Chapter 1 Introduction - University of Virginia

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Chapter 1 Introduction

New lightweight structures capable of supporting a multiplicity of functions are needed as cores for sandwich panel constructions. Currently, stochastic foam materials and prismatic structures such as corrugations and honeycombs are used. Each of these materials / structures possesses certain strengths and weaknesses inherent to their respective topologies. For example, honeycombs provide the best combinations of stiffness and strength, but their closed cell nature makes them susceptible to corrosion and delamination, and requires costly inspection methods when they are used in primary structure applications. Millimeter length scale 3-dimensional lattice truss structures are being created to replace these foams and honeycombs by offering new combinations of stiffness, strength, robust performance and multifunctionality. However, they are currently made from steel and aluminum casting alloys. This thesis explores their extension to high strength (heat treatable) aluminum.

1.1 Background

Efforts to create structural materials that maximize combinations of stiffness and strength

and minimize weight have increasingly turned towards sandwich panel concepts that

utilize engineered cellular materials for their cores. Interest in these materials has

intensified throughout the twentieth century driven largely by the need for improved land,

marine and aerospace vehicles [1,2]. Insights about the mechanical performance of

structural materials can be gained by examining materials selection maps that rank the

stiffness and strength of structural materials as a function of their density [3]. Figure 1

shows two such maps created using software developed by Ashby and colleagues [4].

The charts reveal that in order to escape the lower density bound of bulk solids (the

lightly green shaded region where the density is less than ~1 g/cm3) some form of

porosity must be introduced. However, there is a concomitant loss of stiffness and

strength, the exact extent of which depends upon the topology of the material in the

porous structure and intrinsic properties of the material from which it is made.

Understanding how to shape the porosity to maximize mechanical performance and

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devising methods to affordably fabricate such structures represent two of the most

fundamental challenges facing new lightweight structural materials development today.

Figure 1. (a) Stiffness property chart used for materials selection. (b) Shown is the strength property materials selection chart. In both charts the upper left corner represents goodness. Materials of densities less than ~1 g / cm3 (light green region) all possess cellular topologies.

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1.1.1 Cellular Metals

In bulk metallic structural materials, porosity is invariably avoided since it results in

unacceptable losses of mechanical performance [5]. However, observations of natural

materials (e.g., bone, and wood) have long indicated that weight efficient load bearing

members have a cellular structure consisting of either reticulated ligaments (i.e., open

cell) or encapsulated membranes (i.e., closed cell). Examples from the work of Gibson

and Ashby are shown in Fig. 2 [6]. These observations led to the identification of a class

of materials known as cellular solids. A significant body of research into the structure

and properties of these natural materials has now been accumulated and organized [6].

Figure 2. Examples of cellular solids found in biological systems. These materials generally consist of reticulated (open cell) materials such as sponge and bone, and membrane (closed cell) materials such as the various wood based materials (courtesy L.J. Gibson, Cellular Solids, 1997).

Synthetic cellular materials from polymers, metals, and ceramics have also begun to be

developed. Intentional manufacture of metal foams can be traced to J.C. Elliot (1956)

Sponge Cancellous bone

Coral Cuttlefish bone

Iris leaf Plant stalk

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who describes the method of particle decomposition in molten alloys [7]. Extensive

research and industrial development has occurred since the 1970’s and has led to

numerous methods for making metal foams and several emerging applications [5].

Figure 3 shows examples of the topology of several metal foam products now available.

A recently published design guide by Ashby et al. reviews ways of making metal foams,

their fundamental properties and some of their current applications [5].

Figure 3. Examples of several metal foams currently available. The Cymat, Alporas and ERG Duocel foams are made from aluminum alloys by particle decomposition (Cymat), gas melt injection (Alporas), and pressure casting (ERG Duocel). The nickel based Incofoam is made by an electroless deposition process.

1.1.2 Topological Optimization and Multifunctionality

Natural cellular materials achieved optimal load support and added function (e.g., fluid

transport) over time through growth and mass redistribution processes. This is a form of

(a) Cymat (b) Alporas

(d) Incofoam (c) ERG Duocel

5 mm

5 mm 5 mm

0.25 mm

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natural topological optimization in which the density, location, shape and connectivity of

the pores is optimized to achieve certain functions [8], In nature this phenomenon is

accomplished by an internal dynamic material redistribution around a genetically

encoded blue print of previously successful designs. It has resulted in sophisticated

structures that accomplish a multiplicity of functions (e.g., bone that provides load

support, a calcium repository and a medium for blood cell growth). Figure 4 is an

example of a natural multifunctional topologically optimized cellular structure – an avian

bone. Notice its resemblance to a sandwich panel structure (i.e., a structure with a pair of

widely spaced high strength facesheets kept separated by a low density cellular structure).

Figure 4. Cross-section view of an avian wing bone [9].

The dynamic optimization (adaptation) processes used by nature have yet to be achieved

synthetically. Today, identifying and implementing static optimized topologies for

predetermined loadings is the current state-of-the-art in structural design [8]. Developing

optimal engineered cellular solids represents current research efforts, while successfully

extending this approach to the dynamic regime via nastic-like [10] or other processes

may be the next great leap in scientific understanding and technological mastery of

10 mm

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cellular materials, but for now we focus upon developing a fundamental understanding of

the structure – property relations of the static structures that can be fabricated today.

1.1.3 Classification of Cellular Metals

Metallic foams may be considered the first attempt at creating engineered cellular solids,

but as seen in Fig. 3 the structures are generally stochastic and little if any attempt to

create an optimal load supporting topology has been evident in their development. There

is another class of cellular solids that have characteristic 2-dimensional (2-D) periodicity.

Hexagonal honeycombs providing a common example of a prismatic cellular solid [6]. It

should be noted that honeycomb is a highly efficient 2-D load supporting topology, but

nearly optimal for only a few loading conditions [6], as is the case for all prismatic

cellular solids. Other 2-D topologies have been devised and are widely used for a variety

of applications [2,11]. Figure 5 shows a schematic profile representation of some

common and less common prismatic cellular solids.

Figure 5. Five samples of prismatic cellular topologies, of these only (a) hexagonal honeycomb has seen widespread application as core material for sandwich panel constructions. (b) triangulated, (c) square, (d) Kagomé, (e) Star-hex [12].

A new class of cellular solids has emerged based upon 3-dimensional lattices of trusses

[13,14]. These lattice truss structures offer the means to achieve the mechanical

efficiency of prismatic cellular solids, and the potential to create optimal load supporting

multifunctional materials due to their open cell topologies.

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1.1.4 Lattice Truss Structures

Recent efforts continue to improve the weight to stiffness and/or strength ratios of

multifunctional cellular solids have begun to investigate lattice truss structures [15]. If

the criterion for multifunctionality requires a quasi-isotropic open cell topology [15], then

these open truss structures appear much more promising than closed cell foams,

honeycombs and other prismatic structures.

Historically lattice truss structures are found at large length scales (e.g., civil engineering

structures such as sports stadium roofs). Lattice truss plates gained widespread adoption

in the 1970’s following the patented inventions of the Octet Truss and related structures

by R. Buckminster Fuller, Fig. 6 [16].

Figure 6. Patented Octet Truss by Buckminster Fuller for civil applications [23].

These lattice truss structures reduce the “dead load” and allow for much greater

unsupported spans (hundreds of meters) to be achieved [17]. The improvements are

based on the principle that the trusses within the structures predominately experience

axial stresses (tension or compression) when loaded (i.e., they are stretch dominated)

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[18]. This mode of loading is capable of higher load support than bending dominated

concepts.

M. Lake appears to be the first to apply the techniques of crystallographic symmetry to

develop a wholly deterministic approach for predicting the globalized elastic stiffness and

strength properties of large space filling lattices [19]. More recently V.S. Deshpande et

al., have applied Maxwell rules for assessing the necessary (but not sufficient)

topological requirements for obtaining stretch dominated lattices [20]. After scaling these

structures to millimeter lengths and smaller, substantial improvements in strength and

stiffness compared to metallic foams have been predicted [20]. Indeed, micro-lattice

truss structures are the first to take the concepts of 3-D shaping and make progress

towards the achievement of minimum weight quasi-isotropic materials (which would fill

the desired vacant regions in the top-left portion of Figure 1).

It has been shown that open cell metal foams (the topological sibling of lattice trusses)

have macroscopic mechanical behaviors governed by beam bending [5]. Thus their

stiffness, when scaled by their relative density parameter (i.e., the ratio of the volume of a

unit cell occupied by material to that of the cell volume, or equivalently the unit density

to the solid density from which it is made), , scales by 2 , and the best case the

strength scales with 2/3 . The stiffness and strength properties at low relative densities

( < 0.10) limit their usefulness (due also impart to low measured scaling coefficient

values) in applications requiring high structural load support [5] and provide the impetus

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for seeking lattice truss topologies with improved mechanical properties that are

predicted to scale linearly with in both stiffness and strength.

Utilizing lattice truss structures for the cores of sandwich panels has been suggested as a

means to achieve efficient load support and other functionality such as cross flow heat

exchange or high intensity dynamic load mitigation. The lattice truss topologies (Fig. 7)

of interest include the Octet-truss [21], its derivative mono-layer tetrahedral structure [22-

24], the lattice-block [25-27], and its mono-layer pyramidal derivative [28], and the 3-D

Kagomé structure [29, 30], a variant of the tetrahedral topology.

(a) OCTET TRUSS (c) LATTICE BLOCK (b) TETRAHEDRAL LATTICE TRUSS (d) PYRAMIDAL LATTICE TRUSS

(e) 3D KAGOMÉ

Figure 7. Lattice truss topologies recently investigated. All have been made by investment casting. The tetrahedral (b) and pyramidal (d) trusses have also been fabricated by the folding of perforated sheet. In (b),(d) and (e) the lattice truss structure is bounded by solid face sheets. Development of suitable methods for the manufacture of these lattice truss structures

from high performance light alloys has paced their application [31]. Initial efforts utilized

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investment casting of high fluidity non-ferrous casting alloys such as copper/beryllium

(Cu-2Be wt.%) [29] aluminium/silicon (Al-7Si-0.3Mg wt.%) [21, 22, 26, 27], and silicon

brass (Cu-4Si-14Zn wt.%) [22]. However, the intricacy of the lattice paths made it

difficult to fabricate porosity free structures. These cast lattice materials lack the

mechanical robustness required for many structural applications [31,32]. This has led to

the development of other approaches by Sypeck and Wadley that exploit the ductility of

wrought metals [33]. They include perforated sheet folding [33] and wire lay-up

techniques [34]. These resulting structures can be bonded to each other or to facesheets

by transient liquid phase (TLP) bonding, brazing, micro welding, or other metal fusion

techniques.

