On the Effective Youngs Modulus of Elasticity for Porous Materials
the bound concept using variational methods in which the averaged values result from the principles of the minimum of elastic potential and complementary energy.
the model concept using direct methods in which the averaged stresses and distortions are calculated with the aid of Hooke’s law.
Based on the model concept the present paper treats the problem of the effective elastic modulus of porous materials, considering materials with closed porosity as a limiting case of two-phase composite materials.
which, as a limiting case, exactly holds true for porous materials. — The two phase material then is subdivided into elementary cells (finite elements), where the elementary cell consists of a cube of given elastic materials in which spheroidal inclusion in any orientation are discontinuously embedded in a matrix phase. The mean stresses and strains are calculated for this elementary cell by dividing it into small, disjunct prisms. An effective modulus of elasticity is approximately calculated for each prism. The final effective modulus of elasticity is determined on the basis of a new averaging over all prisms. The resulting analytical formula for the effective Youngs modulus of elasticity depends on the elastic moduli of the phases on the phase concentration as well as on the axial ratio of the spheroid and its orientation. If in this formula the modulus for the inclusion is assumed as equal to zero, then the effective modulus of elasticity of porous materials is obtained as a function of porosity and pore structure. The second part of the paper, published elsewhere in due course , is concerned with the theory of special and limiting cases of the model equation and a comparison between experimental and calculated Youngs moduli of elasticity. The comparison is made for porous glass and porous calciumtitanate ceramic and — summarizing — for experimental data of sintered metals and ceramics with spherical porosity taken from the literature.
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