Mechanical Properties of Miura-Based Folded Cores under Quasi-Static Loads
Xiang Zhou
1*
, Hai Wang
1
, Zhong You
2
1 School of Aeronautics and Astronautics, Shanghai Jiao Tong University, China
2 Department of Engineering Science, University of Oxford, UK
ABSTRACT
Sandwich structures with folded cores are regarded as a promising alternative to conventional
honeycomb sandwich structures in the aerospace industry. This paper presents a parametric study
on the mechanical properties of a variety of Miura-based folded core models virtually tested in
quasi-static compression, shear and bending using the finite element method. It is found that the
folded core models with curved fold lines exhibit the best mechanical performances in
compression and shear while the multiple-layered models outperform the other folded core models
in bending. Furthermore, the folded core models are compared to a honeycomb core model with
the same density and height. In this case, it is shown that the honeycomb core has the best
performance in compression while the folded cores have comparable or even better performances
in the shear and bending cases. The virtual test results reported in this paper can provide
researchers with a general guideline to design the most suitable folded core structure for a certain
application.
Keywords: Miura-based folded core; sandwich structure; quasi-static test; finite element method.
*
Corresponding author, E-mail: xiangzhou@sjtu.edu.cn, phone: +86-21-34207538, address: School of Aeronautics
and Astronautics, No. 800 Dongchuan Road, Shanghai, 200240, China. X. Zhou is an associate professor in the
School of Aeronautics and Astronautics at Shanghai Jiao Tong University.
1 INTRODUCTION
Composite sandwich structures, typically consisting of two thin and stiff faces separated by a thick
lightweight cellular core, have many successful applications in the aerospace industry where
weight-saving is a paramount design goal. In this context, honeycomb cores made of aluminum or
Nomex paper are the most commonly used core type today due to their excellent weight-specific
mechanical properties. However, honeycomb cores are known to suffer from an undesirable
moisture accumulation problem whereby the condensed moisture is trapped inside the sealed
hexagon cells leading to deterioration of the mechanical performance over time [1].
Folded cores, made by folding sheet material into a three-dimensional structure according to the
principle of origami – an ancient art of paper folding, do not have the moisture accumulation
problem because of the existence of open channels in such structures. Moreover, they allow for
tailored mechanical properties with a wide range of possible configurations. Therefore, they
emerge as a promising alternative to conventional honeycomb cores and have seen a surge in
research interest from the aerospace industry in recent years. For example, in the transnational
project CELPACT, the fabrication cost and impact performance of three different advanced
cellular core concepts, i.e. folded core, selected laser melted lattice core were evaluated and
compared [2]. Besides, the aircraft manufacturer Airbus presented a sandwich fuselage concept
VeSCo which incorporates folded cores as a sandwich core material [3] and has made a 4.5 m
2
test
assembly consisting of approximate 165,000 creases [4].
While specimen manufacturing and mechanical testing remain routine procedures, numerical
analysis based on the finite element (FE) method, as an established time- and cost-efficient tool,
has been widely adopted in the development of new composite structures. Besides, FE simulations
can provide analysis details such as the cross-sectional stress/strain data that are usually difficult
to be obtained experimentally. As a result, a number of numerical studies of folded-core sandwich
structures, such as virtual in- and out-of-plane quasi-static compression and shear tests [5-9], low-
and high-velocity impact simulations [10-12], residual bending strength simulations after impact
[13] and macro- and multi-scale modelling [7,11], are available in the literature. However, most
folded cores used in research works are made of two simple Miura-based unit cell geometries with
zigzag and chevron shapes [14]. So far, the authors are not aware of any literature on
computational or experimental study of folded core structures beyond these two simple cases.
Consequently, the mechanical properties of other folded configurations remain unexplored.
This paper presents a parametric study on folded cores with different geometric parameters based
on the standard Miura folding pattern [15] and its variation forms subject to out-of-plane
compression, in-plane shear and bending using the finite element method. To facilitate the
parametric modelling, a new origami geometric design approach, known as the vertex method [16],
is used to generate the various folded core models in this study. Furthermore, the weight-specific
mechanical properties of the folded core models were compared to those of a honeycomb model
with the same density.
