Effects of cohesive interfaces and polymer viscoelasticity on improving mechanical properties in an architectured composite

Accepted Manuscript

Effects of cohesive interfaces and polymer viscoelasticity on improving mechanical properties in an architectured composite Muhammed R. Imam, Trisha Sain PII: DOI: Reference:

S0020-7683(18)30403-7 https://doi.org/10.1016/j.ijsolstr.2018.10.008 SAS 10142

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International Journal of Solids and Structures

Received date: Revised date: Accepted date:

1 May 2018 31 August 2018 10 October 2018

Please cite this article as: Muhammed R. Imam, Trisha Sain, Effects of cohesive interfaces and polymer viscoelasticity on improving mechanical properties in an architectured composite, International Journal of Solids and Structures (2018), doi: https://doi.org/10.1016/j.ijsolstr.2018.10.008

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Effects of cohesive interfaces and polymer viscoelasticity on improving mechanical properties in an architectured composite Muhammed R. Imam, Trisha Sain∗

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Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University Houghton, Michigan, United States, 49931

Abstract

Improving the functionality of composite materials is a key requirement for various aerospace, auto-motive, sports and defense applications. The trend

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is to identify mechanisms, design, constituents, and, preferably, the combination of all of them that can result into better mechanical properties in

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the conflicting domain of interest (e.g. high stiffness and high damping or high stiffness and high toughness) without adding much complexity in the

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analysis and design. In the present work, a naturally inspired “interconnection” is considered within a composite material made of dissimilar mechanical

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properties with an objective to improve stiffness, toughness, and wave attenuation capability. The computational study showed that creation of weak interfaces along with the “interconnection” works two-fold in terms of me-

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chanical property improvement. The interconnection provides an additional load-transfer mechanism through contact-friction between two dissimilar materials, whereas the cohesive (weak) interfaces results in higher toughness ∗

Corresponding author, email:[email protected]

Preprint submitted to International Journal of Solids and Structures

October 11, 2018

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(area under the stress-strain curve) of the material promoting distributed interface failure and delaying bulk material yielding. It was further identified

proposed composite. Keywords:

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that the presence of weak interfaces acts better in wave attenuation for the

composite, viscous polymer, interface, cohesive zone, finite element model, wave mitigation, toughness, bio-inspired

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1. Introduction

The ever-increasing demand for identifying materials with better functionality drives the current trend in material’s research. Several applications in the aerospace, defense, and sports industries require the materials to

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have high stiffness, decent damping, high toughness, and impact resistance, many of which are extremely difficult to achieve within a single material [14].

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Therefore, architectured materials or, more precisely, architectured composite materials find a scope by introducing architectures at different length

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scales (i.e. micro, nano or macro) to obtain material’s conflicting properties [12, 4, 7, 6]. Researcher have reported expanding the boundaries of material

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property space in several domains, which was hitherto impossible [3]. For example, researchers have been trying to gain simultaneously high stiffness

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and damping in a composite for impact protection applications, using various techniques such as varying geometrical arrangement and properties of the constituents, introducing hierarchy or self-similarity within the material architecture at different length scales [17, 36, 18, 24]. Brodt and Lakes [8] investigated metal composites with a relatively high damping matrix com2

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bined with high stiffness tungsten inclusions to demonstrate a combination of high stiffness and damping. Viscoelastic materials with structural hierarchy

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also have been proposed to enhance stiffness and damping in [19]. Meaud et al. [21] developed an integrated computational approach to fabricate 3D

CNT nanotruss layers filled with a viscous polyurethane(PU), to design high stiffness and high damping polymer composites.

The concept of layer-by-layer materials in the composite literature has

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been used to improve the toughness and stiffness of polymer based composites. The advantage of using layered materials has been observed and

explained with reference to natural examples such as bone, teeth, nacre, and mollusk shells. Brick and mortar architecture of nacre has been widely explored to achieve a combination of high stiffness, toughness [31, 29, 11, 5],

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band gap of elastic waves [10, 34], and damping [35] in man-made materials. Helicoidal organization of fiber layers in stomatopod dactyl club has been

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investigated to design composites with reduced damage propagation during impact [33, 16]. Recently, deformable glass with enhanced toughness is

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fabricated by laser-engraved, polyurethane-filled wavy architecture within it [23, 30, 22]. Layered composite materials consisting of alternate layers of stiff

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and soft constituents can simultaneously achieve an optimum combination of stiffness, damping, and energy dissipating nature [9], provided the choice of the constituents and thickness of different layers is done optimally. More-

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over, development of analytical formulae and fabrication are easier for layered systems due to simplicity or absence of microstructure. Meaud et. al. [20] analyzed layered composites made of parallel layers of alternate stiff and lossy polymers subjected to loading from arbitrary directions and demonstrated

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that the high stiffness and damping are due to high normal and shear strains in the lossy material. In another finite element simulations-based study by

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Sain et al. [26], simultaneously high stiffness and damping was achieved by sandwiching a thin PU layer between wavy steel sheets. Tanaz et. al. pro-

posed a concept of mitigating stress waves in the layered material system by

tuning the material and geometric properties of constituent layers [25]. One of the assumptions on designing layered material system is the perfect bond-

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ing (cohesive, ionic or Vanderwaals) between alternating layers, which can be inadequate if there is a high stiffness mismatch between them [28]. While analyzing the wave mitigation capability, one important note to remember

is to withstand the initial impact for which these materials need to be adequately stiff, simultaneously being capable of dissipating the impact energy

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without having a sudden fracture failure. Therefore, the materials need to have high stiffness, toughness, and impact resistance at the same time. To

for decades.

