First Vs. Second Gradient of Strain Theory for Capillarity Effects in an Elastic Fluid at Small Length Scales
Computational Materials Science 50 (2011) 1299–1304
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Computational Materials Science
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First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales
S. Forest *, N.M. Cordero, E.P. Busso
MINES ParisTech, Centre des Matériaux, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France article info abstract
Article history: Mindlin [22] wrote a milestone paper claiming that a second strain gradient theory is required for a con- Received 4 October 2009 tinuum description of volume cohesion and surface tension in isotropic elastic media. The objective of the Received in revised form 28 March 2010 present work is to compare Mindlin’s approach to more standard capillarity models based on a first strain Accepted 30 March 2010 gradient theory and Korteweg’s equation. A general micromorphic model is then proposed as a numerical method to implement Mindlin’s theory in a finite element code. Ó 2010 Elsevier B.V. All rights reserved. Keywords: Capillarity Cohesion Surface tension Nano-particle Free surface Micromorphic continuum Strain gradient Second gradient theory Third gradient theory
1. Introduction tained for both bulk phases separated by a zero-thickness surface [25,8]. The continuum mechanical theory of surface/interface Due to their different local environments, atoms at a free sur- behaviour has been settled by Gurtin and Murdoch [18,19].It face and atoms in the bulk of a material have a different associated introduces a volume stress tensor in the bulk of the material and energy and lattice spacing. This excess of energy associated with a surface stress tensor in the surface or interface modelled as a surface atoms is called surface free energy and gives rise to surface membrane. Both stress tensors fulfill balance of momentum equa- tension. The surface region is only a few atomic layers thin, that is tions. A specific elastic behaviour is attributed to the membrane why the surface tension can be neglected when the characteristic and kinematic constraints ensure that the bulk part and the surface length of the microstructure of the considered material is in the remain coherent. The most common manifestation of surface micrometer range or larger. But in the case of nano-sized materials, behaviour is capillarity effects in elastic fluids. It is described by the ratio between the surface and the volume is much higher and the Young–Laplace equation which states that the internal pres- the surface region behaviour cannot be neglected anymore. There sure, p, in a spherical droplet is proportional to the surface tension, are several ways to introduce the mechanical properties of the sur- T, multiplied by the surface curvature, 1/r: face. If an interface separating two homogeneous bulk phases is considered, one can define the interfacial properties by using an in- 2T ter-phase with a finite volume and assign thermodynamic proper- p ¼ : ð1Þ r ties in the usual way, as in Capolungo et al. [3]. Three phases are considered in this approach and the boundaries of the inter-phase When the size of the considered object is small enough, there is have to be defined more or less arbitrary. no clear way to define a sharp interface or surface. Instead, a con- One can also consider that the two homogeneous phases are tinuum model can be used to describe a transition domain be- separated by a single dividing surface; the thermodynamic proper- tween two bulk regions, or between the bulk and the outer free ties of the interface are defined as the excess over the values ob- surface. Such continuum theories for diffuse surface or interfaces have been developed for a refined description of capillarity in elas- tic fluids and solids. They are based on higher order gradient theo- * Corresponding author. Tel.: +33 1 60 76 30 51; fax: +33 1 60 76 31 50. ries like the Korteweg equation which involves the gradient of E-mail address: [email protected] (S. Forest). density vector, see [30]. A more general strain gradient theory
