Influence of Cell Topology on Mode I Fracture Toughness of Cellular Structures

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.

PHYSICAL MESOMECHANICS Vol. 21  ρ ρ relative density, σapplied load in order to produce a mode I loading,

INTRODUCTION
Considering the fact that cellular materials consist of a solid interconnected network layers like a structure [13], they presents physical, mechanical and thermal properties which are measured by the same methods as those used for fully dense solids [47]. They have great potential as core materials in sandwich construction [810], 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. .urthermore, foams can undergo large compressive strains at almost constant stress, so that large amount of energy can be absorbed without generating high stresses [1215]. 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 .ig. 1 are presented scanning electron microscope images with three different cell shape: rectangular (40 kg/m 3 ), hexagonal (100 kg/m 3 ) and circular (300 kg/m 3 ) 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 (.ig. 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. .inally, 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.

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 * s , ρ ρ where * ρ is the density of the cellular material or foam and s ρ 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 Ic K normalized to the fracture strength of solid material fs σ and cell size l versus relative density is 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 foam cells: 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. .or a regular tetrakaidecahedron cell with the relation between relative density and cell dimensions given as

NUMERICAL METHODS AND RESULTS
.inite 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 .leck [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. .leck and Qiu [24] performed a finite element analysis on hexagonal honeycomb, regular triangular honeycomb and Kagome lattice models using the EulerBernoulli 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: 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 diameter/flat-to-flat distance using statistical ana-lysis as well as determining the approximate shapes of the cells (.ig. 1). The micromechanical models were designed using the SolidWorks TM 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). .racture toughness was predicted based on linear elastic fracture mechanics, taking into account equating the maximum stress y,max σ in the first unbroken strut with the fracture strength of solid material fs .
σ The advantage of this model is that fully describe the stress field in the solid struts.
The mechanical characteristics of the solid material (density s , ρ fracture strength fs , σ Youngs modulus E, and Poissons ratio ν) 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.
.igure 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 (301×501) were used (153 458 elements and 551 540 nodes for the rectangular structure, 144 297 elements and 526 361 nodes for the honeycomb structure, 164 033 elements and 586 429 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 σ until the maximum stress y,max σ in the first unbroken strut reaches the fracture strength of the solid fs . σ 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 fracture toughness of cellular material was determined according to Murakami [29]: .igure 3 shows the deformed meshes for different proposed micromechanical models: rectangular model with a crack length of 6.85 mm (.ig. 3a), hexagonal model with a crack length of 6.92 mm (.ig. 3b) and circular model with a crack length of 5.70 mm (.ig. 3c), while .igs. 3d3f presents the stress distribution y σ 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 Ic K was performed by numerical analysis and revealed that the crack length does not influence the fracture toughness. So the predicted values of Ic K could be considered as a material property. The crack is assumed to be normal to the loading direction and the crack is cre-  Table 2.
.igure 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.
.igure 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 simulations 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/m 3 density) and 0.333 (~390 kg/m 3 density): .racture toughness values are in range of 0.06 MPa m 0.5 for 0.077 relative density (square cells) and 0.59 MPa m 0.5 for 0.333 relative density (hexagonal cells).

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 [3133]. So, they can be treated using fracture criteria of linear elastic fracture mechanics [3437].
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. Re-.ig. 5. Variation of fracture toughness with crack length.
.ig. 6. .racture toughness variations with relative density. cently, 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, .ig. 7 presents a comparison of the finite-element analysis (.EA) obtained results together with the experimental fracture toughness results, with numerical results from Ref. [24] and some of the mentioned analytical micromechanical models [1820].
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 * s ρ ρ = 0.333.
.rom 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 .ig. 1).
As it can be seen in Table 2 and .ig. 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 Ic K for all three proposed micromechanical models revels that the crack length does not influence the fracture toughness (see .ig. 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 of 0.077 (~90 kg/m 3 density) and 0.333 (~390 kg/m 3 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 * s ρ ρ = 0.333 (see .ig. 7).