On the Effective Youngs Modulus of Elasticity for Porous Materials

Part I: The General Model Equation
  • Petrisor Mazilu
  • Gerhard Ondracek
Conference paper
Part of the Lecture Notes in Engineering book series (LNENG, volume 59)


The determination of the effective moduli of elasticity represents one of the most basic tasks of materials science on composite materials nowadays. There are two concepts used to date for this determination, which are
  • 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.

Whereas the bound concept provides upper and lower bounds, between which the effective Youngs moduli of elasticity have to be expected, the model concept results in single approximate values via an effective Hooke’s tensor.

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.

The derivation in this theoretical first part starts by assuming a two-phase material with matrix-type microstructure, where the two phases behave isotropically. In order to simplify the calculations it is presumed that the following proportion is valid between the elastic moduli of the inclusions (v D, ED) and of their matrix (v M, EM):
$${}^VD{}^EM = {}^VM{}^ED$$

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 [2], 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.


Porous Material Elementary Cell Effective Modulus Effective Elastic Modulus Complementary Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ondracek G., Reviews on Powder Metallurgy and Physical Ceramics 3–3 /4 (1987) 205Google Scholar
  2. [2]
    Mazilu P., Ondracek G., Int. Conf. on the Mechanics, Physics and Structure of Materials, Thessaloniki August (1990) in printGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • Petrisor Mazilu
    • 1
  • Gerhard Ondracek
    • 2
  1. 1.Petersenstr. 30Institut für Umformtechnik Technische HochschuleDarmstadtGermany
  2. 2.Mauerstr. 5 Institut für Gesteinshüttenkunde — Glas, Keramik, Binde- und VerbundwerkstoffeRheinisch-Westfalische Technische HochschuleAachenGermany

Personalised recommendations