0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.03.048 1300 S. Forest et al. / Computational Materials Science 50 (2011) 1299–1304 has been proposed by Casal [4–6]. In a unsufficiently known paper, mass density, q, on stress but also that of the density gradient Mindlin [22] claims that a second gradient of strain or, equiva- $q, in the form lently, third gradient of displacement theory is in fact needed to 2 2 describe, in a continuous manner, capillarity and cohesion effects T ¼ pðqÞ 1 að$qÞ 1 b$q $q þ cð$ qÞ 1 þd$ $q; ð4Þ in isotropic linear elastic solids and fluids. Based on a simple where T is the stress tensor and a, b, c and d are material parame- one-dimensional atomic chain model, he identifies the higher or- ters, namely higher order elasticity moduli. The divergence of T is der elasticity modulus that is responsible for the variation of lattice assumed to vanish, in the absence of volume forces. In their account spacing from the free surface into the bulk in a semi-infinite crys- of Korteweg’s constitutive theory, Truesdell and Noll [30] show how tal. This model has not been discussed in literature, so that there it can be used to represent a spherical non-uniform field of mass seems to be no general opinion whether a first or second strain gra- density, q(r), thus allowing for the presence of an interface between dient theory is needed for capillarity effects in linear elastic media. liquid and vapor in a water droplet. The balance equation, On the other hand, Mindlin’s second gradient of strain theory is challenging from the computational point of view to compute 2 T0 þ ðT T Þ¼0; ð5Þ fields of lattice parameters in nano-objects like nano-particles or rr r rr hh nanocrystals. Molecular static simulation provide such non-homo- is combined with the constitutive relation, geneous distributions of lattice spacing in crystals close to free sur- 0 faces or grain boundaries, that could be represented by a suitable 02 q 00 T Thh ¼ bq d þ dq ; ð6Þ continuum model. rr r The objective of the present work is, firstly, to compare Kor- where the prime denotes derivation with respect to r. Integrating teweg’s equation with the first and second gradient of strain theo- Eq. (5) on the interface zone [r1,r2] yields the relation ries in order to highlight the main differences, and, secondly, to Z 0 0 r2 02 00 provide a framework for the numerical implementation of Mind- q ðr2Þ q ðr1Þ bq dq Trrðr2Þ Trrðr1Þ¼2d 2 dr: ð7Þ lin’s second strain gradient theory. Finite element implementa- r2 r1 r1 r tions of the first gradient of strain theory exist in literature. They In order to obtain results appropriate to a thin shell of transi- are based on the introduction of additional strain degrees of free- tion, we calculate the limit of the latter relation as r , r tend to dom in the spirit of Eringen’s micromorphic approach [28,7,12]. 1 2 r . The first term on the right-hand side vanishes whereas, under That is why a second order micromorphic model is formulated in 0 suitable assumptions of smoothness, the second term is propor- the last section of the present work. Then, an internal constraint tional to the mean curvature, 1/r . Accordingly, Eq. (7) can be inter- must be enforced by means of Lagrange multiplier or suitable 0 preted as the diffuse counterpart of Laplace sharp interface penalization, so that the general micromorphic model reduces to equation. Mindlin’s second gradient of strain model. The compatibility of such higher grade constitutive equations, The article is organized as follows. The links between Kor- formulated within the framework of classical continuum mechan- teweg’s equation and the first strain gradient theory are presented ics, with continuum thermodynamics has been questioned by Gur- in Section 2. Arguments pleading for the necessity of a second tin [17] (see also [27]). Gurtin [17] argued that such higher order strain gradient model are provided in Section 3. A micromorphic constitutive statements can be acceptable only if higher order generalization of Mindlin’s model is finally proposed in Section 4, stress tensors are introduced in addition to the usual Cauchy sim- as the suitable framework for a future finite element implementa- ple force stress tensor. To see that, let us now rephrase Korteweg’s tion