Effects of material heterogeneities on the compressive responseof thiol-ene pyramidal lattices

R. G. Rinaldi • J. Bernal-Ostos • C. I. Hammetter •

A. J. Jacobsen • F. W. Zok

Received: 19 March 2012 / Accepted: 19 May 2012 / Published online: 6 June 2012

� Springer Science+Business Media, LLC 2012

Abstract A process of directed UV photo-curing was

previously developed for producing periodic thiol-ene lat-

tices, with potential for use in lightweight structures. The

present study probes the compressive response of two

families of such lattices: with either one or two layers of a

pyramidal truss structure. The principal goals are to assess

whether the strengths of the lattices attain levels predicted

by micromechanical models and to ascertain the role of

lattice heterogeneities. These goals are accomplished

through characterization of the lattice geometries via X-ray

computed tomography and optical microscopy, measure-

ments of the mechanical properties of the constituent thiol-

ene and those of the lattices, and strain mapping on the

lattices during compressive loading. Comparisons are also

made with the properties of the thiol-ene alone, produced

in bulk form. We find two lattice heterogeneities: (i) vari-

ations in strut diameter, from smallest at the top surface

where the incident UV beam impinges on the monomer

bath to largest at the bottom surface; and (ii) variations in

physical and mechanical properties, with regions near the

top surface being stiffest and strongest and exhibiting the

highest glass transition temperature. Finally, we find that

the measured strengths of the lattices are in accord with the

model predictions when the geometric and material

property variations are taken into account in the micro-

mechanical models.

Introduction

The specific stiffness and strength (on a mass basis) of

periodic lattice structures can be much greater than those of

stochastic cellular materials [1, 2]. The differences derive

from the manner in which loads are transmitted. Stochastic

materials respond by bending of the constituent cell/strut

members [3, 4]; lattice structures, when properly designed,

transmit loads solely by axial tension/compression of the

strut members. When failure occurs by yielding, the

strength of a bending-dominated structure follows a scaling

of the form rc=r0 ¼ 0:3�q3=2 where �q is the relative density

and r0 is the yield strength of the solid from which the

structure is made. In contrast, the yield strength of a

stretch-dominated structure scales as rc=r0 ¼ k�q where k

is a non-dimensional coefficient, typically in the range

0.3–0.5, depending on the lattice architecture. Thus, for

low relative densities (say �q � 0:1), the yield strength of a

well-designed lattice can be an order of magnitude greater

than that of a stochastic foam made of the same material

and with the same relative density.

While the mechanics principles governing the strength

of lattice structures have been well established [5], the

routes to their fabrication into useful engineering forms

have proven to be challenging. Some progress has been

made in adapting existing cutting, forming, and bonding

operations for making metallic structures [6–10]. There

have also been significant developments in additive man-

ufacturing methods using, for instance, electron beam

melting of fine metal particles [11] or 3D printing of

R. G. Rinaldi (&) � J. Bernal-Ostos � F. W. Zok

Department of Materials, University of California,

Santa Barbara, CA 93106, USA

e-mail: rrinaldi@engineering.ucsb.edu

C. I. Hammetter

Department of Mechanical Engineering, University

of California, Santa Barbara, CA 93106, USA

A. J. Jacobsen

HRL Laboratories, LLC, Malibu, CA 90265, USA

123

J Mater Sci (2012) 47:6621–6632

DOI 10.1007/s10853-012-6598-5

polymers [12–15]. Although the latter routes have found

utility for rapid prototyping of engineering components,

their use in large-scale production appears unlikely because

of high costs and long processing cycles, except for very

high value-added goods.

A second route for fabrication of polymer lattices

involves directed photo-curing [16]. Here the structure is

created by placing a mask with a periodic array of apertures

on top of a monomer bath and directing collimated ultra-

violet light at prescribed angles through the mask. Provided

the differences in the indices of refraction of the monomer

and the polymer are sufficiently large, polymerization

occurs only along the light path; that is, the polymer acts as

a waveguide during polymerization. The process has the

potential for cost-effective up-scaling. Furthermore, adap-

tations that would enable continuous manufacturing pro-

cesses can be readily envisioned. The process is also

versatile for tailoring the lattice geometry—characterized

by the slenderness ratio, inclination angle, and connectivity

of the constituent struts—over a wide range by altering the

mask design, the depth of the monomer bath, and the

number and angles of the light beams [17].