To date, austenitic stainless steel structures with tetrahedral [33] and pyramidal truss [28]

geometries have been made by this method. However, lattice truss structures made from

austenitic stainless steels remain in an annealed (i.e., low strength) condition after the

bonding process. While they are much more robust than their investment cast

counterparts [31], their low specific strength reduces their desirability for weight

sensitive structural applications. Extensions to light alloys are therefore desired.

1.2 Goals of the Thesis

Previous studies of lattice truss structures manufactured via the investment-casting route

were unable to experimentally probe the mechanical properties at low relative densities

because of manufacturing limitations. This approach also precluded a thorough

investigation of the relationship between alloy mechanical properties and those of the

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lattice truss structure. Here we show that the newly invented perforated sheet folding can

be applied to age hardenable aluminium alloys (e.g., AA6061) and affords an opportunity

to examine the role of the mechanical properties of the parent material upon the specific

strength of the lattice truss structures over a wide range of relative densities.

1.2.1 Thesis Outline

The thesis is organized as follows: Chapter 2 presents the mechanical property

predictions for tetrahedral lattice truss cores. Chapter 3 reports on lattice truss

fabrication technologies developed during the course of the research. Chapter 4 details

the fabrication methodology for making age hardenable Al-Mg-Si alloy tetrahedral lattice

truss sandwich panels. Chapter 5 gives the experimental methods used to determine the

lattice truss mechanical response. Chapter 6 reports the experimental lattice truss

compressive and shear response results for both annealed and age-hardened alloy

tempers. In Chapter 7 the measured stiffnesses and strengths are compared to the model

predictions and to competing cellular metal topologies. Chapter 8 summarizes the

study’s findings while Chapter 9 provides a brief outline of future work.

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Chapter 2 Mechanical Behavior Predictions

Sandwich panels made with thin strong facesheets are being increasingly utilized in

weight sensitive structural applications [2]. The core structures of these panels must

simultaneously meet several requirements to ensure the successful mechanical

performance of the sandwich design [2,11]. These functions include, (i) maintaining the

allowable separation between facesheets (ii) prevention of the facesheets from sliding

past one another, and (iii) ensuring the facesheets remain nearly flat (with respect to their

original configurations) [2]. When the loads exceed the ability of the core to sustain

these requirements, failures of various forms may occur. Whether or not these failures

are catastrophic (i.e., sudden and substantial loss of load support) depends largely upon

the topology for a given core relative density [11]. Moreover, the magnitudes of the

loads that are sustained also depend upon the topology of the core structure. Figure 8

shows some of the more commonly encountered types of core influenced failures, and the

mechanical properties that determine the onset of these failures [11].

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Figure 8. Schematic of the common core dependent failure modes of sandwich panels.

With the wide availability of strong facesheet materials it is the properties of the core that

now drive the performance of many sandwich panels and identifying the best candidate

requires both topological and material considerations [35]. The tetrahedral lattice is one

of the most weight efficient quasi-isotropic topologies known, with predicted strengths

varying less than 15% in the out-of-plane shear directions [22, 28]. This provides the

basis for characterizing the tetrahedral lattice truss performance over a range of practical

relative densities.

The mechanical properties of tetrahedral lattice truss sandwich core constructed from

elastic perfectly plastic and plastic strain hardening materials have been reported by

Desphande and Fleck [22]. The analytical predictions for stiffness and collapse strength

of the tetrahedral lattice are based upon the treatment of truss members as individual

axially loaded columns comprising the tetrahedral unit cell. The mechanical properties

are then globalized via the introduction of the relative density.

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2.1 Relative density

The relative density has been used to characterize the properties (mechanical and

otherwise) of porous materials [6]. It also may be used to characterize the properties of

lattice truss structures, and to make comparisons with other cellular topologies [5,6].

Simple expressions can be derived linking the relative density to the topology of a

representative unit cell. Consider the tetrahedral lattice truss unit cell defined in Fig. 9.

The relative density, of the lattice truss is the volume fraction of the truss members

occupying the unit cell. Appendix A.I derives an expression relating the relative density of

the free (design) parameters of the tetrahedral lattice, shown in Fig. 9. For square cross-

section truss members it has the form:

2

2

1

3

2

l

t

sincos (1)

where is the included angle (the angle between the truss members and the base

tetrahedron) and t and l are the thickness and strut length, respectively.

Figure 9. Tetrahedral unit cell used to derive relative density and mechanical properties.

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2.2 Elastic Stiffness

For lattice truss materials that are stretch dominated the elastic properties are predicted to

scale linearly with the relative density [20]. For a single layer tetrahedral lattice truss two

independent stiffness constants (the out-of-plane compression and shear stiffness) are

important for the mechanical performance of a sandwich panel [22]. The out of plane

compression stiffness, E33, can be written (Appendix A.II):

433 sinsEE (2)

where Es is the Young’s modulus of the parent alloy, is again the included angle

defined in Fig. 9, and is the relative density defined by equation (1). The out-of-plane

shear stiffness (Appendix A.III) of tetrahedral lattice truss is given by [22]:

2sin8

22313

sEGG (3)

2.3 Strength Predictions

Here we specify the compressive and shear strengths of the tetrahedral lattice trusses,

respectively. Of the several sandwich panel failure modes possible, the compressive

strength of the lattice determines the failure loads for flexural crushing, local crushing

and facesheet wrinkling, while the shear strength of the truss controls the transverse shear

panel failure, Fig. 8.

In determining the compressive strength, first consider the tetrahedral lattice truss made

from rigid perfectly plastic material with tensile yield strength, y . In this case yielding

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(of the entire lattice) is coincident with the peak strength, pk33 , and is given by (Appendix

A.IV):

233 sinypk (4)

A linear dependence of compressive strength upon parent alloy yield strength and truss

relative density is predicted.

Next, consider the case where elastic buckling of the constituent truss members controls

core collapse. The peak compressive strength is obtained by replacing y in equation (4)

with the elastic buckling strength of a truss member. Written explicitly, the predicted

peak compressive strength becomes:

22322

33 cossin38

spk E

k (5)

where sE is the Young’s modulus of the solid (parent) material and k is a factor

accounting for the rotational stiffness of the ends of the struts: 1k or 2 for pin-ended or

built-in end conditions, respectively.

Very slender (low ) truss members will fail by elastic buckling whereas stocky trusses

will fail by plastic yielding. By equating (4) and (5) it is possible to define the that

partitions which failure controls the compressive strength of the core. Tetrahedral lattice

trusses made from an elastic ideally-plastic material will collapse by elastic buckling of

the constituent truss members when the relative density:

s

y

Ecossink

222

38 (6)

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Now consider a tetrahedral lattice made from an elastic-strain hardening material.

Compressively loaded trusses begin to plastically deform (yield) at a strength given by

(4), but the lattice can continue to support increased load because the flow stress of the

truss material increases due to parent alloy strain hardening effects. In such cases, as

discussed in [22], the trusses of the tetrahedral lattice eventually collapse by plastic

buckling at a plastic bifurcation stress, cr , given by Shanley-Engesser tangent modulus

theory [36,37]:

222

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l

tEk tcr

(7)

where Et is the tangent modulus defined as the slope d/d of the uniaxial stress versus

strain curve of the solid material at a stress level cr [37]. The compressive strength of

the lattice truss is obtained by replacing y in (4) by cr . Note that in the case of a

material with a linear strain hardening response tE is a constant and pk33 again scales

with 2 , whereas pk33 of a tetrahedral core made from ideally-plastic solid material

scales linearly with relative density.

We proceed to specify the minimum out-of-plane shear strength of the tetrahedral lattice

truss. Consider the load application in the 1-2 plane of Fig.9. Using methods analogous

to the compression case enables expressions for the lattice truss shear strength for an

elastic perfectly plastic parent material to be derived. From symmetry of the tetrahedral

arrangement in the 1-2 plane the lattice truss shear strength is periodic (period of 2/3)

with respect to the load orientation angle (Fig. 9). The out-of-plane shear strength

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therefore has a minimum and maximum strength direction. The minimum out-of-plane

lattice truss shear strength lies in the 1-3 type directions and is given by (Appendix A.V):

2sin413min

yy (8)

where y is the material yield strength, is the included angle and is the relative

density. The maximum out-of-plane lattice truss shear strength is:

2sin32

23maxyy (9)

Next, consider the case for a plastic strain hardening material with tensile strength, TS .

Here the shear strength of the lattice in the 1-3 shear direction is now given by replacing

y in (8) by TS :

2sin413TSpk (10)

whereas in the 3-1 shear direction, Fig. 9, the shear strength is given by replacing y in

(8) by cr :

2sin431crpk (11)

where cr again corresponds to the column (truss member) bifurcation stress. Note that

a difference in TS and cr associated with strain hardening materials will lead to

asymmetric shear strength of the tetrahedral lattice truss in the 1-3 (minimum strength)

type directions [22].

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With these predictions in hand a perforated sheet folding method was devised for

fabricating samples from a heat treatable 6061 aluminum alloy (Chapter 4). A series of

experiments were then designed and conducted (Chapters 5 and 6) to probe the lattice

compressive and shear stiffness expressions (2) and (3), as well as the compressive

strength (4) and the shear strengths corresponding to (10) and (11). First, however, we

explore technologies developed for creating lattice structures then detail the methodology

used to make tetrahedral lattice truss sandwich panels using an age hardenable aluminum

alloy (AA6061).

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Chapter 3 Fabrication Methodologies

3.1 Development of Lattice Truss Fabrication Technologies

The use of lattice trusses as core structures in ultralight sandwich panels has been paced

by the methods of their manufacture. Therefore, an important component of the thesis

research task involved inventing and developing new and improved approaches for

making lattice structures from light alloys. A guiding theme has been the maximization

of the fraction of original starting material incorporated in the structure while maintaining

mechanical performance.

