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The effect of stochasticity in mechanical behaviour of metamaterials is quantified in a probabilistic framework. The stochasticity has been accounted in the form of random material distribution and structural irregularity, which are often encountered due to manufacturing and operational uncertainties. An analytical framework has been developed for analysing the effective stochastic in-plane elastic properties of irregular hexagonal structural forms with spatially random variations of cell angles and intrinsic material properties. Probabilistic distributions of the in-plane elastic moduli have been presented considering both randomly homogeneous and randomly inhomogeneous stochasticity in the system, followed by an insightful comparative discussion. The ergodic behaviour in spatially irregular lattices is investigated as a part of this study. It is found that the effect of random micro-structural variability in structural and material distribution has considerable influence on mechanical behaviour of metamaterials.
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103
E1ν12
E2ν21 G12
11
21
11(¯ω) =
σ1(¯ω)hω)
l(¯ω)+ sin θ(¯ω)sin2θω)
Es(¯ω)tω)
l(¯ω)3
cos θ(¯ω)
21(¯ω) =
2σ1(¯ω) sin θω) cos θ(¯ω)h(¯ω)
l(¯ω)+ sin θ(¯ω)
Es(¯ω)tω)
l(¯ω)3hω)
l(¯ω)+ 2s( ¯ω)
l(¯ω)+ 2 sin θ(¯ω)
¯ω
12 22
12(¯ω) = σ2(¯ω) sin θω) cos θ(¯ω)
Es(¯ω)tω)
l(¯ω)3
22(¯ω) =
σ2(¯ω) cos θω) 2 cos2θ(¯ω)+8s(¯ω)
l(¯ω)3cos2αω)
sin3α(¯ω)+cos2βω)
sin3β(¯ω)+ 2 s(¯ω)
l(¯ω)2
(cot2α(¯ω) + cot2βω))!
Es(¯ω)tω)
l(¯ω)3hω)
l(¯ω)+ 2sω)
l(¯ω)+ 2 sin θ(¯ω)
γ12
γ12(¯ω) =
2τ(¯ω) cos θ(¯ω) 2s(¯ω)
l(¯ω)2
+ 4 s( ¯ω)
l(¯ω)31
sinα(¯ω)+1
sinβ( ¯ω)+h( ¯ω)
l(¯ω)3
+1
2h(¯ω)
l(¯ω)2!
Es(¯ω)t(¯ω)
l(¯ω)3h(¯ω)
l(¯ω)+ 2s(¯ω)
l(¯ω)+ 2 sin θ( ¯ω)
ZUij Z
E1Uij =σ1
11
E2Uij =σ2
22
ν12ij =21
11
ν21ij =12
22
G12ij =τ
γ12
m n
ith jth i j i = 1,2, ..., m
j= 1,2, ..., n
Zeqω)
E1ν12 E2
ν21 G12
E1eqω) = L(¯ω)
B(¯ω)
n
X
j=1
B(¯ω)j
m
P
i=1
2l(¯ω)3
ij (h(¯ω)ij +l(¯ω)ij sin θ(¯ω)ij ) sin2θ(¯ω)ij
Es(¯ω)ij t(¯ω)3
ij
E2eq( ¯ω) = Bω)
L(¯ω)×
n
X
j=1
B(¯ω)j
m
P
i=1
Es(¯ω)ij tω)3
ij (h(¯ω)ij + 2sω)ij + 2l(¯ω)ij sin θ(¯ω)ij )
l(¯ω)3
ij cos2θ(¯ω)ij + 4sω)3
ij cos2α(¯ω)ij
sin3α(¯ω)ij
+cos2β(¯ω)ij
sin3β(¯ω)ij +sω)2
ij l(¯ω)ij (cot2αω)ij +cot2β(¯ω)ij )
1
ν12eqω) = L(¯ω)
B(¯ω)
n
X
j=1
B(¯ω)j
m
P
i=1
(h(¯ω)ij + 2s(¯ω)ij + 2l(¯ω)ij sin θ(¯ω)ij ) sin θ(¯ω)ij
cos θ(¯ω)ij
ν21eq( ¯ω) = Bω)
L(¯ω)×
n
X
j=1
B(¯ω)j
m
P
i=1
sin θ(¯ω)ij cos θω)ij l(¯ω)3
ij (h(¯ω)ij + 2sω)ij + 2l(¯ω)ij sin θ(¯ω)ij )
l(¯ω)3
ij cos2θ(¯ω)ij + 4sω)3
ij cos2α(¯ω)ij
sin3α(¯ω)ij
+cos2β(¯ω)ij
sin3β(¯ω)ij +sω)2
ij l(¯ω)ij (cot2αω)ij +cot2β(¯ω)ij )
1
G12eqω) = B(¯ω)
L(¯ω)×
n
X
j=1
B(¯ω)j
m
P
i=1
Es(¯ω)ij t(¯ω)3
ij (h(¯ω)ij + 2s(¯ω)ij + 2l(¯ω)ij sin θ(¯ω)ij )
2l(¯ω)ij s(¯ω)2
ij +h(¯ω)2
ij h(¯ω)ij +l(¯ω)ij
2+ 4s(¯ω)3
ij 1
sinα(¯ω)ij
+1
sinβ( ¯ω)ij 
1
B(¯ω)1=B(¯ω)2=... =B(¯ω)ns(¯ω)ij =h(¯ω)ij
2α(¯ω)ij =β(¯ω)ij = 90l(¯ω)ij =l θω)ij =θ
i j
t Es
θ= 0,1,3,5,7θ
2500
[θθ, θ + ∆θ]
N N(= n×m)
θ= 0
θ
30h/l
θ
E1E2ν12 ν21 G12
θ= 30h/l
E2
E1G12 ν21 ν12
E2
E2
G12
G12
E2G12 ν21
E1ν12
ν21
ν21
E2G12 ν21
01234567
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Degree of irregularity
Coefficient of variation
E2
ν21
G12
EsE2
Es= 20%
E2G12
ν21 ν21
E2G12
E2E2
ν21
G12
E2
Es
E2
θ= 5E2
E2
103
ν21 G12
t Es
E1ν12 E2ν21 G12

Supplementary resource (1)