The latter process has been demonstrated using a photo-

curing thiol-ene monomer [16]. Its simplicity and rapidity

makes this system particularly attractive [18, 19]. It has

been used to make a wide variety of multi-layered lattices

with feature sizes ranging from tens of lm to several mm

and with macroscopic dimensions comparable to those of a

bread slice. The monomer is photo-cured to the point

where it has sufficient mechanical integrity to be self-

supporting in a matter of seconds or minutes. Upon

removal from the surrounding liquid monomer bath, the

polymerized structure is thermally cured to obtain the

desired physical and mechanical properties [20].

In the present study we probe the compressive response of

two families of such lattices—with either one or two layers of

a pyramidal truss structure—and attempt to critically assess

their structural response in terms of the constituent material

properties and the lattice geometry. The principal goal is to

assess whether the lattice strength attains levels predicted by

micromechanical models. Ascertaining the roles of geo-

metric and material heterogeneities prove to be essential.

The goals are accomplished through characterization of the

lattice geometries via X-ray computed tomography (CT) and

optical microscopy, measurements of the mechanical prop-

erties of the constituent thiol-ene and those of the lattices,

and strain mapping on the lattices during compressive

loading. We find that the lattice properties are strongly

affected by variations in strut diameter as well as variations

in their mechanical properties. In addition, the measured

strengths are in accord with the predictions of microme-

chanical models provided the geometric and material prop-

erty variations are taken into account.

The paper is organized in the following way. First, we

present a synopsis of the method used for lattice fabrication

and the techniques employed for characterizing their geo-

metrical, physical, and mechanical properties. We then

present the results of geometric characterization studies via

optical microscopy and X-ray CT. This is followed by

measurements of the compressive response of the lattices

as well as those of the constituent thiol-ene polymer.

Crucial details of the deformation characteristics of the

lattices are gleaned from full-field strain measurements

during compressive loading. We finish with an assessment

of the predictive capabilities of micromechanical models

for lattice strength. The roles of heterogeneities in strut

geometry and material properties feature prominently

throughout.

Materials and methods

Processing and geometry of lattice structures

The lattices employed in this study were made from of a

proprietary thiol-ene polymer using the directed photo-

curing technique described above. Following UV exposure,

the polymerized lattice was removed from the bath, washed

with toluene to remove remnants of the monomer, and

cured for 8 h at 80 �C in air. They were then subjected to

an additional 24 h treatment at 130 �C in a vacuum oven

before testing to increase the material stiffness and strength

and remove any remaining solvent. (The same treatment

had been used in previous studies on these lattices [16, 17,

20, 21]). The connectivity of the constituent struts

(described below) was varied by altering the depth of the

monomer bath.

Two types of lattice structures were examined. Both

comprised periodic arrays of struts in a pyramidal config-

uration, i.e., with four struts intersecting at each node

(Fig. 1). One of the structures comprised of two layers of

pyramidal cells through-thickness with a strut slenderness

ratio (defined by the ratio of the distance L between nodal

points and the strut diameter D) of about 8–9, a strut

inclination h of 60� and a total lattice thickness H of

25 mm (Fig. 2a). The second was a single-layered structure

with essentially the same strut slenderness ratio and incli-

nation angle, but with a thickness of 12.5 mm, i.e., the

same as the first except for the number of layers (Fig. 2b).

The relative density �q of the lattices was nominally 10 %.

Slight variations in the strut slenderness ratio and hence

relative density obtained between test samples was mea-

sured (as described below) and subsequently used in

rationalizing the measured strength variations. Complete

geometrical descriptions of the lattice samples are given in

Table 1. It should also be noted that, because the apertures

6622 J Mater Sci (2012) 47:6621–6632

123

in the mask are circular in the x–y plane (defined in Fig. 1),

the strut cross-section perpendicular to the strut axis is

elliptical with an area of pD2sinhð Þ=4.

