On the Modeling of Surface and Interface Elastic Effects in Case of Eigenstrains
The constitutive equations of interface elasticity in case of eigenstrains are obtained in terms of interface (surface) values defined as integrals of the excesses of the corresponding volumetric values over the normal to the interface. The equations are
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Abstract The constitutive equations of interface elasticity in case of eigenstrains are obtained in terms of interface (surface) values defined as integrals of the excesses of the corresponding volumetric values over the normal to the interface. The equations are consistently linearized, which corresponds to the case of both elastic strains and eigenstrains being small. It is shown that the obtained equations possess more general form then Shuttleworth equations. The obtained type of equation was confirmed by considered example: an interface formed by a thin layer of constant properties. It was also shown that the type of energetic restrictions on the surface elastic constants may depend essentially on the definition of the position of the surfaces.
1 Introduction The state of matter at or near the surface or interface is generally different from the state of the matter in the bulk. Therefore to describe behavior of finite body (liquid or solid) the specific surface values (such as energy, stress and so on) are introduced in addition to bulk specific values [1]. Historically efforts of studying the surface effects were devoted to liquids rather than solids. That is because the influence of surface effects is much more pronounced in liquids than in solids. However with reducing the sizes of the objects under consideration the relative amount of the matter near the surface increases, raising the role of the surface effects. The present article is devoted to peculiarities in describing surface and interface effects in elasticity. K. B. Ustinov (B) · R. V. Goldstein · V. A. Gorodtsov A. Ishlinsky Institute for Problems in Mechanics RAS, Prospect Vernadskogo 101, Moscow 119526, Russia e-mail: [email protected] R. V. Goldstein e-mail: [email protected] V. A. Gorodtsov e-mail: [email protected] H. Altenbach and N. F. Morozov (eds.), Surface Effects in Solid Mechanics, Advanced Structured Materials 30, DOI: 10.1007/978-3-642-35783-1_13, © Springer-Verlag Berlin Heidelberg 2013
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Fig. 1 Distribution of volumetric value near interface
2 Surface Values Let us first introduce the definition of the surface value. Following [1], the surface density g s (x, y) of any value g at any point (x0 , y0 ) of the surface z(x, y) between two phases A and B (the interface media-vacuum forms a particular case) is understood as the integral of the excess of the volumetric density of the corresponding value g(z) over the normal to the surface intersecting it at the point under consideration (Fig. 1).
zB
g (x, y) ≡ s
g(x, y, z) dz − h A g A (x, y) − h B g B (x, y),
zA
h A = (z 0 − z A ), h B = (z B − z 0 )
(1)
Here g A and g B are magnitudes of the value in question in phases A and B, respectively. There remains, however, a variety in the choice of the position of the boundary, z 0 , unless it is prescribed externally. The variation of the density of the surface energy is δW = s
zB zA
σi j (z)δεi j (z) dz − σiAj δεiAj h A − σiBj δεiBj h B
(2)
It is followed directly from here that the surface energ
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