Can confined mechanical metamaterials replace adhesives?

The subject of mechanical metamaterials has been gaining significant attention, however, their widespread application is still halted. Such materials are usually considered as stand-alone, vis-\`a-vis all characteristic length scales being associated solely with geometry of material itself. In this work we propose novel application of mechanical metamaterials as interface regions joining two materials with potential of replacing bulk adhesives. This idea leads into paradigm shifts for both metamaterials and adhesive joints. In specific, we outline methodology for testing and evaluating confined lattice materials within fracture mechanics framework. The theoretical and numerical approaches are inter-winded, revealing a set of critical parameters that needs to be considered during design process. Lattices that are stretching and bending dominated are explored and failure maps are proposed, indicating susceptibility to a certain failure mode depending on level of confinement and characteristic dimension of each lattice's unit cells.


Introduction
microstructures inspired by three-dimensional unit cells, specifically the tetrahedral-octahedral (octet truss) and the 40 bitruncated cubic honeycomb (Kelvin cell). The three geometries will be further explored in the following sections 41 and further merged into a common framework with the joined materials. Results of theoretical developments will 42 finally be compared with numerical model. Specifically, we aim at building a consistent theoretical framework that 43 unifies metamaterial interface with composite adherents that will result in failure load predictive capabilities.  45 We begin by analysing the rigidity of the selected geometries, as shown in the last column of Fig. 1. These are identified through Maxwell's rule [Maxwell, J. C., 1864] and its generalisation, as formulated by [Pellegrino and Calladine, 1986]. Maxwell's rule in two-dimensions states that a frame is rigid when − 2 + 3 ≥ 0, where and stand for, respectively, the number of bars and joints of the frame in question. This can be further generalised as − 2 + 3 = − , where counts the states of self-stress and the number of mechanisms. For the case of pillars (see first row of Fig. 1), we include one bar ( = 1) and two joints ( = 2) hence, the unit cell is isostatic ( − 2 + 3 = 0). Similarly, for the triangular lattice (2D analogy of the octet truss as shown in the second row of Fig. 1), we include five bars and four joints hence, the unit cell is isostatic ( − 2 + 3 = 0). For the hexagonal lattice (depicted in the last row of Fig. 1), we include six bars and six joints hence, the resulting frame is a mechanism ( − 2 + 3 < 0). Since pillars and triangular lattice are isostatic frames, their members are only loaded axially, and therefore are behaving as stretching dominated geometries. Moreover, when the joints of the mechanism formed by the hexagonal lattice are locked in place, the unit cell is loaded with bending moment, and therefore is behaving as bending dominated geometry [Zheng et al., 2014]. The design variable of a given lattice structure consisting of struts is taken to be the inverse of the slenderness parameter, which is given by the aspect ratiõ = ∕ , where there radius of the strut and the unit cell length are illustrated in Fig. 1. This quantity relates directly with the effective mechanical properties of a specific choice of unit cell geometry; when under fracture processes, the lattices will carry load up to failure and their effective stiffnesses are found to scale with their cross-sectional area of the strut in which are made, i.e. proportional tõ 2 . Moreover, it can be shown that stretching and bending dominated unit cells result into lattices with effective stiffnesses scaling respectively with̃ 2 and̃ 4 [Gibson and Ashby, 1997]. Physically, this means that in the stretching dominating regime tensile stresses in strut are prevalent, whereas coupled tensile-compressive stresses are dominant for bending dominated unit cells. Therefore, we may write that the lattice interface effective stiffnesses of the choices of unit cells are given the following expressions:

Effective mechanical properties of architected interfaces
where represents the effective stiffness of each lattice, the label ∈ {|, △, ⎔} indicates which unit cell makes up the lattice (| = pillars, △ = triangular and ⎔ = hexagonal), and * stands for the stiffness of the base material. The geometrical constants 45 = 3 + √ 2 ∕7 and 60 = 32 √ 3 are obtained for the 45 • triangular horizontal lattice and 60 • regular honeycomb [Gibson and Ashby, 1997]. Notice that, from the theoretical perspective, using 3D formulation would merely result in modified set of constants . Furthermore, we refer to the normalisation with respect to the base material of the interface as̃ ≡ ∕ * . The three lattice interfaces are considered as homogeneous brittle structures carrying uni-axial loading. Therefore, an upper and a lower bound for the localised normal strain is set for each type of geometry. The upper (tensile) bound describes the reach to tensile failure stress at an individual strut, and the lower (compressive) bound corresponds to reaching the Euler buckling load at an individual strut. This last failure mechanism forms an augmented version of the Cohesive Zone Model (CZM) [Barenblatt, 1962;Dugdale, 1960;Hillerborg et al., 1976] incorporating compressive regime which characterises the lattice bondline. Specifically for the pillar unit cell, the tensile fracture external stress is 2 ̃ and the buckling (assuming fixed boundary conditions) external stress is 2 3̃ 4 * , since the maximum internal force and total external force scale with 1∕2. Hence, the bounds for tension ( ) and compression ( ) are respectively given as follows: For the triangular unit cells, the fracture external stress is √ 2 ̃ and, assuming pinned boundary conditions, the buckling external stress is √ 2 3̃ 4 * ∕2, since the maximum internal force and total external force scale with 2 −1∕2 . This results in the following expressions for the bounds in the critical stain: For the last example, the bending dominated hexagonal unit cell, failure is achieved when the ultimate stress is reached as bending stress. The corresponding bending moment at failure in the strut is 3 ∕2, thus providing a stress in the unit cell given by 16 ̃ 3 ∕3. Because the unit cell is bending dominated, its response is symmetric in tension and compression. Hence, we arrive at the following bounds: This formulation of the bounds uncovers a dependence between the lattice failure mode and the geometry. More 46 specifically, it will later determine the critical loads of the stretching dominated cells relate to the geometry at the compressive regime, while the tensile fracture load is geometrically independent. Regarding bending dominated cells, 48 the dependence exists in both fracture modes but appears at the denominator. The present problem can be treated as a particular case of an adhesive joint with discontinuous, architected interface 51 region, i.e. the bondline. We consider a well known, and reliable, Double Cantilever Beam (DCB) configuration under 52 fracture loading [Kanninen, 1973], outlining some differences between homogeneous solids and the lattice (metama- trusses measuring usually nano-to-millimetres, sharp crack assumption cannot be met, and alternative way for studying 64 progressive failure should be proposed.

