Systematic Generation, Analysis, and ... - ACS Publications

Subscriber access provided by University of Florida | Smathers Libraries

Article

Systematic Generation, Analysis and Characterization of 3D Micro-architected Meta- materials Ninad T. Trifale, Eric A. Nauman, and Kazuaki Yazawa ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.6b10502 • Publication Date (Web): 05 Dec 2016 Downloaded from http://pubs.acs.org on December 6, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

ACS Applied Materials & Interfaces is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Applied Materials & Interfaces

Systematic Generation, Analysis and Characterization of 3D Micro-architected Metamaterials Ninad T. Trifale*, Eric A. Nauman, Kazuaki Yazawa Purdue University, West Lafayette, IN 47907. KEYWORDS: Micro-Architected, Anisotropy, Optimization, Porous material, Effective thermal conductivity, Effective elasticity

ABSTRACT

Controlling unit cell topology of micro-lattice structures can enable customization of effective anisotropic material properties. A wide range of properties can be obtained by varying connectivity within the unit cell which then can be further used to optimize structures specific to applications. A methodology for a systematic generation of micro-lattice structures is presented which focuses on controlling discrete topology instead of average porosity (as done in conventional porous media). An algorithm is developed to create valid lattice structures without redundancies from a given set of template nodes. Set of possible permutations of structures from an 8-node cubic octant of a unit cell are generated for evaluation of the degree of anisotropy.

ACS Paragon Plus Environment *Corresponding author email: [email protected]


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 36

Generic models are developed to calculate the effective thermal and mechanical properties as an effect of topology and porosity of the micro-architected structure. The thermal and mechanical anisotropies are investigated for the effective properties of micro-architected materials. A few of the structured materials are fabricated using 3D printing technology and their effective properties characterized. Structures are represented as graphs in the form of adjacency matrices. Effective thermal conductivity is analyzed using a resistance network model and effective stiffness is evaluated using a self-consistent elastic model, respectively. A total of 160,000 structures are generated and compared to porous metal foams where porosity is one of the design variables. The results show that it is possible to obtain a wide range of properties spanning more that an order of magnitude in comparison to porous metal structures. Structures with a maximum anisotropy ratios of 7.1 and 8.2 are observed for thermal and mechanical properties respectively. Preliminary experimental results validated the anisotropy ratio for the thermal conductivity and stiffness.

1. INTRODUCTION: Porous materials have been receiving considerable attention over the years for mechanical and thermal applications such as structural materials1, energy absorbing composites2, thermal interface materials3, heat sinks4,5, and bone scaffolds6,7 to name a few. Their low density, large area moment of inertia, high specific strength and relatively high thermal conductivity make porous materials an appropriate choice for various applications. In spite of their favorable characteristics it is difficult to determine which open cell architecture is optimal for a given set of constraints. For instance, thermal interface materials (TIMs) have a requirement of high thermal conductivity and a large contact area which requires a relatively low mechanical stiffness. Heat sinks require a high thermal conductivity and high permeability for minimizing pressure drop.

ACS Paragon Plus Environment

2

Page 3 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Bone scaffolds require high stiffness and high permeability to encourage vascularization. These inherent constraints often result in trade-offs which cannot be effectively addressed through the use of conventional isotropic porous media. The geometry and thereby the properties of porous structures are constrained by the ‘natural’ manufacturing process8. Nevertheless, porous materials are a first step towards engineering custom designs to selectively enhance specific characteristics of the geometry as per the required application. Taking inspiration from the variety of unit cell models available in literature it is useful to study and design custom anisotropic structures specific to applications and ‘tune’ their effective properties. Additionally, development of additive manufacturing techniques have opened up the possibility of designing three dimensional micro architected meta-materials. It is promising then to consider designing the micro architected meta-material structures with designer properties. An inverse approach is considered in this study wherein unit cell geometries are theoretically generated and analyzed for specific design criterion. Models for predicting effective properties of porous media have been well established in literature and have been verified through experiments and numerical simulation. The methodologies for analyzing properties of the architected structures are developed along the same lines. The challenge in the latter case is to develop generalized models applicable for any possible generated micro-architected metamaterials. Various optimization methods have been employed by researchers to develop lattice structures targeted for specific applications9, 10. These methods have been employed from the standpoint of topological optimization by considering either a grid of frame-like structures or a set of predefined unit cell geometries/primitives. Applications such as transmission tower design and cantilever beam optimization have been chosen in literature to showcase this methodology11, 12.


