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220
УДК 539.42
Influence of cell topology on mode I fracture toughness of cellular structures
E. Linul, D.A. Serban, L. Marsavina
Department of Mechanics and Strength of Materials, Politehnica University of Timisoara, Timisoara, 300222, Romania
A cellular structure is made up by an interconnected network of beams or plates which forms the edges and faces of cells. This paper proposes three different micromechanical models to determine the fracture toughness values of cellular materials such as rigid polyurethane foams using the finite element micromechanical analysis and Abaqus software. This study was carried out for mode I fracture and fracture toughness was predicted based on linear elastic fracture mechanics. Models of two-dimensional cellular solids with square, hexagonal and circular cells were generated for five different relative densities (0.077, 0.105, 0.133, 0.182 and 0.333). A study of the influence of geometrical parameters on fracture toughness was also conducted. Based on the finite-element simulations, three linear correlations are proposed which could be useful for estimation of fracture toughness values if relative densities are in the considered range of 0.077 (90 kg/m3 density) and 0.333 (390 kg/m3 density). Finally, the authors validate their proposed micromechanical models presenting a comparison of analytical, numerical and experimental results of fracture toughness of cellular materials. It was found that at low relative densities (between 0.077 and 0.333), the proposed micromechanical models predict the fracture toughness values similar to experimental and numerical ones, but they must be used according with the real cellular structure.
Keywords: cellular materials, mode I fracture toughness, micromechanical models, finite element analysis, cell topology
DOI 10.24411/1683-805X-2018-11012
Влияние геометрии ячеек на вязкость разрушения ячеистых структур
при нормальном отрыве
E. Linul, D.A. Serban, L. Marsavina
Политехнический университет Тимишоара, Тимишоара, 300222, Румыния
Ячеистая структура представляет собой взаимосвязанную сеть стержней и пластин, которая образует края и грани ячеек. В статье предложены три различные микромеханические модели для определения значений вязкости разрушения ячеистых материалов, таких как жесткие полиуретановые пены, с использованием микромеханического конечноэлементного анализа и программного обеспечения Abaqus. Изучено разрушение нормального отрыва, определены значения вязкости разрушения в рамках линейно-упругой механики разрушения. Разработаны модели двумерных ячеистых тел с квадратными, гексагональными и круглыми ячейками с разными значениями относительной плотности (0.077, 0.105, 0.133, 0.182 и 0.333). Также исследовано влияние геометрических параметров на вязкость разрушения. При моделировании методом конечных элементов получены три коэффициента линейной корреляции, которые могут использоваться для оценки значений вязкости разрушения для значений относительной плотности в рассматриваемом диапазоне 0.077 (плотность 90 кг/м3) и 0.333 (плотность 390 кг/м3). Проведена проверка предложенных микромеханических моделей путем сравнения аналитических, численных и экспериментальных результатов для вязкости разрушения ячеистых материалов. Показано, что при низких значениях относительной плотности (от 0.077 до 0.333) значения вязкости разрушения, полученные с помощью предлагаемых микромеханических моделей, аналогичны экспериментальным и численным значениям, но их использование возможно только с учетом реальной ячеистой структурой.
Ключевые слова: ячеистые материалы, вязкость разрушения при нормальном отрыве, микромеханические модели, метод конечных элементов, геометрия ячейки
Nomenclature
a—crack length; E—Young's modulus;
Fmax —maximum load from the load-displacement recordings;
F^a/W) —mode I nondimensional stress intensity factor; H—height of the micromechanical model; KIc —mode I fracture toughness; l—cell length; t—cell wall thickness;
© Linul E., Serban D.A., Marsavina L., 2018
W—width of the micromechanical model; p* —density of rigid PUR foam;
ps —density of the solid material of which the foam is made; p*/ps —relative density;
a—applied load in order to produce a mode I loading; afs —fracture strength of the solid material;
—maximum stress in the first unbroken strut; v—Poisson's ratio.