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achieve all these properties together in a real material has been a challenge

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In the present work, we demonstrate how the interplay between material building blocks, geometry of the architecture, and properties of constituents

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influence the stiffness, toughness, and impact resistance in a biologically inspired interconnected architectured material. Biological materials and structures have been a source of inspiration for solving challenges in material sci-

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ence and mechanics [13]. The objective of the present study is to identify the pathway through computational analysis, for improving the aforementioned mechanical properties in a composite material. To compare how much improvement can be made possible, the baseline is considered as the layered

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material system since it is known for impact resistance and better toughness. The material architecture considered in the present work is motivated by

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natural examples of geometrically interlocking elements and materials [15]. Nature enables numerous species of animals and plants with a wide range of

attachment devices such as a mechanism that allows an organism to attach to a substrate surface. Gorb [15] mentioned eight fundamental attachment

devices that involve physical effects such as chemical bonding, viscous forces,

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mechanical interlocking, and friction. Among various mechanisms of attachment, a typical “hook” like interlocking mechanism as shown in Figure 1 (a)

a)

b)

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is widespread.

Figure 1: a.) Hook type attachment system; b.) temporary locking mechanism between fore- and hindwings in sawfly (Hymenoptera) [15].

Inspired by the “hook” type attachment, the present work considered 5

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an architecture that includes the geometric ”interconnection” as shown in Figure 2 (a). This interconnection is assumed to act better in load transfer

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between two dissimilar materials by providing an additional restraint against horizontal and vertical sliding by virtue of the interlocking mechanism. In

the present study, the composite consists of a relatively stiffer material as

the backbone and a soft visco-elastic elastomer as filler. It is assumed that the stiffer backbone material was predominantly providing the stiffness of the

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composite and the interconnection assuring load transfer between two materials during the event of interfacial sliding. The composite’s performance has

been evaluated computationally in terms of its quasi-static stiffness, toughness, and impact mitigation capability under dynamic loading. In Sec. 2 and 3, the geometry of the architecture and the materials considered for the

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analysis are detailed, respectively. In Sec. 4, the details of the finite element simulations are explained followed by the results in Sec. 5. Finally,

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the study has been concluded by identifying the mechanisms, combination of materials, and geometry that can eventually guide future multifunctional

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materials design for achieving simultaneously high stiffness, toughness, and impact mitigation capabilities.

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2. An interconnected bio-inspired architecture

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As mentioned earlier in the present work, an interconnected architecture

is considered within the polymer composites, with the intention that the additional “hook” type design would act as a locking mechanism against sliding between two materials and result in higher stiffness compared to conventional layered composite. In addition, creation of a large number of weak 6

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a)

b)

Figure 2: (a) Geometry of the interconnected architecture in the polymer composite in a 2D representation; (b) A representative unit cell for the geometry showing the characteristic

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dimensions that describe the geometry (58% volume fraction of the stiff component).

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interfaces and allowing them to fail in a distributed manner will also increase toughness of the composite. The interfaces are assumed to be weaker

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than the stiffer material; forcing them to fail under external loading would introduce a gradual and more controlled failure mechanism within the ar-

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chitecture rather than the localized yielding of the stiffer material, which is more rapid. Figure 2 (a) shows a 2D representation of the proposed interconnected architecture design within the composite. The “grey” portion

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consists of the stiff material and frames the backbone architecture, whereas the “green” material consists of the soft viscous filler and is confined by the stiff part. The interface is basically representing the contacting surface (with zero thickness) between the stiff phase (grey) and the soft material phase

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(green) that has been modeled using mechanical contact and cohesive zone in the later sections of the article. A representative unit cell of (W) 5mm by

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(H) 10mm, with the characteristic dimension is as shown in Figure 2 (b). An intermediate volume fraction (58%) is chosen for the stiff constituent such that the resulting material remains sufficiently stiff and reasonably ductile at the same time. Inspired by the “hook” like interlocking mechanism shown in

Figure 1, the stiff members are rotated inward by 180◦ . Therefore, the inter-

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connected stiff and soft members stay connected without use of any extra adhesive at the interfaces. In the design, L1 , L2 , L3 , w1 and t1 are considered as independent variables and they had been varied in the parametric optimization step, while keeping the volume fraction of the materials constant. As a result, the other two dimensions t2 and t3 become dependent variables

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as: t2 = 0.5H − L1 − L2 − L3 − 0.5(t1 + t3 ) and t3 = W − 2w1 respectively. It is seen that the ultimate strain does not depend on the fillet radius (r) as

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stress concentrations around the corners are avoided incorporating the fillets. Therefore, r is not considered as a variable in this study. It is also observed

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that the architecture is highly anisotropic with thin, continuous segments of the stiffer constituents along the X direction, making it relatively brittle for

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X directional loading. Therefore, the composite’s mechanical properties are computed along the Y direction unless otherwise specified. The interconnections provide resistance against normal separation and tangential sliding at

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the interfaces between the stiff and soft constituents. Moreover, splitting the contacting surfaces (rather than being straight) with the help of interconnections increases contact conformability. Hence the surface cohesion increases due to more frictional resistance generated across the interfaces. Therefore,

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the stiffness and the toughness of the composite material increased for the proposed architectured composite compared to a layered stiff-soft material

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as shown later. 3. Materials

As mentioned, for the backbone materials, materials with high stiffness

such as steel, aluminum and polymethylmethacrylate (PMMA) are consid-

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ered in the present study. These stiffer materials are modeled as elasticplastic, considering J2 flow theory of plasticity with Von Mises yield condition. ABAQUS/Standard [1] material module is used to model their elasticperfectly plastic response. The material parameters for the stiff phases, used in the FE simulations are given in Table 1. Local yielding of the stiff mate-