3.1.1 Multilayer/Full-Occupancy Tetrahedral Lattice Trusses

Four new processes for creating lattice truss structures have been developed. The first,

Figure 10, is an extension of the perforating and folding of metal sheets developed by

Sypeck and Wadley for pyramidal and tetrahedral lattices [33]. Here a 2-dimensional

perforation pattern was created which, upon folding resulted in a tetrahedral lattice with

all sites occupied. Previous tetrahedral lattice trusses (as shown in Fig. 7(b)) have only

50% of available tetrahedral sites occupied by trusses. Figure 9 shows the two available

sites per unit cell. Folding was accomplished with an interleaving comb punch/die (Fig.

10). The resultant lattice may take the form of either two 50% occupancy tetrahedral

layers, or a fully occupied single layer depending upon the configuration of the punch/die

tooling [38]. Figure 11 shows an example of the double layer lattice truss made from

AA6061.

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Figure 10. Schematic of the manufacturing process of a generalized perforation and folding process for making multilayer tetrahedral lattice truss structures. The punch and die sets may be reconfigured such that the bottom tetrahedral layer is folded upwards resulting in a fully occupied single layer lattice.

Figure 11. Photograph of the double layer tetrahedral lattice made from AA6061.

10mm

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Simple estimates can be made for the fraction of material utilized by the perforated sheet

method and its dependence upon This is done by relating the relative density of the

flat perforated sheet, D2 to the relative density of the folded lattice structure, . To a

first approximation (i.e., ignoring overlapping truss member volumes at the nodes) the

general relationship for fraction of material utilized is given by:

CU D 2 (12)

where C is a topology specific proportionality constant. The materials utilization

expression for a tetrahedral lattice (assuming regular tetrahedrons, = 54.7°) as shown

in Fig. 9, becomes:

97.0U (13)

If we apply the same considerations to the multilayer / full occupancy tetrahedral lattice

approach of Fig. 10 we may write:

94.1U (14)

Appendix B details the derivations for eqns (13) and (14). Figure 12 shows the utilization

improvements garnered by multilayer / full occupancy lattice compared to the tetrahedral

lattice. From Fig. 12 we observe that for a tetrahedral lattice with a relative density of

1% only 10% of starting material is used, but for the same relative density the full

occupancy method will double the amount of material utilized to 20%.

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Figure 12. Materials utilization improvement between the single-layer folding method (Figure 15), and the multilayer/full occupancy folding method shown in Figure 10.

3.1.2 Lattices from Expanded Metal Sheet

A second method for manufacturing pyramidal lattices was devised that yields close to

~100% materials utilization. In this approach, a metal sheet was slit and expanded by

stretching [39] (Fig.13). A cold rolling step was used to flatten the expanded sheet.

Finally, the bending locations were assigned and the sheet was folded with a finger-brake

to create a 3-D pyramidal lattice.

Figure 13. Schematic of the manufacturing process for the expanded pyramidal lattice truss cores. The primary steps involve slitting, flattening, and folding the metal sheet.

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A recent study by Kooistra and Wadley shows that this method results in comparatively

larger core-to-facesheet bonding nodes, and the subsequent compressive and shear

mechanical performance of the trusses are reduced in scale by the mass fraction of metal

located at the nodes [39]. Figure 14 shows examples of the expanded metal truss structure

at various stages of processing.

Figure 14. Expanded pyramidal lattice trusses made from AA3003. (a) Expanded and flattened sheet (b) pyramidal lattice after folding and (c) shows the lattice after being brazed to aluminum alloy (AA6061) facesheets.

(a)

10mm

(b)

10mm

(c)

10mm

25

Significant refinements appear feasible by exploring the limits of truss and node mass

apportionments possible with the expansion process [39]. Moreover, other researchers

have reported potential issues in core robustness due in part to the node bond size [40],

indicating a trade-off between lattice truss and node bond mechanical properties.

3.1.3 Lattice Before Expansion Method

A third high materials utilization method for creating lattice structures has been

motivated by the process used to make honeycomb sandwich core structures, Figure 14

[11]. In this approach metal sheets were coated with a bonding agent (either a polymeric

adhesive or braze alloy) or strip welded to create a block of periodically joined material.

A wire EDM (electric-discharge machining) method is then used, prior to the expansion

that would normally result in honeycomb, to cut a zigzag pattern from the block.

Stretching of the zigzag block then yielded a tetrahedral lattice as shown in Fig. 15. This

method has both high materials utilization and a node area similar to the perforated sheet

method. The approach could be used to create tetrahedral or pyramidal topologies, Fig.

16.

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Figure 15. Schematic for the lattice before expansion (LBE) process. The method is similar to HOBE (honeycomb before expansion) with the addition of the pattern cutting step.

Figure 16. Effect of varying bond line width in (LBE) process. (a) bond widths equal to truss lengths result in the tetrahedral topology upon expansion, (b) small bond widths produce the pyramidal topology.

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3.1.4 Slot Truss Method

A fourth method for fabricating pyramidal lattices from any sheet material has also been

developed, Fig. 17. A primary attractiveness of this approach is that sheet material

ductility is not required as the materials employed need only satisfy the criteria involved

in cutting and bonding, and once again nearly all of the initial sheet material is utilized.

Figure 17 illustrates the two steps needed to create assembled pyramidal lattices. The

approach proceeds as follows: First the desired cutout pattern is generated. This may

include gradations of truss member included angles to achieve a final contour of the

constructed lattice. Truss member cutout cross-section profiles may be varied according

to structural requirements, but the width of the node slot must be equivalent to the

materials sheet thickness at the node to ensure a good fit.

Figure 17. Schematic for the slot truss fabrication method.

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Cutouts may be made by appropriate methods such as machining, wire electrical

discharge machining, laser cutting or abrasive water jet cutting, or chemical milling. The

length scale of lattices made by this method is limited only by the minimum slot width

achievable which can be on the order of 50m if chemical milling is applied and as large

as tens of centimeters by more conventional cutting methods. The fitted node locations

must be bonded / joined in a suitable fashion so as to minimize the effects of stress

concentrations associated with the node slots. For metallic materials this may involve

fusion welding, brazing or diffusion bonding. The approach appears extendable to

polymer composites. In this case adhesive bonding with metallic bridging pins (z-pins)

may be necessary [40]. Figure 18 shows an example of a slotted and assembled

pyramidal lattice made from 3-ply birch veneer.

Figure 18. Pyramidal lattice made using the slotted truss approach. Prototype non-ductile material is a 3-ply birch wood laminate of 1.5mm thickness.

25mm

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Chapter 4 Tetrahedral Lattice Truss Test Samples

The remainder of the thesis focuses upon the tetrahedral lattice truss sandwich panels

manufactured by the newly invented perforated sheet folding process [33]. The method

is applied to the age hardenable Al-Mg-Si aluminum alloy (6061) system. The formed

lattices are brazed to solid facesheets of similar alloy compositions, and these sandwich

panels will then be mechanically tested in compression and shear for relative densities

between 2.0 – 10.6% for both annealed and peak age hardened heat treatments.

4.1 Lattice Truss Fabrication

A folding process was used to bend elongated hexagonal perforated AA6061 (Al-0.6Si-

1.0Mg-0.28Cu-0.20Cr wt.%) sheet to create a single layer 50% occupancy tetrahedral

truss lattice [41]. Figure 19 schematically shows the process. The folding was

accomplished node row by node row using a paired punch and die tool with the sheets

folded so as to form regular tetrahedrons (that is the angle = 54.7º).

Figure 19. Schematic of the manufacturing process of the tetrahedral lattice truss cores involving perforation and folding.

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An example of an elongated hexagonal perforated sheet (with open area fraction of 0.82)

that would create a tetrahedral lattice with predicted relative density, = 0.067 is

shown in Fig. 20(a). Figure 20(b) shows a folded tetrahedral lattice truss with a

(measured) pre braze relative density, . The relative density of the truss cores

was varied by using different perforated sheet thicknesses and appropriately spacing the

perforating punches to maintain a square truss cross section. This required only one

punch/die set to produce the five relative density lattices investigated.

(a) Perforated sheet (b) Tetrahedral lattice truss

Figure 20. (a) Photographs of the perforated sheet used to form a 4.8 % relative density core and (b) the corresponding tetrahedral lattice truss after the folding operation.

4.2 Sandwich Panel Construction

Sandwich panels were constructed from the folded truss structures by placing a

tetrahedral lattice core between AA6951 alloy sheets clad with AA4343 braze alloy.

Alloy compositions, melting and brazing temperatures are shown in Table I. The

assembly was then coated with proprietary metal-halide flux slurry (Handy Flo–X5518,

Lucas Milhaupt Inc., Cudahy, WI), dried, and placed in a muffle furnace for brazing.

Each assembly was heated to between 595±5ºC for approximately 5-10±1 minutes

(depending on sandwich mass) to minimize joint weakening associated with silicon

10mm

t l

10mm

31

interdiffusion from the brazing alloy [42-44]. After air-cooling to ambient temperature,

one set of sandwich panels was solutionized at 530ºC for 60 minutes and subsequently

furnace cooled to place the alloy in the annealed (O) condition. A second set of panels

was water quenched from the solutionizing temperature and then aged at 165ºC for 19

hours. This achieved the peak strength (T6 temper) for the AA6061 alloy [45].

Table I. Aluminum alloy compositions and temperatures for melting and brazing [43].

Si Cu Mg Zn Mn Fe Cr Ti Melting Brazing

6061 0.4-0.8 0.15-0.40 0.80-1.2 0.25 0.15 0.7 < 0.35 0.15 616-652

6951 0.2-0.5 0.15-0.40 0.40-0.8 0.20 0.10 0.8 -- -- 616-6544343 6.8-8.2 0.25 -- 0.20 0.10 0.8 -- -- 577-602 593-616

Temperatures (ºC) Composition (wt%)AA designation

It must be noted that this thermal history was designed given the constraints of the

laboratory equipment, and less thermally expensive routes are possible. No visible

distortion was observed after water quenching. Tensile test coupons of AA6061

accompanied the cores through each thermal process step and were later used to

approximate the mechanical properties of the parent material used in the tetrahedral

lattice trusses. Figure 21 shows the schematic laboratory brazing/heat-treatment process

flow.

32

Figure 21. Furnace brazing and heat treatment process used to create AA6061 tetrahedral lattice truss sandwich panels.

Two orthogonal views of a sandwich panel with a post braze 3.0% relative density core

are shown in Figure 22. The brazing step results in an increase in the relative density of

up to 0.8% and depended on the AA4343 clad thickness.