... Though most of the analyses on multi-material lattices are developed in the framework of numerical approach [28][29][30], some analytical treatments of the problem are also reported that provide more physical insights [31]. Another broad aspect of research on metamaterials is the quantification of spatial irregularities [32] and estimation of effective physical properties including its effect. The practically inevitable spatial irregularities exist in lattices due to manufacturing uncertainty, microstructural defects, pre-stressing, etc. [33]. ...
... Most of such studies are experimental and purely numerical (finite element), which suffer from a lack of physical insights in addition to high computational time and cost. A few analytical studies providing an in-depth understanding of the physics of disorder and irregularities are found in recent literature [32,33]. ...
Article
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Over the last decade, lattice-based artificial materials have demonstrated the possibility of tailoring multifunctional capabilities that are not achievable in traditional materials. While a large set of mechanical properties can be simultaneously modulated by adopting an appropriate network architecture in the conventional periodic lattices, the prospect of enhancing global specific stiffness and failure strength has become rather saturated lately due to intense investigation in this field. Thus there exists a strong rationale for innovative design at a more elementary level in order to break the conventional bounds of specific stiffness and failure strength that can be obtained only by lattice-level geometries. Here we propose a novel concept of anti-curvature in the design of lattice materials, which reveals a dramatic capability in terms of enhancing the elastic failure strength in the nonlinear regime while keeping the relative density unaltered. A semi-analytical bottom-up framework is developed for estimating the onset of failure in honeycomb lattices with the anti-curvature effect in cell walls considering geometric nonlinearity under large deformation. The physically insightful semi-analytical model captures nonlinearity in elastic failure strength of anti-curvature lattices as a function of the degree of curvature and applied stress together with conventional microstructural and intrinsic material properties.
... With signicant potential for a wide range of engineering applications, mechanical metamaterials form a domain of cutting-edge research in the present age [4,5,6,7,8]. One of their many dening features is the denition of mechanical properties of the material at a global scale through geometric attributes of the microstructure, rather than only intrinsic material properties of constituent members [9,10,11,12,13,14]. This enables meeting specic engineering demands and achieving unusual (not attainable in orthodox natural materials) yet useful mechanical properties, proving benecial for various multifunctional systems [15,16,17]. ...
... Recently auxetic metamaterials with negative Poisson's ratios [18,19,20], which can be realized through articial microstructuring [9,21], are attracting increasing attention due to their enhanced mechanical performances in multiple applications. Over the last two decades, it has been convincingly demonstrated that negative Poisson's ratio could lead to enhanced impact and indentation resistance [22,23,24], higher energy absorption [25,26], increased shear stiness and fracture toughness [27,28], variable permeability, creation of synclastic doubly curved panels, modulation of wave propagation, shape modulation, development of novel actuators and sensors, improved structural designs in terms of stress distribution and deformation control. ...