3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 36

The methods stated above impose constraints on the objective functions from the given loading conditions, which are typically in the form of stress or force with emphasis on minimizing massmaximizing strength. This process generally requires lot of control through constraints and is only applicable for specific boundary conditions. There are few unit cell based topology optimization approaches for architected materials as well. Valdevit et al13,

14

have developed protocols for design process of the architected structures.

Schaedler et al15 proposed ultralight designs for strength constraints. Lu et al16 targeted applications such as active cooling using architected structures. Gu et al17 discuss manufacturing and testing of octet unit cell structure. In the present work, the analysis focuses on systematic generation at the unit cell level. This process results in creation of a database of material properties which can then be used for optimization process. Essentially, set of effective material properties corresponding to a structure is considered as a discrete design variable as opposed to considered topological connections as design variables (conventional approach). The set of effective material properties is generated beforehand by analyzing properties of various unit cell geometries. The objective of the current study is to generate and analyze an entire class of engineered ‘materials’ and study their relative properties. Additionally, models need to be developed to bridge the material properties at unit cell level and continuum scale (assuming the microarchitected material to be a continuum). The objectives are defined with an ultimate goal of setting up a tool set to efficiently design and choose the best structure for on individual application and defined set of constraints.


4

Page 5 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2. ANALYSIS: 2.1 Generation A template, ‘super-set’ of a collection of nodes was defined in order to create a set of architected-structures. The node template was a simple cubic unit cell with 27 nodes (3 nodes per edge) as shown in Fig. 1. Additionally three symmetry conditions were imposed with the planes of symmetry passing through the centroid of the defined set of node points. This symmetry

condition reduced the analysis to single octant cube of the unit cell. (a)

(b)

(c)

Figure 1. Template of nodes and a sample meso-structure a) A cubic template comprising of 27 nodes (3x3x3) unit cell was chosen for generating mesostructures. b) For the structures, an additional symmetry constraints were imposed along the X-Y and Z plane. c) The 1/8th unit cell needs to be analyzed with regards to structure generation and material properties A 27-node cubic unit cell was chosen for a two reasons. First, the simple cubic structure provided a large number of mathematically trackable potential architected-structures. Second, a majority of the unit cell models previously studied are a subset of the template. The nodes of the 1/8th unit cell were numbered from 1 to 8 as shown in the figure below. The structure was


5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 36

represented by a symmetric square matrix (Fig. 2). The 8 nodes correspond to a total of total of 28 connections (8C2). Since every connection had two possibilities, the total number of possible structures for this template is 228.

Figure 2. Node numbering used to represent the structure in the form 8x8 matrix (Adjacency matrix). Matrix element 1 if the connection existed between the column node to the row node. The matrix was symmetric and diagonal elements were 0 The problem for generating all possible sub graphs within an undirected graph is a subset of the Clique Problem in computer science. It has been shown that such a problem is an np-complete problem and takes exponential time to solve18. In this study, efficient strategies are developed with the goal of maximizing the number of structures generated instead of solving for all subgraphs (clique) in a given graph. It should be noted that these strategies by themselves are not used to create an exhaustive list of structures but rather used in combination to maximize the number of structures generated. Three strategies to create permutations for the 8 node unit are outlined below:


6

Page 7 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. Strategy 1: Increment 28 bit binary number from 0 to 228 and populate symmetrical adjacency matrix 2. Strategy 2: Define maximum number of connections (number of 1s) and permute the 28 bit string. Increment maximum number of connections. 3. Strategy 3: Create set of simple paths with a specific start and end point. Add combinations and rotations of all simple paths. The first two strategies discussed are computationally expensive above maximum 8 connections in the structure. The third strategy of adding simple paths allows for generation of structures with high number of connections. The three different algorithms provided are computationally suitable for different range of number of connections. The difference in the process of generation also influences the general starting orientation of the network. The set of structures generated in this study is certainly not an exhaustive set of structures but the combination of the three different algorithm enables for the creation of a large number (160,000) of structures. The double counted structures in various orientations are filtered out to get the resultant set of structures. After generating permutations, checks were performed to ensure that the resultant structure was realizable. Algorithms for three following checks were developed and implemented: 1. Repetition: the cube unit cell has 13 axes of symmetry. Rotations of every new generated structure were checked for uniqueness 2. Repeatability: Generated structures had to be a single continuous network after mirroring and repeating the unit cell arrangement.


7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3.

Page 8 of 36

Continuity: Graphs with disconnected ‘island’ struts were not valid structures and were eliminated

These strategies along with the models were implemented using a custom code developed in MATLAB. Using the combination of the three strategies for generating structures and eliminating redundancies and invalid structures outlined above, a total of 160,000 different micro-structures were generated. All the structures were then analyzed for the effective bulk properties (Fig. 3).

