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Parametric Experimentation on Pantographic Unit Cells
Reveals Local Extremum Conguration
N. Nejadsadeghi, M. de Angelo, R. Drobnicki, T. Lekszycki, F. Dell’Isola, A.
Misra
To cite this version:
N. Nejadsadeghi, M. de Angelo, R. Drobnicki, T. Lekszycki, F. Dell’Isola, et al.. Parametric Exper-
imentation on Pantographic Unit Cells Reveals Local Extremum Conguration. Experimental Me-
chanics, Society for Experimental Mechanics, In press, �10.1007/s11340-019-00515-1�. �hal-02268895�
Experimental Mechanics
https://doi.org/10.1007/s11340-019-00515-1
Parametric Experimentation on Pantographic Unit Cells Reveals
Local Extremum Configuration
N. Nejadsadeghi
1
· M. De Angelo
2,3
· R. Drobnicki
4
· T. Lekszycki
5,6
· F. dell’Isola
7,8
· A. Misra
3
Abstract
Pantographic metamaterials are known for their ability to have large deformation while remaining in the elastic regime. We
have performed a set of experiments on 3D printed pantographic unit cells to parametrically investigate their response when
undergoing tensile, compression, and shear loading with the aim of i) studying the role of each parameter in the resultant
mechanical behavior of the sample, and ii) providing a benchmark for the mathematical models developed to describe
pantographic structures. Results show the existence of local extrema in the space of the geometrical parameters, suggesting
the use of optimization techniques to find optimal geometrical parameters resulting in desired functionalities. We have also
performed tensile relaxation tests on the samples, with the results indicating the complexity of the dynamic behavior and
the
existence of multiple relaxation characteristic times. Such results can be used to for calibrating mathematical models
describing pantographic structures u
nder dynamic loadings.
Keywords Pantographic metamaterials · Parametric investigation · Additive manufacturing · Complex structures ·
Experimental tests
Introduction
Since the advent and continuous progress of additive manufac-
turing techniques, new horizons in terms of analysis of the
mechanical behavior of materials and design of new mate-
rials have been opened. By employing the most advanced
processes it is now possible to create objects with complex
geometries at very small length scale. Arranging such com-
plex objects in a periodic pattern, playing the role of micro-
or nano- scale building block, results in micro-structured
materials, also called metamaterials, whose macroscopic
mechanical behavior is strongly dependent on the charac-
teristics of the kinematic field of the given micro-structure.
Barchiesi et al. [1, 2] This influences the research in the field
of innovative materials as we are now able to endow them
with a micro-structure, which is potentially customizable
and tunable according to the applications to be addressed.
However, the new promising possibilities require the def-
inition of more refined mathematical models, involving
A. Misra
amisra@ku.edu
Extended author information available on the last page of the article.
detailed kinematic fields and higher order gradient terms in
their equations of motion to capture their unusual behavior.
Examples are represented by pantographic structures [3–7],
for which their exotic behavior has to be described with
higher gradient continuum theories [8–10] or micromorphic
theories [11, 12], granular media [13–19], auxetic material
[20, 21] and functionally graded material [22, 23].
Pantographic structures have recently been investigated
particularly from a theoretical, mechanical and manufac-
turing point of view because of their peculiar capability to
undergo large tensile deformation while remaining in elastic
regime [3]. The design of such metamaterial comes from a
mathematical understanding of the related mechanical prob-
lem and considerations on the insufficiency of classical
theories to describe materials with complex micro-structure,
and it has been conceived so that its deformation energy
depends on the second gradient of the displacement [7,
24–26]. Morphologically, the pantographic system can be
described as a double array of mutually orthogonal beams,
also called fibers, interconnected by elastic cylinders at
intersection points, called pivots [27].
Earlier investigations can be found in a series of papers
(see e.g. [5, 27–31]) in which the authors address the
problem of determining the mechanical properties of panto-
graphic metamaterials. Various mathematical models have
Exp Mech
also been proposed which can be classified into two-types i)
discrete, as in [27, 29, 32–36], wherein a finite set of rota-
tional and extensional springs are introduced to describe the
complex deformation phenomenon occurring in the micro-
structure; or ii) continuous, as in [27, 33, 37, 38], which
can be further distinguished into purely continuous ones,
wherein a generalized plate model is used (of the same
kind of those proposed e.g. in [11, 12], and into hybrid
discrete/continuous ones, wherein beams (i.e. 1D continua)
form lattices by means of interconnecting pivots placed
at finite distances (see [39–42]). The main motivation of
these works is to find a reasonable compromise between
predictive capacity and computational feasibility in analyz-
ing pantographic structures. Both continuous and discrete
models demonstrate their own peculiarities in facing the
problem and some comparisons have been recently pre-
sented where the validation of the models is demonstrated
by using experimental results [29, 43, 44]. It is noteworthy
that the behavior observed during tensile test [36, 44, 45],
shear test [45–47] and also torsion test [45] conducted on
specimens of pantographic structures (as the kind described
in [27]) is well represented by the aforementioned models.