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rial has been assumed as a failure mode for the bulk and the simulations are terminated right at that point. It is to note that rate dependent behavior of

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PMMA is ignored in the analysis. For the filler materials, a class of softer viscoelastic elastomers such as

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Table 1: Parameters for the stiffer materials

Materials

E(GP a) ν

σy (M P a)

Steel

200

0.30

340

Aluminum 69

0.30

95

PMMA

0.30

39

2.0

Polyurethane (PU) and a nanocomposite gel (as presented in [32]) are considered in the present study. These classes of soft polymers are highly stretchable and show nonlinear response even at small deformation. In the present 9

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study, these materials are assumed to be responsible for dissipating energy because a significant amount of viscous dissipation occurs locally within

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the polymer. Therefore, a large deformation hyper-viscoelastic constitutive model has been used to predict the response of these polymers following [27]. For the sake of clarity, the important equations of the constitutive model are

restated. A 1-D schematic representation of the constitutive model is shown

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in Figure 3. The long-term response of the material is modeled using the

Figure 3: 1-D schematic representation of the constitutive model for the soft viscous

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polymers.

hyperelastic strain energy density function as represented by the nonlinear

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spring in Figure 3. Also, the well known incompressible Arruda-Boyce energy density function has been used, with the stress-strain response of an

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elastomer given by [2],

√   n1 kθ N −1 λchain ¯ 0 √ T = L B 3J λchain N n

(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 a constant at room temperature and J = det(F). N represents the length of the chains in the 10

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polymer network and

√

N represents the limiting stretch of each chain. Con-

cisely G0 = n1 kθ, the rubbery modulus or initial stiffness.

T

¯n ¯ =F ¯ nF B

¯ 0 is the deviatoric component of B ¯ as given by and B ¯0 = B ¯ − tr(B/3) ¯ B

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¯ is the isochoric right Cauchy Green strain given by In Equation 1, B (2)

(3)

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

(4)

In Equation 1, L−1 is the inverse Langevin function, given by its Pad´e approximation

3 − ξ2 (5) 1 − ξ2 is the stretch on each chain in the network as given by

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L−1 (ξ) = ξ

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and (1), λchain p ¯ the first invariant of B. ¯ λchain = I1 /3, with I1 = tr(B),

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A classical Zener-type configuration is used to model the rate dependent viscoelastic deformation, where multiple relaxation mechanisms are considered by several linear spring-dashpot combinations (Figure 3). In a finite

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deformation setting, each of these spring-dashpot branches is subjected to

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the deformation gradient F, which is multiplicatively decomposed as F = Fe Fv

(6)

where, Fe and Fv are the elastic and viscous part of F. Following which, the velocity gradient can be written as ˙ −1 = Le + Fe Lv Fe−1 L = FF 11

(7)

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where e Le = F˙ Fe−1

(8)

v

Lv = F˙ Fv−1

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and where (9)

As is standard in the plasticity literature, assuming the material to be isotropic and thereby the viscous spin being zero, we get

and

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Lv = Dv + Wv = Dv ; as Wv = 0

v Dv = F˙ Fv−1

Therefore, the flow rule becomes

(11)

(12)

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v F˙ = Dv Fv

(10)

A simple linear viscous rate equation has been further assumed to model the

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viscous shear strain rate as given by

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γ˙d v Dv = √ Tv0 2τv

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The equivalent deviatoric stress (τv ) is given by  1/2 1 e0 e0 (τv ) = T T 2

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The reference shear strain rate γ˙d v is given by √ 2(τv ) v γ˙d = 2Gtr

(13)

(14)

(15)

The Cauchy stress in the linear spring of each branch is given by the Hookean spring model as Te =

1 Re−1 GEe ReT (detFe ) 12

(16)

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where G is the shear modulus of the spring; Fe = Re Ue is the polar decomposition of the elastic deformation gradient and the elastic Hencky strain is

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defined by Ee = ln(Ue ). Finally, the total Cauchy stress for the material is given by the sum of stresses from each branch n

T=T +

nv X

Tei

(17)

i=1

where Tn is the Arruda-Boyce stress component as given by 1 and Tei is

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the elastic stress carried by the ith viscoelastic branch, as given by 16. The

constitutive model is implemented in ABAQUS/Explicit by writing a user material subroutine VUMAT for simulating the response of the soft, viscous filler within the composite.

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3.1. Model parameter estimation

Theoretically, a viscous polymer has multiple relaxation mechanisms in

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its macromolecular configuration, that are characterized by a number of relaxation times tr . The theoretical model is limited to consider only the physically measurable characteristic relaxation times (tr ) that are important for

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the quasi-static and dynamic loading cases considered in the present study. In the constitutive model described before, each of the spring-dashpot as-

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semblies as shown in Fig 3 is characterized by a pair of material parameter (G, tr ), where G is the shear modulus of the spring and tr is the relaxation

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time for the dashpot, which is proportional to

1 , η

where η is the measure

of viscosity. In other words, the relaxation time represents the rate of viscous flow within the material. To use the constitutive model for response predictions, we need to estimate the material parameters (G, tr )’s and the long -term modulus characterizing the nonlinear spring. In [27] the authors 13

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have presented a numerical technique to estimate the model parameters from Dynamic Mechanical Analysis experiments for a PU sample. Following a lin-

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ear viscoelastic equivalence of the finite deformation model, the time varying shear modulus of the material can be written in terms of Prony series approximation as ∗

G (t) = G0 +

nv X

Gi exp(−t/tri )

i=1

in the time domain and as

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nv X Gi Iωtri G (ω) = G0 + 1 + Iωtri i=1

(18)

∗

(19)

in the frequency domain. Using the above linearized temporal modulus, the viscoelastic parameters can be estimated through a least-square curve fitting

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as explained in [27]. In the present study, we have used the parameters for PU as one of the softer fillers as determined in [27] and reported in Table 2.

