## 1 Introduction and modeling  
### Introductory remarks  
A wide class of theoretical and technological problems are related to the mechanical description and the practical use of bodies endowed with a large number of microcracks scattered throughout the volume. Reference is made to [7] for a detailed review of these problems. When microcracks are dilute in the sense that the interactions between them are not prominent and also the microcrack distribution is approximately periodic, standard homogenization procedures can be advantageously applied to describe the influence of the microcracks on the gross behavior, see e.g. [14]. Conversely, when microcracksare dense in a way that the interactions between them are prominent and also their distribution is not homogeneous, non-negligible gradients are involved and standard homogenization procedure cannot be used in standard way. Precisely the microcracked body has to be considered strictly as a complex body and its description falls naturally within the setting of multi-field theories representing complex bodies. Here we pay attention to bodies with a dense population of microcracks scattered throughout a \"soft\" matrix of material. In particular we try to describe the influence on the gross behavior of the deformation of microcracks, considered both as sharp defects (pieces of planes not interpenetrated by interatomic bonds) and/or as elliptic voids with one dimension very small with respect to the others. The following treatment is restricted to elasticity, since any irreversible growth of microcracks (damage evolution) is not accounted for. The basic aim of the paper is to represent in some way sub-structural interactions due to microcrack changes in a scattered sense. Since we consider dense populations of microcracks we follow here a multi-field model of microcracked bodies that has been already formulated in a series of papers (see [9] and references therein). This approach seems to predict verifiable non-usual phenomena, namely strain localization that straightforwardly arise from the adopted model notwithstanding the linear elastic regime. The model takes into account two vectorial components of a global displacement field, namely macro-displacements and micro-displacements. The constitutive relationships described in Section 1.2 allow to link their gradients to the relevant macro and micro stress fields.  
The numerical investigations discussed in [9] and [10] refer to displacement-based finite element methods that adopt post-processing techniques to derive the stress fields. The present paper proposes instead ad hoc mixed formulations that interpolate independently stresses and displacements. The mixed scheme is adopted both at the macro and micro level, thus providing a numerical framework that allows a direct discretization of displacements and stresses that arise at the two scales. Suitably extended Hellinger-Reissner principles of primal and dual type are derived to this purpose in Section 2.. Two different solutions are herein discussed. The first adopts displacements as main variables, while stresses play the role of Lagrangian multipliers. The second, the so-called \"truly-mixed\" formulation [2], implements a dual approach where the stresses are main unknowns of the elastic problem. Peculiar attention is paid to the interactions between a macrocrack and diffuse microcracks. Details of the implementation are discussed in Section 2.3., while two numerical examples are illustrated in Section 3.. The first is devoted to assess the capabilities of the proposed method to predict localization phenomena not only in the micro displacement field, as already shown in [9], but also in the micro stress one. Afterwards, the second example allows to discuss the effects of the presence of a single macrocrack within the diffuse microcracks peculiar to the adopted multifield model. Particular attention is devoted in this case to the convergence features of the proposed method. Section 4 concludes the paper introducing the ongoing developments.