Figure 3. Sample microstructures generated from the process outlined above. Both of the structures contain 5 complete unit cells in along all three directions 2.2 Modeling The effective mechanical and thermal properties of foam structures such as elastic modulus, yield stress and thermal conductivity are functions of independent geometric parameters, including pore size, pore density and porosity. The thermal and mechanical models developed


8

Page 9 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


for porous media have been well established. Similar modelling strategies were used to model each generated structures. A local co-ordinate system was defined for every one of the 28 struts or connections in order to develop a models that could analyze any structure generated from the original template. The local co-ordinate system was specified such that the local x-axis was oriented along the strut, local y-axis was parallel to the global X-Y plane, and the local z-axis defined using the right hand rule. The x-axis of the local co-ordinates system was defined using two angles with respect to the global system, a. Angle with respect to the X-Y plane and b. Angle of projected local xaxis on global X-Y plane with respect to the global X-axis. Strut length normalized by unit cell edge length was assigned in addition to the two defined angles. These two angles and the normalized length together uniquely represent a strut and were defined for all of the 28 possible struts. 2.2.1

Thermal Model

There have been various approaches to analytically determine the effective thermal conductivity of foams, most of which differ in the choice of representative volume element. Paek et al19 analyzed the effective thermal conductivity considering a simple cubic structure unit cell with orthogonal struts as a first order estimate. As an extension, a fraction of material struts was considered in the same and normal to the heat flow direction, and the conductivity was evaluated as a weighted sum20. Leong et al21 developed a rectangular shell model with quarter spherical pockets at the vertices and analysis was sub divided into multiple layers and the conductivity in the layers was evaluated separately. The radiation effect was found to be an important factor to be considered for further improving the prediction but only at very high temperatures22.


9


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 36

Boomsma et al23 proposed a geometrical model employing a repeating tetrakaidecahedron (TKDH) cell structure with cubes at the nodes and cylinders as the struts. Owing to the symmetry, 1/16th part of the TKDH was used to evaluate for 4 different sub layers. The same model was then extended24 to incorporate the effect of the orientation of the struts. Bauer et al22 used a more comprehensive approach to evaluate the effective thermal conductivity, by considering perturbations in the continuous medium with a governing differential equation instead of analyzing a representative volume element. They found that the ratio of effective thermal conductivity to the bulk solid was directly proportional to the relative density raised to 1/nth power, where the value of n is a semi-empirical constant. In most of the aforementioned studies, the medium was assumed to be homogenous and rule of mixtures applied to a representative volume element. This coupled with the approximation that the temperature at any given layer is constant, in solid as well as fluid, reduces the problem to 1D conduction. The common aspect of the various unit cell choices is that all of them employ the resistance network model wherein every connection is modelled as an individual resistance and the effective thermal conductivity is evaluated from the effective resistance of the unit cell. A similar model is developed for analyzing thermal conductivity of the generated structures. Additionally, existing models do not take into account heterogeneity, density gradients, and anisotropy, caused by manufacturing process. Natural convection and radiation are neglected in most of the studies, because of relatively minor contributions. This study is based on similar assumptions which have also been supported by experimentation in literature25. The models discussed above predict a linear relationship between effective thermal conductivity and relative density. Such dependence has been confirmed through experimental results4, 19. The same linear dependence is expected for the micro-architected structures as well.


10

Page 11 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The model was developed by approximating every strut or connection as a cylindrical beam. Actual cross section of the beam will depend on the nature of the manufacturing process used to create the structures. The diameter of the struts (cylinders) was calculated from the porosity of the structures. Individual resistance of every strut was evaluated using the expression below. =

(1)

Cross-section perpendicular to the direction of conduction was used to evaluate individual resistances. Subsequently, the effective resistance of the structure was found out by considering that all of the resistance in parallel. The resistances could be considered to be connected in parallel on the basis of initial set of assumptions that conduction is one dimensional for a given orientation and that convection and radiation effects were negligible. Matrix operators were defined which when multiplied by the graph extracted strut lengths and projected areas for individual struts. Three operators were defined for each of the three directions (X-Y-Z). The effective thermal conductivity was evaluated for the three orthogonal directions. 2.2.2

Mechanical Model

Porous structures have been widely studied for their mechanical virtues. The light weight nature and large area moment of inertia make them ideal for use in beams and structures for saving weight without compromising the strength26, 27. There has been considerable amount of research on determining the effective mechanical properties of porous structure based on the seminal work done by Gibson and Ashby1. They carried out substantial experimental as well as analytical work and developed empirical relationships to determine the Young’s modulus and yield stress


11


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 36

for foams. In case of elasticity, it has been shown that the effective Young’s modulus is directly proportional to the square of relative density:

= (1 − )

(2)

Similar results have been reported by various authors on the basis of experimental data28. The empirical relationship has also been verified through numerical studies. Wicklein et al29 generated a CAD model from an actual sample of foam and analyzed the mechanical behavior using FEA tools. In other studies, finite element method was applied to the TKDH model to obtain the resultant stiffness matrix and subsequently evaluate the effective Young’s modulus28. Zhu et al30 considered the TKDH model and used force and moment balance at specific nodes to analytically derive an expression for effective Young’s modulus as a function of the relative density (1-ε). Work was conducted to extend the same model to incorporate anisotropy31. The method of calculating effective elastic modulus by applying force and moment balance equations for a representative volume element has been shown to effective predict the elastic modulus of open cell foam geometries. This model is suitable and applicable to the current set of unit cell geometries. The elastic properties of porous structures on a continuum scale have been shown to vary with the number of unit cells used for the analysis. The properties converge to a constant value as the number of unit cells are increased such that continuum approximation is valid. As a general rule, at least 5 unit cells are required for the continuum assumption to be valid. For analyzing a large number of engineered structures efficiently it would be required that the analysis be restricted to single unit cell. In order to eliminate the dependence on the number of unit cells chosen for the analysis and enforce continuum scale considerations as well as symmetry conditions, the unit cell was assumed to be surrounded by material having properties same as the effective properties as


12

Page 13 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the unit cell(Fig. 4). Such modeling strategies are common for modeling composites and were first developed by Hill et al32. The effective properties of the engineered structures were evaluated in the form of stiffness. Stiffness of individual beams was known in axial and transverse directions. The values of the stiffness ka (axial) and kt (transverse) are given by the following relationships: = =

(3)

3 4

(4)

Where r is the radius of struts, s is the share factor which accounts for the fraction of strut completely part of unit cell, and Lstrut is the normalized strut length. The elastic modulus term of the solid strut forming material is not included for the stiffness calculation and the calculated final stiffness will be essentially equal to α, where, α depends on porosity () and topology. The value of α then can be used to calculate the effective stiffness using the following relationship.

= () ×

(5)

The calculation of the normalized effective elastic modulus (α) is carried out in terms of stiffness. Transformation matrices were defined for every connection using the set of local coordinate systems. The angles ϕ and ψ, represent the orientation of strut with respect to global co-ordinate system, cos % cos & = ! cos(& + /2) cos(% + /2) cos &

cos % sin & sin(& + /2) cos(% + /2) sin &

sin % 0 sin(% + /2)

(6)


13


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 36

A local stiffness matrix was defined for every individual strut and assembled into a global stiffness matrix after transforming from local to global coordinates. ./0120 = 3

4

× .0150 ×

(7)

The force and moment balance equations were applied to every individual node and summed over every individual strut. The unknown effective stiffness’ in three directions were added to the diagonal elements of the stiffness matrix. 6./0120 78∆:; = 8 : @ =BC = 3? A ? A

@: :

(9)

Where xi is a variable affecting quantity E and U is the uncertainty of measurement of that variable. The following uncertainties from the sensors and measurements was used to calculate the uncertainty of the measure elastic modulus. These values were obtained from the specifications provided by the manufacturer of the sensors. Table 2: Measurement accuracies of the sensors used in setup Variable Length Force Displacement Uncertainty 0.001 m 22.24 N 10µm The impact of the sensor uncertainties on the stress- strain calculations and consequently slope of linear fit was evaluated. It was found that the uncertainty in calculated elastic modulus varied between 6-11 % of the measured force values for different test samples. 2.3.2.3 Results from Experiments The experimentation was carried out for 3 different samples of both #658 and #829. The typical stress-strain curve for the samples is provided in the Fig. 9 below. The two curves indicate the response for two different directions of loading.


20

Page 21 of 36

8.00E+06 7.00E+06 Orientation 1 6.00E+06

Orientation 2

5.00E+06

Stress(Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.00E+06 3.00E+06 2.00E+06 1.00E+06 0.00E+00 -1.00E+06 0

0.05

0.1

Strain

0.15

0.2

Figure 9. Stress-strain curve for two orientations of the sample #658. The stiffness for one of the directions is much larger than the stiffness for the other direction. The point of collapse of the structure is evident from sudden drop of stress levels

It can be observed that the sample tested (#658) is stiffer in one of the directions as compared to the other. The sudden drop in the stress was associated with the collapse of the structure. The increase in stress following the drop corresponds to the collapsed layer coming in contact with other intact layers making the sample stiffer. The process continued till all of the 4 exterior layers collapsed onto each other. Elastic modulus was then evaluated from the slope of a linear regression in the elastic region of the curve. Similar behavior was observed with the other samples (#829). A cyclic collapse and ‘hardening’ was observed corresponding to sequential failure of individual layers. The elastic modulus was normalized by the raw material properties of PLA (polylactic acid) filament and the values were compared with the analytic model as well as finite element simulations. There is reasonable agreement with the measured values, analytic model and FE


21


simulations (Fig. 10). A large deviation was observed for the second sample for both the directions.