The layout of the paper is arranged as follows. First, the mechanical behaviors of eight folded core
models with the standard Miura origami folding pattern are simulated and compared. Second,
eight folded core models with curved fold lines are virtually tested. Third, further two folded core
models with multiple layers are considered. Fourth, the mechanical performances of the folded
core models are compared with those of a honeycomb core model. Finally, a brief discussion
concludes the paper.
2 STANDARD MIURA FOLDED CORES
2.1 Geometric models
Using a set of geometric parameters to define the folded configuration of a unit cell is a commonly
employed modelling technique of folded cores in the literature [4]. However, this approach lacks
flexibility in that a new set of geometric parameters and their relationships must be established
when a different type of folded core is studied. In this paper, an alternative modelling technique,
known as the vertex method for designing developable origami structures, is used to generate the
geometric models of folded cores, in which input points in the x-z plane, denoted by their
position vectors
0
, 1,2,…,, and 2 input points in the y-z plane,
denoted by their position vectors
0
, 0,1,…,1 are first specified in a
Cartesian coordinate system, respectively and then vertices
,
of the target folded core
geometric model are obtained using the following equation
,
,
,
,
,1,2,…,;1,2,…,, (1)
where
is a 33 matrix given by
A
10 0
001
001
, (2)
where the angular variable
is determined by
sin
∙
, (3)
cos
∙
, (4)
where
010
and
001
are the unit vectors of the y and z axis,
respectively, and
‖
‖
denotes the norm of a vector .
Figure 1 shows the input points in the x-z and y-z planes used to generate the models in this
section, which are defined by four parameters, i.e. ,
, and
. By fixing
to 10 mm
and choosing different combinations of values for the other three parameters, eight unit cell
models known as standard Miura origami structures are obtained, as shown in Fig. 2. The core
density
can be obtained by
, (5)
where
,
and
are respectively the thickness, total area and material density of the sheet
from which a unit cell of the core is folded and
is the spatial volume of the unit cell, defined
by
, (6)
where
is the base area of the unit cell and
is the core height, as illustrated in Fig. 3. Since
the weight-specific mechanical properties of the folded cores are concerned, a unified core density
equal to 0.05
is used for all models studied in this paper. According to Eqn. (5), the thickness
of the sheet material is given by
0.05
. (7)
Table 1 summarizes the geometric properties of the eight unit cell models considered in this
section. It is found that
is not affected by
given the other input parameters are fixed. The
larger
is, the larger the amplitude of the flatwise zigzag fold lines is. With the increase in
or ,
becomes smaller whereas the folded core becomes denser in the x- or y-direction.
2.2 FE models
The finite element analysis was performed in the FE solver ABAQUS/Explicit (SIMULIA Inc.,
USA) due to its good capability to cope with large nonlinear deformations, post-buckling
behaviors and complex contact conditions. Because the main purpose of this paper is to study the
structural influence on the mechanical properties of folded cores, both the faces and the core are
assumed to be made of 5052-O aluminum alloy and a bilinear isotropic plastic material model [17]
is employed for simplicity. For quasi-static loading cases, the strain rate effect is not considered.
The detailed material parameters are summarized in Table 2.
S4R, the four-node quadrilateral shell element with reduced integration and hourglass control, is
the element of choice in the simulation. With this particular element type, the mesh density has a
strong influence on the accuracy of the simulation results. Although a coarser mesh reduces the
computational time, it is not able to accurately represent the post-buckling behavior of the facets.
Therefore, convergence testing of different element sizes ranging from 0.15 mm to 0.4 mm was
firstly performed for all eight unit cell models in Fig. 2 subject to compressive loads in the
thickness direction. The results converged for element sizes below 0.2 mm. Therefore, the 0.2 mm
element size is used for all subsequent analysis unless otherwise specified. In the virtual tests,
each folded core model consists of four unit cells in a 22 array, as shown in Fig. 4. The numbers
of elements in the eight folded core models used in the virtual tests range from 29696 for M17 to
98832 for M12.
Three types of virtual tests, i.e. compression, shear and bending, were considered. In the virtual
