A novel high symmetry interlocking micro-architecture ...

Viewer
Transcript

ARTICLE IN PRESS

JID: SAS

[m5G;July 10, 2017;14:54]

International Journal of Solids and Structures 0 0 0 (2017) 1–15

Contents lists available at ScienceDirect

International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties Sandip Haldar a, Trisha Sain b,∗, Susanta Ghosh b,∗ a b

IMDEA Materials Institute, Getafe, Madrid, Spain Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA

a r t i c l e

i n f o

Article history: Received 8 December 2016 Revised 20 June 2017 Available online xxx Keywords: Interlocking Micro-architecture Damping Viscoelasticity Polymer composites Bio-Inspired High symmetry Multifunctional Impact

a b s t r a c t Development and design of novel materials based on their micro-structural arrangements are being intensely investigated due to their wide range of real and potential applications. One of the key objectives of such design is to achieve multiple properties, which are often competing in nature (such as high stiffness-high damping, high strength-high toughness etc.), within a single composite material. In the present work we propose a novel interlocking micro-architecture design to achieve high symmetry in a plane within a composite. The present study shows that constraining a very low volume fraction of high damping polymer within this micro-architecture, together with a stiff polymer, results in a simultaneously high stiffness and high damping polymer composite. The proposed micro-architecture design possesses high symmetry that is not commonly found in ﬁber-reinforced polymer composites. The interlocking feature avoids use of extra adhesives for holding two adjacent building blocks. Finite element simulations are performed by considering the micro-architecture made of two widely used polymeric materials such as polymethyl methacrylate (PMMA as the stiffer building block) and polyurethane (PU as the soft viscous material). Our numerical predictions show remarkable stiffness and damping properties for the design over a wide range of frequencies. Simulations are also performed considering contact between two polymer interfaces to establish the fact that the geometric interlocking works well without any use of adhesive for holding the blocks together. Further simulations are performed under high frequency impulse pressure loading to study dynamic wave propagation through the micro-architecture that shows the near-isotropic response of the proposed micro-architecture. Published by Elsevier Ltd.

1. Introduction Incorporating intricate micro/nano-architecture at various length scales within materials is a highly promising methodology to improve the performance of the constituent materials. In a recent review on micro-architectured materials, Fleck et al. (2010) summarized the principles of designing micro-architectures at different scales and the ways these new materials were able to occupy the material property space (such as stiffness, strength, toughness) that were so far empty in Ashby (1993). In the domain of micro-architectured materials, introducing 3D lattice structures within a material’s microstructure has been identiﬁed as a promising technique for imparting multifunctional applications (Kooistra et al., 2004). These materials show promise on exceeding the theoretical bounds in cross-property correlations

∗

Corresponding authors. E-mail addresses: [email protected] (S. Haldar), [email protected] (T. Sain), [email protected], [email protected] (S. Ghosh).

(such as high stiffness and toughness with low density). Multifunctional materials are designed for improved overall performance. Thus their performance metrics are inherently different (or better) than their individual constituents as reported in Veedu et al. (2006). Some of the microstructure design strategies to improve stiffness and/or damping properties are brieﬂy described in the following. In Lakes (2002) adding spherical inclusions of different size in hierarchical fashion produced viscoelastic composites with high stiffness and damping. By putting spherical inclusions of stiff silicon carbide within a high damping matrix, Kim et al. (2002) found that the most favorable condition in terms of high damping and stiffness was reached at high volume fraction of inclusions. To improve the vibration damping in wind turbine blades special star-shaped inclusions were designed within a macro-composite (Agnese and Scarpa, 2014), which resulted in higher loss factor of the material. Negative Poisson’s ratio materials or auxetic materials (Murray and Gandhi, 2013) are also being used by combining a hard and soft phase to achieve multiple properties together. Chen and Lakes (1993) tested auxetic copper foams both with and

http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030 0020-7683/Published by Elsevier Ltd.

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS 2

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

without ﬁller material. The auxetic (negative Poissons ratio) foam ﬁlled with the polymer had a measured loss factor greater than the regular (positive Poissons ratio) foam. Different strategies to improve the mechanical properties in the realm of auxatic materials is provided in the review (Alderson and Alderson, 2007). In a recent work by Meaud et al. (2014), 3D carbon nanotube-based micro-architecture was used and ﬁlled with a lossy polymer to enhance stiffness and energy dissipation of the resulting composite. The intricate and time consuming manufacturing of the CNT nanotruss might limit the practical use of the designed material. Estrin et al. (2011) have utilized the concept of topological interlocking, which introduces an unusual property (such as negative stiffness) within a material without damaging or buckling the stem. They mentioned that the topological interlocking has an advantage over conventional interlocking as it does not require high precision machining like the latter. In Ravirala et al. (2007), an interlocking rough particle model is proposed to predict behavior of the auxetic ﬁbers and ﬁlms produced using the partial melt extrusion process. Their model consists of an array of rigid hexagonal particles of unequal sides connected through geometrically matched rectangular locking keys. Each hexagon has three keys and three holes, with a key and a hole placed in adjacent sides. The model considers a liner spring to represent the bonding between two adjacent particles at the interlock. Their model could predict Poisson’s ratio for auxetic polypropylene ﬁlms and axial Young’s modulus of the auxetic ﬁbres. In the work by Khandelwal et al. (2014), topologically interconnected materials were deﬁned as a class of 2D “mechanical crystals” made by a structured assembly of polyhedral elements. They had mentioned the mechanical properties of the topologically interconnected materials emerged as a combined deformation of the individual unit and their contact interaction. Topologically interlocked materials are reported to have attractive mechanical properties such as high damage tolerance, quasi-ductile response and negative stiffness (Dyskin et al., 2003; Estrin et al., 2011). However, often they require an additional tension bearing element such as steel wire, which might nonetheless limit their applicability. Though interlocked materials are understandably weaker in strength than their solid counterparts due to the load transfer mechanism solely governed by geometric interlocks (similar to bond-like), they have several advantages. Interlocked materials have inherent advantage over monolithic materials as the later class of materials can be pre-fabricated and assembled according to need to obtain a desired size and does not need cutting or machining. For the same reason these materials need not be replaced, but can be repaired when a small part is damaged. Besides, interlocked materials are more suitable for applications, where compliant behavior is preferred. Besides their aforementioned man-made counterpart, microstructure interlocking techniques are also available in nature with varieties of structural diversities and features; see Fig. 1 for some examples. Structural materials found in nature incorporate a variety of microstructures within them that improve their mechanical performances and sustainability in a great scale. The most commonly talked about natural material, nacre, is 30 0 0 times tougher than its extremely brittle mineral content due to its sophisticated micro-architecture, as discussed by Katti et al. (2005). Nacre is made of microscopic tablets of calcium carbonate with some glucose holding them together, as shown in Fig. 1. The alternating brick-mortar arrangement provides additional locking between layers against shear failure. Another common example is tooth enamel in which minerals form 95% volume of the material. The microstructure is made of long rods perpendicular to the surface of the tooth and 4 − 8 micron (Mirkhalaf et al., 2014; OLIVEIRA et al., 2010) in diameter and held together by a small fraction of proteins. In general, the structure of these materials is organized

