 2) A second distortion gradient variable, say ηi(jkl = ∂i ∂ j hkl denotes,
as stated before, a deformation state of a volume element determined by a distortion hkl = ∂k ul varying bi-linearly in the direc2) tions xi and xj . The related work-conjugate triple stress, say τi(jkl , is endowed with two lever arms, the directions of which are specified by the index pair (i, j) related to the gradient directions of the inherent distortion hkl . Therefore, in analogy to the double stress 2) τi(jk1) , the triple stress τi(jkl can be interpreted as a quadrupole force of direction xl , distributed over a plane of normal xk and having lever arms in the directions xi and xj . Also, similarly to τi(jk1 ) , the triple 2) stress τi(jkl can be thought of as the limit of the product (ci cj σ kl ),
where σ kl is a standard stress the value of which tends to infinite, whereas ci and cj are the lengths of two line segments of directions xi and xj , respectively, both of which tend to zero, that is (2 )
τi jkl = lim (ci c j σkl ) for ci , c j → 0 and σkl → ∞
(12)
Fig. 4. Schematics of a cubic volume element of unit edges under uniform second (2 ) (2 ) (2 ) (2 ) = η2223 = k and related triple stresses τ2123 = τ2213 . distortion gradient η2123
1 2 2 kx1 (x2 ) , (2 ) forces τ2123 =
determined by the displacements u1 = u2 = 0, u3 = hence h23 = ∂2 u3 = kx1 x2 . The related quadrupole (2 ) τ2213 are also graphically illustrated.
2) while the axis directions are all taken fixed. τi(jkl has as many in-
4. Strain gradient variables and related higher order stresses
As a consequence of the symmetry of the index triple (i, j, k) at2) tached to τi(jkl , the role of the related indices can be interchanged.
In this section the physical interpretation of the higher or2) der stress variables σi(jk1 ) and σi(jkl , work-conjugate of the strain
2) dependent components as ηi(jkl .
This implies that either i, or j, may be interpreted as the normal to the loaded plane and k as a lever arm direction, which constitutes Mindlin’s rule extended to triple stresses. The graphical representation of the triple stresses is similar to the one used for a double stress, either by an arrow, but equipped with two transversal strokes to denote the presence of two lever arms, or by a discrete geometric sketch whereby the quadrupole force is specified within the element boundary layer. An illustrative example similar to that of Fig. 1 is reported in Fig. 3, where the unit cubic volume element is subjected to (2 ) (2 ) a second distortion gradient η1221 = η2211 = ∂1 ∂2 ∂2 u1 = k. The resulting deformation pattern is determined by the displacements u1 = 12 kx1 (x2 )2 , u2 = u3 = 0. The deformation of the element may be considered either as the result of a distortion h21 = kx1 x2 , or as a consequence of a dilatation h11 = 12 k(x2 )2 . The work-conjugate (2 ) (2 ) triple stresses, τ1221 = τ2211 are also reported. In Fig. 4 an example similar to that of Fig. 2 is reported, but the cubic volume element is here subjected to a second distortion (2 ) (2 ) gradient η2123 = η2213 = ∂1 ∂2 ∂2 u3 = k. The deformation pattern is
1) 2) gradient variables εi(jk and εi(jkl , respectively, is addressed. In
the present context, though not strictly necessary, the double and triple stresses may be evaluated by means of the extended Aifantis constitutive law (10) starting from the (known) ordinary stresses σ ij . This will be done in the following applications to make more evident the inapplicability of Mindlin’s rule in SGE. 4.1. First strain gradient variables and related double stresses 1) As stated previously, the strain gradient variable εi(jk = ∂i ε jk de-
notes a deformation state of the volume element determined by a strain ε jk varying linearly along the xi direction. The first in1) dex attached to εi(jk is still elected to designate the gradient di-
rection, that is, the direction in which the strain variation does occur, whereas the other two indices specify the strain component. As asserted in Polizzotto (2013), the double stress σi(jk1 ) , work-
1) conjugate to εi(jk , has to be interpreted in exactly the same way
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(1 ) Fig. 5. Cubic volume element of unit edges under a constant strain gradient ε121 =
(1 ) (1 ) (1 ) ε112 = k. The deformation pattern and of the associated double stresses σ121 = σ112 are schematically reported.