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In order to capture interaction between the lattice interface and joined material the DCB configuration, as represented in Fig. 2 is adopted. Mechanical behaviour of such material system originates effectively from a bonded structure under fracture loading, and it can be seen and modelled as a generic beam on elastic foundation [Dillard et al., 2018]. The following equilibrium equation gives us the minimiser for the elastic strain energy of such system: where and = ℎ 3 ∕12 are the Young's modulus and moment of area of the beam, respectively. The relevant geometric parameters associated to the beam are its width and thickness ℎ, assuming a rectangular cross section.
On the right hand side of Eq. (5), ( ) and ( ) are forces per unit-length, respectively, representing the external applied load and the stresses due to the elastic foundation. Since ( ) yields a restoring force, and assuming the elastic foundation to be linear, we have that ( ) = − ( ) ( ), where ( ) is the elastic foundation stiffness (force per unit length square). This stiffness is expressed as follows, where is the crack length, is the effective stiffness of each choice of lattice, given by Eq.
(1), ℎ is the thickness of each interface, and ( − ) is the Heaviside step function determining the location of the crack front-notice that the domain in, 0 ≤ < ∞, is such that for 0 ≤ < the beam is debonded and the foundation only acts for ≤ < ∞. Finally, the applied load is simple shear at the boundary, which is expressed by ( ) = ( ), where ( ) is the Dirac delta function necessary to localise the applied load (units of force) acting on the boundary = 0. Before proceeding with the solution to the Eq. (5), we acknowledge the fact that this equation gives rise to a characteristic wave-length for the region over which crack front stress can be distributed, which is here derived from the length scales in the problem as well as stiffnesses' ratio: where 0 ≡ ℎ∕6 1∕4 is the resulting wave-length when the interface thickness is negligible [Kanninen, 1973],h ≡ ℎ ∕ℎ is the thickness ratio used as one of our design parameters, and̄ ≡ ∕ * is the stiffness' ratio of the beam and the lattice's base material. The above characteristic wave-length, along with the definition of dimensionless quantities 1 , gives us the following dimensionless form of Eq. (5): Ensuring that the edge of the beam is moment free, i.e. d 2̃ (0)∕d̃ 2 = 0, as well as that the solutions are at least  3 at̃ =̃ -thus ensuring continuity of the solution, slop, moment, and shear-we find the following piece-wise solutioñ Contrary to solid interfaces for which in the present DCB configuration compressive failure will never be observed, confined lattice interfaces will likely be sensitive to the direction of loading [Heide-Jørgensen et al., 2020]. In order to highlight these differences, we shall here define a tension-compression amplification factor. Let us look at the first two extrema of Eq. (9) in the interface region (̃ ≥̃ ), which are found at̃ =̃ for the maximum and at =̃ + − tan −1 (1 + 2̃ ) for the minimum. The corresponding deflection values are found: Therefore, a tension-compression amplification factor is can be defined as follows: where the bounds are given by ∕2 <̂ < √ 2 3 ∕4 .
We now propose to solve a modified version of Eq. (5) by including a CZM [Barenblatt, 1962;Dugdale, 1960;Hillerborg et al., 1976] in both tensile and compressive loading directions. More specifically, we distinguish between three possible regimes of the interface behaviour. In the elastic regime of the interface the stresses are simply proportional to the beam deflection, i.e.̃ ≡ ∕ * = 2 (̃ −̃ )̃ (⋅) ̃ . When the largest negative deflection is less than some critical value ( ) , it is assumed that the interface fails instantaneously (i.e. through brittle failure) and no longer carries any stress. Hence, this compressive failure scenario is avoided when 2̃ > ( ) . When the largest positive deflection is larger than critical value ( ) , the stress carried by the interface is locally zero and the interface fails gradually. All of these facts are summarised by the following conditional statement: Crack initiation due to tension happens when the upper bound of the unit cell strain is reached, 2̃ max = ( ) , while failure due to compressive loads is observed when the lower bound of the unit cell strain is reached, 2̃ min = ( ) . From Eq. (10), we may write the critical reaction force due to tension and compression of each unit cell, respectively: Similarly to Eq. (11), a critical strain factor can be defined as:̂ = | ( ) ∕ ( ) |. Since the tension-compression amplifi-67 cation factor,̂ , is independent of the normalised force, the failure mechanism is instead imposed in comparison with

68̂
. This follows the failure criterion due to tension, which happens when̂ >̂ , and the one due to compression, when 69̂ ≤̂ . Bernoulli beam elements. We also assume that the interface's base material properties is such that its behaviour is 75 brittle. Hence, all the failure cases were considered instantaneous and the behaviour of the material was modelled as 76 linear elastic until failure occurs.

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For stretching dominated unit cells, we assume failure under tension occurs when the ultimate tensile stress is 78 applied to the struts. Therefore, a maximum tensile strain is obtained by ( ) = ∕ * . On the other hand, when in 79 compression, the strut is considered slender thus buckling occurs before reaching the maximum compressive stress.

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The critical buckling load, considering built-in ends, is given as = 3 * 4 ∕ 2 . Therefore, the maximum compres-  The FEM analyses were carried using COMSOL Multiphysics (version 5.5). The brittle failure of the struts was implemented manually in the definition of the Young's modulus of the lattice material [Vermeltfoort and Van Schijndel, 2013]. In specific, the Young's modulus was defined as a function of the appropriate strain for each case, resulting into  Figure 4: (a) Example,̃ = 4.5 × 10 −2 andh = 1 of the normalised deflection field for the pillar interface under tensile failure mode. Both models indicated similar deflection at failurẽ = ( ) | and length of the process zonẽ | and deflection field distribution -red background for tensile and blue for compressive deflections. (b) Failure map for pillar geometry where both power-law are visible as well as the effects of the interface thickness. Two regions are found: In low aspect ratios, the compressive displacement related to the buckling is reached first and the critical force recorded is related to the buckling load. In high aspect ratios, the tensile displacement related to the yielding of the pillars is reached first and the critical force recorded is related to the yield stress of the interface material.
In the Figs. 4, 5, and 6 we discuss the effects the choice of unit cell topology has on the type of failure, which is  Figure 5: (a) Example,̃ = 4.5 × 10 −2 andh = 1 of the normalised deflection field for the octet-like interface under tensile failure mode. Both models indicated similar deflection at failurẽ = ( ) △ and length of the process zonẽ △ and deflection field distribution -red background for tensile and blue for compressive deflections.) Failure map for octet truss geometry. Two regions are found: In low aspect ratios, the compressive displacement related to the buckling is reached first and the critical force recorded is related to the buckling load. In high aspect ratios, the tensile displacement related to the yielding of individual struts of the unit cell is reached first and the critical force recorded is related to the yield stress of the interface material.
the slenderness parameter, namely the aspect ratiõ . Both stretching dominated geometries, pillars shown in Fig. 4-130 (b) and the triangular unit cells Fig. 5-(b), show a qualitative similar behaviour. For low aspect ratiõ , the struts are 131 slender and the structure is more susceptible to failure under collapse; on the other hand, at higher values of̃ , i.e. 132 stubbier elements, the power-law changes and the onset of failure occurs in tension. The failure map results for the 133 hexagonal lattice are shown in Fig. 6. As expected, the hexagonal unit cells being bending dominated, will only fail in 134 tension since any compression will be uptake by moments on the hinges.  136 One of the characteristic feature of materials with confined interfaces is existence of the dominating length scale 137 , which describes interaction between the joined material and the interface. In Fig. 7, the relation between the 138 and̄ is depicted. In particular,̄ is defined as the fraction area occupied by the lattice microstructure over the area of

