on surface elasticity theory for plane interfaces

9
ISSN 1029-9599, Physical Mesomechanics, 2014, Vol. 17, No. 1, pp. 3038. © Pleiades Publishing, Ltd., 2014. Original Russian Text © R.V. Goldstein, V.A. Gorodtsov, K.B. Ustinov, 2013, published in izicheskaya Mezomekhanika, 2013, Vol. 16, No. 4, pp. 7583. On Surface Elasticity Theory for Plane Interfaces R. V. Goldstein*, V. A. Gorodtsov, and K. B. Ustinov Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia * [email protected] Received July 3, 2013 AbstractA closed system of surface elasticity equations was derived in terms of surface quantities defined as integrals of respective excess bulk quantities normal to the interface. The equations were consistently linearized for the case of small strains. It is shown that these equations are more general than the Shuttleworth equations. Equa- tions of this type were also derived for the particular case of an interface formed by a thin layer with constant prop- erties. The derived equations were used to consider bending of a plate under pressure applied to both sides. DOI: 10.1134/S1029959914010044 Keywords: surface elasticity, interface, plate compression 30 1. SURACE QUANTITIES. SURACE KINEMATICS. SURACE CONSTITUTIVE RELATIONS. GENERALIZATION O SHUTTLEWORTH EQUATIONS OR INTERACE INTERACTIONS ollowing the definitions [1], the surface density s (, ) g xy of an arbitrary quantity g at a certain point 0 0 ( , ) x y of the surface z(x, y) is taken as an integral of excess bulk density of the respective quantity g(z) along the surface normal through the point considered (ig. 1). Although we restrict ourselves to small strains, all reasoning below follows the Lagrangian description, i.e., all quantities are related to a material surface site. We consider a rather general case of the interface which is the interface between phases A and B (phase Ava- cuum interface or phase Bvacuum interface can be considered as a particular case when the quantity under study is equal to zero or infinity). Strictly speaking, the integration limits should thus cover the entire examined region. However, in fact, if the excess of the quantity un- der study decreases rather rapidly with distance from the surface, the integration limits can be reduced to certain small finite sizes A z and B z for which the values of the quantity are A g and , B g respectively: s 0 0 (, ) (, , )d (, ) (, ), , . B A z A A z B B A A B B g xy gxyz z hg xy hg xy h z z h z z = = (1) ormally, in surface density expression (1) for an ar- bitrary quantity, the choice of the interface point coordi- nate 0 z remains arbitrary. This arbitrariness vanishes if the interface position is chosen from consideration exter- nal to the problem studied. In this context, let us consider the expression for va- riation in the surface energy density: s () ( )d . B A z A A B B ij ij ij ij A ij ij B z W z z z h h δ = σ δε −σ δε −σ δε (2) The surface energy, in this case, need not be positive definite. There should be positive definite energy of the solid as a whole, rather than the excess associated with the surface and defined by Eq. (2). In other words, posi- tive definiteness is required of the energy in the domain within the integration limit in Eq. (2), i.e., of the integral term in this formula. Similarly, there is no constraint on the surface elastic moduli when defined by Eq. (1). Here, a remark should be made. In considering the solidvacuum interface or, in some approximation, the solidgas interface, the choice of an external boundary of the solid is more arbitrary. Let in the parti- cular case in ig. 1 the solid to the left of the point 0 z is absent. Then, with the boundary passing through the point 0 , z we arrive at the description given above, and with the boundary passing through the point , B z which may be preferable not at the maximum of g(z) like in ig. 1 but, e.g., at the minimum of this function (deno- ting, in our case, the elastic modulus), an ordinary con- straint takes place requiring positive definiteness of the surface moduli. The stress and strain tensors, ij σ and , ij ε are conve- nient to decompose into perpendicular and parallel com-

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Page 1: On surface elasticity theory for plane interfaces

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