In the subsequent discussion of properties of the constit-

uent thiol-ene, we distinguish between various regions

within the lattice on the basis of their locations with respect to

the incident UV beam. For the two-layer lattice, the regions

are denoted as the top node (nearest the surface where the UV

beam impinges on the monomer bath), the top strut (imme-

diately below the top node), the middle node, the bottom

strut, and the bottom node. For the single-layer lattice, they

are denoted the top node, the strut, and the bottom node.

Processing of bulk thiol-ene

For the purpose of characterizing the constituent thiol-ene,

flat sheets, about 3 mm thick, were also fabricated. These

were made following a process nominally identical to that

used for fabricating the lattices, with the exception that

neither the UV mask nor the toluene wash were employed.

We show subsequently (through measurements on both the

neat thiol-ene sheets and samples extracted from the lat-

tices) that these seemingly minor differences in processing

along with the differences in thicknesses of the lattices and

the neat thiol-ene sheets are manifested in measureable

differences in material properties. Consequently, some

caution must be exercised in using bulk properties in

rationalizing the response of a lattice.

Characterization of lattice geometry

The strut geometries were characterized by quantitative

analysis of optical images taken normal to the strut axes, as

illustrated in Fig. 3. Using ImageJ,1 computational routines

were generated to grab edge contour points along each strut

and the data were then converted to strut area, assuming the

cross-section to be circular. In some cases, the lattices were

imaged by X-ray CT. X-ray scans were performed at the

Advanced Light Source at the Lawrence Berkeley National

Laboratory. Data from the X-ray scans were processed

using the NormalizeStack832 functionality of the ALS

MicroCT plug-in for FIJI. The Octopus software package

from inCT was then used to reconstruct the data into two-

dimensional slices. Three-dimensional images were created

using Avizo Standard 7.0.

Compression testing of lattices

Through-thickness uniaxial compression tests were per-

formed on specimens comprising a 2 9 2 array of unit

cells (Fig. 2). Aluminum disks were adhesively bonded to

the two broad faces of each specimen. The tests were

performed at room temperature at displacement rates that

yielded nominal strain rates ranging from 2 9 10-4 to

Fig. 1 a Schematics of 2 9 2 arrays of unit cells of the two types of

lattice structures of present interest and b definitions of geometric

parameters

Fig. 2 Optical images of compression specimens with 2 9 2 arrays of

unit cells of a two-layered and b single-layered lattice structures, with

aluminum plates bonded to the two loading faces (NB: black and white

paints have been applied to obtain a black speckle on a white background)

1 Java-based image processing program; http://rsbweb.nih.gov/ij/.

J Mater Sci (2012) 47:6621–6632 6623

123

2 9 10-2 s-1. In some instances, full-field strain mea-

surements were made by 3D digital image correlation (Vic-

3D, Correlated Solutions, Columbia, SC). The purpose of

these measurements was to ascertain the uniformity of

strain through the two-layer lattice. This was motivated by

ancillary observations of non-uniformities in strut dimen-

sions and by correlations between observed buckling pat-

terns and the direction of UV curing. Strain measurements

were made at three types of locations: (i) within the central

portions of the top struts, each over a length of about

2 mm; (ii) within the middle nodes, over a length of 1 mm;

and (iii) within corresponding central portions of the bot-

tom struts. The measured displacements within the struts

were analyzed in a manner that yielded strains with respect

to a local Cartesian coordinate system, defined by the strut

axis and its normal. The strains at the node were defined

with respect to the global coordinate system, characterized

by the compression direction and its normal. Furthermore,

the rotation of the middle node was calculated based on the

orientation of a vector joining two material points 1 mm

above and below the plane of minimum cross-section.

Measurements of constituent properties

The glass transition temperature Tg was measured by dif-

ferential scanning calorimetry (DSC). Small (ca. 10 mg)

specimens for DSC were cut from the lattices at each of the

nodes and within the struts. Each was then subjected to

three consecutive thermal cycles from -50 to 100 �C at

heating/cooling rates of 10 �C/min; Tg was measured on

the third cycle.