of higher order gradient theories. equation within the linear elasticity framework. For that purpose, For the sake of conciseness, the small strain framework is we define the dilatation as the trace of the small strain tensor, adopted. Volume forces are not considered throughout the work. The analysis is limited to the static case. We follow Mindlin’s nota- q $q $q D ¼ trace e ¼ 1 0 ; $D ¼ q ’ ; ð8Þ tion as closely as possible. However, we adopt an intrinsic notation 0 2 q q q0 where zeroth, first, second and third order tensors are denoted by a; a; a; a, respectively. The simple, double and triple contractions within the small strain approximation with respect to a reference g . are written ., :, and ., respectively. In index form with respect to mass density q0. The Korteweg Eq. (4) can therefore be written as an orthonormal Cartesian basis (e1, e2, e3), these notations corre- T ¼ pðDÞ 1 að$DÞ2 1 b$D $D þ cð$2DÞ 1 þd$ $D; ð9Þ spond to Tij ¼ pðDÞdij aD;kD;kdij bD;iD;j þ cD;kkdij þ dD;ij: ð10Þ . a:b ¼ aibi; a : b ¼ aijbij; ga . gb ¼ aijkbijk; ð2Þ If we neglect the quadratic terms in the previous expression, we obtain the linearized Korteweg equation where repeated indices are summed up. The tensor product is de- 2 noted by . The nabla operator with respect to the reference con- T ¼ pðDÞ 1 þcð$ DÞ 1 þd$ $D: ð11Þ figuration is denoted by $. For example, the component ijk of a $ 2 It involves higher order gradients of the dilatation. This suggests is aij,k. In particular, $ is the Laplace operator. Index notation is also used at places to avoid any confusion. For instance, we give the cho- that it could be derived from a strain gradient theory as proposed sen intrinsic and index notations for the second gradient of a scalar by Toupin [29], Casal [4] and Mindlin and Eshel [23] in the early field and of a second rank tensor: sixties. As shown by Toupin, the second gradient of displacement theory is equivalent to the first strain gradient theory. Such theo- $ $q ¼ q;ij e i e j; e $ $ ¼ eij;kle i e j e k e l ð3Þ ries introduce higher order stresses, as advocated by Gurtin [17]. The link between strain gradient theory and capillarity or surface tension in fluids or solids has been recognized by Casal [5]. The 2. Korteweg’s equation and first strain gradient model strain gradient theory can be limited to the gradient of density ef- fects and therefore compared to Korteweg’s equation, as it was The Van der Waals and Korteweg equations are the first at- done by Casal and Gouin [6]. The relation between the stress tensor tempts to introduce capillary effects in a continuum mechanical T in Eq. (4) and the higher order stress measures of strain gradient theory. For an elastic medium, they include not only the effect of theories is discussed in the next section. More recent works have S. Forest et al. / Computational Materials Science 50 (2011) 1299–1304 1301 developed the concept of a fluid with internal wettability based on question of the definition of a first strain gradient elastic fluid has modified Korteweg equations within the strain gradient frame- been only tackled very recently by Fried and Gurtin [14] and Po- work, also called Cahn–Hilliard fluid in [20,2]. dio-Guidugli and Vianello [26]. Clearly, Korteweg and Mindlin’s def- This idea of considering higher gradients of density to describe initions of an elastic first strain gradient fluid differ. Mindlin’s the material behaviour close to free surfaces or interfaces has been formulation is less general than that given by Eq. (11). Following implemented for phase transformations like in water droplets in Mindlin’s definition, the free energy potential of an isotropic second vapor [9]. This represents a diffuse interface model for liquid–gas strain gradient linear elastic fluid takes the form interfaces. k qWðD; $D; $ $DÞ¼ D2 þ a ð$DÞ2 þ b $2D þ b ð$2DÞ2 2 1 0 1 3. Second strain gradient theory 2 þ b2ð$ $DÞ : ð$ $DÞþc1D$ D: ð17Þ
