1 Predicting failure initiation in nonlinear composite materials, often referred to as metamaterials, is a 1 fundamental challenge in nonlinear solid mechanics. Microstructural failure mechanisms encompass fracture, 2 decohesion, cavitation, compression -induced contact, and instabilities, affecting their unconventional static and 3 dynamic performance s. To fully take advantage of these materials, especially in extreme applications, it is 4 imperative to predict their nonlinear behavior using reliable, accurate, and computationally efficient numerical 5 methodologies . This study presents an innovative nonlinear homogenization -based theoretical framework for 6 characterizing the failure behavior of periodic reinforced hyperelastic composites induced by 7 reinforcement /matrix decohesion and interaction between contact mechanisms and microscopic instabilit ies. 8 Debonding and unilateral contact between different phases are incorporated by employing an enhanced 9 cohesive/contact model, which features a special nonlinear interface constitutive law and an accurate contact 10 formulation within the context of finite strain continuum mechanics. The theoretical formulation is 11 demonstrated using periodically layered composites subjected to macroscopic compressive loading conditions 12 along the lamination direction. Numerical results illustrate the way in which debonding phenomena, in 13 conjunction with fiber microbuckling, may influence the critical loads of the examined composite solid . The 14 sensitivity of the results obtained through the proposed contact -cohesive model at finite strain with respect to 15 its implementation is also explored. 16 1 1.Introduction 1 The onset of failure prediction in finitely strained nonlinear composite materials, such as fiber or particle - 1 reinforced elastomers with periodic microstructures frequently embedding voids or imperfections , is a 2 challenging problem of fundamental interest in nonlinear solid mechanics. Since their microstructures can be 3 tailored to obtain unusual as well as enhanced static and dynamic properties , and often incorporat ing 4 multifunctional properties arising from coupling between mechanical and electrical, thermal and/or chemical 5 characteristics (see [1,2] , for instance), they are frequently referred to as metamaterials and adopted in a 6 2 Preprint version of AAM for the publication on Phil. Trans. R. Soc. A. considerable number of advanced technological applications in large deformations contexts . In the last decade , 7 the role of additive manufacturing in metamaterial research has gained significant importance enabling the 8 production of digitally designed complex microstructures by using metal, polymeric , ceramic , biocompatible, 9 and bioinspired materials with highly versatile performances [3–5]. 10 The adoption of biocompatible and bioinspired meta materials is often associated with medical applications , but 11 not limited to, where highly deformable biocompatible materials are typically used for prosthetics, implantable 12 medical devices, artificial tissues, and more. Biocompatibility is important in these applications but the ability 13 of these materials to be highly deformable is essential since the materials need to conform to the shapes and 14 movements of the human body. For instance, in implantable medical devices or prosthetics, the adopted 15 meta materials must be capable of withstanding large deformations and mechanical stresses without undergoing 16 damage . In this context, the combination of advanced numerical models, able to investigate the nonlinear 17 behavior of highly deformable metamaterials, together with the capabilities of additive manufacturing to 18 produce custom -designed metamaterial samples that can incorporate controlled defects and features , is the best 19 way to bridge the gap between theory and practical experimentation. This convergence among theoretical 20 approaches, innovative methodologies, and advanced manufacturing techniques is essential for realizing the 21 full potential of metamaterials and harnessing their unique properties for a multitude of engineering 22 applications. 23 In light of these considerations, the theoretical knowledge of the involved phenomena and the capability of 24 modeling the potential damage mechanisms , that these materials can manifest when subjected to large 25 deformations , becomes a crucial engineering aspect in numerous fields of application . Such nonlinear 26 phenomena, frequently occurring at the microstructural level and drastically influencing the metamaterial 27 macroscopic behavior consist of fracture phenomena , decohesion at the interfaces between differen t 28 micro constituents, matrix cavitation, contact under compression, and instabilitie s that can compromise the 29 unconventional characteristics and multifunctionality of the metamaterials . Moreover, there has been a growing 30 interest in investigating the dynamic and wave propagation characteristics of periodic microstructured 31 metamaterials [6,7] . This interest arises from the periodicity of the microstructure, which can intrinsically 32 influence the propagation of elastic waves. However, it is worth noting that the combination of high 33 deformations and the onset of microscopic instabilities makes it necessary to carefully address the complexities 34 associated with contact mechanics and damage/fracture mechanics under conditions of finite deformations, 35 from both static and dynamic points of view. 36 In the past few years , to effectively address these issues from the numerical point of view and avoid 37 computationally demanding numerical simulations , researchers proposed reliable and accurate modeling 38 strategies based on nonlinear homogenization [8,9] and multiscale [10] techniques in