1

2

3

4

5

tr (sec)

22.95

1.12

0.16

0.023

0.0023

G(MPa)

1.85

4.04

15.04

43.28

107.45

Go = nkθ(MPa)

1.85

N

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i

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Table 2: Material Properties for Polyurathene

The other soft material used in the present study is a silica nanoparticle

reinforced nanocomposite gel (NanoGel) as experimentally characterized by [32]. The NanoGel has a quasi-static stiffness of 0.07 MPa, which is nearly 55 times lower than the PU considered. The constitutive model as explained 14

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before is used to model the NanoGel’s mechanical response. To determine the relaxation parameters, the experimental stress relaxation data from [32] has

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been used. The relaxation parameters (G and tr ) obtained by the least-square curve fitting of Eqn. 18 are reported in Table 3. To validate the predictability of the constitutive model and accuracy of the estimated material parameters for the NanoGel, uniaxial tensile simulation has been performed under a

quasi-static condition and compared with experimental data. As seen in

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Figure 4, the model can predict experimental stress-strain response up to 30% strain for the material constants listed in Table 3.

Table 3: Material Properties for nanocomposite gel(NanoGel)

1

tr (sec)

0.50

G(MPa)

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i

0.03

2

3

30.0

65.0

0.00783

0.000195

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N

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Go = nkθ(MPa) 6.5 × 10−3

4. Methods: finite element simulations for response predictions

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To predict the response of the proposed interconnected composite for

various stiff-soft material combinations, a series of finite element (FE) simu-

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lations are performed using the commercial package ABAQUS(Abaqus 2016). A 2D plane strain model of the representative unit cell as shown in Fig 2 (b) is considered for the analysis. In the case of composite materials with a repeating architecture, a representative unit cell corresponds to the repeat unit in which the stress distribution represents a spatial snapshot of the 15

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0.01 0.005 0

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0.015

Experiment (Yang et al., 2013) Model Prediction

0

0.1

0.2

0.3

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Stress (MPa)

0.02

Strain

Figure 4: Uniaxial quasi-static response of the NanoGel: comparison between FE predicted data and experimental measurement.

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stress profile, which repeats within the entire volume of the material and far away from the physical boundary. Periodic boundary conditions have been

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applied at the edges of the unit cell to represent the repeating nature of the architecture. In ABAQUS, the model has been meshed by using 4-noded

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plain strain elements (CPE4R). A mesh convergence study was performed to obtain a converged mesh with the number of elements being ≈ 96, 000.

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Simulations are performed by combining each of the stiff materials with all of the soft fillers as mentioned later. It is important to note that the choice

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of the representative volume element is only limited for the quasi-static analysis; for simulations under dynamic impact loading a larger 10 × 5 geometry

has been considered to take into account the reflections and transmittance of the propagating wave across the finite number of material interfaces, as described later in Sec. 5.4. 16

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5. Results and Discussion 5.1. Parametric study of the design based on mechanics of deformation

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FE analysis were performed on the RVE considering “no-slip” condition across the interfaces and choosing the stiff material as PMMA and the soft

material as PU, to do a parametric optimization for the design. Based on

the mechanics of deformation and stress transfer within the RVE, the critical

dimensions had been re-iterated while keeping the volume fractions of the

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materials constant with the goal to increase the stiffness (E) and the strain at which local yielding (y ) of PMMA initiates. It is to be noted that the initiation of local yielding of the stiffer material is considered as the failure criteria for the analysis. The rate dependent behavior of the stiff materials

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is not considered in the present study. In the present study, toughness refers to the deformational energy that is computed as the area under the stress-

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strain curves and it is entirely different from fracture toughness. The volume fraction of PMMA (vf ) was 58% for the unit cell (denoted now on as PD1) as shown in Fig 2 (b) and kept constant for all the design. Thus the problem was

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formulated as: maximize {E, y } as a function of F(L1 , L2 , L3 , w1 , t1 ), with constraints as (vf = constant). Quasi-static uniaxial tension simulations

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were done at a strain rate of 0.2%/sec to obtain E and y . The stressstrain curve and the Von Mises stress distribution are plotted in Figure 5(a)

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and Figure 5(b) respectively for one typical design realization (referred to as PD1). The stiffness for this particular design was computed as 270 MPa and yield strain as 3%. As obvious from the architecture, the majority of the axial load was carried by the vertical PMMA sections and yielding started due to localization along them. Therefore, the dimensions were re-iterated to 17

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6

4

2

0

0

0.02 0.04 Engineering Strain

a)

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Engineering Stress (MPa)

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b)

Figure 5: (a) Uniaxial tensile stress-strain response of PD1 unit cell design under quasistatic condition; (b) Von Mises stress contour showing local yielding of PMMA within the

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unit cell (PU was switched off for better visualization).

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distribute stresses more effectively across the stiffer sections. Table 4 shows how the various independent dimensions are varied across designs PD2 to

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PD5 to increase the stiffness and yield strain of the composite while keeping the volume fractions the same; Fig 6 (a) shows how the different parametric

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designs were obtained with the adoption of parameters as listed in Table 4. For each of these design realizations, the quasi-static stiffness and the

global yield strain for the unit cells are calculated and plotted in Fig 6 (b).