0.09 Model

0.08

Effective Elastic Modulus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 36

Experimentation

FEA

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Orientation 1

Orientation 2

Sample 658

Orientation 1

Orientation 2

Sample 829

Figure 10. Comparison of numerical values of the elastic modulus obtained from the experiments,developed model and finite element simulations. Sample 658 provides reasonable agreement between the experiments and model.

3. RESULTS Thermal and mechanical properties of all generated structures were evaluated using the generic analytical models developed. Properties of all the structures were normalized by the properties of the solid strut forming material. The effective characteristics not only depended on porosity and unit cell size but also varied greatly with respect to geometry. Fig. 11 shows the effective normalized stiffness plotted against the effective thermal conductivity. The continuous curve represents the effective normalized properties for open cell foam structures. It is important to note that all the points in the plot correspond to the same porosity of 0.85. The curve on the other


22

Page 23 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


hand represents variation of porosity from 0.7 to 0.99. The mechanical and thermal behavior of the foam is modeled as a function of porosity using the pre-existing models available in the literature, Krishnan et al33 for thermal conductivity and Zhu et al30 for elastic modulus.

Figure 11. Stiffness plotted against thermal conductivity for 160,000 the generated structures corresponding to 0.85 porosity. Properties of solid strut forming bulk material used as normalizing factor. The black curve represents metal foam for varying porosity

On analyzing the distribution more closely, it can be observed that structures corresponding to a high thermal conductivity and stiffness, in general, had fewer number of connections as compared to the structures having lower stiffness and conductivity. The structures with fewer struts had a larger diameter as compared to the ones having more number of struts since the porosity was constant. Fig. 12 shows the variation of diameter for different number of strut depending on whether it is an edge, face or body diagonal strut. The upper limit of the properties for the architected-structures was a line with slope 1. This implied that the normalized effective thermal conductivity could not be larger than the normalized effective stiffness for a given


23


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 36

architected-structure. This can be verified by considering a hypothetical structure with only vertical struts. The ratio of normalized stiffness to conductivity for such a structures is 1. Radius of struts is a dominant factor in defining the material characteristics in addition to the topology of the structure. The radius of struts for various structures can vary significantly (more than 3 times) even for the same porosity depending on the number of connections and the relative location of the connections (edge, face or body diagonal). Anisotropy index was calculated for the thermal conductivity and is plotted in Fig. 12. As expected, majority of structures had anisotropy index close to 1. The number of structures exponentially decreased for a higher index.

Figure 12. Histogram plot of number of structures against thermal and mechanical anisotropy index. The scale for number of structures is logarithmic. The number exponentially drops for higher indices. Similar results were observed for the mechanical anisotropy index. The number of structures with higher anisotropy exponentially decreased. It was also evident that the number of structures with relatively higher anisotropy was larger for mechanical stiffness as compared to the thermal conductivity. The similarity of thermal and mechanical characteristics with regards to the distribution of anisotropy index is more obvious in the plot below (Fig. 13). The structures were sorted by increasing mechanical anisotropy index and the corresponding thermal anisotropy index was plotted alongside.


24

Page 25 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 13. All the structures were sorted in order of increasing mechanical anisotropy and the corresponding thermal anisotropy was plotted alongside. Thermal and mechanical anisotropy indices show roughly similar trends with some outliers

Like foams, variation of porosity directly affects the properties of the architected-structures. Lower porosity structures in general tend to have a higher thermal conductivity and stiffness due to larger radius of the struts. Fig. 14 shows the effect of variation of porosity on the characteristics of the generated architected-structures.

(a) 0.9 porosity

(b) 0.85 Porosity


25


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 36

(c) 0.7 Porosity Figure 14. Variation of porosity ((a) 0.9 porosity , (b) 0.85 porosity and (c)0.7 porosity)affecting the distribution of properties relative to the metal foam curve. Lower porosity structures have larger range of properties. Foams are more conductive than generated structures at higher porosity

Plotting mechanical anisotropy index against thermal anisotropy index provides some insight about relative range of properties achievable (Fig. 15). Various fitness functions (combination of objective parameters to be minimized) can be defined and a pareto front can be generated to pick an optimum structure for specific application.