However it is not clear how the geometric features of the
pantographic micro-structure affect the mechanical behav-
ior of the specimens tested yet. An understanding of the role
of geometrical parameters on the macroscopic behavior of
pantographic structures seems indispensable for the purpose
of designing of such structures with desired functionalities.
Indeed, regardless of the nature of the approach (continu-
ous or discrete) used to forecast the behavior of the system
under various load and boundary conditions, it is necessary
to estimate the relevant constitutive parameters on which the
deformation energy function depends. Also, an investigation
on pantographic beam structures with relatively small num-
ber of cells could be, according to the theory developed in
[48], useful to determine information about the extent to
which continuum modeling can approximate more spatially
extended structures. We believe that better identification
schemes have to be devised for the aim of identifying the
mechanical properties and achieving structural optimiza-
tions. Such identification schemes require experimental
results sufficient for calibrating mathematical models.
In this paper we present a series of results obtained from a set
of experimental tests performed on specimens of pantographic
structure. Rather than using larger samples, the specimens
considered herein can be defined as the elementary module
of the pantographic micro-structure. Tensile, compression,
shear and tensile relaxation tests are performed on a large
set of polyamide specimens whose geometric features
are presented in “Fabrication and Experimental Setup”.
In the same section we present the prescriptions of the
tests and the tools employed. “Results and Discussion”is
devoted to the results and corresponding discussions of the
experimental data. In the conclusion section an outline
about the results is given with considerations regarding
future studies.
Fabrication and Experimental Setup
The specimens considered herein represent the unit cell of
the micro-structure employed to build a larger pantographic
structure (Table 1). The geometry of the objects was ini-
tially generated employing the CAD software SolidWorks
(Dassault System SolidWorks Corporation, Waltham, MA,
USA). The specimens were thereafter printed using 3D
printer Formiga P 100 (EOS GmbH, Munich, Ger-
many) at the University of Technology, Warsaw, Poland.
The 3D printer uses a selective laser sintering (SLS)
technology to produce the pantographic structure out of
polyamide powder (PA2200), where the average grain size
of the powder used was 56μm. The complete set of speci-
men contains thirty three objects characterized by different
values of the geometric parameters (see Fig. 1). In order
to easily refer to each specimen, depending on the value
assumed by the parameter p, we arrange the specimens with
p = 5 mm in a group A and the specimens with p = 8
mm in a group B. Furthermore, an index i = 1, 2, 3, 4is
assigned to each specimen depending on the length of the
pivot, which indicates sample type. Details about the sam-
ple type, number and geometric features can be found in
Table 2. For each sample type a tensile, compression, ten-
sile relaxation and a shear test were performed. A Bose
ElectroForce 3200 testing-device controlled by the software
WinTest Material Testing System was used to perform all
the experiments presented in the following. The load cell
used to measure the reaction force values has a measure-
ment range of ±22 N, a measurement uncertainty of 0.1%
and precision of 0.001 N. All the tests were performed
in displacement control, which is measured with the built-
in transducer with the range of ±6.5 mm, a measurement
uncertainty of 0.1% and precision of 0.001 mm. For evalu-
ations on the deformation fields by means of digital image
correlation technique (DIC), pictures were taken during the
tensile, compression and shear tests. DIC along with rele-
vant discussions on the deformation field, similar to [35],
will be pursued in a separate paper in the future. To this end
the specimens were sprayed on their surface with a black
spray -to make a speckle pattern- prior to the testing, and
a NIKON D300 digital camera was used to take consecutive
Table 1 Constant parameters shared by all the specimens
abdLθφ
(mm) (mm) (mm) (mm) (rad) (rad)
1.2115
π
2
π
4
Exp Mech
Fig. 1 Parameters describing
the pantographic structure
pictures at fixed time intervals with image resolution of
4288 by 2848 pixels. The tensile test is experimentally car-
ried out by keeping fixed one side of the specimen and
applying an axial displacement on the other side. For this
test, the imposed values of displacement are 4 mm and 6
mm, for groups A and B respectively, at the rate of 0.1 mm/s.
A similar setup has been used to conduct the compression
test but in this case, obviously, the imposed displacement
has opposite sign. The compression displacements applied
to the specimens of group A and group B are of 10 mm
and 12 mm respectively, at a rate of 0.1 mm/s. The experi-
mental realization of the shear test is carried out by fixing
the specimen on one side while on the other side a tan-
gential displacement is applied. The displacement occurs
in the plane on which the specimen lies. The specimens
of both groups A and B are displaced up to 10 mm at the
rate of 0.1 mm/s. The tensile relaxation test, whose setup
is similar to that of the tensile test, prescribes the applica-
tion of three successive load levels, starting from 2 mm for
group A and from 4 mm for group B, being the increase
of displacement of each load step equal to 1 mm. Every
load level is maintained constant for eight hours. This dura-
tion has been selected after a previously conducted test on
a specimen for two days which showed a great portion of
relaxation occurs in the first eight hours. One has to remark
that the fulfillment of all the devised experiments required to
design and 3D print some support structures (see Fig. 2)in
order to hold the specimens during the tests. For the tensile,
compression and tensile relaxation test, two clamps have
been printed employing a Mojo 3D printer (FDM Fused
Deposition Modeling, by Stratasys) using as constituent
material an acrylonitrile butadiene styrene (ABS) thermo-
plastic (Fig. 2(a)). Similarly, for the shear test a rigid support
structure and two different clamps have been created by
employing the same 3D printer and constituent material.
