Bu çalışmada hasarlı malzemeler mikrogermeli ortam teorisi ile modellenerek tanımlanmış ve ortaya çıkan bilinmeyen malzeme katsayıları, kompozit malzemeler ve hasarlı malzeme arasında kurulan analoji ile hesaplanmaya çalışılmıştır. Bu amaç için kullanılan homojenleştirme yöntemlerinin çoğunda ilgili Eshelby tansörlerinin bulunması öncelikli problemlerden biridir. Bu nedenle bu çalışmada da Eshelby tansörlerinin küresel katkı maddeleri içeren mikrogermeli ortam için bulunması problemi ele alınmıştır. Bu amaca ulaşmak için öncelikle mikrogerme teorisi ile modellenen bir ortamın yerdeğiştirme, mikrodönme ve hacimsel mikrogenleşme büyüklükleri için temel çözümler verilmiştir. Bu temel çözümler kullanılarak katkı maddelerinin varlığından kaynaklanan elastik alanlar Green fonksiyonları yardımıyla elde edilmiştir. Son olarak bu elastik alanların mikrogermeli ortamın bünye denklemlerinde kullanılması ile klasik teoride verilen Eshelby tansörleri mikrogermeli ortama genelleştirilmiştir. Klasik elastisitede benzer problem için tek bir Eshelby tansörü elde edilirken, mikrogermeli ortam için dördü dört, ikisi iki boyutlu ve biri de skaler olmak üzere toplam yedi Eshelby tansörü elde edilmiştir. Klasik Eshelby tansörü elipsoidal katkı maddele-ri içinde homojen olmasına rağmen, mikrogermeli ortam için elde edilen Eshelby tansörleri küresel katkı maddeleri içinde bile homojen değildir. Çalışmada, mikrogermeli ortamın mikrodönme ile ilgili malzeme sabitlerinin sıfır alınması durumunda mikrogenleşen ortam için Eshelby tansörlerinin elde edilebildiği gösterilmiştir. Ayrıca mikrogermeli ortamın mikrogenleşme ile ilgili malzeme sabitlerinin sıfır alınması durumunda mikropolar ortamın Eshelby tansörlerine ve her iki grup mikro ortamın malzeme sabitlerinin sıfır alınması durumunda ise klasik elastisite için elde edilen Eshelby tansörlerine indirgendikleri de görülmektedir.

Eshelby tensors for the materials modelled by micropolar theory including spherical inclusions are given by Cheng and He (1995). Later on similar work is carried on for microelongation theory by Kiris and Inan (2005). It is assumed that the material particles undergo only microrotation in micropolar theory and volumetric microelongation in microelongation theory in addition to the classical components. Although it is more meaningful to model damaged materials with microelongation theory rather than micropolar theory, because growth of the voids reflects the damage phenomena better than their rotation, both theories are not good enough to model the damaged materials. The reason of the improperness of modelling also with the microelongation theory is that the voids volumetric ratio is very low and thus modelling with microelongation theory accounts only volumetric micro elongation of the whole body. Therefore, to model the damaged materials with microstretch theory is a better approach (Kiris and Inan, 2006).In the present work, the damaged materials are modelled by the microstretch theory, developed by Eringen (1990) by assuming that the material particles can make both microrotation and volumetric microelongation, in addition to the classical bulk deformation. Then, the concept of Eshelby tensors which constitutes the basis of most of the homogenization methods, is extended to microstretch theory to use the analogy between damaged materials with composites which is proposed by Inan (1998). Then we present the fundamental solutions for the displacement and the microrotation vectors and the microelongation scalar in static case, and extend the “eigenstrain” and “microeigenstrain” concepts to the microstretch theory. With the help of these solutions and Green’s functions technique, the elastic fields of the material defined by microstretch theory due to the existence of the spherical inclusions are calculated. By substituting these elastic fields to the strain expressions of the microstretch theory, Eshelby tensors for microstretch medium having spherical inclusions, are obtained.In this work, seven Eshelby tensors, as four of them in four dimension, two in two dimension and one scalar are obtained for the microstretch theory, as oppose of the four Eshelby tensors all in four dimension for micropolar theory, four Eshelby tensors as one in four dimension, two in two dimension and one scalar for microelongation theory and lastly one Eshelby tensor in four dimension for the classical theory of elasticity. Furthermore, Eshelby tensors obtained for microstretch theory reduce to Eshelby tensors for micropolar theory, while all microelongational material constants are taken as zero, and correspondingly reduce to Eshelby tensors for microelongation theory, while all micropolar material constants are taken as zero, and finally reduce to the classical Eshelby tensors while all micro material constants are taken as zero. These tensors are not homogeneous in even spherical inclusions as of micropolar and microelongation theories, unlike their classical forms. Besides, the results of the microstretch theory show that, the field quantities related with both micropolar and microelongation theories have contribution to Eshelby tensors corresponding to “eigenstrain” due to macro deformations, but it is observed that the field quantities related with microelongation theory do not have any contribution to Eshelby tensors corresponding to “microeigenstrain” due to microrotation. And in the same way the field quantities related with micropolar theory also do not have any contribution to Eshelby tensors corresponding to “micro-eigenstrain” due to micro elongation.The main objective of obtaining Eshelby tensors for these micro theories is to determine the effective properties of the materials modelled with micro elastic theories by use of these tensors in the homogenization methods. Although our starting point was to use these tensors in damage mechanics, all the obtained results are quite general. Namely, Eshelby tensors obtained for the micro elastic theories may be used to determinate effective properties of composite materials as well as the damaged materials. Since Eshelby tensors obtained for all the micro theories are not homogeneous, one can not express the homogenization process analytically, unlike the Eshelby tensors obtained for classical case. It is expected that the homogenization methods like Mori-Tanaka’s method can be extended to these micro theories by taking average processes and integrating over the inclusions.