26

Page 27 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 15. The mechanical anisotropy index plotted against the thermal index for various generated structures. The points roughly lie close to the line with slope 1. There are outliers of interest that need to be evaluated in detail

4. DISCUSSION The large variation of properties for the generated class of architected-structures is desirable to create a ‘material library’.

Generated architected-structures have a wide range of possible

properties in comparison to parametrically designed foam structures for a given porosity as can be seen from fig 14. It should be noted that there are inherent trade-offs associated with the material properties for various structures generated. For instance a structure can have relatively high stiffness only by compromising stiffness in other two directions. Similar trends can be observed for the thermal properties as well. Additionally the stiffness has been shown to vary with the square of porosity whereas the thermal conductivity varies linearly. These differences provide for a significant variation of material properties in addition to the 160,000 structures generated through topology variation. The wide variety of possible structures can be used to potentially replace conventional porous media with a specially designed engineered structure.


27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 36

The applications can range from heat sinks, heat spreaders to bone scaffolds and orthopedic fixation devices. It can be observed that the overall trends for variation of the thermal and mechanical properties and anisotropy indices is similar as can be seen from fig 13. It is important to note that there are numerous exceptions and variations which can be exploited to address the trade-off when it comes to the thermal and mechanical response. Generating a set of diverse thermal and material properties is only the first step for achieving a customized optimum topology. The implication of the results is that for a given porosity, instead of having only one design point, there are now 160,000 other design points that can be used for design selection. The database of the generated material properties can be merged with various optimization algorithms to pick out the best suited architected-structure. A fitness function can be defined on the basis of multi-physics constraints that incorporate the thermal and mechanical trade-off. Adding an optimization layer in combination with a commercial finite element solver enables the analysis of a wide range of applications involving combined thermal and mechanical loads. The effective properties serve as input material properties to the finite element solver and allow the modelling of porous structures as a continuum. To showcase such a methodology, a mock heat spreader application was chosen for implementing the system to evaluate best suited structure for maximizing performance. A constant heat flux of 0.09W was applied at the bottom 1/8th corner surface. Symmetry conditions were imposed on two sides and adiabatic conditions on two sides. A convective boundary condition was imposed on the top surface. ANSYS was used to evaluate the fitness function. Input file was generated for the finite element model shown below (Fig. 17).


28

Page 29 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 16. Boundary conditions used for the finite element model developed in ANSYS. Input file was generated and imported into MATLAB for evaluation of objective function at every iteration Genetic algorithm was used for the actual optimizing process since it could easily handle discrete variables (structure number and corresponding material property). The material number index and porosity were two design variables for the problem. 90% bit string affinity was used as the stopping criterion for the genetic algorithm calculation. The fitness function was defined with objective to minimize average temperature and standard deviation of temperature on the top surface. An additional objective was set to minimize the elastic modulus in the Z-direction (along the direction of applied flux). Such optimization constraints are often encountered in heat spreader applications. Objective function φ: ∅=

EF/

EI + + GH EF/

(10)


29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 36

where, Tavg and Tσ are the mean and standard deviation of temperature, Eeff and Es are the effective and solid elastic modulus corresponding to the architected-structure and complete solid respectively. ND is a normalizing factor used to ensure that the relative contributions of all three terms was in the same range. Copper was used as the strut forming material for the analysis. The mean and standard deviation of the temperatures were calculated form the nodal results provided through ANSYS. The analysis was run multiple times and the last generation of possible solutions were consistently composed of the structure shown in the Fig. 18 with different porosity value lying in the range of 0.78 and 0.8.

Figure 17. The best final results from the last generation population processed through the genetic algorithm. A porosity value of 0.8 and 0.78 was found to be the optimum for the two structures respectively Table 3: Comparison of properties of two of the optimum structures with porous metal foams Porous metal Structure A(0.8 Structure B(0.78 (0.8 porosity) porosity) porosity) Normalized Thermal Conductivity