over several distinct hierarchical length scales, from the nanoscale to the macroscale. Gorb’s work (Gorb, 2008) summarized the diversity in biological attachments or interlocking mechanisms found in natural materials and classiﬁed them based on their physical mechanism, function of the attachment device, etc. These bio-inspired concepts are being used in the ﬁeld of engineering materials design and synthetic composites resulting in a paradigm shift from materials by chance to materials by choice. Recent works by Barthelat and co-workers Mirkhalaf et al. (2014); Valashani and Barthelat (2015); Funk et al. (2015) are noteworthy; following the arrangement of nacre microstructure, Barthelat has developed a fully tunable multiobjective optimization scheme to maximize strength, stiffness, and toughness simultaneously in a staggered composite (Valashani and Barthelat, 2015). In another work, Mirkhalaf et al. reported signiﬁcant improvement in energy dissipation and toughness in glass by introducing weaker polymer interfaces. In other words, brittle glasses with no-microstructures, when modiﬁed with bio-inspired interfaces, had become more deformable and nearly 200 times tougher. Therefore, design of architectured materials is one of the key approaches to improve performance in the material property space. In the present work, a novel interlocking micro-architecture is used to design a polymer-polymer composite material with high stiffness and damping. The micro-architecture is based on load-bearing elements made of stiffer polymer, namely polymethyl methacrylate (PMMA) and thin layers of highly viscous polyurethane (PU) between the load-bearing elements. The loadbearing structure consists of hexagonally arranged (circular) elements, which are connected to nearest neighbors via (triangular) interlocking keys. The proposed micro-architecture is inspired by the concept of hexagonal crystal structure. Due to the hexagonal symmetry, the micro-architecture introduces nearly-isotropic response in the composite. This nearly-isotropic behavior is demonstrated for both quasi-static and dynamic loading conditions. The specially designed interlocks hold two different building blocks together without any additional use of adhesives. Numerical predictions of the mechanical response of the composite under various loading conﬁgurations show promising behavior in terms of stiffness and damping at various rates of loading. The paper is organized as follows: Section 2 describes the design and geometry of the micro-architecture. The numerical modeling and material models are detailed in Section 3. The numerical results are presented in Section 4. Finally, Section 5 summarizes the main conclusions. 2. Microarchitecture design The proposed micro-architecture design is motivated by the near-isotropic nature of planar hexagonal geometry as described by Metrikine and Askes (2006) and later used by Ghosh et al. (2014). The geometry possesses a near isotropic feature in its stiffness evaluation, which imparts an additional advantage compared to other (natural/bio) interlocked materials (Katti et al., 2005) or polymer nano-composites (Meaud et al., 2014; Zhang et al., 2015) that often lack isotropic character and are expected to be much weaker in one direction. The proposed micro-architecture design is also motivated by the isotropic character of planar Graphene due to its regular hexagonal geometry (see Arroyo and Belytschko, 2004; Zhou and Huang, 2008; Ghosh and Arroyo, 2013). The major fraction of the present micro-architecture design consists of interlocking building blocks made of a stiff polymer that predominantly contributes to the stiffness. In addition, a thin layer of viscoelastic soft polymer is placed at the interface of the stiff blocks. The interfacial shear within the thin layer of soft polymer contributes to the viscous damping of the material. The details of the geometry and arrangement of the microarchitecture for the material are shown in Fig. 2. The geometry of

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

3

Fig. 1. Examples of different attachment systems: (a) temporary locking mechanism between fore- and hind wings in sawﬂy (Hymenoptera) (Gorb, 2008); (b) Schematics of two kinds of top and bottom microtrichia (of adult Odonata/dragonﬂy), separated on the top and attached on the bottom (Gorb, 1999; Dragonﬂy, 2016); (c) Inner nacreous layer of a red abalone shell (Barthelat et al., 2007); (d) Scanning electron micrograph showing arrangement of a tablets (brittle ceramic made of Aragonite) in nacre obtained from red abalone (Katti et al., 2005). The inside layer of a two layer system of some species from the gastropod and bivalve families is nacre (Barthelat et al., 2007); (e) A close-up view of (d) showing interlocking between platelets. This interlocking system endows the nacre high toughness and strength Katti et al. (2005). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

the micro-architecture consists of two distinct shapes as building blocks- one approximately circular and the other triangular. The triangular elements act as interlocking keys between adjacent circular gear elements and prevent separation of the two adjacent gear elements under tension. Both the circular gear elements and triangles form hexagonal lattices, as shown in Figs. 2 and A.19. The circular gear elements, the upward triangles, and downward triangles form three separate hexagonal lattices (marked as red, pink, and magenta respectively). These lattices are shifted by a constant distance with respect to each other1 . In Fig. 2, two rectangular unit cells are shown. It is clear that these cells repeat themselves at an angle of 60°. Enlarged ﬁgures for these two unit cells are shown in Fig. 3. The thickness of the soft polymer layer is denoted as tPU therein. Due to the non-rectangular nature of the micro-architecture and relative shifts between the hexagonal lattices of the building blocks, it is diﬃcult to ﬁnd a smaller rectangular unit cell suitable for the stress analysis. Even though the unit cells are much larger than the primitive cells of the repeating hexagonal lattices, they are chosen mainly because of the convenience in stress analysis. Theoretically, one quarter of the unit cell -A is suﬃcient to capture the geometry due to in-plane symmetry. It can be noted that unit cell A in Fig. 3 (a) fully encloses a circular gear element and unit cell -B in Fig. 3 (b) encloses the intersection between two circular gear elements. Further details on the geometry of the micro-architecture are included in Appendix A

1 Since, we have more than one type of elements (triangular and hexagonal), it is not like a Bravais lattice. However, the relative shifts between different shapes are similar to Bravais multi-lattices (see Sec 3.6 of Bhattacharya, 2003).