1. Introduction
Considering the fact that cellular materials consist of a solid interconnected network layers like a structure [1-3], they presents physical, mechanical and thermal properties which are measured by the same methods as those used for fully dense solids [4-7]. They have great potential as core materials in sandwich construction [8-10], with application in heat exchangers and thermal protection systems, in military and commercial aerospace structures, in large portable structures and flotation devices [11]. The low stiffness makes foams ideal for a wide range of cushioning to absorb the energy of impacts without subjecting the contents to damaging stresses. Furthermore, foams can undergo large compressive strains at almost constant stress, so that large amount of energy can be absorbed without generating high stresses [12-15]. The strength of foam can be adjusted over a wide range by controlling its relative density. The low density means the package is light, reducing handling and shipping costs [16].
The physical and mechanical properties of cellular materials may be adjusted using different cell shape [17]. In Fig. 1 are presented scanning electron microscope images with three different cell shape: rectangular (40 kg/m3), hexagonal (100 kg/m3) and circular (300 kg/m3) cells. As it can be seen in these figures each typical closed PUR foam cell is surrounded by connected faces.
In this work we propose a different approach in modeling of cellular materials, replacing beams with parametrical solid structures of different geometries (square, hexagonal and circular cells), chosen in accordance with the mentioned real foams microstructures (Fig. 1). Thus, this study proposes three different micromechanical models and investigates the influence of geometrical parameters (such as cell
wall thicknesses, cell dimensions and cell shapes) on the mode I fracture toughness values for rigid polyurethane foam materials using the finite-element analysis and the Abaqus software. Finally, in order to validate the proposed models a comparison of the experimental fracture toughness results together with numerical results and some of analytical models are presented.
2. Analytical methods
Different models have been developed to predict the mechanical behavior of cellular materials, to find and analyze the failure mechanism by which cell walls deform under load. Unit cell models have proven to be useful analytical tools for understanding some of the key aspects of the mechanical behavior of cellular solids, such as effective elastic stiffness and the dependence of failure properties on relative density and on the failure mode of individual cells [18].
The most important parameter of a cellular material is the relative density p*/ps, where p* is the density of the cellular material or foam and ps is the density of the solid strut or ligament material. The relative density is a measure of solidity and most of the material properties depend on the relative density.
While analytical methods for predicting thermal and thermomechanical properties of cellular media are well documented, research on fracture behavior of various types of cellular materials is not fully understood. Gibson and Ashby [18] summarized the formulations for mode I fracture toughness. They assumed that each time the row of cell walls along a crack front fractures, the crack advances with one cell length. Comparing with experimental data for cellular materials with open cells the relationship between fracture toughness KIc normalized to the fracture strength of solid material afs and cell size l versus relative density is
= 0.65
V \
ps
1.5
(1)
Green [19] derived a similar result (but with different exponent) by treating the elastic deformation in terms of simple shell theory using a hollow sphere model for the
Fig. 1. Scanning-electron microscopy images of the used rigid PUR foams with square (a), hexagonal (b) and circular cell shapes (c)
foam cells:
K Ic
Table 1
Cfs^Kl
= 0.28
V >
Ps
1.3
(2)
Another model was proposed by Choi and Lakes [20]. They considered that the stress field is nonsingular in view of the nonzero size of the foam cells. For a regular tetrakai-decahedron cell with the relation between relative density and cell dimensions given as p */ps = 1.06(t//)2, the fracture toughness relationship is
K
GfsJnl
= 0.19
Ps
(3)
3. Numerical methods and results