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This paper develops kirigami-inspired modular materials with programmable deformation-dependent stiffness and multidirectional auxeticity. Mixed-mode deformation behaviour of the proposed metastructure involving both rigid origami motion and structural deformation has been realized through analytical and computational analyses, supported by elementary-level qualitative physical experiments. It is revealed that the metamaterial can transition from a phase of low stiffness to a contact-induced phase that brings forth an extensive rise in stiffness with programmable features during the deformation process. Transition to the contact phase as a function of far-field global deformation can be designed through the material's microstructure. A deformation-dependent mixed-mode Poisson’s ratio can be achieved with the capability of transition from positive to negative values in both in-plane and out-of-plane directions, wherein it can further be programmed to have a wide-ranging auxeticity as a function of the microstructural geometry. We have demonstrated that uniform and graded configurations of multi-layer tessellated material can be developed to modulate the constitutive law of the metastructure with augmented programmability as per application-specific demands. Since the fundamental mechanics of the proposed kirigami-based metamaterial is scale-independent, it can be directly utilized for application in multi-scale systems, ranging from meter-scale transformable architectures and energy storage systems to micrometer-scale electro-mechanical systems.
... Though analyses for the majority of multi-material lattices are developed in the framework of numerical techniques [40,41,42], some insightful analytical treatments are also reported recently [43]. Quantication of spatial irregularities and estimation of eective physical properties, including their eect, is another signicant area of research on metamaterials [44]. Due to manufacturing uncertainty, microstructural aws, pre-stressing, and other factors, spatial irregularities in lattices are practically unavoidable [45]. ...
... The majority of this research is experimental and solely numerical (nite element) and thus suers from a lack of physical insight and high computational times and costs. Recent literature has a few analytical investigations that provide an in-depth understanding of the physics of disorder and irregularity [44,45]. Because mechanical metamaterials include scale-free mechanics of periodic forms over a wide range of length scales (nano, micro, and macro), the research area of eective mechanical property estimation is relevant to a wide range of structures, from macro-level (such as honeycomb cores in sandwich structures) to nanomaterials with regular honeycomb-like congurations (such as graphene and hBN) and microstructures of various woods and bones [54,55,56]. ...
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Shear modulus assumes an important role in characterizing the applicability of different materials in various multi-functional systems and devices such as deformation under shear and torsional modes, and vibrational behaviour involving torsion, wrinkling and rippling effects. Lattice-based artificial microstructures have been receiving significant attention from the scientific community over the past decade due to the possibility of developing materials with tailored multifunctional capabilities that are not achievable in naturally occurring materials. In general, the lattice materials can be conceptualized as a network of beams with different periodic architectures, wherein the common practice is to adopt initially straight beams. While shear modulus and multiple other mechanical properties can be simultaneously modulated by adopting an appropriate network architecture in the conventional periodic lattices, the prospect of on-demand global specific stiffness and flexibility modulation has become rather saturated lately due to intense investigation in this field. Thus there exists a strong rationale for innovative design at a more elementary level in order to break the conventional bounds of specific stiffness that can be obtained only by lattice-level geometries. In this article, we propose a novel concept of anti-curvature in the design of lattice materials, which reveals a dramatic capability in terms of enhancing shear modulus in the nonlinear regime while keeping the relative density unaltered. A semi-analytical bottom-up framework is developed for estimating effective shear modulus of honeycomb lattices with the anti-curvature effect in cell walls considering geometric nonlinearity under large deformation. We propose to consider the complementary deformed shapes of cell walls of honeycomb lattices under anti-clockwise or clockwise modes of shear stress as the initial beam-level elementary configuration. A substantially increased resistance against deformation can be realized when such a lattice is subjected to the opposite mode of shear stress, leading to increased effective shear modulus. Within the framework of a unit cell based approach, initially curved lattice cell walls are modeled as programmed curved beams under large deformation. The combined effect of bending, stretching and shear deformation is considered in the framework of Reddy’s third order shear deformation theory in a body embedded curvilinear frame. Governing equation of the elementary beam problem is derived using variational energy principle based Ritz method. In addition to application-specific design and enhancement of shear modulus, unlike conventional materials, we demonstrate through numerical results that it is possible to achieve non-invariant shear modulus under anti-clockwise and clockwise modes of shear stress. The developed physically insightful semi-analytical model captures nonlinearity in shear modulus as a function of the degree of anti-curvature and applied shear stress along with conventional parameters related to unit cell geometry and intrinsic material property. The concept of anti-curvature in lattices would introduce novel exploitable dimensions in mode-dependent effective shear modulus modulation, leading to an expanded design space including more generic scopes of nonlinear large deformation analysis.