In plane

0.0667

0.0749

0.0747

Out of plane

0.0667

0.0136

0.0230

0.04

0.0184

0.0217

Normalize elastic modulus in Z


30

Page 31 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The objective was to achieve isothermal top surface through control of conductivity in various directions. Additionally a larger compliance is desirable in order to reduce the contact resistance. Compliance results in a larger contact area and thereby smaller contact resistance. The anisotropy of the resultant structure is evident from the geometry. Intuitively, lower thermal conductivity and stiffness is expected along Z direction since the structure has fewer struts oriented along Z direction (only the body diagonal). Whereas, the lower conductivity in Z ensures that the heat diffuses over the entire cross-section. The final structures provided not only have a larger compliance but also have tuned conductivity along various directions aligning with the objectives of the application. 5. CONCLUSION A methodology has been presented to systematically generate effective new micro-lattice structures (micro-architected structures). A total of 160,000 structures were generated and evaluated for a 27 node unit cell template using the methodology presented. Algorithms were developed and implemented for generation of structures and filtering out repeated symmetric structures as well as structures without continuity and repeatability. Effective thermal and mechanical properties of the generated structures were evaluated using low fidelity analytic models. The effective properties of the generated structures were evaluated by modelling them as orthotropic materials. The models were validated through experimentation as well as numerical simulations. The results indicated that a wide range of properties is achievable for the microarchitected meta-materials relative to the effective properties of porous metal foam structure. The maximum thermal and mechanical anisotropy was found out to be 7.1 and 8.2 respectively.


31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 36

The generated structures can be potentially used to replace conventional porous media in thermomechanical applications. To showcase the possible enhancements, an optimization framework was developed to determine best possible structure for given application. The framework was coupled with a commercial FE solver to simultaneously incorporate thermal as well as mechanical response. The optimization framework was applied on a mock heat spreader problem to demonstrate optimization process. For the same porosity (mass), the micro-architected design achieved 12% higher in-plane thermal conductivity and 55% lower elastic modulus compared conventional foams. The optimum structure extracted is better suited for application in terms of complying with the objective functions which is defined in a manner to ensure uniform heat distribution across the heat spreader. 6. REFERENCES [1] Gibson L J, Ashby M F. Cellular Solids: Structure and Properties [M]. 2nd ed. Cambridge: Cambridge University Press, 1997. [2] Yi, F.; Zhu, Z.; Zu, F.; Hu, S. and Yi, P. Strain Rate Effects on the Compressive Property and the Energy-absorbing Capacity of Aluminum Alloy Foams. Mater. Character. 2001, 47(5), pp.417-422. [3] Py, X.; Olives, R. and Mauran, S. Paraffin/porous-graphite-matrix Composite as a High and Constant Power Thermal Storage Material. Int. J. Heat Mass Transfer. 2001, 44(14), pp.2727-2737. [4] Calmidi, V.V. and Mahajan, R.L. The Effective Thermal Conductivity of High Porosity Fibrous Metal Foams. J. Heat Transfer. 1999, 121(2), pp.466-471. [5] Bhattacharya, A.; Calmidi, V.V. and Mahajan, R.L. Thermophysical Properties of High Porosity Metal Foams. Int. J. Heat Mass Transfer. 2002, 45(5), pp.1017-1031. [6] Shimko, D. A.; Shimko, V. F.; Sander, E. A.; Dickson, K. F.; & Nauman, E. A. Effect of Porosity on the Fluid Flow Characteristics and Mechanical Properties of Tantalum Scaffolds. J. Biomed. Mater. Res., Part B. 2005, 73(2), 315-324


32

Page 33 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[7] Sander, E. A., & Nauman, E. A. Permeability of Musculoskeletal Tissues and Scaffolding Materials: Experimental Results and Theoretical Predictions. CRC Crit. Rev. Bioeng. 2003, 31(1&2). [8] Banhart, J. Manufacturing Routes for Metallic Foams. Jom. 52.12, 2000: 22-27. [9] Chang, P.S. and Rosen, D.W. The Size Matching and Scaling Method: A Synthesis Method for the Design of Mesoscale Cellular Structures. Int. J. Comp. Integ. M. 2013, 26(10), pp.907-927. [10] Stanković, T.; Mueller, J.; Egan, P. and Shea, K. A Generalized Optimality Criteria Method for Optimization of Additively Manufactured Multimaterial Lattice Structures. J. Mech. Design. 2015, 137(11), p.111405. [11] Chu, J.; Engelbrecht, S.; Graf, G. and Rosen, D.W. A Comparison of Synthesis Methods for Cellular Structures with Application to Additive Manufacturing. Rapid. Prototyping. J. 2010, 16(4), pp.275-283. [12] Shea, K. and Smith, I.F. Improving Full-scale Transmission Tower Design through Topology and Shape Optimization. J. Struct. Eng. 2006, 132(5), pp.781-790. [13] Valdevit, L.; Jacobsen, A.J.; Greer, J.R. and Carter, W.B. Protocols for the Optimal Design of Multi‐Functional Cellular Structures: From Hypersonics to Micro‐Architected Materials. J. Am. Ceram. Soc., 2011, 94(s1). [14] Asadpoure, A. and Valdevit, L. Topology Optimization of Lightweight Periodic Lattices under Simultaneous Compressive and Shear Stiffness Constraints. Int. J. Solids. Struct. 2015, 60, pp.1-16. [15] Schaedler, T.A.; Jacobsen, A.J.; Torrents, A.; Sorensen, A.E.; Lian, J.; Greer, J.R.; Valdevit, L. and Carter, W.B. Ultralight Metallic Microlattices. Science, 2011, 334(6058), pp.962-965. [16] Lu, T.J.; Valdevit, L. and Evans, A.G. Active Cooling by Metallic Sandwich Structures with Periodic Cores. Prog. Mater. Sci. 2005, 50(7), pp.789-815. [17] Gu, X.W. and Greer, J.R. Ultra-strong Architected Cu Meso-lattices. Extreme Mechanics Letters, 2015, 2, pp.7-14. [18] Yannakakis M. Node-and Edge-deletion NP-complete problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, 1978, (pp. 253-264). ACM. [19] Paek, J.W.; Kang, B.H.; Kim, S.Y. and Hyun, J.M. Effective Thermal Conductivity and Permeability of Aluminum Foam Materials. Int. J. Thermophys. 2000, 21(2), pp.453-464.


