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If there is a surface energy associated with a material, and the material has a very large surface area, a change in that surface energy can cause a measurable expansion or shrinkage of the material. This has been seen in Vycor glass, at very low relative partial pressures of an absorbed gas [39,40]. As the partial pressure of the absorbed gas increases, the layer of gas molecules on the internal surface of the Vycor increases in thickness, and thus lowers the specific surface free energy. This lowering in surface energy allows an expansion to take place, which trades volume elastic energy for surface free energy. This expansion is linear in the change in surface free energy [41,39,40].
If a 3-D model of the material is available in a digital form, then a surface energy can be applied in finite element form. All surfaces are pixel faces, and are initially flat. One then must write, for general small strains of the pixel face, what the new area is in terms of the nodal displacements. One can then linearize this when the surface distortion is small (usually the overall strains are a percent or much less), to give an energy that can be added to the elastic energy, which is linear in the displacements.
The key to deriving a surface energy, γSA, where SA is the surface area of the digital image found by counting pixel faces and γ is the specific surface free energy, is knowing that each pixel surface is originally flat, before any strains are generated. Consider the z=0 face (1-2-3-4 face) of a pixel labelled like that shown in Fig. 1. The coordinates of the number 1 node are (x1 + u 1, y1 + v1, z1 + w1), where (x1, y1, z1) are the coordinates before any strain has occurred, and (u1, v1, w1) are the x, y, and z displacements. There are similar formulas for the other nodes. No matter how the pixel face has been strained (assuming small distortions), the area of the face can be defined as the area of the triangle 1-2-3, and the triangle 2-3-4, since any three points are co-planar. The area of these two triangles can be written as one half the sum of the magnitude of the cross-products of the vectors making up their sides. The resulting formula contains the square root of the squares of various combinations of the differences of the nodal coordinates. The formula is somewhat simplified for this face by remembering that z1 = z2 = z3 = z4 = 0. Making the assumption of small differences between nodal displacements (small strains), the area of this face can be reduced to a linear form,
Consider a strain of 0.1 in the x-direction only. If the pixel has node 1 at the origin, then u1 = u4 = u5 = u8 = 0, u2 = u3 = u6 = u7 = 0.1 r, and all the other displacement components are zero. Computing the above equations with these displacements gives SA = 1.1 r2 for the z = 0 face, 1.1 r2 for the y = 0 face, and SA = r2 for the x = 0 face, as expected.
A global vector can be built out of these displacements, which will also contain terms for the macrostrains, via the periodic boundary conditions, which are picked up at the boundaries. This vector is constant with respect to the macrostrains, and so will simply add to the gradient vector, and will not come into the subroutine DEMBX at all, since only the second derivative of the energy is used in that subroutine.
A simple example will suffice to show the accuracy of this technique [41]. Consider a solid block of material, with a dilute volume fraction c of spherical holes of radius R. In the dilute limit the holes can be treated separately, so the analytical problem can be carried out for a single hole. The solid material has bulk modulus K and shear modulus G, and is at zero strain. Now suddenly add a surface energy per unit area γ to the surface of the hole. The material will shrink in order to reduce the surface area of the hole and thus the surface energy stored. The reduction in surface energy from the shrinkage will be exactly balanced by the increase in volume stored elastic energy, giving the result:
For a 15 pixel diameter spherial hole in a 403 unit cell, the agreement with the exact result for the coefficient of c in eq. (78) was 4.7% [41]. Better agreement would be obtained using a larger system.
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