Mindlin [22] claims that a second strain gradient theory is nec- All terms are quadratic except the linear term that involves the essary to account for capillarity and cohesion effects in elastic material parameter b0. The stress–strain relations follow media, instead of the first strain gradient theory reported in the @W @W p ¼ q ¼ kD c $2D; p ¼ q ¼ 2a $D; ð18Þ previous section. We recall here his arguments in order to show @D 1 @$D 1 the fundamental difference between Mindlin’s approach and the @W 2 previous one. The second strain gradient theory is presented and p ¼ q ¼ ðb0 þ c1D þ 2b1$ DÞ 1 2b2$ $D: ð19Þ @ð$ $DÞ contains the first strain gradient model as a special case. It is based on the assumption that the stress state at a material point depends It contains, in general, a self-equilibrated higher order stress b 1, in the absence of any external load. The effective stress on the values of the strain, the first and the second strain gradients p ¼ 0 at that point. Following the method of virtual power [15,1], the vir- then becomes tual work density of internal forces is a linear form with respect to T ¼ ð kD þ 2ða c Þ$2D 2ðb þ b Þ$4DÞ 1 ð20Þ all strain measures: 1 1 1 2 . Tij ¼ ð kD þ 2ða1 c1ÞD;kk 2ðb1 þ b2ÞD;kkllÞdij: ð21Þ ðiÞ . w ¼ r : e þ gS .ðe $ÞþS :: ðe $ $Þ; ð12Þ The corresponding effective stress for the first strain gradient where r is the simple force stress tensor and S; S are the so-called model is obtained by setting b1 = b2 = 0. It can be directly compared g 2 hyperstress tensors as required by the mentioned thermodynamical to Korteweg’s Eq. (11). Both relations have the Laplacian term, $ D, consistency. Mindlin [22] shows that the stresses must fulfill the in common. Korteweg’s equation contains an additional non- following balance equation spherical term, as already mentioned. Mindlin’s second gradient of strain theory introduces an additional contribution of fourth T :$ ¼ 0; with T ¼ r ðS S :$Þ:$: ð13Þ order. g Combining the balance and constitutive equations, the follow- The stress tensor T is an effective stress tensor whose diver- ing fourth order partial differential equation is derived gence vanishes. It is expressed in terms of the stress tensors at 2 2 2 2 all orders. This partial differential equation is accompanied by ð1 l1$ Þð1 l2$ ÞD ¼ 0; ð22Þ three sets of complex boundary conditions that involve the surface that involves two characteristic lengths such that curvature and the normal and tangent derivatives of stress quanti- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ties. For the sake of conciseness, they are not recalled here and the 2 2 kl ¼ a1 c1 ða1 c1Þ 2kðb1 þ b2Þ; i ¼ 1; 2: ð23Þ author is referred to Eqs. (18a–c) in Mindlin [22]. Corresponding i expressions of the boundary conditions for the first gradient of Note that Eq. (22) follows from a first integration of the fifth or- strain theory can be found in [23,15]. The free energy density func- der balance Eq. (13). Accordingly, a constant term should be added tion, Wð e; e $; e $ $Þ, is then a potential from which stresses in the right-hand side of the equation, leading to a homogeneous are computed: pressure field to be superimposed to the solution. The Eq. (22) @W @W @W can be referred as a bi-Helmholtz equation. It can also be derived r ¼ q ; gS ¼ q ; S ¼ q : ð14Þ from a non-local elasticity law as done by Lazar et al. [21]. @ e @ðe $Þ @ðe $ $Þ Let us now consider capillary effects in a spherical homoge- neous material made of a second strain gradient isotropic linear When the terms S and e $ $ are dropped, the first strain elastic fluid. We look for a dilatation field D(r) and equations are gradient theory is recovered. Korteweg’s stress T can then be inter- preted as the effective stress of the first gradient of strain theory, as solved in spherical coordinates. In that specific case, Mindlin shows done in Casal and Gouin [6]. that the dilatation field is then of the form Let us specify the constitutive equations in the case of an elastic C l r C l r D ¼ 1 1 sinh þ 2 2 sinh : ð24Þ fluid. Mindlin considers that, in an elastic fluid, all stress tensors r1 l1 r2 l2 are spherical so that The integration constants C1 and C2 must be determined from r ¼ p 1; S ¼ 1 p; S ¼ 1 p; ð15Þ g the boundary conditions. A key point of the analysis is that the out- er surface of the droplet of initial radius r0 is assumed to be free of where Mindlin’s notations for the stress tensors p, p, and have p traction forces. Mindlin shows that these traction-free boundary