One of the key aspects of the proposed micro-architecture (and any of its kind) is that a larger material can be made in a nonmonolithic way by using pre-manufactured building blocks. Materials consisting of many interlocked elements are more compliant than their monolithic counterpart and can withstand considerable deformations. Besides, the micro-architectures of regular shapes are easier to manufacture as multiple building blocks can be cast separately with high precision manufacturing and assembled as required. Further, any damage occurring in the building blocks of the proposed micro-architectured material can be easily repaired or the design of a speciﬁc building block may be modiﬁed without dissembling the entire material. 3. Numerical prediction of stiffness and damping for the proposed design 3.1. Finite element simulations of the proposed design To predict the response of the proposed interlocked microarchitectured material under mechanical loading, ﬁnite element (FE) simulations are performed using commercial package ABAQUS(V6.14) (Abaqus/Standard, 2015). In the case of composite materials with a repeating micro-architecture, a representative unit cell corresponds to the repeat unit in which the stress distribution represents a snapshot of the stress proﬁle, which repeats within the entire volume of the material and far away from the physical boundary. The choice of the unit cells has been described in Section 2. Periodic boundary conditions have been applied at the edges of the unit cells to represent the repeating micro-architecture. In ABAQUS, the model has been meshed by using 4-noded plain strain elements (CPE4R). A mesh convergence

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS 4

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

3.2. Material models for the constituent polymers

Fig. 2. Interlocking material with micro-architecture for enhanced stiffness and damping behavior. The grey circular gear elements and triangular components are made of stiff polymer and the green inter-layer consists of a soft viscoelastic polymer. Hexagons (in red, pink and magenta colored) show geometry of lattices for the gears and the triangles. Rectangles (Blue and black colored) denote unit cells used in analysis later. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

In the present study, PMMA is modeled as a rate independent elastic-perfectly plastic material with Young’s modulus E = 3 GPa and yield stress σy = 50 MPa. We consider ABAQUS inbuilt elastic-perfectly plastic material model for that. The soft, extremely lossy polymer PU has been chosen (loss factor tan δ = 0.6), as used in Sain et al. (2015). This particular PU has been well characterized by combining a rate dependent ﬁnite deformation constitutive model and several experimental results in Sain et al. (2015). In the present simulations, the PU has been modeled as a hyper-viscoelastic material to incorporate its large deformation and rate dependent constitutive response. In ABAQUS (Abaqus/Standard, 2015), for a hyper-viscoelastic material in ﬁnite strain theory, the instantaneous response of the material follows the hyper-elastic constitutive equation. We have used Arruda– Boyce 8-chain polymer model to capture the fully relaxed long term response for the PU with parameters G = 1.115 MPa and limit stretch N = 20 with Poisson’s ratio 0.48. The Arruda–Boyce hyperelastic material model is a commonly used constitutive model to describe large deformation elastic response of polymer materials Arruda and Boyce (1993). The basic equations considering Arruda– Boyce energy density function are presented in the Appendix B. To incorporate the rate dependent viscous damping of the PU, we have used ABAQUS’s (in-built) ﬁnite strain visco-elastic material model. The ﬁnite strain viscoelasticity in ABAQUS(V6.14) considers a time domain generalization of the hyperelastic models. The model implements a thermodynamically consistent (Abaqus/Standard, 2015; Simo, 1987) hereditary integral formulation of the linear isotropic viscoelasticity. The relaxation moduli are presented in terms of the Prony series. The important equations describing the viscoelastic material model in ABAQUS have also been presented herein. To implement linear viscoelasticity for a compressible material, a common approach is to split the current stress measure in volumetric and deviatoric parts such as:

τ (t ) = τ D (F¯ (t )) + τ H (J (t ))

(1)

where τ D is the deviatoric part of the Kirchoff stress and τ H is the pressure part; F¯ = J1F/3 is the distortional deformation gradient and

J = det (F ) is the volume change; As in small-strain viscoelasticity, the relaxation (shear) moduli in terms of Prony series are given by:

G(t ) = G0 1 +

Nb

gi e

−t τi

(2)

i=1

Fig. 3. Representative unit cells of the interlocking microarchitecture used in ﬁnite element simulations. These ﬁgures correspond to thickness (of PU), tPU =0.01 mm.

To characterize the rate dependent behavior of a viscoelastic polymer, the relaxation behavior is approximated as a combination of several discrete relaxation spectra. G0 is the initial/instantaneous shear modulus of the material, and gi , i = 1, .., Nb are the (relative) shear moduli of the elastic springs as gi = Gi /G0 and τ i are the relaxation times associated with the rate-dependent chain motions of the polymer macromolecules. The hereditary integral in the reference conﬁguration for large strain response is given by

study was performed to obtain a suﬃciently ﬁne mesh with number of elements being ≈ 25, 0 0 0. Two different representative unit cells are considered for the FE analysis as shown in Fig. 3. It is explained later that the uniaxial responses of the two unit cells are nearly identical and it is worth considering either of them for analyzing the behavior of the micro-architecture under different loading conditions. Case studies are performed considering polymethyl methacrylate (PMMA) as the stiffer material and polyurethane (PU) as the soft, viscous material. In this work, the stiffness and damping of the micro-architectured composite are predicted over a wide range of frequency (10-100Hz) under sinusoidal loading.

τ D (t ) = τ0D (t ) + dev

Nb gi i=1

τi

−1

F

(t )

t 0

−t τi

τ D (t − t )e dt .F (t ) −T

(3) We assumed the pressure part of the Kirchoff stress is rate independent. The Prony series parameters used in this study, in terms of the relaxation times and the associated shear moduli are reported in Table 1. Having multiple relaxation times is a common characteristic of a viscoelastic polymer. These relaxation times represent various motions in macromolecular network within a polymer activated at different rates of external loading. This particular

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

ARTICLE IN PRESS

JID: SAS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15 Table 1 Material parameters for polyurethane (PU). i

1

2

τ i (sec)

22.95 1.85

1.12 4.04

Gi (MPa) G = n1 kθ N

3

4

0.16 0.023 15.04 43.28 1.115 MPa 20

5 0.0023 107.45

Fig. 4. Global stress-strain response of two unit cells under uniaxial tension in vertical (Y) and horizontal (X) directions. The cells correspond to PU layer thickness of 0.01 mm.

feature of a viscoelastic polymer makes it applicable as an effective damper for a wide range of frequencies. 4. Results and analysis 4.1. Response under uniaxial tension The quasi-static tensile response of the interlocking microarchitecture has been computed to determine the effect of the PU layer thickness on the modulus and strength of the composite. The thickness of the PU layer has been varied between 0.5% to 6% of the side of the base hexagon, i.e. tPU = 0.005 mm to tPU = 0.06 mm. The volume fraction of PU are 7.25% and 44% corresponding to tPU = 0.005 mm and tPU = 0.06 mm respectively. The quasi-static simulations are performed at a strain rate of 10−3 /s. At ﬁrst the mechanical responses of the two unit cells (as in Fig. 3) are compared under uniaxial tension. Stress-strain plots for the two cells along X and Y directions are plotted in Fig. 4. The mechanical response for both the cells are nearly identical at small strains but the difference increases at higher strain. Nonetheless, the responses for both the cells are same in small strain; hence unit cell -A will be analyzed for subsequent case studies. The rotational symmetry of the micro-architecture at each 60° is shown earlier in Fig. 2. To demonstrate the near isotropic nature of the proposed microarchitecture, a comparison between tensile response for both X and Y direction is shown in Fig. 4 for the two unit cells. The similarity in responses along X and Y directions conﬁrms that the microarchitecture indeed exhibits high rotational symmetry. Note that at strains smaller than 0.01, the difference between the response in X and Y directions is quite small, negligible for all practical purposes. Though this difference increases at higher strain, it remains less than 5% even at strain as high as 0.03. The quasi-static tensile response for the micro-architectured composite has been plotted in Fig. 5(a) for different thicknesses