(1 ) Fig. 6. Cubic volume element of unit edges under uniform strain gradient ε123 =
as τi(jk1 ) , that is, as a double force of direction xk , distributed over
(1 ) (1 ) ε132 = k. The deformation pattern and the work-conjugate double stresses σ123 = (1 ) (1 ) (1 ) are schematically reported along with the double stresses σ213 and σ312 . σ132
a plane surface element of normal xj and having a lever arm in the direction xi . Namely, the same rule as in DGE holds to specify the indices indicating the lever arm direction and the normal to the loaded plane. Due to the lack of symmetry in both the index pair (i, j) and (i, k), Mindlin’s rule cannot be applied to strain-gradient based double stresses σi(jk1 ) . The latter circumstance
4.2. Second strain gradient variables and related triple stresses
seems to be unknown to most literature; in particular it is in contrast to Polizzotto (2013), Appendix B. In the latter paper, the author, though guided by the (correct) idea that double (and triple) stress tensors should be interpreted in the same way for both distortion-gradient based and strain-gradient based approaches, arrived at the wrong conclusion that Mindlin’s rule may be applied in both cases. Indeed, this is not correct, except in particular cases. Therefore, the author suggests the following erratum to the paper (Polizzotto, 2013). Namely, the last four paragraphs of Appendix B of the latter paper have to be ignored by potential readers due to some misleading assertions therein reported; also, the last paragraph of Section 4.1 of the same paper, where the latter considerations were anticipated, has equally to be ignored. The mathematical representation as a limit given by (11) for τi(jk1 ) holds good also for σi(jk1 ) .
For comparison purposes, let us discuss the same examples of the previous section, but replacing the distortion gradients with the strain gradients as the pertinent deformation sources. In Fig. 5 a unit cubic volume element is considered under the (1 ) (1 ) action of a constant strain gradient ε121 = ε112 = k, all other deformation modes being zero. Then, the deformed configuration of the element is determined by the displacements u1 = kx1 x2 , u2 = 12 k(x1 )2 , u3 = 0, hence ε12 = kx1 , ε11 = kx2 , ε13 = ε23 = 0 and ε22 = ε33 = 0. The stress state can be computed as σ11 = [(1 − ν )/((1 + ν )(1 − 2ν ))]Ekx2 , σ22 = σ33 = [ν /(1 − ν )]σ11 , σ12 = (1 ) (1 ) 2μkx1 , σ13 = σ23 = 0. The double stresses σ121 = σ112 can be (1 ) computed using the extended Aifantis law (10), namely, σ121 =
(1 ) (1 ) (1 )2 ∂1 σ21 = 2(1 )2 μk, but σ211 = (1 )2 ∂2 σ11 = σ121 . In Fig. 5 the (1 ) (1 ) double stresses σ121 and σ112 are reported.
In Fig. 6 the unit cubic element is subjected to the strain (1 ) (1 ) gradient ε123 = ε132 = k. The deformed configuration of the element is determined by the displacements u1 = 0, u2 = kx1 x3 , u3 = kx1 x2 , such that ε23 = kx1 , whereas ε11 = ε22 = ε33 = 0, ε12 = kx3 /2, ε13 = kx2 /2. By the Hooke law, the stresses are computed as: σ11 = σ22 = σ33 = 0, σ23 = 2μkx1 , σ12 = μkx3 , σ13 = μkx2 . The double stresses can also be computed by the extended (1 ) (1 ) (1 ) Aifantis law, that is: σ123 = σ132 = (1 )2 ∂1 σ23 = 2(1 )2 μk, σ312 =
( 1
)2 ∂
3 σ12
= ( 1
)2 μk,
(1 )
σ213 = (1
)2 ∂
2 σ13
= ( 1
) 2 μk .
The latter two
(1 ) (1 ) double stresses happen to be equal to each other, but σ123 = σ213 (1 ) (1 ) and σ132 = σ312 .