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In this work, we attempted to bring together the concept of mechanical metamaterials into adhesive bonding tech-182 nology and answer the question "can mechanical metamaterials replace adhesives?". Aiming at answering or at least 183 shed light at this query, we have developed a theoretical and a numerical framework for studying and analysing systems 184 in which mechanical metamaterials are confined between the two joined materials. An amalgamation of competing 185 effects-metamaterial, interface and joined materials-leads to the relevant characteristic length scales in the problem 186 to be intertwined, which is a situation reminiscent to adhesive joints. Contrary to bulk bondlines, for which charac-187 teristic lengths scales are uniquely related to the material of choice, the metamaterials approach unveils a spectrum of 188 possibilities behind geometrical manipulations and modern manufacturing technologies. Thus, accounting for failure 189 criterion can be shifted to a design process that encompasses a wider properties-on-demand philosophy-this also 190 leads to the specification of elastic properties, failure modes and types of load to be carried.

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Three 2-dimensional lattice structures, composed of unit cells refereed to as pillar, triangular and hexagonal, have been used to form interface metamaterial-bondline. Despite the fact that we have worked with two-dimensional ex-physical models. In our theoretical approach, we have considered a homogenised model of lattice interfaces-playing 195 the role of an elastic foundation-which is then sandwiched between two beams to form the metamaterial inspired 196 adhesive joint. Such formulation allowed a fast and reliable prediction of failure onset loads with the insights to the 197 micro-structural details of the unit cell. This model has also proven to be very robust and accurate in outlining the 198 importance of the different length scales responsible for the mechanical performance of these new types of adhesive 199 joints. Relationships amongst unit cell geometry, density, process zone length and the failure load, can be readily ob-200 tained. To verify results of the theoretical analysis, a numerical model based on custom FEM formulation was devised.

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In specific, the trusses constituting metamaterial microstructure were modelled as beams with piece-wise strain-stress 202 relation during loading. Such formulation proved very efficient in solving problem at hand and considerably reduced 203 computational costs related to nowadays often seen formulations using two and three-dimensional FEM. A very good 204 agreement, both quantitative and qualitative between the two approaches is recorded. Both models predict remarkably 205 accurate failure loads and stress fields. In addition, both models predicts the same power-law relations between the 206 failure load and aspect ratio of unit cell truss elements for all the configurations investigated. The results obtained 207 indicate a strong relation between the failure loads and the process zone length. The process zone length is then very 208 sensitive to truss element aspect ratio and unit cell geometry. This novel microstructure geometry driven materials 209 offers a new tool to designers for shaping properties of structure with interfaces.

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This work indicates that metamaterial interfaces can potentially lead to a new approach for designing of adhesive 211 joints. In particular, different unit cells can be used to mitigate stress fields over different regions of the joint-thus, the 212 size effects could be controlled. Very low weight-to-volume ratios could be achieved with controlled lost of mechanical 213 performance. However, the present work is by no means conclusive, as it is our intention and hopes that this novel 214 approach will provoke fresh discussion on alternatives into design of adhesive joints and interfaces between materials.