5

of the PU layer. The elastic modulus of the composite for various PU layer thickness has been computed using the initial linear part of the stress-strain response (corresponding to a strain of 0.2%). The ultimate strength of the composites has been considered as the (global) stress value corresponding to the very ﬁrst appearance of local Mises stress as 50 MPa in PMMA, which corresponds to local plastic yielding in PMMA. The normalized stiffness and strength of the composite with respect to the stiffness and yield strength of PMMA have been plotted in 5(b). The normalized modulus and strength decreases from 55% to 8% and 27% to 5% respectively as the volume fraction increases from 7.25% (tPU =0.005 mm) to 44% (tPU = 0.06 mm). As expected, the stiffness and the strength of the composite decrease along with an increase in the thickness of PU layer. Though at higher volume fraction of PU the composite retains only a tiny fraction of stiffness of the PMMA, it is still much higher than PU stiffness. To understand the stress pattern within the micro-architecture, the von Mises stress distribution within the unit cells having 0.01 mm thickness of PU layer is shown in Fig. 6. It is clear from the stress proﬁle that the unit cell possesses symmetry with respect to the X and Y axes passing through its center. It also has symmetry under 180° rotation with respect to its center point. In general a diagonal symmetry is also expected from the hexagonally symmetric microstructure under symmetric loading; however, the present stress ﬁelds do not exactly maintain such symmetry. The reasons for such can be attributed to the difference in shape between non-rectangular primitive cell and rectangular unit cell, periodic boundary condition, lack of symmetry in loading, relative shifts between hexagonal lattices (Fig. 2), and geometric and material nonlinearity in PU at high strain. It is to be noted that the triangular keys on both sides participate in the load transfer, in addition to the central one. It is also observed from Fig. 6, that stress concentration occurs around the circular heads and the gear teeth made of PMMA. While the circular heads yield due to tension, the gear teeth yield due to bending. These portions of PMMA yield ﬁrst and continue to deform plastically until the material separates completely with failure in the interlocking mechanism. It is also observed that at global strain approximately 1.7%, PMMA starts to yield locally in several regions. The value of this global strain corresponding to local yielding of PMMA increases with increase in PU layer thickness which indicates that the composite becomes more stretchable due to an increase in the volume fraction of soft PU. In order to understand the variation of the load transfer mechanism against the thickness of PU layer, the von Mises stress ﬁelds for the unit cell A are shown in Fig. 7 for different thicknesses (tPU ) of PU layer. The size of the triangular element is reduced to increase tPU while the size of the circular gear element is maintained the same. Hence, the triangular neck becomes weaker with increasing tPU . This weakness of triangular interlocking keys is reﬂected in stress transfer pattern with increasing tPU under uniaxial tensile load. At tPU = 0.005 mm the central triangular key heads (in the middle of the X-axis) transfer a signiﬁcant portion of the load (along Y-axis) and they start to yield nearly at the same time as the circular gear element. However, as the tPU increases the load transfer through these central triangular key heads decreases. For tPU greater than 0.04 mm the circular gear element does not yield at all but some of the triangular key heads yield while resisting rotation of the circular gear element they are locked with. The rotation of the circular gears and triangles on both sides of the unit cell are contained due to the periodic boundary condition, which represents the effect of conﬁnement from the neighboring unit cells. Note that at tPU = 0.04 mm the size of the triangular head is slightly smaller than the enclosing gear; hence in absence of PU the interlock should not hold two adjacent elements together when pulled in the normal direction. However, using the stress

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS 6

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

Fig. 5. (a) Global stress-strain response of PU and the composite under uniaxial tension for various thickness of PU layer; (b) Normalized modulus and strength of the composites as a function of PU layer thickness.

Fig. 6. von Mises stress ﬁled under uniaxial tension along Y- direction at the onset of yield in PMMA in (a) Unit cell A and (b) unit cell B, both having 0.01 mm thickness of PU layer. (c) A close–in view at the intersection of two circular gears for unit cell A is showing the location of onset of yield and the FE mesh.

ﬁelds for tPU ≥ 0.04 mm, it can be concluded that even if the interlock does not transfer the load normal to it, the presence of PU and the hexagonal orientation of the interlock system endows the micro-architecture to hold together and transfer a smaller amount of load. This is another advantage to be noted for the present hexagonally oriented interlock system. 4.2. Uniaxial tensile response considering contact/friction across the interfaces To accurately represent the interfacial layering between the stiff PMMA and the soft PU, ﬁnite element simulations are performed considering contact/friction conditions. A surface-to-surface contact algorithm is used in ABAQUS/Standard where the stiffer PMMA ´ surfaces were considered as master’ and softer PU surfaces were considered as slave surfaces. To characterize the interface property for normal constraint conditions, a hard contact condition was implemented to ensure impenetrability between the surfaces under compression. To describe the normal behavior of the interface under tensile loading, a cohesive zone based traction-separation law was implemented between the surfaces. The tensile stiffness of the interface cohesive zone is assumed to be greater than the elas-

tic stiffness of the PMMA and PU to ensure exact stress transfer across the interface before damage initiates. A maximum stress based damage initiation criterion was implemented and the corresponding damage initiation stress was considered as 50 MPa (same as the yield stress of PMMA) and fracture energy of the interface is also assumed to be the same as PMMA GIc = 2.84 N/mm. To describe the tangential behavior of the interface, a frictional contact algorithm is implemented to resist tangential sliding between the surfaces with coeﬃcient of friction as 0.5. Simulations were performed under uniaxial tension and the stress-strain response was compared with the result for the perfect bonding case as shown in Fig 8(a) for tPU = 0.005 mm for the PU thickness. It is to be noted that the initial stiffness corresponding to the contact/friction simulation is not very different (≈ 5% difference) than that of the perfect-bonding case until 0.1% strain. Once the surfaces start opening up at strain > 0.1% due to interfacial separation as shown in Fig 8(b) by the deformed interfacial mesh, the stiffness reduces compared to the perfect bonding case; however the composite continues to carry load due to the geometric interlocking still remaining functional. It is to be noted that the ability to load transfer for the composite is solely governed by the interface strength. The interface opening depends on the fracture criteria de-

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

7

Fig. 7. von Mises stress ﬁeld (MPa) under uniaxial tension in vertical (Y) direction in the unit cell with different PU layer thickness.