It has been already explained that a second strain gradient vari2) able, say εi(jkl = ∂i ∂ j εkl , indicates a deformation state of the volume element determined by a strain ε kl varying bi-linearly in the directions xi and xj (but quadratically if i = j). The fourth order tensor
2) εi(jkl is symmetric with respect to the first index pair (i, j) indicat-
ing the gradient directions, as well as with respect to the second index pair (k, l) indicating the strain component. 2) 2) The triple stress tensor, say σi(jkl , work-conjugate to εi(jkl , pos2) sesses the same symmetry features of εi(jkl , hence in particular
it also contains 36 independent components (but only 30 on imposing the compatibility conditions, i.e. εkl = u(k,l ) ) . As stated in Polizzotto (2013), the interpretation of the indices of a triple stress 2) tensor σi(jkl is the same as for the analogous distortion-gradient 2) 2) based triple stress tensor τi(jkl ; that is, σi(jkl can be interpreted as a
quadrupole force of direction xl , distributed over a plane element of normal xk and having two lever arms in the directions xi and xj , respectively. The mathematical representation as a limit like that 2) 2) given by (12) for τi(jkl is valid also for σi(jkl . Like for the strain-gradient based double stresses σi(jk1 ) , the
2) Mindlin rule does not hold for the triple stresses σi(jkl , since the
latter stresses are not invariant for interchanges of anyone of the first two indices (i, j) (associated to the lever arms) with anyone of 2) 2) the last index pair, therefore it is σi(jkl = σk(jil = σik(2jl) . The same examples considered in Section 3.2 are addressed again, but this time the cubic volume element is subjected to a specified second strain gradient. In Fig. 7, a constant strain (2 ) gradient ε1221 = ∂1 ∂2 ε21 = k is applied to the volume element. The displacements are given by u1 = 12 kx1 (x2 )2 , u2 = 12 k(x1 )2 x2 , u3 = 0, such that ε12 = kx1 x2 , ε23 = ε13 = 0, ε11 = 12 k(x2 )2 , ε22 = 1 2 2 k (x1 ) , ε33 = 0. By the Hooke law, we can obtain: σ12 = 2με12 = 2μkx1 x2 , σ13 = σ23 = 0, σ11 = 12 Ek[m1 (x2 )2 + m2 (x1 )2 ), σ22 = 12 Ek[m1 (x1 )2 + m2 (x2 )2 ), where m1 := (1 − ν )/((1 + ν )(1 − 2ν )) and m2 := ν /((1 + ν )(1 − 2ν )). By the extended Aifantis law, (2 ) (2 ) the triple stress σ1221 work-conjugate to ε1221 is computed as (2 ) (2 ) (2 ) σ1221 = (2 )4 ∂1 ∂2 σ12 = 2(2 )4 μk. The triple stresses σ1221 = σ1212
are reported in Fig. 7 using the already explained graphical rules. The other example is in Fig. 8, where the cubic element (2 ) is subjected to the uniform second strain gradient ε1223 = ∂1 ∂2 ε23 = k. The deformed configuration of the element is determined by the displacements u1 = 0, u2 = kx1 x2 x3 , u3 =
C. Polizzotto / International Journal of Solids and Structures 90 (2016) 116–121
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discussed with a particular focus on their physical significance. A distinction has been made between distortion gradient elasticity (DGE) in which the gradient effects are carried in by the gradients of the (compatible) distortion tensor hi j = ∂i u j , and the strain gradient elasticity (SGE) in which instead the gradients of the strain tensor εi j = u( j,i ) are used for the same purpose. This distinction substantiates in a different behavior of the related sets of double and triple stresses. A same rule holds to specify, among the indices of a double (or triple) stress, which one(s) indicate the lever arm direction(s), which one the normal to loaded plane, whereas the alternative Mindlin’s rule holds good only within the DGE approach. Double and triple stresses have been separately discussed with the aid of simple illustrative examples. The presented results improve and in part amend previous results given in Polizzotto (2013). In the author’s belief, this note provides a useful contribution for a better understanding of higher order strain and stress tensors within the mechanics of generalized continua. (2 ) Fig. 7. Unit cubic volume element under uniform second strain gradient ε1221 = (2 )
∂1 ∂2 ε21 = k. Schematics of the deformation pattern and of the related triple (2 ) (2 ) = σ1212 . stresses σ1221
(2 ) = Fig. 8. Unit cubic element subjected to a constant second strain gradient ε2123
(2 ) ∂2 ∂1 ε23 = k. Schematics of the deformation pattern and the work-conjugate triple
(2 ) (2 ) stresses σ2123 = σ2132 .
1 1 1 2 2 2 kx1 (x2 ) , such that ε23 = kx1 x2 , ε13 = 2 k (x2 ) , ε12 = 2 kx2 x3 , ε11 = ε33 = 0 and ε22 = kx1 x3 . By the Hooke law we obtain: σ23 = 2μkx1 x2 , σ13 = μk(x2 )2 , σ12 = μkx2 x3 , σ22 = m1 Ekx1 x3 , σ11 = σ33 = m2 Ekx1 x3 , where m1 and m2 are the same scalar quan-
tities used within the example of Fig. 7. Then, by the ex(2 ) (2 ) tended Aifantis law, we can obtain: σ1223 = σ1232 = (2 )4 ∂1 ∂2 σ23 =
(2 ) (2 ) (2 ) (2 ) 2(2 )4 μk, σ2213 = σ2231 = (2 )4 ∂2 ∂2 σ13 = (2 )4 μk, σ2312 = σ2321 =
(2 ) (2 ) (2 )4 ∂2 ∂3 σ12 = (2 )4 μk. The double stresses σ1223 = σ1232 are re-
ported in Fig. 8. 5. Conclusion
In the present short note, the higher order strain and stress tensors encompassed within gradient theories of elasticity are
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