The measurement error introduced by the deformability
of these support objects was considered negligible follow-
ing some numerical simulations of the experiments in FE
software ABAQUS.
Results and Discussion
The results presented here comprise a complete set of
tests sufficient to calibrate pantographic models, and
therefore can be used as benchmarks for the developed
theoretical models. In the following, we briefly comment
the peculiarities observed. Note that in the following we
will use the term stiffness to refer to the trend of the values
generally assumed by the ratio between force and imposed
displacement, but no specific force-displacement relation is
meant.
In Fig. 3 the results of the tensile test for the two groups
of specimens are shown. In the plots related to group A
in Fig. 3(a) one can notice that the stiffness of the sample
is inversely proportional to length of the pivots. Indeed
for a given value of displacement, among those considered
in the experiments, the force decreases progressively from
the maximum value, measured for the sample A1, to the
minimum value, measured for the sample A4. Furthermore,
observing singularly the data for each specimen, the curves
related to A2, A3 and A4 have almost a linear behavior
while for the specimen A1 a hardening tract arises after 2
mm of displacement. This could be attributed to the plastic
deformation of the pivots or increased axial extension of the
Table 2 Type of specimen, number and values of the distinctive
parameters for each type of specimen
Specimen type Number of specimens ph
(mm) (mm)
A1 4 5 1
A2 3 5 1.5
A3 4 5 2
A4 4 5 3
B1 5 8 1
B2 4 8 1.5
B3 5 8 2
B4 4 8 3
Exp Mech
Fig. 2 Experimental setup of the
tests for a tensile, compression
and tensile relaxation tests b
shear test
central beams for the specimen A1. In Fig. 3(b) the plots for
the tensile test related to the specimen of the group B are
shown. Also in this case the data depicts an overall inverse
proportionality between the stiffness and the length of the
pivots, but for the specimen B1 the behavior is different.
The reaction force values decrease gradually from B2 to
B4 but the curve of B1 overlaps the plot of B4 for the
most part and it has a hardening part starting from about 4
mm of elongation. With a more detailed look at the data,
the specimens B3 and B4 seem to have linear behaviors
while for the specimens B1 and B2 an incipient hardening
behavior can be seen for higher values of displacement.
The plots in Fig. 4 give information about the force trend
depending on the length of the pivots, for the tensile test.
Specifically, in each plot the abscissa axis reports the sample
type while the ordinate axis reports the measured force,
which is evaluated for four different values of imposed
displacements. In Fig. 4(a), which shows the data related
to the group A, it can be observed that for a given applied
displacement, the force exerted is decreasing with the length
of the pivot, which represents the inverse proportionality
mentioned above. On the contrary, the specimens of group
B do not follow the aforementioned inverse proportionality.
Indeed, in Fig. 4(b), it is noticeable that there is a
maximum force value corresponding to the specimen B2.
This aberration deserves mathematical justification and will
be further investigated in future works.
In Fig. 5 the experimental data of the compression tests
are plotted for both group A and group B. In Fig. 5(a), which
shows the data regarding the group A, it is hard to recognize
a linear stage as the behavior seems to be nonlinear even
for small values of the imposed displacement. Up to 6
mm of compression, all the curves, whose values are close
to each other, describe a softening behavior. After 6 mm
of compression, the behavior of the samples changes and
reaction forces increase rapidly, describing a hardening
behavior. An interpretation of this observation is attempted
in the following with the help of the pictures taken during
the test. We can remark that the group A observes the
inverse proportionality relation between the stiffness of the
sample and the length of the pivots. Similar considerations
can be made looking at Fig. 5(b). Indeed, the plots of
the compression test for group B describe a softening
behavior with a not easily recognizable initial linear stage
and a hardening behavior starting from about 10 mm of
displacement. Similar to the results of tensile test, the
sample B2 leads to the highest values of reaction force. We
remark that the curve related to sample B1 is closer to the
values assumed by sample B4 up to 10 mm of compression
while starts to rise fast over this value, overlapping the plot
of sample B2 for higher value of compression. The higher
stiffness of sample B2 can be observed in the Fig. 6(b),
where the force values recorded at several displacement
values are plotted with respect to the length of the pivots. It
Fig. 3 Plots of the data from
experimental tensile test for a
group A b group B