ﬁned on the cohesive interaction between the surface. Further, the simulation is continued to estimate the ultimate failure stress for the composites. It is to be noted that no failure criterion is used for the bulk PMMA and PU in the current simulation. It is assumed that the initiation of (local) plastic yielding in PMMA will dictate bulk PMMA failure. For the present study, it was observed that local yielding of PMMA starts at the neck regions of the tab, prior to interfacial failure, as shown in Fig 8(c). Once yielded the bulk PMMA continues to more pronounce yielding eventually leading to failure of the tabs. The reason for the localized tab failure governing the ﬁnal failure of the microarchitecture can be attributed to high co-eﬃcient of friction (μ = 0.4 ) and relatively higher strength of the interface used in the simulation. It is anticipated that (as explained in Malik and Barthelat, 2016), for lower friction value and weaker interface, failure will be governed by interfacial separation, but with a reduction in stiffness of the microarchitecture. Therefore, in the proposed micro-architecture, the geometric interlocking works well in terms of stress transfer and the initial stiffness of the micro-architecture geometry is independent of the interface characteristics. In the following sections, simulations are carried out to determine the low frequency damping characteristic of the proposed micro-architecture design by considering a frequency dependent sinusoidal input. Simulations are carried out considering

a small strain amplitude (0.1%) periodic input to avoid interfacial debonding at higher strain. 4.3. Rate dependent damping response of the composites FE simulations are performed to obtain frequency/rate dependent damping behavior of the composite. As the hysteresis response of viscous PU is both frequency and strain amplitude dependent, simulations are performed under sinusoidal loading over a wide range of frequency and strain amplitude. However, the global strain amplitude is chosen such that local strain within PMMA does not go beyond its elastic limit to avoid any plastic yielding during deformation. Following the standard theory of viscoelastic damping, the stress response within a viscoelastic material under the input of a sinusoidal strain (Eq. (4)) is accompanied with a phase lag δ as given by Eq. (5):

= 0 sin (ωt )

(4)

σ = σ0 sin (ωt + δ )

(5)

The dynamic modulus of the viscoelastic material is given as,

E ∗ = E + iE ,

(6)

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS 8

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

Fig. 8. (a) Uniaxial tensile response of the microarchitecture considering contact/friction based interface modeling; (b) Zoom-in view of the deformed mesh showing interface separation at global strain 0.2%; (c) Stress contour at the neck region of the circular tab, showing initiation of yielding (zoom-in view).

where the in-phase component E is the storage modulus, which designates elastic stiffness of the material and the out-of-phase component E is loss modulus of the material, which contributes to dissipate energy. The tangent of the phase lag δ is deﬁned as the measure of viscoelastic damping in a material. In general for a nonlinear material response, an energy based deﬁnition is used to calculate damping. A typical stress-strain response of the composite under cyclic loading has been shown in Fig. 9 by the hysteresis loop. To characterize the damping response of the proposed interlocking micro-architectured composite, we calculated the speciﬁc damping capacity ψ as

ψ=

WD , WS

(7)

where WD is the dissipated energy over a cycle and WS is the stored energy of the material over a quarter cycle (Martz et al., 1996). Once the response becomes stabilized (by ignoring initial few cycles), area inside the hysteresis loop is the dissipated energy (WD ) and the area OBC (Fig. 9) is computed as the stored energy (WS ) over a quarter cycle. The point B corresponds to the maximum strain state ( 0 ). The loss factor tan δ is then related with

speciﬁc damping as (Martz et al., 1996):

tan

δ=

ψ

2π

(8)

The dynamic modulus, E∗ is determined as σ / 0 (Eqs. (4) and (5)), where σ and 0 are stress and maximum strain, respectively. To estimate the composite’s frequency dependent stiffness, an equivalent storage modulus is computed from the stored energy under the applied strain amplitude ( 0 ) similar to the concept of strain energy density in linear elastic material. This modulus represents the elastic stiffness of the composite. The equivalent storage modulus is given by,

Eeq =

Wu , 0.5 × 02

(9)

where Wu is the area under the unloading curve that represents the strain energy density of the architectured composite. The response of the two unit cells under cyclic loading is shown in Fig. 10. Similar to the quasi-static response, the responses of both the unit cells in two normal directions are very close in terms of equivalent storage modulus and loss factor. Therefore, the following parametric study has been performed using the unit cell-A under Y direction loading.

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

Fig. 9. A representative stress-strain response of the micro-architectured composite under cyclic loading.

The equivalent storage modulus (Eq. (9)) and the loss factor (Eq. (8)) for the micro-architectured composite is plotted in Fig. 11 (a) and (b) for different thickness of the PU layer. Two different strain amplitudes as 0.1% and 1% are considered in the analysis. The effect of strain amplitude is negligibly small in the response. As the thickness of PU layer increases, equivalent storage modulus of the composite drops down, accompanied by an increase in material damping. It is also observed both the composite stiffness and damping are functions of loading frequency, which is a manifestation of the rate-dependent properties of PU. The rate-dependent effect is more pronounced within lower frequencies, 10–50 Hz; beyond 50 Hz the response saturates. This phenomenon can be explained in light of the viscoelastic time constants τ i of PU. As explained in Sain et al. (2015), the relaxation constants of PU had been found using Dynamic Mechanical Analyzer under a frequency sweep set up in which the polymers are commonly tested between 1 Hz-100 Hz frequency. Therefore, the time constants extracted using DMA data are limited within ≈ 0.01 Sec and FE simulations using these constants cannot capture anything beyond ≈ 100 Hz. It can be also noted that the amplitude of the cyclic strain does not signiﬁcantly inﬂuence the stiffness or the damping of the composite.

9

To compare the performance of the proposed microarchitectured composite with reference to the existing materials with multifunctionalities, the predicted rate dependent stiffnessloss value is plotted in the stiffness-loss map given by Chen and Lakes (1993) as shown in Fig. 12 and compared with the WangLake line given by (|E ∗ | tan δ = 0.6 ). The frequency dependent (|E∗ |tan δ ) for PU is also plotted for comparison. It can be noted that the micro-architectured composite has signiﬁcantly improved stiffness with a slight reduction in loss factor over PU. Though the predicted stiffness-loss combinations do not cross the Wang-Lake line at present, the values for the proposed micro-architectured composites are reasonably good when compared with polymer composites (such as glass ﬁber reinforced composites, SiC-InSn and Cast-InSn whose E ∗ tan δ ≈ 0.1 − 1.0 (Chen and Lakes, 1993)). Given the stiffer PMMA does not have any damping (tanδ ≈ 0) and the stiffness of the viscoelastic PU is considerably low E ≈= 40 MPa, in this study and with signiﬁcantly low volume fraction of lossy PU (7 − 44%), the ﬁnal composites show signiﬁcant damping in lower frequencies. Therefore, the present micro-architecture design is indeed a step-forward for designing polymer composite materials with high stiffness and high damping [in comparison with values reported in recent literature; such as for the nanoengineered PU/clay nanocomposite ﬁlled material as studied by Meaud et al. (2014) the product of stiffness-loss modulus value was predicted as 1 (by FE analysis) and experimentally measured as 0.2]. In the future the micro-architecture can be improved by optimizing its critical dimensions to achieve better stiffness-loss combinations. Further, the response can be improved by selecting different constituent materials, such as considering a soft gel with higher damping than PU and the stiffer polymer as Polyvinyl Acitate (PVA). Fig. 13 shows the contour map of the dynamic modulus, loss factor and product of stiffness-loss factor (|E∗ |tan δ ) as a function of thickness of the PU layer and frequency of the sinusoidal loading. Due to the rate dependent character of the PU the composite’s stiffness increases and damping drops down with the increase in frequencies for a ﬁxed thickness. The loss factor tan δ is higher for thicker PU layer and at lower frequencies. At lower frequencies (around 10 Hz), the composite attains its peak value of |E∗ |tan δ about 0.35 for PU thicknesses, approximately tPU = 0.025 to 0.04 mm. However, the value of |E∗ |tan δ decreases due to the decrease in damping for tPU ≤ 0.025 mm and decrease in stiffness for tPU ≥ 0.04 mm. Therefore, to achieve optimal |E∗ |tan δ for the present geometric proportions the thickness can be chosen from Fig. 13(c) for a given loading frequency. The present analysis for different

Fig. 10. Comparison of the equivalent storage modulus and damping response of the micro-architectured composite under cyclic load along vertical (Y) and horizontal (X) directions. The properties correspond to the composite with PU layer thickness of 0.01 mm.