been kept. The notation S ¼ 1 p stands for S = d p .Asa ijkl ij kl conditions lead to two equations for the unknowns C1 and C2: result, the effective stress is spherical and reads X2 2 T p p: : 1; 16 Ciri 2ðb1 þ b2Þ ð2a1 c1Þl ðri cosh ri sinh riÞ¼0; ð25Þ ¼ ð $ þ p ð$ $ÞÞ ð Þ i i 1 ¼ X2 the divergence of which vanishes. It is in contrast to Korteweg’s 2 2 medium, see Eq. (11), which can transmit shear stresses. The con- Ci ri 2b2 1 þ þ 2b1 þ c1li sinh ri 4b2 cosh ri ri cept of an elastic fluid has been clearly defined by Noll for simple i¼1 2 materials [30] and leads to a spherical Cauchy stress tensor. The ¼ b0r0; ð26Þ 1302 S. Forest et al. / Computational Materials Science 50 (2011) 1299–1304
where ri = r0/li. When the radius of curvature is large in comparison where Germain’s development has been truncated after the second with the characteristic lengths li (i.e., ri 1), Mindlin derives a rela- order micromorphic terms. In addition to the displacement u, the tion between the mean dilatation of the droplet and its radius of microdeformation v and the second order microdeformation v are g curvature introduced as independent degrees of freedom, together with their Z first gradients, 3 6c T D ¼ DdV ¼ 1 ; ð27Þ 4 r3 b k r p 0 V 0 0 K ¼ v $; K ¼ v $: ð32Þ g g where T can be identified with the surface tension in the Laplace– Young Eq. (1) and takes the value The simple force stress tensor, r, the relative stress tensors, s and gs, the double and triple stress tensors, gS and S, are dual quan- T 0 tities of the strain measures in the virtual work form. They must T ¼ 2 ; 1 þð8b2T0=b0r0Þ fulfill the balance equations, ð28Þ b ðl2 l2Þ with T ¼ 0 1 2 : 0 2 2 2 2 2 2 ðr þ sÞ:$ ¼ 0; S :$ þ s ¼ 0; S :$ þ s ¼ 0; ð33Þ 2 2 g g 2kðl1ðl2 þ c1=k Þ l2ðl1 þ c1=k Þ Þ
The identification between the diffuse and sharp interface mod- in the absence of volume forces. The corresponding three sets of els for surface tension shows that the relevant material parameters boundary conditions are that intervene in the relation are c1, b2 and b0 which are all related to the contributions of the second strain gradient in the elastic po- ðr þ sÞ:n ¼ t ; gS :n ¼ t ; S :n ¼ gt ; ð34Þ tential (17). One should note the key rôle of the initial higher order stress b . Indeed, if b = 0, the system of Eqs. (25) and (26) is homo- 0 0 where t ; t , and t are, respectively, contact simple, double and triple geneous and leads to the trivial solution C = C = 0 and to a homo- g 1 2 tractions. When the microdeformation v is forced to coincide with geneous mass density field. As claimed by Mindlin, the internal the macrodeformation gradient, 1 þu $, by an internal constraint, hyperstresses account for material cohesion which is destroyed and when the second order terms are neglected, the micromorphic at a free surface. theory is known to reduce to the strain gradient theory [10,12]. The corresponding balance equations and solution of the spher- Similarly, when v and v are constrained to coincide with the defor- ical droplet can also be obtained for the first gradient of strain the- g mation gradient, 1 þu $, and its gradient, u $ $, respectively, ory, in the same way. For, the first strain gradient theory can be Germain’s second order micromorphic medium degenerates into formally obtained as a special case of the second strain gradient Mindlin’s second strain gradient continuum. In contrast to strain model by setting that the fourth order stress tensor S vanishes in gradient theories, the general micromorphic models, especially Eq. (12), or, equivalently, that the second characteristic length l 2 the boundary conditions (34), are quite straightforward to imple- vanishes in (22). The Eq. (22) then reduces to ment in a finite element code. Penalty factors or Lagrange multipli- 2a ers can then be introduced to obtain a numerical model for strain ð1 l2$2ÞD ¼ 0; with l2 ¼ 1 ; ð29Þ k gradient media. We illustrate the second order micromorphic model and make which involves only one characteristic length l. There is still a solu- the link with Mindlin’s theory in the case of an elastic fluid at small tion of the form deformation. The degrees of freedom are limited to the displace- C r ment, u, a microdilatation, vD, and