The viscoelastic behavior was measured through

dynamic mechanical analysis (DMA). Test specimens were

cut from the lattices using scissors. These specimens were

tested in a EPLEXOR� apparatus by forced oscillatory

tensile loading at a frequency of 1 Hz, over a temperature

range of -100 to 100 �C, with heating rates of 1 �C/min.

Corresponding DSC and DMA measurements were also

made on specimens of the bulk thiol-ene.

Specimens for compression testing were prepared from

struts using a thin (0.3 mm) diamond wire saw. These had

an aspect ratio (length/diameter) of *1. Optical images

were taken from each end of the latter specimens and the

images were then analyzed using ImageJ to ascertain the

cross-sectional area. The compression tests were conducted

in a servo-hydraulic mechanical test apparatus at ambient

temperature (22 ± 2 �C) over a strain rate range of 10-4 to

10-1 s-1. The load–displacement data were subsequently

converted to true stress–true strain on the assumption that

the deformation is uniform and volume conserving.

Lattice geometry

Three-dimensional CT renderings of the one- and two-layer

lattices are shown in Fig. 4. Corresponding transverse cross-

sections and measurements of cross-sectional areas are

shown in Figs. 5 and 6. These results (along with others not

Table 1 Geometrical descriptions of tested lattice specimens

Label h (�) H (mm) q (kg/m3) q (%) Top strut Bottom strut

L (mm) D (mm) s = L/D L (mm) D (mm) s = L/D

Double-layer geometry

BF1 59.3 25.5 166 12.4 13.7 1.9 7.2 14.2 2.3 6.2

BF4 59.5 24.3 111 8.3 14.0 1.6 8.8 13.9 1.9 7.3

BF6 60 25.4 122 9.1 14.2 1.6 8.8 14.2 2.1 6.8

BF6 60 25.4 122 9.1 14.2 1.6 8.8 14.2 2.1 6.8

Single-layer geometry

BH1 61.2 14.0 92 6.8 14.1 1.5 9.4 – – –

1 mm

Topstrut

Bottomstrut

5 mm

Topstruts

Bottomstruts

(a) (b)

Middlenode

Bottom nodes

Top nodes

Fig. 3 a Representative optical image used to measure variations in

strut diameter, b cross-sections of compression cylinders

6624 J Mater Sci (2012) 47:6621–6632

123

shown) confirm that the struts are straight and consistently

oriented at 60� ± 1� with respect to the plane of the nodes. In

addition, the nodal regions are larger than that expected of an

ideal structure: that is, the one obtained if each of the four UV

beams were aligned to achieve perfect convergence at the

nodes and each acted independently in the curing process

(see, for example, schematics in Fig. 1). In the ideal case, the

node area would be � of the combined area of the four struts

remote from the nodes; the measurements show that the

actual area ratio is about �. The difference might be attrib-

utable to enhanced polymerization in nodal regions where

the UV beams intersect and hence where the material is

subjected to a light intensity essentially four times that within

a single strut. It might also be due in part to slight misa-

lignments of the incident beams. Although not introduced by

design, the larger-than-expected node area should have a

beneficial effect (if any) on the compressive strength of the

lattices, albeit at the expense of increased relative density.

The CT images further reveal two geometric imperfections

within the struts. First, in the single-layer lattice, the strut area

increases gradually from the top node downward. This result

suggests some dispersion in the UV beam during polymeri-

zation: the polymerized struts not acting as perfect wave-

guides. A similar gradient is obtained in the top struts of the

two-layer lattice and continues into the bottom struts (beyond

the middle node). Typically, the minimum area of the bottom

struts is about 50 % greater than that of the top struts. In

Fig. 5 Transverse cross-

sections through a single-layer

lattice at various through-

thickness locations and the

measured cross-sectional areas

obtained from X-ray CT. The

dotted squares in a–d denote the

boundaries of the pyramidal unit

cell

Fig. 4 3D renderings of CT scans of a one-layered and b two-layered

lattices. Lattices are 12.5 and 25 mm tall, respectively

J Mater Sci (2012) 47:6621–6632 6625

123

contrast, the strut area appears to be uniform within each plane

perpendicular to the through-thickness direction. This is evi-

denced by the CT images in Figs. 5a–d and 6a–f as well as the

measured cross-sectional areas (also from CT images) plotted

in Figs. 5f and 6h. The resulting average geometric charac-

teristics of the lattices are summarized in Table 1.