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS 10

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

Fig. 11. (a) Equivalent storage modulus of the composites as a function of frequency for different thickness of PU layer at strain amplitudes of 0.1% and 1.0%. (b) Loss factor (tan δ ) of the composites as a function of frequency for different thickness of PU layer at strain amplitude of 0.1% and 1.0%.

Fig. 12. Stiffness-loss plot of the micro-architectured composite for all thicknesses of PU along with Wang-Lakes line. The responses correspond to a strain amplitude of 0.1% and frequencies from 10 Hz (right) to 100 Hz (left).

thickness is only a preliminary study and a detailed analysis considering all the geometric parameters is required to ﬁnd the optimal parameters. 4.4. Near-isotropic response under impact loading The wave propagation through the proposed microarchitectured material is analyzed under impact loading to validate the near-isotropic behavior of the proposed design under (high) frequency dynamic loading. Taking advantage of symmetry, only a quarter of the actual geometry is modeled, which reduces the computational effort. It is assumed that analyzing only a quarter of the actual geometry is suﬃcient in spite of the microarchitectural features and nonlinearity present in the sense of geometry and materials. The assumption has been justiﬁed by the numerical results given below, which depict circular symmetry. The geometry considered for the analysis has 7 × 4 unit cells in the horizontal and vertical directions respectively with each unit cell of dimension 1 mm × 1.732 mm. A circular hole of radius 0.1 mm is introduced at the top-right corner of the geometry to impose an impulse pressure loading as shown in Fig 14 [following Casadei and Rimoli (2013)]. The top and right boundaries are

free to move along horizontal and vertical direction respectively. Displacements at the left and bottom boundaries are restricted. The input impulsive pressure load used in the simulation has broadband frequency content. The loading time history is shown in Fig 15. In order to avoid reﬂection at the boundary of the structure, the simulation is stopped once the very ﬁrst wavefront reaches the other side of the rectangular domain. The propagating wavefronts are shown in terms of displacement magnitudes in Fig. 16 and von Mises stress ﬁelds in Fig. 17 at different time instants. From Figs. (16) and (17) it is evident that the proposed microarchitecture exhibits circular wavefronts, hence closely replicates the wave propagation characteristic of an isotropic solid. In the structure under investigation, there is no symmetry at 45° angle. The symmetry lines are at 0° and at 60° measured anti-clockwise from downward-vertical line, which would lead to some directional effect. Further, it is evident from Fig. 16(b–d) that the wavelengths of the propagating waves are in the order of length-scale of the micro-architecture. Such wavelengths produce some degree of reﬂected waves from the internal boundaries as can be seen behind the main wavefront in (Figs. 16(b–d) and 17(b–d)). Despite these facts, the isotropic feature is preserved as the wave propagates through the structure, which is a remarkable property of the proposed micro-architecture design. It is clear from Figs. 16 and 17 that even though the wavefront remains circular the peak amplitude of the wave decays rapidly. It may be noted that though the displacements vary smoothly across the boundaries of the elements of the micro-architecture (Fig. 16), the von Mises stress (Fig. 17) shows a jump at those internal boundaries due to vastly different stiffness of the constituent materials. Therefore, the proposed microarchitecture shows near isotropic behavior both at the long-wavelength limit (at zero frequency), as shown in quasi-static simulations (in Section 4.1) and at ﬁnite frequency short-wavelength propagating waves, as shown in this section. 5. Summary and conclusions In this work, an interlocking micro-architecture is proposed for a polymer composite material to achieve combined stiffness and damping properties (product of stiffness and loss factor) of much higher magnitude than the constituent materials. The interlocking building blocks have been developed to have a hexagonal symmetry. Thus, a key novelty in the micro-architecture is that the structural response is near isotropic by virtue of its symmetry unlike the interlocking material systems found in natural, biological, or

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

11

Fig. 13. Contour plot of (left) |E∗ | in MPa, (centre) tan δ , and (right) |E∗ |tan δ as a function of external frequencies and thickness of the PU layer.

Fig. 14. Simulation domain.

Fig. 15. The impulse pressure loading in (left) time and in (right) frequency domain. Here, P0 = 30MPa.

synthetic materials. Simulations have been performed to verify the symmetry of the composite and it has been observed that the composite exhibits high rotational symmetry under both quasi-static and dynamic wave propagation events. As per the authors’ knowledge, such demonstration of near-isotropic behavior in an interlocked design for nonlinear materials (under quasi-static and dynamic impact loading) is a remarkable improvement in the domain of micro-architecture design. The proposed micro-architectured composite consists of the stiffer PMMA building blocks and soft, viscoelastic PU layer in between the building blocks. By using a thin viscoelastic layer of PU at the interface of the building blocks, the damping performance is signiﬁcantly improved. To investigate the performance

of the proposed interlocking design, detailed mechanical analysis has been performed to characterize the response under different loading conﬁgurations. By varying the volume fraction of PU in the architecture from 7% to 30% it is found that the elastic modulus and strength of the composite varies between 55% to 25% of the stiffer material (PMMA) respectively. The resulting stiffness of the composite is considerably high given that it does not use any adhesive material. A contact-friction based ﬁnite element analysis is also performed by considering the interfacial interaction between the two polymer surfaces. It is observed that for lower strain limit, the interfacial interaction does not play a role in the stiffness of the composites and the geometric interlocking governs the load-bearing ability of the micro-architecture. However, at

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS 12

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

Fig. 16. Magnitude of displacement vector (in mm) ﬁeld under impact load at different instants of time.

higher strains, interlocks start opening up, causing the polymers to debond from each other resulting in a stiffness drop and reduced strength (compared to perfect bonding case), which is dictated by the interfacial strength. The damping property of the composite has been assessed under sinusoidal loading with frequency ranging from 10 Hz to 100 Hz, for PU volume fractions ranging from 7% to 30%. By increasing the volume fraction of the PU from 7% to 30%, it is found that the damping of the composite increases from 10% to 50% of PU at 10 Hz and 9% to 45% of PU at 100 Hz. Given the product of stiffness and damping properties (E∗ tan δ ) of the constituent materials are negligibly low, the proposed composite outperforms the constituent materials by several orders of magnitude. By varying several geometric parameters of the proposed micro-architecture the macroscopic properties can be tuned over a wide range. Currently used geometry for the load bearing elements of the micro-architecture are carefully chosen but not obtained from an optimization framework (such as topology optimization). In the future, the relative geometric proportions of the elements of the micro-architecture would be sought to obtain optimal material properties through an optimization scheme. With the advent of 3-D printing technology, fabrications of the proposed microarchitecture can be done in future and tested experimentally to validate the numerical predictions as presented here. The proposed micro-architecture may ﬁnd potential applications as robust shock absorbing material, compliant multi-functional material and lightweight sandwich panels. The concept of interlocking structure with higher symmetry can be useful as a guide for the design of other novel material microstructures leading to signiﬁcant improvement in useful material properties.