second order microdilatation, D ¼ sinh ð30Þ v r l1 K. The associated strains are the dilatation, D, the microdilatation, v v v for the problem of the spherical droplet made of an linear elastic D, its gradient, $ D, the second order microdilatation K and its v isotropic medium. However the condition of vanishing simple and gradient K $. The work of internal forces (31) then reduces to double traction at the free outer surface leads to a homogeneous wðiÞ p vpv vp: v v :vK v : vK : 35 equation so that the integration constant is identified as C = 0. This ¼ D D þ $ D p þ p ð $Þ ð Þ leads to the trivial solution D = 0 and homogeneous mass density The generalized pressure tensors satisfy the balance of general- within the droplet. Accordingly, the description of capillarity effects ized momentum in the form in a linear elastic medium requires the introduction of the second derivatives of the strain in the free energy density. This fact was $p ¼ 0; vp:$ þ vp ¼ 0; v p :$ þ vp ¼ 0: ð36Þ not mentioned in the subsequent works on the links between cap- illarity and strain gradient theories. The corresponding Neumann boundary conditions are
4. Micromorphic approach p p ; vp:n vp ; v :n v 37 ¼ 0 ¼ 0 p ¼ p 0 ð Þ Higher order strain gradient theories can be viewed as special where p , vp and vp are given generalized pressures at the bound- cases of the general micromorphic medium introduced by Germain 0 0 0 ary. Dirichlet conditions can also be stated and correspond to the [16]. The material point is treated as a material volume with a prescription of D,vD and vK at the boundary. small, but finite, size. Additional degrees of freedom are introduced The free energy potential, WðD; vD; vK ; vK $Þ, is used to for- to describe more accurately the relative motion of this particle mulate the state laws with respect to its center of mass. Following the method of virtual work, the work density of internal forces is introduced as a linear @W @W @W form with respect to the degrees of freedom and their first p ¼ q ; vp ¼ q ; vp ¼ q ð38Þ @D @vD @$vD gradient: @W @W vp ¼ q ; v p ¼ q : ð39Þ . . @vK @ðvK $Þ wðiÞðu; v; v; K; KÞ¼ðr þ sÞ : ðu $Þ ðs : v þ s . vÞþS . K þ S :: K; g g g g g g The free energy density is taken as an isotropic quadratic func- ð31Þ tion of its arguments: S. Forest et al. / Computational Materials Science 50 (2011) 1299–1304 1303
v v k k a1 b found to vanish so that the material density remains constant qW ¼ D2 þ ðD vDÞ2 þ $vD:$vD þ ð$vD vK Þ2 2 2 2 2 and uniform. Only the consideration of the second order microdila- v a 2 b tation K can lead to a non trivial solution due to the initial cohe- þ ðÞtraceðvK $Þ þ ðvK $Þ : ðvK $Þ 2 2 sion stresses associated with the material parameter b0, in the c þ ðvK $Þ : ð$ vK Þ same way as in Mindlin’s original theory. 2 v v þðb0 þ c1D þ c2 DÞtraceð K $Þð40Þ v v In the previous expression, the parameters k and b can be re- 5. Conclusion garded as penalty terms ensuring that the microdilatations vD and v K are sufficiently close to the macrodilatations D and $vD, respec- The continuum description of capillarity effects in elastic bodies tively. The constitutive equations follow, is possible based on the introduction of higher order gradients of the strain tensor. The Korteweg equation can be incorporated into p ¼ kD þ vkðD vDÞþc traceðvK $Þ; 1 the framework of a first strain gradient theory. It can be used to v v v v p ¼ kðD DÞþc2traceð K $Þ; ð41Þ model capillarity effects at the interface between two phases of v v v v v v v v v p ¼ a1$ D þ bð$ D K Þ; p ¼ bð$ D K Þ; ð42Þ different densities like droplets in vapor, particularly when it is v v v embedded in a phase field simulation. However we have shown, p ¼ðb0 þ c1D þ c2 D þ atraceð K $ÞÞ 1 following Mindlin’s arguments, that the linear elastic isotropic first v v þ b K $ þ c$ K ð43Þ strain gradient theory is not sufficient to describe internal strains and stresses that develop close to free surfaces. Their existence re- The stress tensors vp, vp, vp and v p can be eliminated from the quires initial third order stresses accounting for cohesion forces. previous relations by means of the balance Eqs. (36) in order to ob- The cohesion material property in an isotropic elastic second gra- tain