Second, although the struts in the single-layer lattice and

the top struts in the two-layer lattice appear to be nearly cir-

cular in cross-section (consistent with the circularity of the

holes in the curing mask), the bottom struts in the latter lattice

exhibit somewhat scalloped edges (compare, for example,

Fig. 6b, e). Such effects are also evident in the optical images

of the cross-sections of the compression specimens machined

from lattice struts (Fig. 2b). The fact that the scalloped edges

are observed only in the bottom struts suggests that the

intersection of the UV beams at the nodal locations alters the

beam shape as it passes through the bath. One of the conse-

quences is that the area estimates obtained from micrometer

measurements or from optical images taken transverse to the

strut axis (e.g., Fig. 2a) are slightly greater than the true

Fig. 6 a–f Transverse cross-

sections obtained by CT through

a two-layer lattice at various

through-thickness locations and

g the measured cross-sectional

areas obtained from both CT

and optical images. The dottedsquares in a–f denote the

boundaries of the pyramidal unit

cell

6626 J Mater Sci (2012) 47:6621–6632

123

values. In turn, the computed stresses in the mechanical tests

may be slightly underestimated. In contrast, the measured

areas of the struts in the single-layer lattice and in the top struts

of the two-layer lattice were consistent with one another using

the two measurement techniques (Fig. 5e).

Compressive response of lattices

Representative stress–strain curves for the lattices are

plotted on Fig. 7. They exhibit essentially linear behavior

up to a peak stress, at a strain of about 0.025–0.030. The

peak is attributable to the onset of strut buckling (results

presented later in the article show that this event occurs

prior to the onset of strut yielding). The lattices exhibit a

strong sensitivity to strain rate (Fig. 7b). Specifically, for

the two-layer lattice, both the stiffness and the peak stress

essentially double as the strain rate is increased from

2 9 10-4 to 2 9 10-2 s-1. The correlation between the

elevations in modulus and peak stress is consistent with

elastic (Euler) buckling of the struts, as detailed later.

Similar effects are obtained for the single-layer lattice,

although the peak stress is somewhat higher at a prescribed

strain rate. The differences in peak stress are subsequently

rationalized (in ‘Constituent properties’ section) on the

basis of the magnitude of the end-constraints exerted on the

struts during the buckling process as well as non-unifor-

mities in geometry and properties.

The progression of events leading to buckling is most

readily gleaned from the strain measurements obtained by

DIC. Figure 8a shows the evolution of axial (eyy), trans-

verse (exx) and shear (exy) strains within the top and bottom

struts and those in the central node with macroscopic

strain. In the strut regions, the shear strains in the local

coordinate systems, defined by the strut axes, were very

close to zero before buckling, confirming that the struts

experience essentially axial loading. This feature is also

obtained from finite element analyses of these lattices [27].

The distribution in cross-sectional area of the struts is

shown in Fig. 8c. The results reveal that the magnitude of

the strain in the bottom strut initially increases more rap-

idly than that in the top strut, despite the fact that its cross-

sectional area is greater than that of the top strut (by

40–50 %). This is the first clear indication of non-unifor-

mity in the mechanical properties of the lattice material.

Furthermore, non-linearities in the local strains in the

bottom struts are first detected at a macroscopic strain of

about 0.015–0.020, coincident with the first signs of non-

linearity (albeit slight) in the macroscopic stress–strain

curve and shortly before the peak stress (Fig. 8e). This

marks the onset of strut buckling. It is accompanied by

rapid changes in local strains: either increasing or

decreasing, depending on the shape of the buckled strut at

the surface location of measurement. It is also accompa-

nied by the onset of rotation of the middle node (Fig. 8d).

The rotation is a manifestation of cooperative buckling of

the top and bottom sets of struts, evident in the optical

images at larger macroscopic strains (see, for example, the

inset images in Fig. 7a).