Acknowledgments SH acknowledges Miguel HerrÃ!‘ez and Fernando Naya for useful discussions. TS and SG acknowledge Nancy Barr (Michigan Technological University) for a thorough proof-reading of the manuscript.

Appendix A. Details of geometry of the proposed micro-architecture Fig. A.18 shows the (regular) hexagonal arrangement of the circular elements, which are the main load-bearing elements in the micro-architecture. As shown in Fig. A.18, instead of placing the interlocking elements (the triangles) at the point of contact between the two circular gear elements, the triangular elements are placed at the empty space between three adjacent circular gear elements to reduce the voids among them. The side of the base hexagon is taken to be of length, a, and the side of each triangle has been considered to be of length, a/2, aligned at the mid-point of each side of the hexagon. The diameter of the circular heads at the vertices of the triangles is chosen to be around, a/8. A layer of soft polymer of thickness, tPU , surrounds the triangular building blocks and introduces damping in the composite. Since, the circular heads of the triangular block provide interlocking mechanism, these are the key components in the design upon which the strength of the composites greatly depends on. It is evident (and also shown in our ﬁnite element analysis) that the narrow neck region connecting the circular head and the triangle would be one of the weakest links in terms of stress concentration (under tension). In order to achieve better strength at this region, the circular head is shifted more in-

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

ARTICLE IN PRESS

JID: SAS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

13

Fig. 17. von Mises stress ﬁeld (MPa) ﬁeld under impact load at different instants of time.

Fig. A18. Schematic diagram showing hexagonal arrangement of circular gear elements and triangular elements.

A.192 .

ward to the triangle as shown in Fig. To avoid stress concentration, all the sharp corners appearing at the gear teeth and triangle-circular head junctions are made smoother by an arc of

2 Remark: The geometric proportions of the elements of the micro-architecture used in the present work are not obtained via any optimization strategy from the set of all possible combinations. However, a series of simulations are done to check the ensuing stiffness for a wide variety of geometric proportions, in order to reach the currently used option. The pursuit of the optimal geometric proportions of the elements of the micro-architecture would be a part of future study.

Fig. A19. Dimensions (to the scale) of the interlocking-system in the microarchitecture. All dimensions are in mm.

radius 0.02 mm. Fig. A.19 shows the micro-architecture design by considering the hexagon with each side as 1 mm, and the thickness of the soft polymer layer as 0.01 mm. For brevity, the size of the base hexagon (a) and the triangular blocks (a/2) are kept constant throughout this study. In order to increase the thickness of the soft polymer layer, (tPU ), the size of the circular gear element has been reduced accordingly (from the size of the circles shown in Fig. A.18). The choice of circular gear element inherits

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

ARTICLE IN PRESS

JID: SAS 14

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15

and B¯ is the deviatoric component of B¯ as given by, B¯ = B¯ − tr (B¯ /3 )

(B.3)

The isochoric deformation is developed by neglecting the volume change as: n F¯ = J −1/3 Fn

(B.4) L−1

In Eq. (B.1), is the inverse Langevin function, given by its Padé approximation,

L−1 (ξ ) = ξ

3 − ξ2 1 − ξ2

(B.5)

and (B.1), λchain is

the stretch on each chain in the network as given by λchain = I1 /3, with I1 = tr (B¯ ), the ﬁrst invariant of B¯ . References Fig. A20. Uniaxial tensile response comparison between unit cell and a 10 × 10 RVE.

isotropy in the geometry. By choosing some other gear elements such as elliptical, we may need triangular interlocking keys of different size. The resulting system would have less symmetry than that of the current regular hexagon. However, that might be useful for some special applications where directional features are preferred over higher symmetry. It is worth mentioning that the size of the micro-architecture can be much smaller than what is currently used as long as it is manufacturable and does not interfere with the inherent length-scales (see Maranganti and Sharma, 2007; Ghosh et al., 2013) of the constituent polymeric materials. We note that the hexagonal arrangement of circles is known to yield the densest packing of circles in the plane with a packing √ density of 16 π 3 ≈ 0.907. Therefor when the soft polymer layer is very thin or absent (i.e. tPU → 0 or tPU = 0) most of the material volume is occupied by the stiffer material. Hence, for a very thin layer of soft polymer, the proposed micro architecture should retain a signiﬁcant portion of the stiffness of the material that is used in the circles. In the predictive simulations we have used a representative unit cell for analyzing the mechanical behavior of the microarchitecture. Choice of the unit cell with the periodic boundary conditions (PBC) applied can actually represent the inﬁnite geometry; we have compared the results of a uniaxial tensile simulation between the unit cell with the PBC applied and a 10 × 10 RVE as shown in Fig A.20. It is seen that both the responses are identical; thereby we considered the unit cell for further analysis. Appendix B. Arruda-Boyce model Following the well known Arruda-Boyce 8-chain potential, the stress-strain response of an elastomer is given by Arruda and Boyce (1993),

n1 kθ T = 3J n

√ N

λchain

L

−1

λchain √ N

B¯

(B.1)

with n1 being the chain density (number of molecular chains per unit reference volume) of the underlying macromolecluar network, k the Boltzmann’s constant, θ the absolute temperature, currently taken as constant at room temperature and J = det (F ). N √ represents the length of the chains in the polymer network and N represents the limiting stretch of each chain. Concisely G0 = n1 kθ , the rubbery modulus or initial stiffness. In Eq. (B.1), B¯ is the isochoric right Cauchy Green strain given by n n B¯ = F¯ F¯

T

(B.2)