33

Figure 22. Views of the 3.0% relative density brazed tetrahedral lattice (a) as seen from the 2-3 direction (b) as seen from the 1-3 direction, see Fig 9. Table II compares the measured (pre and post brazed) and predicted relative densities.

The first order model (eqn. 1) over predicts pre-braze relative densities due to the

“double-counting” of the nodal volumes [22], especially at the higher relative densities

where the nodal volumes become significant.

Table II. Predicted and measured relative densities of tetrahedral lattice truss structures reported at a confidence level of 95%.

t/l

PredictionPre-braze measurement

Post-braze measurement

0.063 0.017 0.017 ± .003 0.020 ± .001 0.079 0.027 0.025 ± .004 0.030 ± .002 0.099 0.042 0.029 ± .004 0.039 ± .001 0.125 0.067 0.048 ± .003 0.069 ± .002 0.178 0.136 0.083 ± .002 0.106 ± .003

Relative density

10mm

10mm

1

3

2

3

34

Post-braze relative densities increase at lower relative densities due to the volume of

braze fillets being proportionately large enough to counter the effects of double counting

of the nodes, Fig. 23. In this study the lattice trusses will be identified by their post braze

(experimentally determined) relative densities, and subsequent data normalizations are all

based on these measurements.

Figure 23. The percent change in relative density compared to the predictions after brazing. At the lowest relative densities the increases become more pronounced due to the relatively larger volume of the core occupied by the braze fillets.

35

Chapter 5 Experiments

Three types of experiments were performed to characterize the mechanical response of

the aluminum tetrahedral lattice trusses and the parent alloy mechanical properties. Here

we explain the methods used for gathering the data. The first experiments measure the

uniaxial tensile response of the aluminum 6061 alloy, which is used to make predictions

of the lattice truss stiffness and strength properties. Next, the compression experiment

methodology is shown, and we conclude by detailing the tests for obtaining the lattice

truss shear mechanical property data.

5.1 Parent Alloy Uniaxial Tensile Tests

Experiments to measure the uniaxial stress vs. strain response of the AA6061 followed

ASTM E-8 guidelines [45]. Uniaxial tension specimens were machined from AA6061-

T651 (6.35mm plate thickness). A servo-electric universal testing machine (Model 4208,

Instron Corp., Canton, MA) with self-aligning grips was used to test each specimen at

ambient temp (~25°C). The applied nominal strain rate was 0.2mm/min (10-3 s-1), and

the strain measurements were made using a linear variable differential transformer

(LVDT) clip-on extensometer with an accuracy of ±0.5% of the gage length of 50mm.

5.2 Laser Extensometer Validation

A matrix of 50 lattice truss compression and 40 shear tests were needed for this study.

Therefore, it was decided to use a laser extensometer (Model LE-01, Electronic

Instrument Research, Irwin, PA) to obtain nominal strain data. The laser extensometer

36

has a displacement resolution of 0.001±0.001mm. Before testing the sandwich panels a

check on the quality of the laser displacement data was performed. This was done by

comparing it to LVDT displacement data. A tensile test (6061 tensile bar) was performed

that recorded both clip-on LVDT and laser extensometer displacements simultaneously.

The 6061 elastic modulus served as the standard. The tensile Young’s modulus of

AA6061 has an accepted value of 68 GPa at 25°C using ASTM test methods [47]. The

compressive Young’s modulus is reported to be 2% greater (69 GPa). Figure 24 shows

the elastic Cauchy stress vs. logarithmic strain using both measurement methods. The

plotted LVDT data for Es is in agreement within < 1% of the literature. The laser

extensometer agreed with the literature to within < 1.5%. From this test we were

confident the laser extensometer would be sufficiently accurate for the tetrahedral lattice

experiments. Further calibration of the laser data was not performed.

Figure 24. The comparative measurements of AA6061-T651 elastic stress vs. strain behavior. Data acquisition is by laser and LVDT extensometry.

37

5.3 Compression Experiments

The sandwich panels were tested in compression following the guidelines of ASTM STP

C 365-00 [48] for sandwich panel constructions. A screw driven testing machine (Model

4208, Instron Corp., Canton, MA) was used at an applied nominal strain rate 10-3 s-1.

Figure 25 shows a photograph of the experimental setup. The sample was marked with

retro-reflective tape on both the upper and lower facesheets to set the gage length and

placed between two polished 150mm diameter steel platens. The test samples were not

constrained in the directions normal to the applied load. Measured load cell force was

used to calculate the stress applied to the lattice truss core. The nominal macroscopic

truss strain was calculated by dividing the displacement data by the original sandwich

core height.

Figure 25. Photograph of the compression test setup in the Instron 4208 test machine. Compression platens are 150mm in diameter. The load cell is rated at 300kN.

laser extensometer

150mm platens

300kN load cell

Sandwich panel sample

38

5.4 Plate Shear Experiments

Previous studies have identified the tetrahedral lattice truss orientations corresponding to

minimum core out-of-plane shear strength [22]. These orientations (loading in the 1-3

type directions) are shown schematically in Figure 9, and are predicted to fail by either a

single unit cell truss member in compression (the 3-1 direction) or in tension (1-3

direction). The tests were designed to probe these strength minimums. The brazed

sandwich panels were tested according to ASTM STP C-273-00 using a compression

shear plate configuration [49], as shown in Figure 26. The samples were sheared in the

Instron Model 4208 testing machine at an applied nominal strain rate 10-3 s-1. The

measured load cell force was used to calculate the stress applied to the sandwich, and the

displacement data was obtained by the laser extensometer. The height of core was used to

set the retro-reflective tag gage length.

Figure 26. (a) Compressive plate shear test fixture schematic showing the fixation details. (b) A 3.9% relative density sample in place. The bolt-on gages shown on the front face hold the retro-reflective laser tags used to mark the gage length.

retro-reflective laser tags

10mm

shear plate

(a) (b)

39

Previous studies have found good fixation of the test samples critical to obtaining

accurate data, especially at high strains and for high strength cores [22,31]. Here the test

samples were attached using four techniques in concert to prevent premature debonding

and/or movement relative to the shear plates (Fig. 26(a)). Fixation was accomplished by

epoxy bonding (Loctite Hysol® E-120HP), with mating surfaces grit blasted and

degreased with acetone. The facesheets were also tapped and threaded to accept machine

screws (ANSI #10-32) fastened through the shear plates. Further mechanical holding

was provided by a leading edge stop machined into the shear plate and a trailing edge

adjustable clamping bar (Fig. 26(a)).

40

Chapter 6 Results

6.1 Parent Alloy Constitutive Response

To utilize the predictions in Chapter 2 that relate lattice truss topology to mechanical

properties, the constitutive response of the parent material must be known. Data for the

uniaxial elastic stiffness, yield strength, tensile strength and work hardening

characteristics are required. This can be obtained from tensile test data directly or by

fitting a modified power law relation such as the Ramberg-Osgood equation [50]. Since

the mechanical response of AA6061 is very sensitive to heat treatment, it was decided to

make direct measurements. From a uniaxial stress strain curve we may obtain the

effective values of the Young’s modulus, Es, yield strength, y, tensile strength, TS, and

use d/d to compute values of the bifurcation stress, cr (Et) for a given relative density.

The underlying assumption of the direct measurement technique is that the stress strain

response in both tension and compression are sufficiently symmetric to provide valid

predictions.

6.1.1 Effects of Heat Treatments

Five heat treatments were designed to explore the range of mechanical properties

available from the 6061 aluminum alloy. The first tensile specimen was solutionized at

530°C for 90 minutes (specimen temperature) and allowed to furnace cool to place it in

the annealed (O) condition. The four remaining samples received the same solutionizing

treatment, but were subsequently cool water (15°C) quenched. Three of these samples

were then given artificial aging treatments at 190°C for 1, 5 and 100 hours, and the fourth

41

given an industry practiced treatment at 165°C for 19 hours [45]. The artificial ageing at

190°C was done to achieve over ageing at practical time scales. Figure 27 shows the

Cauchy (true) stress vs. logarithmic strain responses.

Figure 27. The monotonic uniaxial stress vs. strain response of AA6061 for three artificial aging treatments.

For this study the annealed (O) and (T6) constitutive responses were chosen for

application to the lattice truss experiments. Table III shows the mechanical property

values obtained from the uniaxial Cauchy stress – logarithmic strain curves. The yield

strength, y, is taken at the 0.2% offset per convention, and the tensile strength, TS, is the

peak stress prior to fracture of the specimen. The fracture strain is denoted by f.

42

Table III. The measured elastic and strength properties of AA6061 for various heat treatments.

Heat Treatment Es (GPa) y (MPa) TS (MPa) f (%)

190°C 1hr 67.0 222 302 20.9190°C 5hrs 68.3 289 346 18.9190°C 100hrs 67.7 290 333 16.3165°C 19hrs (T6) 69.1 268 306 15.1Annealed (O) 68.6 70 208 21.2

6.2 Compressive Stress Strain Response

The representative stress strain responses of tetrahedral truss cores loaded in compression

at each relative density and heat treatment are shown in Figure 28. The lattice trusses

exhibit similar compressive stress strain behavior to that of many cellular metals [5].

After some initial bedding-in there is a region of linear type loading. Following the linear

response, gradual core yield occurs followed by a peak in the compressive stress.

Continued loading resulted in “softening” followed by a stress plateau until densification

(at a strain of 0.5 to 0.6) where upon the core exhibited greatly increased load resistance.

Figure 28 reveals that the degree of softening depended both on the relative density of the

truss core and the metallurgical state of the parent alloy. For impact energy absorption

applications, a stress versus strain response with little or no softening after yield is

desirable [6]. The highest relative density annealed samples, Fig. 28, exhibit this

behavior.

43

Figure 28. The combined compressive response of (a) age hardened and (b) annealed lattices.

44

Photographs of the 3.0% relative density annealed core at various stages of deformation

are shown in Fig. 29. This figure reveals that bending of the truss members occurs at

loads just prior to the peak strength, Fig. 29(b), with the softening coinciding with the

formation of a plastic hinge in the middle of the truss member, Fig. 29(c). Neither truss

member fracture nor node failure was observed during any (annealed or age hardened

cores) of the compression experiments performed.

(a) = 0.00 33 = 0.0 MPa (b) = 0.04 33 = 0.58 MPa

(c) = 0.10 33 = 1.15 MPa (d) = 0.20 33 = 0.68 MPa Figure 29. Photographs of the 030. annealed tetrahedral lattice truss at four selected levels of compression. Formation of a plastic hinge in the middle of a truss member is clearly seen in (c). Subsequent deformation results in bending about this hinge leading to softening (d).