Constituent properties

The glass transition temperatures Tg obtained by DSC are

plotted on Fig. 9. The glass transition temperature of the

bulk material (*56–59 �C) is consistently higher (by at

least 10 K) than the values obtained on the specimens

extracted from the lattice. Furthermore, in the two-layer

lattices, Tg decreases progressively into the depth of the

lattice, from a peak value (44–48 �C) at the top node to a

minimum (35–37 �C) at the bottom node. Similar varia-

tions are obtained in the single-layer lattice. The through-

thickness variations are attributable to attenuation and

Fig. 7 Compressive stress–strain curves for the lattices, showing

a effect of the number of layers and b effect of strain rate on the two-

layer lattice. Inset images in a show the buckling patterns in the

lattices at 10 % strain

J Mater Sci (2012) 47:6621–6632 6627

123

dispersion of the UV beam, leading to a reduction in the

efficacy of the beam in polymerizing the thiol-ene mono-

mer as it penetrates into the monomer bath. Similar effects

have previously been reported by others [22, 23]. Fur-

thermore, the difference in Tg of the bulk specimens and

that of the top nodes of the lattices is attributable to dif-

ferences in light intensity in the absence and the presence

of a mask. It is not yet clear, whether the toluene wash

leaches out unreacted monomer from the photo-cured lat-

tice material, thereby reducing the maximum attainable

crosslink density upon subsequent thermal curing. Current

studies are addressing these issues. Regardless of the origin

of these differences, the results clearly demonstrate that the

physical properties of the thiol-ene in lattice form differ

from those in bulk form. Thus, some caution must be

exercised in using the bulk properties to rationalize the

lattice response.

The DMA measurements (presented in Fig. 10) reaffirm

the conclusions drawn from the DSC results. In this case,

the storage modulus of the bulk thiol-ene at low tempera-

tures is of the order of 2 GPa. The main relaxation asso-

ciated with the glass transition, as manifested in the

precipitous drop in the storage modulus, begins at about

50 �C. The peak damping (characterized by the ratio of the

loss modulus E00 to the storage modulus E0) occurs at about

68 �C. Thereafter, at higher temperatures, up to 130 �C, the

storage modulus increases approximately linearly with

temperature. The latter result is consistent with the theory

of entropic elasticity for rubbery polymers [24], wherein

the modulus scales as NkT where k is the Boltzmann

Fig. 8 Results of DIC strain measurements. a Distributions in true

strains in each of the three regions within a two-layered lattice;

b optical image of the regions of interest; c strut areas obtained from

optical images; d evolution of rotation near the middle node;

e macroscopic stress–strain curve

6628 J Mater Sci (2012) 47:6621–6632

123

constant and N is the number of active chains per unit

volume. The corresponding measurements on the top struts

of the lattices reveal that the peak damping occurs about

20 K below that of the bulk material, again consistent with

the DSC measurements. Thus, at room temperature, the

lattice thiol-ene is near the glass transition and is thus

expected to exhibit greater sensitivity to temperature and

strain rate relative to the bulk thiol-ene. The higher rubbery

modulus measured for the bulk thiol-ene further demon-

strates that the bulk material exhibits a greater crosslink

density relative to the lattice material, which, again, is

consistent with the DSC results. Based on the measured

glass transition temperatures, yet higher discrepancies

would be expected from the bottom struts of the two-layer

lattices.

The uniaxial compression tests on the bulk thiol-ene are

shown in Fig. 11a. The stress–strain curves are reminiscent

of those obtained for other glassy polymers [25]: notably,

yield initiation at a strain of about 0.05 followed shortly

thereafter by a peak stress (at a strain of 0.06–0.08,

depending on strain rate) and strain softening up to strains

of about 0.2. In addition, the variation in the peak stress

with strain rate follows the usual Eyring-type relationship

[26] wherein strength scales logarithmically with strain rate

(Fig. 11d). There is no detectable effect of strain rate on the

Young’s modulus over the strain rate range probed by the

present experiments. This is consistent with the minor

temperature insensitivity of the storage modulus near room

temperature (Fig. 10).