Finite element modeling methods are used to describe the behavior and mechanical properties of cellular structures [21, 22]. An analytic model of the fracture toughness has been validated with finite element simulations for the diamond-celled honeycomb by Alonso and Fleck [23]. Also, a finite element based method developed by Choi and Sankar [11] has been used by Wang [2] to study the fracture toughness of two types of foams: foams with rectangular prism unit cells, including homogeneous foams and functionally graded foams, and tetrakaidecahedral foams. He obtained the plane strain fracture toughness of the foam by relating the fracture toughness to the tensile strength of the cell struts. He studied the effects of various geometric parameters that describe the cell. Two crack propagation criteria, one at the microscale and one at the macroscale were used. The fracture toughness of the brittle foam is calculated based on the stress intensity factor and the corresponding maximum tensile stress in the struts ahead of the crack. Fleck and Qiu [24] performed a finite element analysis on hexagonal honeycomb, regular triangular honeycomb and Kagome lattice models using the Euler-Ber-noulli beam elements with cubic interpolation functions, considering each beam with thickness t and length l. They present the prediction of fracture toughness related to the solid material in the form: f p* \d
K T,
GfsJnl
= D
(4)
where D is 0.212 (the Kagome lattice), 0.5 (regular triangular honeycomb) and 0.8 (hexagonal honeycomb), while exponent d equals 2 for the hexagonal honeycomb, 1 for the regular triangular honeycomb and 0.5 for the Kagome lattice loaded in mode I. Recently, a novel 2D solid rectangular micromechanical finite element model was proposed by Linul and Marsavina [25] for predicting the fracture toughness of cellular polymers for both modes I and II of loading.
Prior to geometrical model design, the cellular structure of polyurethane foams was investigated using scanning-electron microscopy images, determining the pore di-
Material properties of solid polyurethane
Ps, kg/m3 f MPa E, MPa V
1170 130 1600 0.4
ameter/flat-to-flat distance using statistical analysis as well as determining the approximate shapes of the cells (Fig. 1). The micromechanical models were designed using the SolidWorks™ software.
This section presents three different micromechanical models to determine the fracture toughness values for cellular materials using the finite element analysis and the Abaqus software. Models of two-dimensional cellular solids with square, honeycomb and circular cells were generated. The investigated micromechanical models present five different relative densities (0.077, 0.105, 0.133, 0.182, and 0.333). Fracture toughness was predicted based on linear elastic fracture mechanics, taking into account equating the maximum stress aymax in the first unbroken strut with the fracture strength of solid material afs. The advantage of this model is that fully describe the stress field in the solid struts.
The mechanical characteristics of the solid material (density ps, fracture strength afs, Young's modulus E, and Poisson's ratio v) considered for fracture toughness determination are listed in Table 1. Both the fracture toughness and the tensile strength of brittle foams depend on the fracture strength of the solid material.
Figure 2 presents the plane strain micromechanical models and imposed boundary conditions used in this analysis. A quarter of a central cracked plate was modeled using plane strain conditions and the 2D solid models of 1500 cells (301 x 501) were used (153 458 elements and 551540 nodes for the rectangular structure, 144297 elements and 526361 nodes for the honeycomb structure, 164033 elements and 586429 nodes for the circular structure) approximating an infinite array of cells: the stress distribution along the outermost layer of cells is uniform and the stress field far away from the crack tip is undisturbed [26].
The symmetric boundary conditions were imposed, and the applied load was imposed perpendicular to the crack, in order to produce a mode I loading. The crack is created by breaking the ligaments of the cells [18]. Mode I fracture toughness was obtained by progressive loading of the model with increasing the applied load a until the maximum stress aymax in the first unbroken strut reaches the fracture strength of the solid afs.