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Since increasing the toughness of the composite is one of the objectives for the present study, we considered PD4 as the best one, where the toughness was maximized among all the designs. Therefore in the subsequent analysis, unit cell PD4 is considered. To compare the properties with layered geometry,

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Table 4: Design parameters for the parametric optimization

Design L1 (mm) L2 (mm) L3 (mm)

w1 (mm) t1 (mm)

PD1

1.5

1.5

1.2

2.3

PD2

1.5

1.9

0.5

2.3

PD3

1.1

1.9

0.5

2.4

PD4

0.5

2.13

1.0

2.4

PD5

0.5

0.5

1.2

2.45

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0.4 0.7 0.8

0.8

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1.1

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Stiffness (GPa)

0.5

a)

0.4

PD1

0.3

PD2

0.2

PD3 PD4

0.1

0.01

PD5 0.02

0.03 0.04 Yield Strain

0.05

0.06

b)

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Figure 6: (a) Parametric design PD1 to PD5 (b) Stiffness-yield strain plot for all the

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parametric design.

simulations have been conducted considering “Voigt” and “Reuss” configu-

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ration for same volume fraction of the constituents. It is noted that PD4 demonstrates lower stiffness but higher yield strain than the corresponding “Voigt” geometry, whereas the stiffness is four times higher than the “Reuss” configuration. While the present parametric study shows an effective way of altering the design with improved quasi-static properties, a rigorous opti19

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mization study to obtain a better design is still needed. Fig 7 (a) shows the uniaxial tensile response for PD4 considering steel,

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aluminum, and PMMA respectively as stiffer material combining with PU as the softer material; Fig 7 (b) shows the response of the same PD4 when the softer material is considered as the NanoGel. It is important to note that

the stiffer material’s property does not influence the composite’s stress-strain behavior, when the softer phase is extremely soft (in this case, the NanoGel)

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compared to the stiff phase (as seen in Fig 7 (b)). The reason behind this

as identified by analyzing the strain contour for all three material combinations with NanoGel as plotted in Fig 8. Since the softer phase is extremely soft compared to the stiff one, the deformation remained constrained mostly within the soft material volume. Thus the stiffer materials do not contribute

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any stress transfer and hence the stiffness of the composite remains unaltered

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with changes in stiffer material properties. 5.2. Cohesive zone model for the material interfaces As evident in Fig 2 (b), the interconnection between the stiff and soft

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material introduces multiple interfaces within the RVE; many of which are non-straight. Under static loading, these bi-material interfaces are respon-

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sible for controlling the toughness of the composite. If the interfaces are weaker than either of the bulk materials, significant energy gets dissipated to

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separate them and thus result in higher toughness of the composite. Assumption of perfect bonding between two materials across the interface therefore underestimates the toughness and overestimates the stiffness of the material. Moreover, under dynamic impact loading, these bi-material interfaces act as wave-reflectors and significantly decay the transmitted wave amplitude. 20

PMMA-PU Al-PU Steel-PU

6 4 2 0

0

0.005 Engineering Strain

0.03 0.02

PMMA-NanoGel Al-NanoGel Steel-NanoGel

0.01 0

0

0.005 Engineering Strain

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a)

0.01

0.04

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Engineering Stress (MPa)

Engineering Stress (MPa)

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0.01

b)

Figure 7: (a) Uniaxial tensile response of design PD4 considering PU as soft material with three different stiff constituents (b) Uniaxial tensile response of design PD4 considering NanoGel as soft material with three different stiff constituents.

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Therefore, the interfaces need to be modeled accurately for predicting the response of the composites. In the present work a cohesive zone-based inter-

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face model is used to characterize the constitutive response of the stiff-soft material interfaces. In a cohesive zone model (CZM), a set of cohesive sur-

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faces is introduced in the FE discretization and a traction-separation law is used to phenomenologically describe the constitutive relation of the interface.

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In the present study, a bilinear traction-separation law is used, in which the normal and tangential surface traction are modeled in an uncoupled manner

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as:





    t K Kns δ  n  =  nn × n  ts Ksn Kss δs

(20)

where the traction {t} = {tn ; ts } consists of normal (tn ) and tangential component (ts ); the direct stiffness components are Knn (normal) and Kss (tan21

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Figure 8: Logarithmic strain contour plot for PD4, considering steel-NanoGel, Al-NanoGel and PMMA-NanoGel as material combination, respectively

gential); δn and δs represent the normal and tangential separation, respectively. Since the two tractions are un-coupled in nature, we get [Kns = Ksn = 0].

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The initial response of the cohesive zone remains elastic (following Eqn.20) until a damage initiation criterion is met by the surfaces. In the present

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study, an initiation criterion based on the effective separation (δef f ) is used as

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δef f ≥ δ0 (21) p p 2 2 = δn2 + δs2 ; and δ0 = δn0 + δs0 is the effective separation at

where δef f

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damage initiation with δn0 and δs0 being the normal and tangential separation at the damage initiation point, respectively. Once damage initiates the cohesive surfaces start to open and the constitutive response becomes a

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softening type as given by {t} = (1 − D)[K]{δ}

(22)

The scalar damage variable 0 < D < 1 represents the overall damage in the cohesive interaction. The evolution law for the damage variable is also given 22

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(23)

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in terms of the effective separation by a linear relation as
δ f δef f (t) − δ0
D˙ = δef f (t) δf − δ0

where δf is the effective separation at complete failure and δef f (t) is the

effective separation calculated at current time t. The interface fracture energy is defined by the area under the traction separation curve GC . For a linear

traction-separation law as described above, GC and δ0 are uniquely related

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with δf . 5.2.1. Estimation of cohesive zone parameters

To estimate the cohesive zone parameters for the bi-material interfaces, we have followed an iterative approach. The assumption of perfect bond-