a finite deformation 39 framework . Specifically, the application of multiscale techniques in the context of microstructural arrangements 40 was taken into account to address the limitations of the first -order homogenization approaches . For instance, in 41 [11,12] the instability phenomena at various length scales in hyperelastic composite materials together with the 42 effects of constitutive and geometrical nonlinearities were investigated. In [13] was highlighted that the 43 prediction of the onset of local instability in composite materials often demands large computational effort. This 44 arises from the direct finite element discretization of a unit cell assembly of unknown size or the use of advanced 45 techniques like Bloch wave stability analysis [14,15] , except when the instability mode has a significantly longer 46 wavelength compared to the characteristic size of the microstructure, for which a simplified analysis based on 47 loss of ellipticity condition can be applied. In [16] a multiscale finite element schem e was applied to analyze the 48 coupled effects of microcracks and instability in the context of heterogeneous solid materials . Furthermore, 49 failure arising from fiber microbuckling is a common issue in unidirectional fiber -reinforced composite 50 materials under compression along the fiber direction (see [17,18] , for additional details ). This type of failure 51 may induce microcrack initiation and propagation that significantly reduces th eir compressive strength due to 52 3 Preprint version of AAM for the publication on Phil. Trans. R. Soc. A. increased energy release at crack tips especially in combination with instability phenomena. A similar 53 phenomenon, known as buckling -induced debonding or delamination buckling, can occur in structural 54 components reinforced with composite materials or composite laminates [19,20] . Therefore, it is important to 55 conduct a thorough examination of the interplay between microcracks and local fiber buckling to mitigate their 56 adverse effects on the failure behavior of fiber -reinforced composites. To the best of the author s’ knowledge, the 57 study of the nonlinear behavior of 3D printed microstructured metamaterials, considering damage and 58 instability phenomena, represents a relatively new and emerging research field. This underscores the potential 59 for further research and innovation in this area, as it holds significant promise for applications across various 60 scientific disciplines. To this end, this paper proposes an original theoretical approach for modeling the failure 61 behavior of hyperelastic layered composites as a consequence of fiber/matrix decohesion and contact 62 mechanisms in coupling with microscopic instability phenomena. 63 64 65 2. Theoretical setting of the microstructural failure problem 66 The examined representative volume element ( RVE ) of a periodic composite microstructure occupies the 67 domain ()iV (with boundary ()iV∂) in its initial undeformed configuration and contains internal cohesive/contact 68 interfaces ()ch iΓ comprising pairs of coincident physical surfaces . The deformation process is described by the 69 time- like parameter t, family of deformations () ( , ) ( ): it t VV= +→ xX X uX mapping points X of the initial 70 configuration ()iV into points x of the current configuration tV, with 0t= corresponding to ()iV and ()uX being 71 the displacement field vector. 72 73 Figure 1: Kinematical description of the cohesive -contact interfaces and related surface pairs during the 74 deformation process: a) initial undeformed configuration; b) Lagrangean description; c) Eulerian description. 75 (A,A’) and (B,B’) are cohesive points pair while (A,B’) is a contact points pair. 76 77 As reported in Figure 1, according to a Lagrangean description , the set of cohesive interfaces in tV are denoted 78 by ()() ch it Γ (in the reference configuration () () ( 0) ch i ch i t Γ= Γ = ) and comprise pairs of coincident lower and upper 79 reference cohesive surfaces ()()l ch it Γ and ()()u ch it Γ , respectively placed on the positive and negative side of ()() ch it Γ 80 oriented by its unit normal ()l in, separating after deformation into the disjoint current cohesive surfaces ()l chtΓ 81 and ()u chtΓ . On the other hand the portions ()()l ch it Γ and ()()u ch it Γ where contact occurs a t time t are generally 82 disjoint and respectively denoted by the reference surface pair ()()l cit Γ and ()()u cit Γ , since pair of points in contact 83 in the current configuration are generally not coincident in the reference one ; in the current configuration the 84 intersection of physical surfaces pairs where contact occurs, coinciding with a single interface, is denoted by the 85 symbol ()ctΓ . In the following , the dependence on t will be omitted from explicit consideration in order to 86 simplify the notation unless where it is needed to avoid ambiguity. 87 88 4 Preprint version of AAM for the publication on Phil. Trans. R. Soc. A. 2.1 Contact -cohesive interface modeling 89 The displacement jump vector across the cohesive- contact interface at a generic cohesive point pair 90 ) ) () ( ( } (, ) { , |lu l l uch il uu ch c i hi = ∈Γ = Γ∈ XX X X X X is defined as : 91 ( ) () (), ,] ( ,) ( ,) ( ,)lu u l ch i ch it t ttΓ[= − ∆= − X X uX uX uX . (1) 92 With reference to the current unit normal ()ltn to ()l chtΓ , obtained as the transform ation by the deformation 93 mapping of ()l in, ∆ can be decomposed, respectively, into normal and tangential component vectors ln∆n and 94 s∆ according to: 95 ( ) ,l l ll n sn s , ∆∆ + =−⊗ ∆= ∆ ∆ ∆ ∆ n = n 1n n . (2) 96 The following continuity- like condition for the nominal cohesive traction vector across the undeformed cohesive 97 interface ()ch iΓ is usually assumed (see, for instance, [21–23] and the