This study considers only 2D plane isotropic geometries [27] which can be considered in the plane (flow) direction of foam forming. Our previews experimental results [28] highlighted that the in-plane properties are isotropic. Thus, considering only 2D plane isotropic geometries, the frac-
t mtmmmmmmmm
Fig. 2. Micromechanical model dimensions and boundary conditions for square (a), honeycomb (b) and circular cells (c)
ture toughness of cellular material was determined according to Murakami [29]:
K Ic = KI = aJmFl (a/W), (5)
where a [mm] is the crack length, W [mm] is the width of the model and FI (a/W) is a nondimensional function [29]:
Fi(a/W ) =
1 - 0.5 a/W + 0.37(a/W)2 - 0.044(a/W)3 a/W
(6)
Figure 3 shows the deformed meshes for different proposed micromechanical models: rectangular model with a crack length of 6.85 mm (Fig. 3, a), hexagonal model with
a crack length of 6.92 mm (Fig. 3, b) and circular model with a crack length of 5.70 mm (Fig. 3, c), while Figs. 3, d-f presents the stress distribution ay in the first unbroken strut for mode I loading.
A combined tensile with bending stresses occurs in the first unbroken strut for all proposed models which confirm the hypothesis from the micromechanical model of Choi and Sankar [11].
A study of the influence of crack length on KIc was performed by numerical analysis and revealed that the crack length does not influence the fracture toughness. So the
a
a
Fig. 3. Deformed meshes and stress distribution (contour plot) oy in the first unbroken strut for mode I loading
Fig. 4. Deformed meshes in mode I loading for different crack length and different micromechanical models: rectangular model with W = = 30.95 mm (a), hexagonal model with W = 30.31 mm (b), circular model with W = 24.50 mm (c)
predicted values of KIc could be considered as a material property. The crack is assumed to be normal to the loading direction and the crack is created by breaking the ligaments of the cells. Figure 4 shows the deformed mesh for proposed micromechanical models with different crack lengths, while Fig. 5 presents the influence of the crack length on fracture toughness for investigated structures. The geometrical parameters (t, l, H, W, a), the number of elements and nodes for mentioned models are presented in Table 2.
Figure 5 shows that the crack length does not influence the predicted fracture toughness values (fracture toughness results could be considered independent on the crack length), obtaining a relative differences in fracture toughness about 0.78% for square cells, 0.56% for mode honeycomb cells and 0.82% for circular cells.
Figure 6 presents the fracture toughness variations with relative density and in Table 3 is listed the mode I fracture toughness values obtained from the finite-element simula-
Table 2
Micromechanical model dimensions
Model Square Hexagonal Circular
Relative density 0.333
Number of cells 1500 (301 x 501)
/ 0.60 0.35 0.50
Model t 0.10 0.10 0.10
dimensions, 0.035
mm W 30.95 30.31 24.50
a 4.45, 5.05, 5.65, 6.25, 6.85 4.48, 5.09, 5.70, 6.31, 6.92 3.70, 4.20, 4.70, 5.20, 5.70
Number of elements 153458 144297 164033
Number of nodes 551540 526 361 586 429
Crack ]
Fig. 5. Variation of fracture toughness with crack length
Fig. 6. Fracture toughness variations with relative density
tions for three studied cell geometries: square, hexagonal and circular cells.
Based on the finite-element simulations, three linear correlations (Eqs. (7)) are proposed for all investigated cellular structures which could be useful for estimation of fracture toughness values if relative densities are in the considered range of 0.077 (~90 kg/m3 density) and 0.333 (~390 kg/m3 density):
Kic,sq - !-251
KIc.hex - 2.012
KIc,circ - 1'685
v ^
vPsy
V '
VPs,
y '
- 0.021, R = 0.992, 0.086, R2 = 0.996, 0.053, R2 = 0.997. Mode I fracture toughness results
(7)
Table 3
Relative density P*/ Ps Cell geometry Fracture toughness K Ic,MPam05
0.077 Square 0.063
Hexagonal 0.071
Circle 0.067
0.105 Square 0.112
Hexagonal 0.133
Circle 0.129
0.133 Square 0.161
Hexagonal 0.194
Circle 0.187
0.182 Square 0.214
Hexagonal 0.263
Circle 0.241
0.333 Square 0.392
Hexagonal 0.593
Circle 0.517
Fracture toughness values are in range of 0.06 MPa m05 for 0.077 relative density (square cells) and 0.59 MPa m05 for 0.333 relative density (hexagonal cells).
