Computational Materials Science 50 (2011) 1299–1304 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci 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- $q, in the form lently, third gradient of displacement theory is in fact needed to 2 2 describe, in a continuous manner, capillarity and cohesion effects T ¼pðqÞ 1 Àað$qÞ 1 Àb$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 À 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 À Thh ¼bq À 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 À dq Trrðr2ÞTrrðr1Þ¼2d À À 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 À 0 ; $D ¼ q ’ ; ð8Þ tion as closely as possible. However, we adopt an intrinsic notation 0 2 q q q0 where zeroth, first, second and third order tensors are denoted by a; a; a; a, respectively. The simple, double and triple contractions within the small strain approximation with respect to a reference g . are written ., :, and ., respectively. In index form with respect to mass density q0. The Korteweg Eq. (4) can therefore be written as an orthonormal Cartesian basis (e1, e2, e3), these notations corre- T ¼pðDÞ 1 Àað$DÞ2 1 Àb$D $D þ cð$2DÞ 1 þd$ $D; ð9Þ spond to Tij ¼pðDÞdij À aD;kD;kdij À bD;iD;j þ cD;kkdij þ dD;ij: ð10Þ .