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The 9th European Congress on Computational Methods in Applied Sciences and Engineering 
ECCOMAS Congress 2024 
3 – 7 June 2024, Lisboa, Portugal 
 
 
 
COMPUTATIONAL ANALYSIS OF PENTAMODE STRUCTURES 
UNDER COMPRESSIVE LOADING 
 
PANAGIOTIS N. LYMPEROPOULOS EFSTATHIOS E. THEOTOKOGLOU AND 
KONSTANTINOS KALFOUNTZOS 
*
 Department of Mechanics, Laboratory of Testing and Materials, School of Applied Mathematical 
and Physical Sciences, National Technical University of Athens, 5, Heroes of Polytechnion Avenue, 
Theocaris Building Zografou Campus, PO Box 157 73 Athens, Greece 
e-mail: panagiotislymperopoulos@mail.ntua.gr, stathis@central.ntua.gr 
 
 
Key words: Metamaterials, pentamode structure, computational mechanics, finite element 
method, compressive loading, buckling. 
 
Summary. Pentamodes mechanical metamaterials are lattice structures that are applied in 
many engineering applications. Buckling of pentamodes lattice structures is a phenomenon 
that may be occurred when high compressive loadings are considered. Due to the lattice 
structure of pentamodes, buckling analysis may be a complicated phenomenon. In our study, a 
numerical investigation of pentamodes under compressive loading conditions has taken place. 
Several pentamodes structures have been analyzed. Von Misses stresses have been calculated 
and maximum stresses are determined. Finally, a buckling analyses considering material non 
linearity has also taken place.   
 
1 INTRODUCTION 
Buckling is a significant nonlinear deformation that can occur to beams under high 
compressive loading. Pentamodes mechanical metamaterials are lattice structures, composed 
of bi-cone beams and are designed to have high bulk modulus [1-3]. They can confront 
different high compressive loading conditions and consequently buckling phenomena may 
occur. In addition, due to the non-homogeneous nature of pentamodes, the response of 
pentamodes under different loading conditions is significantly affected.  Their wide 
applications field, from aerospace structure to vibrations isolation structure (antiseismic 
design) [2,4,5]] contribute to further investigation of buckling characteristic of pentamode 
structures. 
In addition, pentamodes with increased height have also been used in order to confront 
different loading conditions [5-10]. For this reason, buckling analyses should be taken place, 
in order to analyse the buckling characteristics of pentamodes structures.  
The present paper deals with the buckling response of pentamodes structures. More 
specifically, the main objective of this study is the investigation of failure mechanism of 
pentamodes structures under high compressive forces, which results buckling phenomena, 
using the Finite Element Method (FEM). The aforementioned procedure is implemented using 
FEM package ANSYS [11] which incorporates nonlinear FEM analyses and buckling 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 2 
analysis. The elastoplastic material behaviour of Clear V4 polymer [10] is modelled using a 
bilinear stress-strain relationship. In addition, large displacement analyses have taken place. 
The convergence of the numerical solution is also tested for different pentamode structures.  
Load displacement curves have been created using the results from the FEM analyses. In 
addition, Von Misseses stresses contours have been developed to determine the maximum 
stress fields and their amplitude taking into consideration material and geometric non 
linearity. It is also found that pentamodes appear to rotate under compressive loading, due to 
their non-homogenous nature. 
2 THEORETICAL CONSIDERATION 
2.1 Pentamode structure 
In our study, a typical pentamode unit cell (Figure 1) is used, 
 
Figure 1: Unit cell pentamode structure [4,5] 
where d is the small diameter, D is the wide diameter and  is the main dimension of 
pentamode structure. In Table 1, the dimensions of pentamodes are presented.  
 
Table 1: Pentamoded dimensions 
α [mm]
 
d [mm]
 
D [mm]
 
30 2.4 5.4 
 
The pentamodes structures considered in this study have unit cells towards x-axis ( xn ) and 
respectively for y- and z-axis ( ,y zn n ), as presented in Table 2. 
Table 2: Pentamoded Structures 
Case Unit Cell 
1 2, 1x y zn n n    
3 6, 1x y zn n n    
5 10, 1x y zn n n    
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 3 
2.2 Materials 
In our analyses, pentamodes are considered to be made from the polymer resin Clear V4 
[12]. The materials stress strain curves are given in Figure 2. 
 
Figure 2: Material properties [12] 
From Figure 2, Clear V4 [12] is considered to be the standard resin that has been used in 
our analyses. The Young modulus of the material is 2.8[GPa], the ultimate tensile strength is 
65[MPa] while the elongation at failure is 6% [12]. 
 