33


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 36

[20] Singh, R. and Kasana, H.S. Computational Aspects of Effective Thermal Conductivity of Highly Porous Metal Foams. Appl. Therm. Eng. 2004, 24(13), pp.1841-1849. [21] Leong, K.C. and Li, H.Y. Theoretical Study of The Effective Thermal Conductivity of Graphite Foam based on a Unit Cell Model. Int. J. Heat Mass Transfer. 2011, 54(25), pp.5491-5496. [22] Bauer, T.H. A General Analytical Approach toward the Thermal Conductivity of Porous Media. Int. J. Heat Mass Transfer, 1993, 36(17), pp.4181-4191. [23] Boomsma, K. and Poulikakos, D. On the Effective Thermal Conductivity of a Threedimensionally Structured Fluid-saturated Metal Foam. Int. J. Heat Mass Transfer. 2001, 44(4), pp.827-836. [24] Dai, Z.; Nawaz, K.; Park, Y.G.; Bock, J. and Jacobi, A.M. Correcting and Extending the Boomsma–Poulikakos Effective Thermal Conductivity Model for Three-dimensional, FluidSaturated Metal Foams. Int. Commun. Heat Mass Transfer. 2010, 37(6), pp.575-580. [25] Dukhan, N.; Quinones-Ramos, P.D.; Cruz-Ruiz, E.; Vélez-Reyes, M. and Scott, E.P. Onedimensional Heat Transfer Analysis in Open-cell 10-ppi Metal Foam. Int. J. Heat Mass Transfer. 2005, 48(25), pp.5112-5120. [26] Bart-Smith H.; Hutchinso J.W. and Evans, A.G. Measurement and Analysis of the Structural Performance of Cellular Metal Sandwich Construction. Int. J. Mech. Sci. 2001, 43(8), pp.1945-1963. [27] Chen, C.; Harte, A.M. and Fleck, N.A. The Plastic Collapse of Sandwich Beams with a Metallic Foam Core. Int. J. Mech. Sci. 2001, 43(6), pp.1483-1506. [28] Kwon, Y.W.; Cooke, R.E. and Park, C. Representative Unit-cell Models for Open-cell Metal Foams with or without Elastic Filler. Mater. Sci. Eng., A. 2003, 343(1), pp.63-70. [29] Wicklein, M. and Thoma, K. Numerical Investigations of the Elastic and Plastic Behaviour of an Open-cell Aluminium foam. Mater. Sci. Eng., A. 2005, 397(1), pp.391-399. [30] Zhu, H.X., Knott, J.F. and Mills, N.J. Analysis of the Elastic Properties of Open-cell Foams with Tetrakaidecahedral Cells. J. Mech. Phys. Solids. 1997, 45(3), pp.319-343. [31] Sullivan, R.M.; Ghosn, L.J. and Lerch, B.A. A General Tetrakaidecahedron Model for Open-celled Foams. Int. J. Solids. Struct. 2008, 45(6), pp.1754-1765. [32] Hill, R. Theory of Mechanical Properties of Fibre-strengthened Materials—III. Selfconsistent model. J. Mech. Phys. Solids. 1965, 13(4), 189-198.


34

Page 35 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[33] Krishnan, S.; Murthy, J.Y. and Garimella, S.V. Direct Simulation of Transport in Open-cell Metal Foam. J. Heat Transfer. 2006, 128(8), pp.793-799.


35


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

TOC graphic 44x23mm (300 x 300 DPI)


Page 36 of 36

Systematic Generation, Analysis, and ... - ACS Publications

Recommend Documents