From the stress strain response we measured four compressive properties. Figure 30

shows schematically where these are located for a generic stress strain curve. From the

curve, the compressive stiffness,initE33 , is taken from the slope of the initial elastic-like

loading region. The stiffness at the peak compressive load,pkE33 , was found by unloading

15mm

45

and reloading the samples, and fitting a regression to find the value of the linear slope.

Also recorded is the apparent compressive yield strength, yield33 , at the 0.2% offset, and

the peak compressive strength, pk33 .

Figure 30. A schematic compressive stress strain curve showing the initial lattice

stiffness,initE33 , the lattice stiffness at the peak stress,

pkE33 . Also denoted is the yield, yield33 , and peak, pk

33 , strengths.

Table IV presents the values of the stiffness and strength properties of the lattice trusses

loaded in the compressive 3-3 direction. The table reports this for each relative density

tested, and for both heat treatments.

46

Table IV. The measured elastic and strength properties of the tetrahedral lattice loaded in compression. All values are reported for a 95% confidence interval. mean relative density

heat treatment

(MPa) (MPa) (MPa) (MPa)

0.020 O 35.1 ± 1.9 307.4 ± 46.6 0.61 ± 0.15 0.77 ± 0.140.030 O 47.3 ± 12.7 396.8 ± 118.7 1.19 ± 0.13 1.49 ± 0.210.039 O 62.6 ± 19.1 700.1 ± 51.2 1.47 ± 0.23 1.93 ± 0.170.069 O 144.3 ± 22.1 1414.6 ± 182.5 2.83 ± 0.38 4.17 ± 0.440.106 O 315.0 ± 30.3 2580.4 ± 337.8 4.79 ± 0.56 9.17 ± 0.42

0.020 T6 138.3 ± 24.3 280.3 ± 47.3 2.51 ± 0.23 2.58 ± 0.250.030 T6 139.7 ± 11.4 738.3 ± 90.9 3.93 ± 0.25 4.35 ± 0.230.039 T6 172.8 ± 20.3 921.2 ± 71.2 5.20 ± 0.32 5.63 ± 0.150.069 T6 334.4 ± 58.0 1932.1 ± 231.0 8.87 ± 0.53 10.51 ± 0.580.106 T6 415.5 ± 11.8 3621.9 ± 94.9 16.13 ± 0.60 20.05 ± 0.68

yield33pkE 33

initE33pk

33

6.2.1 Compressive Stiffness

By unloading / reloading the samples during the compression experiments we obtained

stiffness behavior of the lattice truss for macroscopic core strains up to 30%. Figure 31

shows a representative plot of the normalized, sEE /3333 , compressive elastic

behavior for the five relative density samples tested. The compressive stiffness at the 0.0

strain level represents the slope an initial loading line. The data shown in Figure 31 is

from samples with the T6 temper. In most cases the lattice trusses exhibited stiffening

behavior until the peak load was reached. At greater strain levels the stiffness of the

lattice truss decreased consistent with the geometric softening (i.e., development of a

plastic hinge) as observed in Fig. 28. The exception to this behavior was for the 10.6%

relative density lattice truss, where the truss members are rather stocky. Here a well

47

defined plastic hinge was not observable for all truss members; instead a mixture of

bulging and buckling truss members were found.

Figure 31. The elastic stiffness plotted for a large range of compressive strains. The 0.0 strain values are from the slope of the initial loadings. Ashby suggested normalization of stiffness and strength properties of cellular materials

as a means to examine topological efficiencies [6]. In Figure 31 the stiffness data

presented in Table IV is normalized and plotted verses relative density. We show the

initial compressive stiffness and the peak stiffness. The initial stiffnesses are observed to

scale linearly with relative density. The measurements at the peak load levels are much

greater and follow a more non-linear trend. The T6 lattices also have the appearance of

being slightly stiffer than the annealed lattices.

48

Figure 32. The normalized compressive modulus as seen for the five relative density lattices tested. Data is shown for the initial load stiffnesses, and for the peak stiffnesses.

6.2.2 Compressive Yield Strength

A proportional limit was experimentally observed in the stress strain behavior prior to the

peak compressive loads sustained by the lattice truss. Figure 33 shows the normalized,

49

)/(33 Yyield , 0.2% yield strength measurements for both heat treatments for the five

relative densities tested.

Figure 33. The normalized apparent compressive yield strengths are plotted vs. relative density for both the (a) annealed and (b) age hardened lattice trusses.

50

6.2.3 Compressive Peak Strength

During the compression experiments the lattice trusses of each heat treatment were

observed to continue supporting increased loads until a peak stress was reached. These

non-dimensional peak strengths )/(33 Ypk are plotted in Fig. 34.

Figure 34. Predictions and measurements of the peak compressive strength of the (a) annealed and (b) T6 lattice truss cores. Elastic buckling and plastic buckling are included. The plastic buckling predictions are for both pinned (k = 1) and fixed (k = 2) truss member end conditions.

51

6.3 Shear Response

Shear properties of the lattice truss core control the onset of sandwich panel failure

modes such as shear crimping, general buckling and transverse shear failure [11,22].

Lattices that maximize shear stiffness and strength are therefore important for optimal

(minimum weight) panel design [50]. Here we show the shear stress strain response and

report the measurements for the shear stiffness, yield and peak strengths of the tetrahedral

lattice truss loaded in the minimum strength directions.

Figure 35 shows the representative shear stress-strain behavior for T6 and annealed

samples loaded in the 3-1 direction. On a unit cell basis, Fig. 9, this corresponds to one

truss member being loaded in compression and other two in tension. The lattice truss

displayed elastic-like behavior during the initial loading. This was followed by a

macroscopic yielding of the core, whereupon the lattice truss load support continued to

increase in until a peak stress was reached, which was observed to correspond with

buckling of the compressed truss members. Exceptions to this behavior were found in

the = 0.069 and 0.106 samples where bulging of truss members was the predominant

form of deformation. In the case of the T6 samples some node debonding of the tensile

truss members was also observed. The strain level at which this occurred is indicated in

Fig. 35a.

52

Figure 35. Representative combined shear response in the 3-1 loading direction for the (a) age hardened and (b) annealed tetrahedral lattice trusses. The annealed response inset shows truss yielding occurs after small strains followed by continued increase in load support.

Lattice trusses loaded in the 1-3 directions are shown in Figure 36. This loading

orientation places one unit cell truss member in tension and the other two in compression.

T6 samples (Fig. 36a) show defined linear elastic behavior followed by observable plastic

53

yielding of the tensile truss members. Load support continued to increase until a peak

was reached. Initiation of truss member rupture occurred shortly after the peak stress

level, Fig. 36a, for the T6 samples. These ruptures were observed to occur at various

locations along the truss member length independent of relative density. Figure 37 shows

photographs of a T6 = 0.039 sample at various stages of deformation.

Figure 36. Representative combined shear response in the 1-3 loading direction for the (a) age hardened and (b) annealed tetrahedral lattice trusses.

54

(a) = 0.00 13 = 0.0 MPa

(b) = 0.036 13 = 1.65 MPa

(c) = 0.069 13 = 1.52 MPa

(d) = 0.076 13 = 1.39 MPa

(e) = 0.195 13 = 0.40 MPa Figure 37. Photographs of the 039.0 T6 tetrahedral lattice truss at five selected levels of 1-3 shear. Arrows indicate the loading direction. In (b) the truss is at the peak stress level. Observable rupturing of truss members is seen to initiate by (c). At (d) only the compressed truss members continue to support the shear load.

rupture initiation

55

The annealed shear samples tested in the 1-3 direction (Fig. 36b) sustained large

macroscopic strains without initiating rupture of the tensile truss members. For these

samples there was a small linear response during initial loading (Fig 36b inset), followed

by yielding and increased load support. After large deformations (10 – 40%, increasing

with sample relative density) the tensile truss members displayed strain localization

(necking) followed by rupture and a decrease in the macroscopic load support. Figure 38

shows photograph of truss member necking (at 34.1% macroscopic strain) seen in the

annealed = 0.069 sample.

Figure 38. Photograph of the 069.0 annealed tetrahedral lattice truss at a strain level of 34.1%.

6.3.1 Shear Stiffness

The lattice truss shear stiffness was obtained for various stages of elastic and plastic

macroscopic deformation during each experiment. An initial stiffness,initG13 , was

measured from the elastic-like portion of the shear stress strain response. The lattice truss

was also periodically unloaded and reloaded to measure the stiffness for increasing

macroscopic strain levels. The representative shear stiffness behavior of the lattices is

strain localization10mm

56

shown in Figure 37 for the 3-1 and 1-3 test orientations. The data is from annealed

samples. The normalized shear stiffness, sEG /13 vs. the nominal shear strain is

plotted with Es = 69 GPa. Stiffness values at the 0.00 strain level are the values obtained

frominitG13 . Stiffening behavior similar to the compression experiments was generally not

observed for tests with either loading direction or heat treatment. An exception to this

was measured for the 10.6% relative density samples, Fig. 39.

Figure 39. The normalized lattice truss stiffness measurements shown vs. macroscopic strain. Modulus values at 0.00 strains represent the slopes of the initial loading curves.

In Figure 40 the normalized shear stiffness is plotted vs. relative density. The test

measurements appeared independent of trends based upon test direction or heat treatment.

57

Therefore these measurements were combined to generate the statistical result

(population size of 8 per relative density) shown.

Figure 40. The normalized shear stiffness measurements vs. relative density.

Experimental data represents the initial shear stiffness,initG 13 , for all test orientations and

heat treatments. Error bars are given at the 95% confidence level.

6.3.2 Shear Yield Strength

From the plate shear experiments the lattice trusses exhibited proportional limits in their

stress strain responses. A 0.2% offset was used to define yield strength values. In Figure

39 we plot the normalized macroscopic shear yield strengths tested in the 3-1 orientation

for the T6 (Fig. 41a) and annealed (Fig 41b) samples. Figure 42 shows the results for

samples loaded in the 1-3 direction. The yield strengths are normalized by the AA6061

y = 70 MPa for the annealed condition or y = 268 MPa for the T6 condition.

58

Figure 41. The normalized shear yield strengths from samples tested in the 3-1 orientation are plotted vs. relative density for both the (a) T6 and (b) annealed heat treatments.