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ing (no-slip, no-opening) along the interfaces gives an upper bound estimate for the elastic stiffness of the interfaces. On the other hand, considering a

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friction-less (no resistance against tangential slip) contact formulation across the mating surfaces results in the lower bound measure of stiffness. Our assumption to estimate the CZM parameters is that the interface stiffness

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and strength would be such that the overall response of the composite is softer than the “no-slip” condition but stiffer than the friction-less contact

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simulation. We chose the parameters by assuming the surface characteristic (bonding) to be strong enough to result in a composite stiffness close to

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the “no-slip” case. To estimate a reasonable combination for the parameters (Knn ), tnc , and (GC ) of the CZM, iterations are performed until the stiffness of the composite was closer to the perfect bonding case. Fig 9 shows how reasonable choices are decided comparing the CZM-based FE results with the “no-slip” and friction-less contact cases for PMMA-PU and PMMA-NanoGel 23

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combinations respectively. All six material interface properties are listed in Table 5. The tangential stiffness Kss and fracture energy were estimated as

6 4 2 0

0

0.05 Engineering Strain

a)

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Engineering Stress (MPa)

no-slip contact CZM

8

no-slip contact CZM

0.06 0.04 0.02

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Engineering Stress (MPa)

1/3rd of the normal components. The cohesive properties for the interfaces

0.1

0

0

0.02 0.04 Engineering Strain

b)

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Figure 9: (a) Stress-strain plots for design PD4 by considering PMMA-PU with cohesive zone interfaces, no-slip and friction less contact conditions (b) Stress-strain plots for design

contact conditions.

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PD4 by considering PMMA-NanoGel with cohesive zone interfaces, no-slip and friction less

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considering NanoGel as the soft phase are the same for all three cases. This is because the stiff materials do not contribute to the response of the com-

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posite if the softer material is extremely soft as explained earlier. Further, it is important to mention that the cohesive zone parameters are commonly

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estimated by conducting interfacial fracture testing for bi-materials, which is not considered in the present study. However, the numerical estimation of the CZM parameters as done here considering the theoretical bounds of the interface strength can be used to predict the trend of the composite’s response and failure. 24

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Table 5: Parameters for interface cohesive zone models

tnc

GC

(MPa/mm)

(MPa)

(N/mm)

PMMA-PU

0.9

4.5

Steel-PU

3.0

4.5

Al-PU

3.0

4.5

PMMA-NanoGel

9

0.03

Steel-NanoGel

9

0.03

Al-NanoGel

9

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Knn

1.5

1.0

1.5

7.5e-04

7.5e-04

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Interface

0.03

7.5e-04

5.3. Quasi-static stiffness and toughness estimation of design PD4 considering cohesive interfaces

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To estimate the toughness of the architectured composites, FE simulations are done considering the cohesive zones along the interfaces. It is

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important to note that the toughness computed here is the area under the stress-strain curve until failure and is different than the fracture toughness.

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No initial damage or notch is considered in the present study; and failure is assumed either due to failure in the weak interfaces designated by a drop

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in load carrying capacity or due to yielding of the stiffer material whichever occurs first. To understand which stiff-soft material pair results in the best

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combination of stiffness and toughness values, all six material combinations as listed in Table 5 are considered in the simulations. Quasi-static stiffness and toughness are computed for all six material combinations and normalized against the density of the composite to obtain specific stiffness and toughness respectively and plotted as shown in Fig 10. The density of the composite 25

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is computed as ρcomposite = ρstif f ∗ vf + ρsof t ∗ (1 − vf ); where vf is the volume fraction of the stiffer material. The stress-strain plots are not shown for

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comparison because it is difficult to plot them in the linear scale, for such orders of magnitude difference in the modulus. It was observed from Fig 10, PMMA-PU gives the maximum toughness with a reasonable value of stiff-

ness for PD4. Use of extremely soft NanoGel neither shows good toughness nor good stiffness. The ultimate stress becomes much lower when NanoGel

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is used compared to PU, which reduces the stiffness of the composite, and the composite behaves in a more brittle manner than its PU counterpart, as seen in Fig 9, which reduces the toughness as well. Therefore, effective stress transfer between the constituent material is the key to improving the stiffness of the composite whereas gradual failure of the interfaces helps to improve

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the toughness of the composite. If the elasticity of the two constituents are orders of magnitude different (such as steel-NanoGel), they result in poor

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performance in terms of quasi-static properties. To understand the effect of interconnection, the composite’s performance

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is further compared with a layered material design considering a unit cell of layered material with the same volume fraction of the stiff-soft phases

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as in PD4 as shown in Fig 11 (a). Interfaces are introduced using cohesive zone models across stiff-soft material boundaries and FE simulations are performed to compute stiffness and toughness values for the PMMA-PU

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combination. Fig 11 (b) shows the uniaxial tensile stress-strain response of the layered design when compared with the interconnected PD4 design. As was seen in the layered design, stiff material (PMMA) carry significant stresses and quickly goes to yielding locally near the interfaces before the

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10

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102

1

PMMA-PU PMMA-NanoGel Al-PU Al-NanoGel Steel-PU Steel-NanoGel

100

10-5

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Specific Stiffness (MPa/gm/cc)

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10-4

10-3

10-2

10-1

100

Specific Toughness (Nmm-1/gm/cc)

Figure 10: Specific stiffness-toughness plot for the design PD4 considering all 6 material

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combinations

interface experiences cohesive failure. Local yielding of the stiff PMMA re-

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sults in a rapid failure of the composite thereby reducing the toughness and stiffness. Therefore, it is shown that the interconnection indeed performs

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better in terms of stiffness and toughness for a stiff-soft composite compared to straight layered materials.