3 COMPUTATIONAL ANALYSIS 
In this finite element analyses, pentamodes have been modeled using the finite element 
program ANSYS[11]. The element that has been used is the BEAM189 [11] which is a 3 
nodes beam element, with 6 degrees of freedom each node. In order to confront the beam bi-
cone geometry, TAPER [11] elements have also been used. At first in order to study the 
convergence of our results, a pentamode with 5D mm , 1.4d mm , 30mm  and with
2x y zn n n    is considered. A compressive load of 1.5kN is applied to nodes at the upper 
side of the pentamodes, whereas the nodes at the bottom side are fully constrained. The 
displacements of the upper side of the pentamodes is plotted in the following Figure 3. 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 4 
 
Figure 3: Convergence of the displacement 
In Figure 3, the results for the displacements of the upper side nodes are presented. It is 
observed that the results convergence after 16 elements in each beam. In our study 20 
elements in each beam are considered. 
The buckling finite element analyses is divided in two parts. At the first part, a static finite 
element analyses is take place, with a compressive load of 1N. Then, with the results from the 
static analyses, an eigenbuckling analyses is considered. The eigenbuckling analyses extract 3 
eigenfrequencies each case. 
4 RESULTS AND DISCUSSION 
In this section, the results of the finite element analyses are presented. In our study two 
cases for the boundary conditions are considered. At first, pentamodes bottom side nodes are 
considered fully constrained, and secondly the bottom side nodes are considered to have only 
the displacements constrained. Then the eigenvalue forces are calculated, and for the first 
eigenvalue each case, the Von Misses stresses are extracted. 
3.1 Eigenvalue Forces  
For the 1
st
 case, the eigenvalue forces are given in the Table 3, 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 5 
Table 3: Eigenvalue forces 
Eigenvalue 
force 
Fully Constrained 
[N] 
Only Displacements 
Constrained [N] 
1
st
 156.38 133.38 
2
nd
 452.86 444.33 
3
rd
 653.73 653.20 
 
 According to Table 3, it is observed that the 1
st
 eigenvalue force in the case of the fully 
constrained bottom side nodes has higher value than in the case of only displacements 
constrained at the bottom side nodes. It should be noted that the 1
st
 eigenvalue force is the 
most useful during the design process.  
For the 2
nd
 case, the eigenvalue forces are presented in the Table 4, 
Table 4: Eigenvalue forces 
Eigenvalue 
force 
Fully Constrained 
[N] 
Only Displacements 
Constrained [N] 
1
st
 67.12 65.80 
2
nd
 566.00 555.06 
3
rd
 718.79 718.17 
 
 As it is also noted from the results of Table 4, the 1
st
 eigenvalue force in the case of fully 
constrained bottom side nodes has higher value than in the case of only displacements 
constrained at the bottom side nodes.  
Finally for the 3
rd
 case, the eigenvalue forces are given in Table 5, 
Table 5: Eigenvalue forces 
Eigenvalue 
force 
Fully Constrained 
[N] 
Only Displacements 
Constrained [N] 
1
st
 42.14 41.72 
2
nd
 371.42 367.90 
3
rd
 566.42 565.96 
 
 From the results of Table 5, it is also observed that the 1
st
 eigenvalue force for the case of 
fully constrained bottom side nodes has higher value than the case of only displacements 
constrained at the bottom side nodes.  
In the sequel, for the case of 6x yn n   and 1zn  , a comparison of the 1
st
 eigenvalue 
force among  different beams dimensions has taken place (Table 6).  
Table 6: Pentamoded dimensions 
 d [mm]
 
D [mm]
 
Eigenvalue force 
[N] 
1 4.8 10.8 907.16 
2 1.2 5.7 4.43 
It is observed that pentamode with wider beam dimensions present higher eigenvalue force 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 6 
than that with the pentamode with thinner beam dimensions.  
3.2 Von Misses Stress 
In order to determine where the maximum stresses will be appeared, the Von Misses 
stresses are calculated. For the 1
st
 case (Table 3), and for both boundary conditions, the Von 
Misses stresses are given in Figure 4. 
 
 
 
(a) Fully constrained bottom side nodes 
 
 
(b) Only displacements constrained bottom side nodes 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 7 
Figure 4: Von Misses stress in the 1
st
 case (Table 3) 
 
It is observed that the maximum Von Misses stresses, are near the beam connections. The 
maximum Von Misses stresses are between 15.02[MPA] and 19.25[MPA] for both cases 
respectively, which is below the ultimate strength of the material (Section 2.2).  
In the sequel for the 2
nd
 case (Table 4), and for both boundary conditions, the Von Misses 
are presented in Figure 5. 
 