59

Figure 42. The normalized shear yield strengths from samples tested in the 1-3 orientation are plotted vs. relative density for both the (a) T6 and (b) annealed heat treatments.

60

6.3.3 Peak Shear Strength

Figure 43 plots the peak shear strengths )/(31 Ypk of the annealed and age

hardened samples. Figure 44 shows the peak shear strengths for samples tested in the 1-

3 orientation.

Figure 43. The normalized peak shear strengths from samples tested in the 3-1 orientation are plotted vs. relative density for both the (a) T6 and (b) annealed heat treatments.

61

Figure 44. The normalized peak shear strengths from samples tested in the 1-3 orientation are plotted vs. relative density for both the (a) T6 and (b) annealed heat treatments.

62

Chapter 7 Discussion

This chapter focuses upon evaluating the tetrahedral lattice truss mechanical

performance. An analysis of the experimental data is made with the stiffness and strength

models presented in Chapter 2. Next, these age hardenable aluminum lattice trusses are

compared to lattice trusses made by casting and to those made from non-heat treatable

alloys. We also compare the tetrahedral lattice truss mechanical properties and behavior

to state-of-art aluminum honeycomb core sandwich panels. In the final section we use

the quasi-static mechanical response of the lattices to estimate their performance as

impact energy absorbers.

7.1 Compressive Properties Analysis

Mechanical tests of the tetrahedral lattice truss out-of-plane compressive response have

been performed for samples of five relative densities and two heat treatments. Stress vs.

strain plots were developed and the values of lattice truss stiffness (the initial loading

stiffness, initE33 , and the stiffness at near the peak load level,

pkE33 ) and lattice truss strength

(the macroscopic truss yield strength, yield33 , and the compressive peak strength, pk

33 )

were measured. We proceed to examine the fit of experimental stiffness and strength to

their model predictions.


In Figure 31 the normalized compressive stiffness vs. the macroscopic compressive strain

is plotted for a representative set of the five relative density samples in the T6 condition.

63

Three distinct trends appear from this figure. Firstly, the initial stiffness, .

33initE , of the

lattice truss collapses to a single value when normalized by the Young’s modulus and

relative density. Secondly, substantial stiffening of the lattice appears to occur, peaking

coincident with the peak load support of the truss. This behavior was found to increase

with relative density. Lastly, beyond the peak load the truss members develop plastic

hinges and the lattice truss stiffness decreases due to the change in the geometry of the

truss core constituents. The exception to this behavior appears in the 10.6% relative

density sample where stiffness remained between pkE33 = 0.45-0.55 even after large

strains. The compressive truss stiffness model given by eqn. (2) predicts that 33 = 0.44

when normalized by the Young’s modulus and the relative density of the truss. Figure 31

plots this prediction and the experimental data for each relative density and heat

treatment. The experiments show that .

33initE obtains normalized values of 0.06 (13.6% of

the predicted value) for the T6 lattice trusses and 0.03 (6.8% of the predicted value) for

the annealed lattice trusses. It is believed this is caused by the lattice truss not

experiencing a uniform distribution of loads to each truss member. This may be due to

non-parallelism between the upper and lower facesheets coupled with a lack of lateral

constraint offered by the ASTM-365-00 test configuration. Experimentally this

hypothesis is supported by the fact that the truss substantially stiffens until the peak load

support is achieved. The stiffening must occur because as the peak load level is

approached the majority of truss members are supporting more-or-less equivalent forces.

However, in Figure 32 we also observe that the T6 lattice trusses appear stiffer than the

annealed lattice trusses. According to the prediction the compressive stiffness of the

64

lattice trusses should be the invariant of their heat treatment. Presently this discrepancy

cannot be explained, but we suspect the ASTM-365 test standard may not be appropriate

for truss structures.

7.1.2 Compressive Yield Strength

When the lattice trusses were compressed as seen in Figure 28 a macroscopic

proportional limit in the stress strain behavior was observed. Consistent with the stress

strain behavior of the parent 6061 aluminum alloy, this limit is not well defined, and

following convention a 0.2% offset yield strength was designated. This yield strength

was then normalized by the yield strength of the parent alloy and by the relative density

of the truss sample. In Figure 33 we compare these values to the normalized yield

strength prediction given by eqn (4). The lattice truss yield strength is predicted to have

linear dependence upon . The experimental results show that a nearly linear trend is

present. This demonstrates the model is fairly capable of predicting the lattice truss

compressive yield strength knowing only the yield strength of the parent alloy. To

increase the fidelity of predicting lattice truss compressive yield strength a power law

regression was performed on the experimental data. For the lattice trusses with the T6

heat treatment we find,

09.133 657.0 yyield (15)

and for lattice trusses in the annealed condition the best fit is given by,

19.133 006.1 yyield (16)

We note the relative density exponents are close to the prediction value of unity.

65


After the proportional limit compressive loads where attained during the experiments, the

lattice truss load support continued to increase until a peak load was reached. Fig. 34

shows the normalized measured compressive peak strengths for the samples of relative

densities between 2.0 – 10.6%. The predictions of the peak strength are also plotted

assuming (i) elastic buckling of the truss members, eqn. (5), and (ii) plastic buckling of

the truss members, eqn. (7) with tE calculated from the measured 6061 tensile stress vs.

strain curve using a finite difference differentiation and iteration scheme for each truss

member thickness to length ratio [37]. In cases (i) and (ii) for the annealed trusses (Fig.

34a), consistent with the deformation mode seen in Fig. 29 we assume that the truss

members are built-in at the faces sheets and thus take 2k . The plastic buckling model

is seen to capture the peak compressive strength of all five relative densities. The T6

lattice truss peak strengths are captured with reasonable accuracy when the truss member

prediction assumes pinned connections to the facesheets (that is 1k ). Moreover, the

lattice peak strengths of the T6 (which has a low strain hardening rate) samples are also

captured by the plastic yielding model as eqn. (15) is nearly the same as the analytical

prediction give by eqn. (2), while a plastic buckling model fully accounting for the strain

hardening of the annealed Al alloy is required to predict the peak compressive strength of

the annealed lattice truss. It is worth mentioning that the models over-predict the

compressive peak strengths of the low relative density age-hardening tetrahedral trusses.

This is thought to be a consequence of two dominant factors. The first is geometric

imperfections in the trusses: recall that the knockdown in bifurcation stress due to

imperfections is greatest at the transition from the elastic to plastic buckling [22], and the

66

second is related to fillet mass apportionment which creates a higher effective measured

relative density thereby reducing the normalized strength values.

7.2 Shear Properties Analysis

The out-of-plane shear response of the tetrahedral lattice trusses has been measured.

These experiments were conducted in the 1-3 and 3-1 loading directions, corresponding

to the predicted minimum shear strength. From the stress vs. strain measurements the

values of shear stiffness (the initial loading stiffness,initG13 ) and lattice truss strength (the

macroscopic truss yield strengths, yield13 , yield

31 , and the peak strengths, pk13 , pk

31 ) were

recorded. Here we examine the fit of experimental stiffness and strength to the model

predictions.


For sandwich panels the shear stiffness of the core governs the maximum load support for

lightweight deflection (i.e., large panel) limited designs [2]. Deshpande and Fleck have

predicted the shear stiffness of tetrahedral lattice trusses to scale linearly with , and to

be isotropic in the 1-3 and 2-3 directions [22]. In Figure 39 the shear stiffness response is

shown for increasing macroscopic core shear strains in both the 1-3 and 3-1 load

orientations. The behavior between the two test directions may be considered the same,

and in nearly all cases significant stiffening was not measured (unlike the compression

measurements) for increasing strains. The 10.6% relative density samples were an

exception. We suspect this is due to plate shear sample sizing. These samples had a core

height to panel length ratio of 11.5 due to the use of a single fixture (for modularity

67

purposes). Tests are normally conducted with ratios 12 or greater to minimize the

compressive loads placed upon the core. We believe a compressive loading component

was captured in the measurements. In the following section we will also see this

manifested in the shear strength data of the 10.6% relative density samples. In Figure 39

the initial shear stiffness values are normalized and plotted vs. relative density. The

normalized prediction given by eqn. (3) is also shown. The experimental data is in good

agreement with the model confirming that shear stiffness of lattice trusses scales linearly

with , and that the measurement technique works well for lattice trusses, so as long as

the samples have a sufficient height to length ratio.

7.2.2 Yield Strength in Core Shear

When the lattice trusses were tested in shear they exhibited yielding behavior in their

shear stress strain responses, Figs. 35 & 36. A 0.2% offset yield was defined and plotted

in normalized form vs. relative density. Figure 41 shows this for the 3-1 loading and

Figure 42 is for the 1-3 orientation. The normalized truss strength prediction based upon

the yield strength of the parent aluminum (eqn. (8)) is also shown. The experimental data

is seen to approach the yield prediction as the relative density increases, but the lowest

= 0.020 samples achieve about 64% of the predicted strength. At such low densities the

truss members are quite slender and their strength is again thought to become sensitive to

surface scratches and defects introduced during fabrication. Again, it is useful to obtain

an empirical relationship for predicting the effective shear yield strength of these

samples. A power law fit was made for the T6 samples. In the 1-3 test direction we find:

28.113 387.0 yyield (17)

68

The corresponding fit for the 3-1 test direction is given by:

19.131 298.0 yyield (18)

7.2.3 Peak Shear Strength

After the lattice trusses yielded they continued to increase in load support, analogous to

what was observed in the compression experiments. A peak lattice truss stress was

attained followed by a reduction in load support generally due to either the plastic

buckling or tensile rupture of truss members depending on load orientation. A

comparison between the measured and predicted normalized peak shear strength is shown

in Fig. 43 for the 3-1 test orientation. The prediction of peak shear strength is plotted

assuming plastic buckling of the truss compressed members (eqn 11) using built-in and

pinned node connections. In this 3-1 direction the plastic buckling prediction captures

the peak strengths for both the annealed and age hardened lattices reasonably well. At

the 10.6% relative density the measurements exceed the predictions. Again, this is

thought to occur because of the presence of increased compressive loads due to the

smaller sample core height to sandwich length ratio.