The compression results for the specimens extracted

from the lattice (shown in Fig. 11a) exhibit some of the

same characteristics, with three notable differences: (i) the

peak stress in the lattice material is only about half that

of the bulk thiol-ene at a prescribed strain rate (Fig. 11c);

(ii) the strengths of the bottom struts are always lower than

those of the top struts (Fig. 11d); and (iii) the Young’s

modulus exhibits a measureable sensitivity to strain rate

(Fig. 11d), consistent with the DMA results in Fig. 10.

These results reaffirm the hypothesis that the curing effi-

cacy of the UV beam diminishes with depth into the

monomer bath. Furthermore, the first indications of non-

linearity in the bottom struts are obtained at strains of about

0.03: about twice the value within the bottom struts at the

onset of strut buckling.

Models of lattice strength

An assessment of the measured strengths has been made

using models of yielding and buckling of periodic lattice

structures [27]. First, a standard statics analysis is used to

obtain the axial stress borne by the struts, assuming that

loads are transmitted via axial compression. The stress rsy

for strut yielding is then obtained by setting the strut stress

equal to the material yield strength. For a lattice compris-

ing struts with circular cross-section perpendicular to the

strut axis, this procedure gives:

rsy

r0

¼ psinh2s2cos2h

ð1Þ

where r0 is the material yield strength and s is the

slenderness ratio, L/D. Alternatively, setting the stress in

the strut equal to the Euler buckling stress yields the

macroscopic stress rsb for buckling, given by:

rsb

E¼ p3

32s4K2

sinhcos2h

ð2Þ

where K is a non-dimensional coefficient dictated by con-

straints at the strut ends. In one limit, where both ends are

constrained from both lateral displacement and rotation

(so-called fixed–fixed conditions), K = 0.5. If, instead, one

Fig. 9 Glass transition temperatures measured by DSC

Fig. 10 Storage modulus and loss factor of bulk thiol-ene and that of

the top struts of a two-layered lattice obtained from DMA

measurements

J Mater Sci (2012) 47:6621–6632 6629

123

node is fully constrained while the other is free to rotate

(i.e., pinned–fixed conditions), K & 0.7.

As noted earlier, the strut cross-section in the present

lattices is circular with diameter D in the plane of the mask.

Thus, the strut area relevant to computing axial stress (i.e.,

that measured perpendicular to the strut axis) is altered by a

factor of sinh and the second moment of area is altered by a

factor of sin2h, relative to the corresponding values for a

circular strut of diameter D. Combining these effects with

the preceding solutions yields modified predictions of the

failure strengths, given by:

rsy

r0

¼ p2s2cos2h

ð3Þ

rsb

E¼ p3sin3h

32s4K2cos2hð4Þ

Figure 12a shows the variation in the measured peak

stress with slenderness ratio for both one- and two-layered

lattices, all from tests conducted at a strain rate of

2 9 10-3 s-1. Also shown for comparison are two sets

of predictions based on Eqs. 3 and 4. In one set (pertinent

to the one-layered system), the nodes are assumed to

provide rigid end-conditions on the struts (K = 0.5); the

Young’s modulus and the yield stress are taken to be those

measured on the struts extracted from that lattice (1.2 GPa

and 45 MPa, respectively). In the second (pertinent to the

two-layered system), the top and bottom nodes are assumed

to provide full constraint whereas the center node is

assumed to be pin jointed (K = 0.7); the Young’s modulus

and the yield stress for this case (taken as those of the

bottom strut, where buckling is seen to initiate) are 0.7 GPa

and 30 MPa, respectively.

The measured and predicted strengths for both lattice

systems are in remarkably good agreement. They reaffirm

the previous assertion that failure occurs by elastic buck-

ling (rather than strut yielding or anelastic buckling). In

addition, although not shown in the figure, the results show

that, had the properties of the bulk thiol-ene been

0.10 0.15 0.20 0 0.05

True strain, εt

Bulk thiol-ene

Lattice thiol-ene

Top strut

Bottom strut

80

60

40

20

0

ε ≈10-3 s-1.(b)

True

str

ess,

σt (

MP

a)

Bulk thiol-ene

Lattice thiol-ene(bottom strut)

10-2 s-110-1 s-1

10-4 s-110-3 s-1

True

str

ess,

σt (

MP

a)

100

80

60

40

20

0

10-2 s-110-1 s-1

10-4 s-110-3 s-1

True strain, εt

(a)

0.10 0.15 0.200 0.05

10-3 10-210-4

Youn

g’s

mod

ulus

, E

(GP

a)

10-1

Bulk thiol-ene

Top strut

Bottom strut

Lattice thiol-ene

2.0

1.0

0

(d)

Nom strain rate, ε (s-1).