4. Discussions and conclusions
Micromechanical models are useful tools to predict the fracture toughness of cellular materials. However, these numerical and analytical micromechanical values must be validated with experimental results [30]. Most of the rigid foams have linear-elastic behavior in tension up to fracture followed by a brittle fracture [31-33]. So, they can be treated using fracture criteria of linear elastic fracture mechanics [34-37].
Mode I fracture toughness of cross-linked polyvinyl chloride and rigid polyurethane foams was examined by Kabir et al. [38] on single-edge notched bending specimens under three-point bending tests. Recently, static three-point bending tests and instrumented impact tests were performed by Marsavina et al. [39] on polyurethane foams of six different densities, using single-edge notched bending specimens and they found that the dynamic fracture toughness results are higher than the static ones. The size effect was also investigated in order to find the minimum width of the specimens for fracture toughness determination. The single edge crack specimen with a mixed mode loading device was adopted by Linul et al. [40] to determine the fracture toughness of rigid polyurethane foams under mixed mode loading. The crack propagation angles were also determined on the fractured specimens.
Two types of specimens (asymmetric semicircular bending and asymmetric four-point bending specimens) were used for determining the fracture toughness in modes I, II and a mixed one and also the loading speed and loading direction were investigated in [41]. They proposed correlations for density, cell orientation and mixed mode loading based on the experimental testing results.
In order to validate the models proposed above, Fig. 7 presents a comparison of the finite-element analysis (FEA) obtained results together with the experimental fracture
Klc/(Gis(nl)°-5)
m
-Ashby-Gibsonmodel [18]
Jsfy^/ -Choi-Lakes model [20]
spa/ / -Green model [19]
A > f / --FEA hexagonal [24]
/ ........FEA triangular [24]
---FEA Kagome [24]
□ FEA square cells [39-42]
❖ FEA hexagonal cells [39-42]
• FEA circular cells [39-42]
A Experimental results [38]
0.1 1 pVps
Fig. 7. Comparison of analytical [18-20], numerical [24] and experimental [38-42] results of fracture toughness
toughness results, with numerical results from Ref. [24] and some of the mentioned analytical micromechanical models [18-20].
It can be observed that at low relative densities (between 0.077 and 0.333) the proposed micromechanical models predict similar values for fracture toughness with experimental, analytical and numerical ones. The experimental values of normalized fracture toughness are little higher than those predicted by micromechanical and numerical models for larges values of p*/ps = 0.333.
From these investigations the following conclusions can be drawn.
Based on linear elastic fracture mechanics, new micromechanical models (square, hexagonal and circular micromechanical models) were proposed for predicting the fracture toughness of cellular materials. The novelty of this work consists in proposing a different approach in modeling of cellular materials, replacing beams with parametrical solid structures of different geometries.
All proposed cell geometries predict similar fracture toughness values, but they must be used according with the real cellular structure (see Fig. 1).
As it can be seen in Table 2 and Fig. 6, the fracture toughness is strongly dependent on relative density of foams and a complex bending with tension stress distribution in the first uncracked struts were observed.
The study of the influence of crack length on KIc for all three proposed micromechanical models revels that the crack length does not influence the fracture toughness (see Fig. 5).
Based on the finite-element simulations, three linear correlations are proposed which could be useful for estimation of fracture toughness values if relative densities are in the considered range of0.077 (~90 kg/m3 density) and 0.333 (~390 kg/m3 density).
At low relative densities (between 0.077 and 0.333) the proposed micromechanical models predict similar values for fracture toughness with experimental and numerical ones. The experimental values of normalized fracture toughness are little higher than those predicted by micromechanical and numerical models for larges values of p*/ps = 0.333 (see Fig. 7).
Acknowledgments
This work was partially supported by the CNCS-UEFISCDI Grant PN-II-ID-PCE-2011-3-0456, contract number 172/2011.
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