... Metamaterials are artificial materials with engineered architecture designed to have physical properties determined by microstructural geometry rather than the chemical composition. [1][2][3][4] The hierarchical topologies in metamaterials have led to unique properties such as ultra-high stiffness, 5 ultra-low density, 6 and ultrahigh surface-to-volume ratios that introduce a significant radiation contribution 7 in the thermal transport phenomenon. The physical properties of these metamaterials arise from the geometry and spatial arrangement of microstructural elements. ...
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Shape memory polymers are gaining significant interest as one of the major constituent materials for the emerging field of 4D printing. While 3D-printed metamaterials with shape memory polymers show unique thermomechanical behaviors, their thermal transport properties have received relatively little attention. Here, we show that thermal transport in 3D-printed shape memory polymers strongly depends on the shape, solid volume fraction, and temperature and that thermal radiation plays a critical role. Our infrared thermography measurements reveal thermal transport mechanisms of shape memory polymers in varying shapes from bulk to octet-truss and Kelvin-foam microlattices with volume fractions of 4%–7% and over a temperature range of 30–130 °C. The thermal conductivity of bulk shape memory polymers increases from 0.24 to 0.31 W m ⁻¹ K ⁻¹ around the glass transition temperature, in which the primary mechanism is the phase-dependent change in thermal conduction. On the contrary, thermal radiation dominates heat transfer in microlattices and its contribution to the Kelvin-foam structure ranges from 68% to 83% and to the octet-truss structure ranges from 59% to 76% over the same temperature range. We attribute this significant role of thermal radiation to the unique combination of a high infrared emissivity and a high surface-to-volume ratio in the shape memory polymer microlattices. Our work also presents an effective medium approach to explain the experimental results and model thermal transport properties with varying shapes, volume fractions, and temperatures. These findings provide new insights into understanding thermal transport mechanisms in 4D-printed shape memory polymers and exploring the design space of thermomechanical metamaterials.
... These alterations constitute a major challenge in the analytical behavior of such structures. Nevertheless, stochastic methodologies have been demonstrated taking into account the several constituent representative unit cell elements and utilizing probabilistic descriptions of asymmetrical structures [285]. These models should be expanded in the 3D elasto-plastic mechanical behavior of metamaterial structures, providing a coherent framework to model and predict their mechanical response. ...
Thesis
Architected materials are considered the state of the art of engineering ingenuity. Specifically, mechanical metamaterials have been accentuated due to their unconventional and augmented responses. They have been gingerly investigated under the context of ultralight-ultrastiff structures for aerospace applications, tailored buckling mechanisms for energy storage, soft robotics and controlled wave propagation and designed anisotropy for tissue engineering. Albeit the plethora of remarkable results promulgating this subject, the analysis of architected materials has many questions that need to be addressed. There is no rigorous explanation for the selection of specific 3D designs that have been thoroughly utilized in the literature (regarding the selection of specific design variables and cost functions). Consequently, in practice specific structures are repeatedly used, without any explanation whether further search of the design space could not provide a substantially improved result. Therefore, the lack of understanding of the design space and the inherent physical phenomena has not elucidated the tools to obtain a globally optimal design. Thus, tailoring mechanical metamaterials is extremely arduous and has led to an obstacle in the progress of this field. This thesis aims to provide an analysis for the design of architected materials by illuminating the physical mechanisms and how to model and optimize such problems. The structure of this thesis is comprised of two main themes. The first method aims to control the mechanical performance through interconnected beam members that enhance the densification of the structure and impede catastrophic failure. The second method is related to geometrical defects that dictate the localized failure and anisotropic behavior. Furthermore, the optimization of specific design examples will be presented, employing low computational power for large design spaces and demonstrate how such design problems can be addressed, setting the framework for the systematic design and characterization of architected materials.
... Thus, for safe and economic workability of MCP, uncertainty quantification should be studied before design it. The probabilistic presentation of dynamical responses in marine structures, especially for lightweight composites has received great attention [4][5][6]. ...
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... In the present study, in addition to the lattice geometry, another design parameter is proposed in terms of pre-existing stress in the constituting beam elements. A unit cell based approach is followed here for deriving the lattice level eective elastic properties [38,39,40,41,42,43,44,45]. ...
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