Eshelby tensor

Note that Co is the equilibrium concentration associated with a flat interface, B is a constant term, the two jump terms correspond to the jump in the Eshelby tensor across the interface and the final term corresponds to the Gibbs-Thomson term and depends upon the interfacial curvature k. 

Once the inclusion assembly has been constmcted, the homogenised stiffness matrix of the composite is calculated as follows. First, calculate Eshelby tensors S,- for the inclusions [97,98] in local coordinates CS,. Transform the result in the global coordinate system GCS. Then calculate the strain concentration tensors for all the inclusions ... 

As the configurational or material forces [4, 87] (note that the density of chemical potential pg has a pressure dimension). An analog of chemical potential is the Eshelby tensor (of chemical potential) f defined as (F = (F ) )... 

Note, that if stress is reduced to pressure P,T = —PI, (usual in fluids) this definition gives the classical result (3.203) F = gl, see (3.199). The Eshelby tensor, e.g. gives the condition of phase equilibria (Maxwell relation—equality of chemical potentials (2.116) in fluid phases), namely equality of f[n on both sides of equilibrated solid phases (n is the normal to phase boundary) and may be also used to describe surface phenomena, dislocations, etc. [1, 4, 87]. Eshelby tensors may also be defined in mixtures [2, 3]. 

This result (3.239) may be generalized for Eshelby tensor (generalization of chemical potential, e.g. for solids, see Rem. 38) as... 

In Sect. 4.6 we shall see that this is the usual definition, cf. (4.194). Its density has a dimension of force in some theories, using partial Eshelby tensors [88] as generalization of (4.161) (cf. Rem. 38 in Chap. 3), the configurational or material forces are introduced instead [127, 128]. 

For a special case that the properties of the matrix are isotropic matrix case, closed form expressions of the Eshelby tensor have been obtained by Tandon and Weng (1984). In general, explicit evaluation of Eij i is a difficult problem, but it can be calculated numerically (Gavazzi and Lagoudas 1990). 

When the matrix is isotropic and the inclusion is spheroidal with the symmetric axis identified as xi, the non-vanishing components of the Eshelby tensor were derived by Tandon and Weng (1984) as... 

Where is the Eshelby tensor (Tucker and Liang 1999), I is the fourth-order unit... 

If the fiber aspect ratio is set to one (i.e., spherical particles), one notes that there is a mathematical singularity in the Eshelby tensor and the Tandon and Weng explicit equations for short fibers cannot be used. However, in such a case, there are some simplifications in the tensor and particularly simple... 

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