Figure 44 plots the peak strengths for tests in the 1-3 (tensile governed failure)

orientation. Here the tensile strength prediction (eqn. 10) substantially exceeds what was

measured from the experiments for both heat treatments. Several explanations have been

posed to explain these results. First, it is possible that braze fillet mass is skewing the

normalized strength values downward, but this effect should also show up in the

normalized stiffness measurements, which it does not. Another possibility is that

69

fabrication defects may account for some portion of this reduced measurement. However,

if we examine Fig. 37(e) we find that the truss member ruptures tend to occur near the

ends of the truss members. This implies that bending moments are present during the

course of the loading, which may place the ends of the truss members in shear instead of

tension, thereby providing a lower stress path to failure. It may be necessary to study the

shear response by other techniques such as 3-point bending of the sandwich panels, or

use finite element methods to simulate the ASTM C-273 test method and quantify the

magnitude of this effect for samples loaded in the 1-3 direction.

For purposes of predicting the peak shear strengths for real tetrahedral lattice trusses we

have determined a power law fit for the T6 samples. Such a fit will also be useful in

determining strength asymmetry in the next section. The best fit power regression for

samples loaded in the 1-3 direction is:

27.113 451.0 ypk (19)

while the best fit for the 3-1 loading orientation is found by:

43.131 760.0 ypk (20)

7.2.4 Shear Strength Asymmetry

The shear experiments were designed to probe the lattice truss strength in the 1-3 and 3-1

directions. This family of directions corresponds to the minimum out-of-plane shear

strength of the lattice truss core, but asymmetry between the 1-3 and 3-1 directions exists

due to the mode of truss member failure for the given load direction (i.e., buckling for 3-1

vs. tensile rupture for 1-3). For lightweight sandwich construction evaluating the degree

70

of shear strength asymmetry is crucial in determining which failure mode will control the

minimum lattice truss strength for a given relative density lattice truss.

In Chapter 2 the predictions for the lattice peak shear strength were given by eqns. (10)

and (11). In Figure 45(a) we take the ratio of these relationships and plot them vs.

relative density. If symmetric strength were to exist between the 1-3 and 3-1 orientations

the ratio of 3113 / is 1. For ratio values < 1 the lattice truss shear strength is controlled

by the tensile rupture strength of the truss members, and for values > 1 the truss strength

is determined by the plastic buckling strength of truss members. The prediction ratio

shown in Fig. 45(a) indicates that plastic buckling (3-1) should control the minimum

lattice truss peak shear strength for relative densities between 2.0-10.6%. To see where

the experimental data fits in Fig 45(a) we show the peak strength ratio of eqns (19) and

(20). Unlike the prediction ratio where plastic buckling governs strength, the

experimental data shows a transition at = 0.04 from plastic buckling governed peak

strength to tensile rupture controlled peak strength. Figure 45(b) shows the case for the

yield lattice truss shear strength. Here the lattice strength should be symmetric, and the

experimental data trend is seen to be nearly symmetric. From Figure 45 we may conclude

that the minimum yield shear strength of the lattice trusses in effectively symmetric, but

the minimum peak shear strength possesses asymmetric tendencies due to the lattice

trusses not attaining higher peak shear strengths in the 1-3 load direction.

71

Figure 45. Shear strength asymmetry in the minimum strength load direction. (a) compares predictions and data of peak strengths, and (b) shows the symmetry of the yielding shear strengths.

7.3 Comparisons to Recent Studies

Two previous studies using investment cast aluminum tetrahedral lattices have been

reported [22, 31]. Chapter 1 noted that structures made for these studies possessed

72

significant casting defects coupled with low strength as-cast alloy microstructures. These

features made it difficult to assess all but the peak strength of the lattices. Y. Sugimura

reported both shear strength and stiffness of cast aluminum samples [31]. In figure 46 we

illustrate what shear behavior refinements are possible with the use wrought alloy

lattices. The results are normalized parent alloy yield strength and relative density. This

comparison shows a well defined stiff elastic region followed by a large region of

macroscopic lattice truss core plastic deformation. These are two critical features desired

for high strength and robust structures. Compared to the as-cast lattice a substantial

increase in shear stiffness (~9X) is shown.

Figure 46. The normalized shear stress strain response of the 6061-T6 tetrahedral lattice truss is compared to a previously studied A356 investment cast tetrahedral lattice truss.

73

We also briefly show how the aluminum tetrahedral lattice truss peak shear strength

compares to other topologies made from stainless steel alloys currently being studied. In

Fig. 47 the minimum peak shear strengths are shown for the 6061-T6 tetrahedral lattice

trusses and for other topologies recently studied which were made from stainless steel.

For a given shear strength of 5 MPa the stainless steel pyramidal lattice truss and square

honeycomb require a 350% greater density than the aluminum lattice truss, and the

stainless steel diamond prismatic would need to have a mass 600% larger. The stainless

steels used (AISI SS304) for these studies were annealed due to the brazing process and

consequently possessed tensile yield strength of only around 200 MPa.

Figure 47. The peak shear strengths of the AA6061-T6 tetrahedral lattice truss compared to recent studies of open and closed cell topologies based upon brazed constructions of AISI SS304 [51, 28].

74

7.4 Comparisons with Commercially Available Topologies

Hexagonal honeycomb cores represent the most structurally efficient core topology

developed for flat sandwich panel constructions to date [35,50]. To assess how well

tetrahedral lattice trusses compete with honeycomb cores in compression and shear on

strictly a topological basis we compare the compressive peak strength, shear stiffness and

minimum shear peak strength in their normalized forms. The tetrahedral lattice is

compared to aluminum hexagonal honeycomb, and a reentrant flexible honeycomb both

available from Hexcel Composites (Stamford, CT) under the trade names of HEXWEB®

and FLEXCORE® [51]. These honeycombs are made from AA5052-H38 alloys for

which the y = 255 MPa and TS = 290 MPa [47]. The ratio of TS / y is an indication of

the alloy strain hardening capacity. The ratio for 5052-H38 is very similar to that of the

AA6061-T6 (1.137 verses 1.127, respectively), and both alloys have similar tensile strain

values (14% verses 17%, respectively) in these high strength conditions. Therefore

normalized lattice truss stiffness and strength are assumed to make for a realistic

comparison on a purely topological basis.


Figure 48 shows the normalized compressive stiffness for the tetrahedral lattice trusses

and for Hexcel honeycombs. For a given relative density the honeycombs effectively

have ~2x the stiffness of the lattice trusses. The reason for the greater stiffness of

honeycomb is simply due to all the webbing being directly aligned to the compressive

loading axis. Analytical scaling models for honeycomb stiffness predict a normalized

value of unity [35] compared to a value of 0.44 for the tetrahedral lattice truss. However,

75

from the experimental data we see that such simple models significantly over predict

compressive stiffness of both the honeycomb and lattice trusses at low relative densities.

Figure 48. Comparison between the normalized compressive stiffness of the aluminum tetrahedral lattice trusses and commercially available hexagonal honeycomb. [56].


The normalized peak strength of the aluminum lattice truss is compared with competing

aluminum alloy honeycomb cores in Fig. 49. A distinct difference in how the peak

strength scales with relative density is evident. The honeycomb data scales nearly

parabolically, while the T6 lattice truss scaling is more linear. Altering the strain

hardening response the parent alloy through annealing results in effective scaling more

similar to that of the honeycomb topologies, Fig. 49. Compared to the closed-cell

honeycomb where all the core membranes are aligned in the compressive load direction

76

we find the 6061 tetrahedral lattice truss cores are competitive, and appear the same or

superior at relative densities below 4.0%.

Figure 49. Comparison between the normalized compressive peak strengths of the aluminum tetrahedral lattice trusses and commercially available competing topologies that utilize aluminum alloys. [52].


Figure 50 compares the experimental stiffness of the lattice trusses to that of regular and

flexible honeycomb. The tetrahedral lattice truss was predicted to have isotropic shear

stiffness in the 1-3 and 2-3 load directions. Honeycombs are anisotropic in these two

directions with a minimum stiffness corresponding to the 2-3 orientation, and maximum

core stiffness in the 1-3 direction. The shear stiffness data represents the minimums for

both topologies. From the plot we find the lattice truss shear stiffness to be nearly

77

equivalent to the hexagonal honeycomb data. The lattice trusses are also flexible, but the

flexible honeycomb is only ~60% as stiff as the lattice truss, Fig. 50. The high shear

stiffness efficiency of the tetrahedral lattice trusses may make them attractive

replacements for deflection limited applications that normally use honeycomb.

Figure 50. The shear stiffness of the AA6061 tetrahedral lattices compared to commercially available AA5052 honeycomb cores [52].

7.4.4 Shear Peak Strength

Figure 51 shows the normalized peak shear strength for the two Hexcel honeycomb

products. The data is representative of loadings in the 2-3 direction which corresponds to

the minimum honeycomb shear strength. The tetrahedral lattice truss data shown is for

the 1-3 and 3-1 load orientations. Given the strain hardening similarity between the two

parent alloys we observe that the laboratory produced lattice truss is remarkably efficient,

78

nearly matching the normalized peak strength performance of the regular hexagonal

honeycomb.

Figure 51. The normalized shear peak strength of the AA6061 tetrahedral lattice truss compared to commercially available AA5052 honeycomb [52].

7.5 Lattice Trusses as Impact Energy Absorbers

The merits of different materials for impact energy absorption can be compared by

determining the strain energy absorbed during their compression up to the onset of

densification [5]. The energy absorbed per unit volume, Wv, is defined as:

D

dWv

0

(12)

79

where the densification strain, D , is taken to be the strain at which the stress re-attains

it’s initial peak strength value. Figure 52(a) shows the volumetric data and compares it

with that for other cellular metal candidates such as the egg-box [53,54], woven metal

textile core [55], the pyramidal truss cores [56] and Al honeycombs [52]. The plot

abscissa shows the peak stress sustained by the sandwich core normalized by the parent

alloy yield strength, which may also be thought of as a measure of impact stress intensity.

At low, peak impact stresses, honeycombs are most efficient, but the tetrahedral lattice

trusses are more suitable at for applications demanding higher values of Wv (high

intensity loadings). Energy absorbers of minimum mass are also important for weight

sensitive applications. The energy absorbed per unit mass is defined as:

s

vm

WW

(13)

where s is the density of the parent alloy. The normalized energy absorption per unit

mass of the tetrahedral core is compared with the competing cores in Fig. 52(b). Again,

the tetrahedral cores appear well suited for applications requiring high values of mW .

These rankings are useful first approximations for determining which topologies should

be studied under more realistic dynamic conditions.

80

Figure 52. Comparison between the energy absorption capacity of the tetrahedral lattice trusses and competing core topologies. (a) Energy absorption per unit volume and (b) energy absorption per unit mass.

