The Infona portal uses cookies, i.e. strings of text saved by a browser on the user's device. The portal can access those files and use them to remember the user's data, such as their chosen settings (screen view, interface language, etc.), or their login data. By using the Infona portal the user accepts automatic saving and using this information for portal operation purposes. More information on the subject can be found in the Privacy Policy and Terms of Service. By closing this window the user confirms that they have read the information on cookie usage, and they accept the privacy policy and the way cookies are used by the portal. You can change the cookie settings in your browser.
The constitutive behavior of ductile materials containing two classes of voids with different size scales is examined by introducing an intermediate material containing small voids and a macroscopic material composed of the intermediate material and large voids. The macroscopic constitutive equations are developed based on a lower bound approach proposed by the present author. The results are then compared with available solutions and the finite element calculation. Good agreements are obtained.