the expression of the effective pressure: dient of strain medium is fully characterized by a single parameter 2 2v v p ¼ kD þ c1$ D ða1 c2Þ$ D þðc1 þ c2Þ traceð K $Þ b0 that can be linked to the surface tension when going to the sharp interface limit. Although the explicit example treated in this work þða þ b þ cÞ$2 traceðvK $Þð44Þ was dedicated to elastic fluids, Mindlin’s second strain gradient When the following constraints are enforced: theory exists for solids at small strains. Also, the balance equations v v of the second order micromorphic model where given for elastic D D; K $D; ð45Þ solids at small strain. the effective pressure (44) becomes Such a situation is encountered for instance in single crystalline nano-particles, as computed from molecular statics simulations 2 4 p ¼ kD ða1 2ðc1 þ c2ÞÞ$ D þða þ b þ cÞ$ D: ð46Þ [11]. Indeed, in such small atomic aggregates, the lattice parameter field is strongly inhomogeneous due to the small distance between This expression can then be identified with the effective pres- the particle core and its free surface. A surface tension model sure (20) in Mindlin’s second gradient of strain theory. Therefore, would be inappropriate since, at that size, the capillary effect is the constrained second order micromorphic model coincides with not confined to an infinitesimal surface. That is why Mindlin’s sec- Mindlin’s theory. ond strain gradient theory seems to be suitable. Corresponding fi- The model can now be applied to the problem of the elastic nite element computations of nano-particles based on the spherical droplet with free outer boundary at r = r . For the sake 0 proposed second order micromorphic model could then be per- of conciseness, the form of the solution is given here only for the formed. The obtained strain distribution could be directly com- first order microdilatation model, i.e. dropping the terms involving pared to atomistic computations. In particular the higher order vK (see [13]). The balance and constitutive equations reduce to the elasticity moduli could be identified in that way from the discrete following system of equations computations, as Mindlin did for the one-dimensional atomic v v v v 2v kD þ kðD DÞ¼0; kðD DÞþa1$ D ¼ 0: ð47Þ chain. Mindlin’s theory and the general micromorphic simulations could also be used to represent grain boundary stresses in nano- Elimination of dilatation leads to the following partial differen- crystals [24], or in nano-objects like wires and layers [31]. This will tial equation for the microdilatation: require extension of the theory to anisotropic cases. a ðk þ vkÞ Finite element simulations based on strain gradient theories are vD 1 $2vD ¼ 0: ð48Þ kvk quite challenging, especially because of the complex boundary conditions. In contrast, Eringen’s micromorphic theory is based For the elastic droplet, a solution of the form on the introduction of independent deformation degrees of free- C r dom and their gradients [10]. It has been shown that the corre- vD ¼ sinh ð49Þ r vl sponding boundary conditions are significantly less involved. The numerical implementation merely relies on the introduction of can be worked out. The characteristic length is found to be additional degrees of freedom and the computation of their first a k þ vk gradient. Lagrange multipliers or penalization method can then vl2 ¼ 1 : ð50Þ k vk be used to retrieve the strain gradient formulation. Such an imple- mentation has already been done for the first order micromorphic The macrodilatation turns out to be proportional to the model in [28,7]. It remains to be done for the second order micro- microdilatation morphic theory presented in this work. vk D ¼ vD: ð51Þ k þ vk When the penalty coefficient vk tends to infinity, the macro and Acknowledgement microdilatation coincide and the previous length becomes l2 = A/k.
Mindlin’s result (23), specified for ci = bi = 0, is retrieved in that This work is part of the NANOCRYSTAL project ANR-07-BLAN- way. The outer boundary being free of forces, the constant C is 0186. Financial support of ANR is gratefully acknowledged. 1304 S. Forest et al. / Computational Materials Science 50 (2011) 1299–1304
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