(b)

(a)

81

Chapter 8 Conclusions

This study has sought to explore the theoretical mechanical property scaling laws for the

tetrahedral lattice truss. This was accomplished through the use of the age hardenable

6061 aluminum alloy. As a result of this work these lattice truss sandwich cores are

shown to offer robust, lightweight, high stiffness and strength properties capable of

replacing current honeycomb constructions. Here we summarize the more salient aspects

of what the study accomplished.

1. Four new methods for manufacturing lattice truss sandwich core structures were

invented / developed.

2. Tetrahedral lattice truss sandwich panels have been made by folding hexagonally

perforated 6061 aluminum alloy, and for the first time, a simple furnace brazing

technique was used to metallurgically bond these lattice trusses.

3. The sandwich panels were successfully heat treated to a high strength (T6)

temper following the brazing process.

4. The compressive and shear response of the tetrahedral lattice truss was

experimentally measured and shown to be sensitive to the heat treatment.

5. Plastic column-buckling and plastic yield models were applied and shown to

capture the lattice truss stiffness and strength properties.

6. Comparisons with other cellular aluminum topologies indicate that 6061

aluminum tetrahedral lattice trusses are superior to aluminum open cell foams and

are competitive to closed cell honeycomb sandwiches in thru-thickness

compression stiffness and strength.

82

7. The shear behavior of lattice trusses has been shown to offer substantial

improvements in both stiffness and ductility compared to previous investment

cast lattice trusses.

8. The shear stiffness of the tetrahedral lattice truss matches that of honeycomb and

the peak shear strength efficiency was competitive with hexagonal honeycomb

and superior to flexible honeycomb.

9. The impact energy absorption capacity of these aluminum alloy tetrahedral lattice

trusses compares favorably with other new competing concepts, especially in

situations where the direction of loading is uncontrolled.

83

Chapter 9 Future Work

For this thesis the quasi-static stiffness and strength properties of age hardenable

aluminum tetrahedral lattice trusses have been characterized. If these structures are to be

utilized in sandwich panels other properties and behavior must be understood, such as the

fatigue performance, corrosion resistance, damage tolerance, and dynamic loading

behavior.

A further investigation of bonding and joining methods for aluminum based lattice

trusses would also be desirable. Since currently, most aluminum based honeycomb

sandwich panels are adhesively bonded. Extending this to lattice trusses should be

explored for cost competitive purposes.

Finally, the effects of truss member length scale and the development of efficient

methods for creating a micro-lattice truss material would offer some of the broadest

implications from a multifunctional load support standpoint. The result would be the

effective creation of a whole new class of structural materials similar to open cell foams,

but with large improvements to mechanical performance due to their high nodal

connectivity.

84

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88

Appendix A Mechanical Property Relationships

A.I. Relative Density

The relative density is defined as the volume fraction of truss members occupying a

prescribed unit cell [22]. We assume square truss member cross-sections and ignore the

detailed geometry located at the nodes. We also assume the cell to have 50% occupancy

of the available tetrahedral sites. Based upon these considerations we may write the

volume of the truss members occupying the unit cell shown in Figure 9 as:

23ltVT (A.1)

where l is the truss member length and t is the cross-section thickness. From the unit cell

geometry the volume of the cell can be written:

sincos4

33)sin)(cos

4

33( 2322 lllVC (A.2)

where is the included angle (the angle between the base plane and a truss member).

Taking the ratio between the truss volume and cell volume we obtain the relative density

expression:

2

2 sincos

1

3

2

l

t

V

V

C

T

(A.3)

A.II. Compressive Stiffness

Here we determine the elastic stiffness properties of the tetrahedral lattice truss in the

through thickness (3-3) compressive direction. We begin by assuming the truss members

of the unit cell are pinned connected, hence we need only to sum the force components in

the through thickness direction:

0sin3 33 FFT (A.4)

89

where FT is the internal force of the truss member resisting an applied force F33. Equating

the applied force, F33, to a stress acting upon the unit cell gives:

22

3333 cos4

332 lF (A.5)

Using the small displacement method [37] to determine the truss member displacement in

the through thickness direction. A displacement, 33, can be written:

sin233s

T

Et

lF (A.6)

where Es is the Young’s modulus of the truss material. Relating this displacement to the

macroscopic strain we may results in:

2

3

233

2233 sin

cos

2

3

sin

t

l

ElEt

lF

ss

T

(A.7)

Upon rearrangement the relative through thickness compressive modulus, E33 / Es

becomes:

2

2

333

cos

sin

3

2

l

t

E

E

s

(A.8)

If we wish to express the lattice stiffness in terms of the relative density (A.3) we find the

following relationship:

433 sinsEE (A.9)

If we assume the lattice truss in comprised of a regular tetrahedral arrangement ( =

54.7°), and normalize this prediction by Es and the relative density, we determine the

ideal tetrahedral lattice truss compressive stiffness coefficient:

44.033 sE

E (A.10)

90

The important feature to note is that the compressive stiffness of the lattice is directly

proportional to the relative density.

A.III Shear Stiffness

The shear stiffness of the tetrahedral lattice truss is determined by applying a point load

F13 in the 1-3 direction at the top 3-point node of the unit truss (Fig. 9). We assume rigid

facesheets and pin connected truss members. Summing the forces in the 1-3 direction we

may write:

0cos2

313 FFT (A.11)

where FT is the internal force of a truss member resisting the applied force F13. Equating

the applied force, F13, to a stress acting upon the unit truss we find:

cos2

33 21313 lF (A.12)

Assuming a small displacement in the 1-3 direction we may write the truss member

displacement:

cos2s

TT Et

lF (A.13)

where Es is the Young’s modulus of the truss material. We now relate the displacement

to the macroscopic out-of-plane shear strain by:

sin13 l

T (A.14)

We obtain the lattice truss shear stiffness, G13, by relating the stress and strain in the

normal Hookian manner. The shear stiffness is written:

91

2

13

1313 sin

3

l

tEG s

(A.15)

Upon relating this expression to the relative density we arrive at the linearly dependent

shear stiffness equation given by:

2sin8

sincos2

22213

EEG s (A.16)

Again, we may normalize this expression to find the predicted shear stiffness coefficient

for a regular tetrahedral lattice truss. Upon doing so we find:

11.013 sE

G (A.17)

A.III Lattice Truss Compressive Strength

From elementary beam theory the “fiber” stress, , at a point, y, from the neutral axis of a

uniform cross section beam that is loaded elastically in bending by a moment, M, obeys

the following relation [37]:

I

M

y

(A.10)

where I is the second moment of area for a given beam cross section. Recall from A.4 the

truss member force resisting an externally applied compressive load is:

sin333F

FT (A.11)

Knowing the force in the truss member (beam), we now write the stress, , in the beam:

sin3233

2 t

F

t

FT (A.12)

92

where, t, is the thickness of a square truss member cross section. Upon letting the neutral

axis distance y = t/2. We rewrite A.10 as:

3

2233

sin

cos3

t

l

I

M

(A.13)

Assuming the truss members are made from a perfectly plastic alloy and assuming the

maximum moment in the beam corresponds to its plastic moment we find:

6

3tMM y

p

(A.14)

where y is the plastic yield strength of the alloy and t is the beam thickness. The second

moment of area for a square cross section is I = t4 /12. Replacing M and I in A.13 we

may write the relative compressive strength of the tetrahedral lattice as:

2

233

cos

sin

3

2

l

t

y

(A.15)

Expressing the truss strength in terms of the relative density (A.3) gives:

233 siny (A.16)

Again, as was the case of the compressive stiffness, we observe that the compressive

strength is predicted to scale linearly with relative density for a perfectly plastic material.

Normalizing this expression by y and relative density gives predicted compressive

lattice truss strength of:

67.033

y

(A.17)

A.V Shear Strength

93

The lattice truss shear strength may be derived from beam theory analogous to the

compressive strength. We apply a point load F13 in the 1-3 direction, sum the forces

assuming equilibrium, and write the force in a truss member:

cos3

2 13FFT (A.18)

The applied force, F13, may also be written in terms of the shear stress, 13, which acts

upon the unit cell given by:

221313 cos

2

33lF (A.19)

We need only assume that truss peak strength occurs when the value of FT attains the

yield force of the truss member:

2tF yT (A.20)

where y is the material yield strength and t2 is the truss member cross-section area.

Substituting these relationships into the beam equation (A.10) and assuming the same

plastic moment (A.14) we write the prediction for the lattice truss shear strength in the 1-

3 direction:

2

13 cos

1

3

l

ty

(A.21)

Written in terms of the relative density the minimum shear strength relationship becomes:

2sin4

sincos213

yy (A.22)

In the normalized form we find the regular tetrahedral lattice truss strength to be:

23.013

y

(A.22)

94

Appendix B. Materials Utilization Functions

Consider the 2D unit cell, Fig. B.1(a). The relative density, the volume fraction occupied

by the truss members, can be written:

l

t

V

V

C

TD 22 (B.1)

where VT and VC are the volumes of the truss members and cell, respectively. The truss

member thickness and length are t and l. Using the unit cell, Fig. B.1.b, we also write the

3D relative density:

2

23 sincos3

2

l

t

V

V

C

TD

(B.2)

The amount of material utilized, U, is related to the 3D relative density by:

DD CU 32 (B.3)

where C is a geometric constant for a give angle. Taking the ratio of relative densities

we find:

2/13

2/1

3

2 sincos3

DD

D

(B.4)

Upon separating terms C becomes:

2/1sincos3

62C (B.5)

For a regular50% occupancy tetrahedral lattice ( = 54.7°) the value of C is ~1 (C =

0.97), and the final expression for material utilization is:

DU 397.0 (B.6)

95

In the case of the full occupancy tetrahedral lattice truss the 2D relative density becomes:

l

t

V

V

C

TD 42 (B.7)

and the 3D relative density is given by:

2

23 sincos3

8

l

t

V

V

C

TD

(B.8)

Making the same method as for the 50% occupancy lattice truss we find the final

materials utilization function for a full-occupancy tetrahedral truss is effectively:

DU 3943.1 (B.9)

This expression shows an increase in starting sheet material usage by a factor of 2.

Figure B.1 (a) 2-dimensional unit cell representation, (b) 3-dimensional unit cell. Both are shown with 50% of available tetrahedral sites occupied.

Chapter 1 Introduction - University of Virginia

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