The shape of Maxwell's equivalent inhomogeneity and ‘strange’ properties of regular polygons and other symmetric domains

Abstract

The Maxwell concept of equivalent inhomogeneity is reexamined in the context of elastic composite, porous or micro-cracked materials of periodic structure. It is demonstrated that accurate estimates for the effective elastic properties can be obtained by the modified Maxwell approach that employs the proper shape of the equivalent inhomogeneity and explicitly accounts for the geometry of the cluster and for the interactions between its constituents. In addition, it is shown that regular polygonal and some other symmetric inhomogeneities possess 'strange'and remarkable properties. Under the action of uniform far-field loads, the averages of certain combinations of stresses inside these inhomogeneities are constant and coincide with those for a circular inhomogeneity.

Comparison of full field and single pore approaches to homogenization of linearly elastic materials with pores of regular and irregular shapes

“…The solutions based on conformal mapping were utilized by Zimmerman (1986), Zimmerman (1991), Kachanov et al (1994), Jasiuk et al (1994), Tsukrov and Novak (2002), Ekneligoda and Zimmerman (2006), Ekneligoda and Zimmerman (2008), Zou et al (2010), Mogilevskaya and Nikolskiy (2015) . In 3D, Garboczi and Douglas (2012) presented a procedure to approximate bulk and shear elastic contribution parameters in the case of randomly oriented inhomogeneities shaped as blocks.…”

On ‘strange’ properties of some symmetric inhomogeneities

“…The coefficient in the first term of expansion of equation (6.1) represents the resultant force acting on the boundary of the inhomogeneity, while the coefficient in the second term (dipole coefficient) of that expansion is equal to [1] …”

“…(ii) A part of L is a straight segment with the beginning at the point a = a 1 + ib 1 and the end at the point b = b 1 + ib 1 .…”

“…Under the action of uniform far-field stresses, the averages of the fields inside the inhomogeneities preserve the structure of the far-field loads: hydrostatic far-field load results in hydrostatic average stresses within the inhomogeneity and deviatoric far-field load produces deviatoric average stresses. In [1], it was suggested that those properties might be related to the properties of the corresponding Eshelby inclusions (inhomogeneity whose properties are the same as those of the matrix, see [2][3][4]). The latter properties were studied in [5][6][7][8][9][10][11][12][13][14] where they were referred to as 'strange'.…”

“…Recently, it was numerically discovered in [1] that regular polygonal and other symmetric inhomogeneities possess some remarkable properties. Under the action of uniform far-field stresses, the averages of the fields inside the inhomogeneities preserve the structure of the far-field loads: hydrostatic far-field load results in hydrostatic average stresses within the inhomogeneity and deviatoric far-field load produces deviatoric average stresses.…”