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5.4. Response of the architectured composite under dynamic loading To predict how the architectured composite mitigates pressure and ki-

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netic energy of an incoming stress wave caused due to impact, we need to understand how various (stiff-soft) combinations of materials and the architecture interact with the propagating waves of certain frequencies. In reality, impact loading such as a blast contains multiple frequency components. The amplitude of the wave gets attenuated due to reflection-refraction along the 27

3 2 1 0

0

0.02 0.04 0.06 Engineering Strain

b)

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a)

PD4 design layered

4

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Engineering Stress (MPa)

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Figure 11: a.) A simple layered geometry showing alternate layers of stiff-soft material with periodic boundary condition on either sides; b.) Uniaxial tensile response of the PD4 design and layered design comparison.

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interfaces because of the high impedance mismatch between two different materials in the composite. Along with the impedance mismatch, the soft,

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viscoelastic materials also contribute in energy dissipation. To analyze how the composite effectively attenuates or amplifies each of these frequency com-

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ponents as the wave fonts pass through multiple interfaces, FE simulations are done in a frequency domain and a time domain analysis, as explained

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below.

5.4.1. Transmissibility in finite dimension geometry

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A frequency domain steady state dynamic response analysis is performed

considering a finite dimension geometry of the architectured composite. The simulation results provide a simplified understanding of the dynamic response of the composite over a wide range of frequencies for different material com-

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binations. The finite dimension RVE is chosen, as shown in Fig. 12, where the external force is applied at the top surface and the bottom surface is

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considered as fixed. Periodic boundary conditions are applied along the left and right edges. A harmonic force of amplitude 0.1N is applied at the top

surface and output responses are measured at the bottom surface. The frequency range of interest for this study is considered as 0-0.3 MHz to simulate

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most of the dynamic loading encountered in reality. To estimate the attenu  ation of the input force, transmissibility is calculated as T = 20 log FFout db; in where Fout and Fin are the output and input force amplitude, respectively.

Therefore, negative transmissibility indicates wave attenuation in the material, and the corresponding frequency range indicates the band-gap for the architectured composite. For each of the stiff-soft material combinations,

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transmissibility is calculated as a function of input frequency and plotted in Fig. 13(a) and 13(b) respectively for PU and NanoGel cases. It is seen when

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PU is used as softer phase in combination with any of the stiffer materials, the composite does not show a wide band-gap precisely for the chosen fre-

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quency range (Fig. 13 (a)). Among the three material cases, Steel-PU gives higher impedance mismatch compared to the others; therefore transmissibil-

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ity becomes slightly negative for that combination in the higher frequency range. However, no wide band gap has been observed. Conversely, when NanoGel is used with the stiff constituents, a wide range of band-gap has

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been observed (Figure 13 (b)) for all three cases. This is due to the higher impedance mismatch in the interfaces when PU is replaced by the even softer NanoGel. Therefore, for the given architectured composite, NanoGel is more effective compared to PU in stress wave attenuation under a random dy-

29

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namic loading. Interestingly, this finding is completely opposite to what has been demonstrated for quasi-static performance. For maximizing the tough-

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ness and stiffness of the composite, PMMA-PU has been identified as the best material combination, as effective stress transfer between two dissimilar materials and a controlled failure of weak interfaces are the two responsi-

ble mechanisms for stiffness-toughness improvement, respectively. However, for high frequency wave mitigation, impedance mismatch is the predomi-

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nantly responsible factor rather than the architecture or interfaces. Since the steel-NanoGel combination has the highest impedance mismatch among

the materials sets considered here, the steel-NanoGel pair turned out as the

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CE

PT

ED

M

best for transimissibility and band gap calculation.

Figure 12: A finite dimension geometry for steady state dynamic analysis comprising of 10 by 5 PD4 unit cells

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20 0 -20 PMMA-PU Al-PU Steel-PU

-40 0

0.1 0.2 Frequency, f (MHz)

-50 -100 -150

PMMA-NanoGel Al-NanoGel Steel-NanoGel

-200 -250

0

0.1 0.2 Frequency, f (MHz)

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a)

0.3

0

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Transmissibility, T (db)

Transmissibility, T (db)

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0.3

b)

Figure 13: Transmissibility curve calculated from steady state dynamic analysis by considering (a) PMMA, Al and Steel as stiffer materials with PU as soft constituent (b) PMMA, Al and Steel as stiffer materials with NanoGel as soft constituent.

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5.4.2. Response of the composite under transient impulse

The steady-state dynamic analysis asserts that if the stiffness mismatch

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between two materials is more, the wave amplitude attenuation is higher. However, the steady state analysis does not consider the presence of cohesive

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interfaces precisely. Therefore, to understand how the proposed composite will perform in the event of a real impact with the cohesive interfaces

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present, response of the composite is analyzed under a transient impulse loading. Since, the steel-NanoGel combination turned out to be the best in

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terms of wave amplitude attenuation, that combination is investigated for the transient analysis. To simulate the impact loading, a trapezoidal pressure pulse as shown in Figure 14 (a), has been considered. The pulse consists of a very short rise time (approximately 10% of total duration of the pulse) to represent the characteristic of an impulse loading. The finite dimension 31

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RVE as shown earlier in Figure 12 has been used for the transient analysis. The input pulse was applied as a uniformly distributed loading over the top

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surface. Since we are interested in the transient response of the composite in this case, the simulations are carried out until 2e − 4 sec (approximately four times than the duration of the pulse) and responses such as displacement,

reaction force and stresses are analyzed at significant locations as shown in Fig 14(b).