Abaqus/Standard, 2015. Theory manual, V6.14. http://abaqus.software.polimi.it/v6. 14/books/stm/default.htm. Agnese, F., Scarpa, F., 2014. Macro-composites with star-shaped inclusions for vibration damping in wind turbine blades. Compos. Struct. 108, 978–986. Alderson, A., Alderson, K., 2007. Auxetic materials. In: Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 221, pp. 565–575. Arroyo, M., Belytschko, T., 2004. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-born rule. Phys. Rev. B 69, 115415. Arruda, E.M., Boyce, M.C., 1993. Evolution of plastic anisotropy in amorphous polymners during ﬁnite straining. Int. J. Plast. 9, 697–720. Ashby, M., 1993. On the engineering properties of materials: overview no. 80. Acta Metall. 37, 1273–1293. Barthelat, F., Tang, H., Zavattieri, P., Li, C.-M., Espinosa, H., 2007. On the mechanics of mother-of-pearl: a key feature in the material hierarchical structure. J. Mech. Phys. Solids 55 (2), 306–337. Bhattacharya, K., 2003. Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect. Oxford Series on Materials Modelling. OUP Oxford. Casadei, F., Rimoli, J., 2013. Anisotropy-induced broadband stress wave steering in periodic lattices. Int. J. Solids Struct. 50 (9), 1402–1414. Chen, C., Lakes, R., 1993. Viscoelastic behaviour of composite materials with conventional-or negative-Poisson’s-ratio foam as one phase. J. Mater. Sci. 28 (16), 4288–4298. Dyskin, A., Estrin, Y., Kanel-Belov, A., Pasternak, E., 2003. A new principle in design of composite materials: reinforcement by interlocked elements. Compos. Sci. Technol. 63 (3), 483–491. Estrin, Y., Dyskin, A., Pasternak, E., 2011. Topological interlocking as a material design concept. Mater. Sci. Eng. 31 (6), 1189–1194. Fleck, N.A., Deshpande, V.S., Ashby, M.F., 2010. Micro-architectured materials: past, present and future. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 466, pp. 2495–2516. Funk, N., Vera, M., Szewciw, L.J., Barthelat, F., Stoykovich, M.P., Vernerey, F.J., 2015. Bioinspired fabrication and characterization of a synthetic ﬁsh skin for the protection of soft materials. ACS Appl. Mater. Interfaces 7 (10), 5972–5983. Ghosh, S., Arroyo, M., 2013. An atomistic-based foliation model for multilayer graphene materials and nanotubes. J. Mech. Phys. Solids 61 (1), 235–253. Ghosh, S., Kumar, A., Sundararaghavan, V., Waas, A.M., 2013. Non-local modeling of epoxy using an atomistically-informed kernel. Int. J. Solids Struct. 50 (19), 2837–2845. Ghosh, S., Sundararaghavan, V., Waas, A.M., 2014. Construction of multi-dimensional isotropic kernels for nonlocal elasticity based on phonon dispersion data. Int. J. Solids Struct. 51 (2), 392–401. Gorb, S.N., 1999. Evolution of the dragonﬂy head-arresting system. In: Proceedings of the Royal Society of London B: Biological Sciences, 266, pp. 525–535. Gorb, S.N., 2008. Biological attachment devices: exploring nature’s diversity for biomimetics. Philos. Trans. R. Soc. London A 366 (1870), 1557–1574. Katti, K.S., Katti, D.R., Pradhan, S.M., Bhosle, A., 2005. Platelet interlocks are the key to toughness and strength in nacre. J. Mater. Res. 20 (05), 1097–1100. Khandelwal, S., Siegmund, T., Cipra, R., Bolton, J., 2014. Scaling of the elastic behavior of two-dimensional topologically interlocked materials under transverse loading. J. Appl. Mech. 81 (3), 031011. Kim, M., Swan, C., Lakes, R.S., 2002. Computational studies on high-stiffness, high-damping sic–insn particulate reinforced composites. Int. J. Solids Struct. 39, 5799–5812. Kooistra, G.W., Deshpande, V.S., Wadley, H.N., 2004. Compressive behavior of age hardenable tetrahedral lattice truss structures made from aluminium. Acta Mater. 52 (14), 4229–4237. Lakes, R.S., 2002. High damping composite materials: effect of structural hierarchy. J. Compos. Mater. 36, 287–298. Malik, I.A., Barthelat, F., 2016. Toughening of thin ceramic plates using bioinspired surface patterns. Int. J. Solids Struct. 9798, 389–399. Maranganti, R., Sharma, P., 2007. Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 98, 195504.

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

JID: SAS

ARTICLE IN PRESS

[m5G;July 10, 2017;14:54]

S. Haldar et al. / International Journal of Solids and Structures 000 (2017) 1–15 Martz, E.O., Lakes, R.S., Park, J.B., 1996. Hysteresis behaviour and speciﬁc damping capacity of negative Poisson’s ratio foams. Cell. Polym. 15 (5), 349–364. Meaud, J., Sain, T., Yeom, B., Park, S.J., Shoultz, A.B., Hulbert, G., Ma, Z.-D., Kotov, N.A., Hart, A.J., Arruda, E.M., et al., 2014. Simultaneously high stiffness and damping in nanoengineered microtruss composites. ACS Nano 8, 3468–3475. Metrikine, A.V., Askes, H., 2006. An isotropic dynamically consistent gradient elasticity model derived from a 2d lattice. Philos. Mag. 86 (21–22), 3259–3286. Microstructures create reversible attachment:, Dragonﬂy http://www.asknature.org/ strategy/1df0f2b33d166b85b3e9064197d8e71c, Accessed: 2016-01-29,2016. Mirkhalaf, M., Dastjerdi1, A.K., Barthelat, F., 2014. Overcoming the brittleness of glass through bio-inspiration and micro-architecture. Nat. Commun. 5, 1–9. Murray, G. J., Gandhi, F., 2013. Auxetic honeycombs with lossy polymeric inﬁlls for high damping structural materials. J. Intell. Mater. Syst. Struct., 1045389X13480569. OLIVEIRA, M.A.D.M., TORRES, C.P., GOMES-SILVA, J.M., CHINELATTI, M.A., MENEZES, F.C.H.D., BORSATTO, M.C., 2010. Microstructure and mineral composition of dental enamel of permanent and deciduous teeth. Microsc. Res. Tech. 73, 572–577. Ravirala, N., Alderson, A., Alderson, K., 2007. Interlocking hexagons model for auxetic behaviour. J. Mater. Sci. 42 (17), 7433–7445.

15

Sain, T., Meaud, J., Yeom, B., Waas, A.M., Arruda, E.M., 2015. Rate dependent ﬁnite strain constitutive modeling of polyurethane and polyurethane–clay nanocomposites. Int. J. Solids Struct. 54, 147–155. Simo, J., 1987. On a fully three-dimensional ﬁnite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60 (2), 153–173. Valashani, S.M.M., Barthelat, F., 2015. A laser-engraved glass duplicating the structure, mechanics and performance of natural nacre. Bioinspir. Biomim. 10 (2), 026005. Veedu, V.P., Cao, A., Li, X., Ma, K., Soldano, C., Kar, S., Ajayan, P.M., Mehrad, N., 2006. Multifunctional composites using reinforced laminae with carbon-nanotube forests. Nat. Mater. 5, 457–462. Zhang, P., Heyne, M.A., To, A.C., 2015. Biomimetic staggered composites with highly enhanced energy dissipation: modeling, 3d printing, and testing. J. Mech. Phys. Solids 83, 285–300. Zhou, J., Huang, R., 2008. Internal lattice relaxation of single-layer graphene under in-plane deformation. J. Mech. Phys. Solids 56 (4), 1609–1623.

Please cite this article as: S. Haldar et al., A novel high symmetry interlocking micro-architecture design for polymer composites with improved mechanical properties, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.06.030

A Novel Technique A Novel Technique for High ...