 
 
(a) Fully constrained bottom side nodes 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 8 
 
(b) Only displacements constrained bottom side nodes 
 
Figure 5: Von Misses stress in the 2
nd
 case (Table 4) 
From Figure 5 it is should be noted that the maximum Von Misses stresses, are near beam 
connections. In addition, according to Figures 4, 5, the maximum Von Misses stresses 
appeared lower value in the 2
nd
 case than in the 1
st
 case. The maximum Von Misses stresses 
are between 2.29[MPA] and 2.24[MPA] for both cases respectively, which is below the 
ultimate strength of the material (Section 2.2). 
For the 3
rd
 case (Table 5), and for both boundary conditions, the Von Misses are given in 
Figure 6. 
 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 9 
 
(a) Fully constrained bottom side nodes 
 
 
(b) Only displacements constrained bottom side nodes 
 
Panagiotis N. Lymperopoulos, Efstathios E. Theotokoglou and Konstantinos Kalfountzos. 
 10 
Figure 6: Von Misses stress in the 3
rd
 case (Table 5) 
 
From Figure 6 it also is observed that the maximum Von Misses stresses are between 
0.89[MPA] and 0.90[MPA] for both cases respectively, which is below the ultimate strength 
of the material (Section 2.2).  
3.3 Non linear buckling 
For the case of 6x yn n   and 1zn  , a non linear buckling analyses has taken place. The 
material is considered to be bi-linear (Figure 2), and the corresponding eigenvalue forces are 
presented in the Table 7. 
Table 7: Eigenvalue forces 
Eigenvalue 
force 
Fully Constrained 
[N] 
1
st
 67.12 
2
nd
 566.00 
3
rd
 718.79 
 
It is observed from Tables 5 and 9 that the eigenvalue forces almost coincide for the linear 
and non linear buckling analyses. 
4 CONCLUSIONS 
In this study, a buckling analyses of pentamodes has taken place. Linear buckling 
characteristics were considered for several pentamodes structures under different boundary 
conditions. According to our study, pentamodes with fully constrained bottom side nodes, 
appeared higher 1
st
 eigenvalue force than pentamodes with only displacements constrained at 
the bottom side nodes. In addition, Von Misses stresses appeared to have decreased maximum 
values in the case of pentamodes only displacements constrained at the bottom side nodes. 
Furthermore, as pentamodes increased their dimensions, the 1
st
 eigenvalue force appeared to 
decrease. In addition, the non linear buckling analyses presented results for the eigenvalue 
forces that almost coincided with those from the linear analyses.  
Finally, for future work, large displacement analyses should take place in more materials 
options. Furthermore, a new fillet type for the beam connections [13] should also be 
considered. 
REFERENCES 
[1] Fleck, A. N., Deshpande, S. V., Ashby, F. M. Micro-architectured materials: past, present 
and future. Proceedings of the Royal Society of London A. 2010;466; 2495–2516 
 
[2] Kadic M., Bueckmann T., Stenger N., Thiel M., Wegener M., On the practicability of 
pentamode mechanical metamaterials, Applied Physics Letters, 2012, 100, p.p. (191901-
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[3] Norris N. A., Mechanics of elastic networks. Proceedings of the Royal Society of London 
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A, 2014, 470 (2172). 
 
[4] F. Fabbrocino, A. Amendola, F. Benzoni and F. Fraternali, Seismic Appilcation of 
Pentamode Lattices, International Journal of Earthquake Engineering, 2015, 62-71.  
 
[5] Amendola A , Carpentieri G , Feo L , Fraternali F . Bending dominated response of 
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[6] Zhang L, Song B., Zha A., Liu R., Yang L., Shi Y., Study on mechanical properties of 
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[7] Lin G., Li J., Chen P., Sun W., Chizhik S. A., Makhaniok A. A., Melnikova G. B., 
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[8] Wang Y., Chi Z., Liu J., On buckling behaviors of a typical bending-dominated periodic 
lattice, Composite Structures, 2021, 258, 113204. 
 
 
[9] Cushing C. W., Kelsten M. J., Su X., Wilson P. S., Haberman M. R., Norris A. N., Design 
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[10] Lymperopoulos P. N., Theotokoglou E. E., Numerical Investigation of Pentamode 
Mechanical Metamaterials, WSEAS Transactions on Applied and Theoretical Mechanics, 
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[11] ANSYS, ANSYS mechanical user' s guide (Release 19.0): Pennsylvania, USA: ANSYS 
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[12] FormLabs. Standard Materials for High-Resolution Rapid Prototyping, 2017, 1–3. 
 
[13] M. Dallagoa, S. Raghavendraa, V. Luchinb, G. Zappinib, D. Pasinic, M. Benedettia, The 
role of node fillet, unit-cell size and strut orientation on the fatigue strength of Ti-6Al-4V 
lattice materials additively manufactured via laser powder bed fusion. Int. J. 
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