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Two different values were considered as the maximum amplitudes of the input pulse as 0.1 MPa and 1.0 MPa, respectively. Fig 14 (c) and (d) show

the reaction force measured at the bottom edge of the geometry for both the no-slip condition and with cohesive interfaces for both the pulses. As seen, there is a significant reduction in the reaction force at the bottom surface,

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when the weak interfaces are present, compared to the no-slip condition. The reason being the architecture in this case induces relative sliding across

ED

interfaces and dissipate energy through the process. Therefore, the force transmitted across the layers eventually reduces compared to “no-slip” case.

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Fig 15 (a) and (b) show the normalized vertical stress measured along points 1, 2, and 3 as shown in Fig 14 (b) for 0.1 MPa and 1 MPa pulse, respec-

CE

tively. There is a significant reduction in the stress pulse as it propagated through the RVE due to reflection along the interfaces. It is to be noted, that due to the extremely slow wave speed through the NanoGel, the pulse

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might not have reached at point 3 as can be seen in Fig 15, for the chosen time period of analysis. Nonetheless there is a significant reduction in stress wave amplitude is observed. The strain contour showed in Fig 16 depicts interface failure for 1.0 MPa case, however the steel has not yielded. Impact

32

0.5

0

0

2 4 Time, t (sec)

6 ×10

×10

3

-5

4

2

M

1 no-slip CZM

0

0.5

1 1.5 2 Time, t (sec) ×10-4

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0

-5

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a)

Reaction force (N)

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1

×10

b)

-4

3 2 1 0

no-slip CZM

0

0.5

1 1.5 2 Time, t (sec) ×10-4

d)

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c)

Reaction force, Rf (N)

Norm. pressure, Pin/Pmax

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Figure 14: a.) Normalized pressure pulse applied at the top surface; b.) Locations along

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the RVE where stresses are measured (shown later); c.) Reaction force measured at the bottom surface for 0.1 MPa pulse considering no-slip and cohesive zones along the interfaces; d.) Reaction force measured at the bottom surface for 1 MPa pulse considering

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no-slip and cohesive zones along the interfaces.

loading is mostly compressive; however, it can be seen that locally NanoGel is subjected to tensile strain due to geometric non-linearities of the architectured composite (Figure 16). Though steel has not deformed plastically, 33

0

-0.1

-0.2

-0.2 -0.3

Location 1 Location 2 Location 3

-0.4 0

-0.4

Location 1 Location 2 Location 3

-0.6 -0.8

0.5 1 1.5 2 Time, t (sec) ×10-4

0

0.5 1 1.5 2 Time, t (sec) ×10-4

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a)

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0 σ yy /Pmax

σ yy /Pmax

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b)

Figure 15: a.) Normalized stress (σyy /P ) measured at point 1, 2 and 3 respectively for 0.1 MPa pulse with cohesive zones; b.) Normalized stress (σyy /P ) measured at point 1, 2

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CE

PT

ED

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and 3 respectively for 1.0 MPa pulse with cohesive zones.

Figure 16: a.) Strain 22 contour at the middle of the top layer; b.) Zoomed view of the ’red’ rectangle showing cohesive layer failure at certain locations

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the top layer interfaces reached failure due to normal separation between the layers resulting from these local tensile strain. Therefore, failure had been

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triggered in the weak interfaces before plastic deformation in the stiff layers even under compressive loading. Such interface failure under random impact

loading conditions eventually delays plastic deformation in the stiff materials and thereby can be used to improve the toughness of the composites. The

main finding from this analysis is that, creation of weak interfaces and al-

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lowing them to slide, open, and eventually fail, which results in better wave mitigation capabilities compared to a “no-slip” condition. 6. Summary and Conclusions

The present work considered an “interconnected” geometry inspired by

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biological materials in a composite material made of very different material properties. Our computational study showed that the interconnection

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was effective in improving the load transfer between two dissimilar materials by the virtue of contact-friction mechanism, compared to layered materi-

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als. Since the study is mainly computational, attention is being made to accurately characterize the materials through the physically motivated con-

CE

stitutive models and to validate the model predictions. The study was based on the assumption that if the interfaces between two materials are weaker

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than the bulk materials and eventually they are forced to fail under external loading, the toughness of the composite is significantly improved. Failure due to yielding of the stiffer materials is often localized and sudden, especially in brittle materials like PMMA, however, opening of the (weak) cohesive interfaces are more controlled, distributed, and gradual. Our study showed, if 35

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the applied energy gets dissipated on failing the weak interfaces, yielding of the bulk materials gets delayed and performance such as quasi-static tough-

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ness and impact wave attenuation improve significantly. Further, our study showed the quasi-static mechanical properties such as stiffness and toughness essentially depend on how well the two materials act together in terms of stress transfer between them; whereas the wave attenuation under random dynamic loading is mostly a function of the impedance mismatch between

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layers. This computational study identifies the possible mechanisms for improving the stiffness and toughness in a composite material for a given set of stiff-soft constituents. Constituents being similar in terms of elastic proper-

ties results in better combination of specific stiffness and energy absorption in the quasi-static regime. Conversely, the constituents with higher stiffness

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mismatch results in more effective stress wave attenuation in the dynamic regime. The identification of the cohesive zone parameters and validations

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with real experiments are considered for future study.

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7. Acknowledgments

The authors thankfully acknowledge valuable discussions with Dr. Julien

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Meaud (Georgia Institute of Technology) and Dr. Susanta Ghosh (Michigan Technological University) during the progress of the work. The authors also

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acknowledge Nancy Barr (Michigan Technological University) for thorough proof-reading of the article. Superior, a high-performance computing infrastructure at Michigan Technological University, was used in obtaining results for the architectured composite under dynamic loading conditions. 36

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