10-3 10-210-4

Yie

ld s

tres

s, σ

y (M

Pa)

10-1

(c)

Nom strain rate, ε (s-1).

100

80

60

40

20

0

Bulk thiol-ene

Top strut

Bottom strut

Lattice thiol-ene

0.25

0.25

Fig. 11 Results of compression tests on the constituent thiol-ene,

showing effects of strain rate on the compressive response of a the

bulk thiol-ene and the lattice struts extracted from a two-layer lattice;

b comparisons of the responses of the bulk and lattice thiol-ene at

comparable strain rates; c effects of strain rate on the peak (yield)

stress; d effects of strain rate on the elongational compressive

modulus

6630 J Mater Sci (2012) 47:6621–6632

123

employed, the predictions would have over-estimated the

measured compressive strengths by a large margin.

Conclusions

Using X-ray CT and optical microscopy along with mea-

surements of physical and mechanical properties of thiol-

ene lattices, we have found: (i) variations in strut diameter,

from smallest at the top surface where the incident UV

beam impinges on the monomer bath to largest at the

bottom surface; and (ii) variations in physical and

mechanical properties, with regions near the top surface

being stiffest and strongest and exhibiting the highest glass

transition temperature. When these variations are taken

into account, the lattice strengths are in accord with pre-

dictions from rudimentary micromechanical models. In the

systems probed in the present study, failure occurs by

elastic buckling. In the two-layered system, buckling

occurs first in the bottom array of struts, due to their lower

stiffness. The variations are attributable to attenuation and

dispersion of the UV beam as it passes through the thiol-

ene bath.

Comparisons of the strengths of the two types of lattices

suggest that the internal nodes in a two-layered lattice

provide less constraint to buckling than those attached to

the face plates. This effect is manifested in differences in

the inferred buckling coefficients: K = 0.5 and 0.7 for one-

and two-layer lattices, respectively. Since the buckling

stress is inversely proportional to K2, a change in K of this

magnitude would lead to a twofold change in strength. This

effect has been confirmed through recent finite element

calculations on both one- and two-layer lattices [27]. The

calculations further show that the inferred value of K con-

tinues to increase as the number of layers is increased,

eventually reaching 1. This result is consistent with buck-

ling under pinned–pinned end-conditions.

The large measured differences in the modulus and the

strength of the lattice material within the top and bottom

struts are due in part to the fact that Tg lies only very

slightly above ambient, by 10–20 K (depending on loca-

tion). These differences would be expected to be smaller if

the test temperature were reduced (below ambient) or if the

glass transition temperature were increased. With this goal

in mind, an ongoing study is assessing the efficacy of

alternative (higher temperature) curing cycles in increasing

the glass transition temperature. The results will be pre-

sented in a future publication.

Acknowledgements This study was supported by the Institute for

Collaborative Biotechnologies through Grant W911NF-09-0001 from

the US Army Research Office. The content of the information does

not necessarily reflect the position or the policy of the Government

and no official endorsement should be inferred. Beamtime at the

Advanced Light Source was acquired with proposal titled ‘‘X-Ray

Tomography of Co-Continuous Polymeric Composite Materials for

Blast Mitigation’’ (ALS-04549). The Advanced Light Source is

supported by the Director, Office of Basic Energy Sciences of the

U.S. Department of Energy under Contract No. DE-AC02-

05CH11231. The authors gratefully acknowledge Dr. Dula Parkinson

for his assistance with the beamline experiments and post-processing

of the data in generating the tomographic images. The authors also

thank Prof. L. Chazeau and Dr. J.-M. Chenal of MATEIS Lyon for

use of their facilities in performing the DMA measurements.

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