A micromechanical method to predict the fracture toughness of cellular materials

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Abstract

The Mode I, Mode II and mixed mode fracture toughness of a cellular medium is predicted by simulating the crack propagation using a finite element model. Displacement boundary conditions are applied such that they correspond to a given value of stress intensity factor in a homogeneous solid that has the same elastic constants as the cellular medium. The crack propagation is simulated by breaking the crack tip strut when the maximum stress in that strut exceeds the strength of the strut material. Based on the finite element results a semi-empirical formula is also derived to predict the Mode I and Mode II fracture toughness of cellular solids as a function of relative density. The results show that the displacements and stresses in the foam near the crack tip are very similar to that in an equivalent homogeneous material, and continuum fracture mechanics concepts can be applied to predict the fracture of a cellular medium. The forces acting in the crack tip strut can be considered as the resultant of stresses over an effective length in the corresponding continuum model. A relation for this effective length has been derived in terms of the relative density of the cellular medium.

Introduction

Cellular materials are made up of a network of beam or plate structures leaving an open space or cell in between. Cellular materials, e.g., carbon and polymeric foams, offer several advantages such as thermal resistance, durability, low density, impact damage tolerance and cost effectiveness. They have great potential as core materials in sandwich construction, which has application in heat exchangers and thermal protection systems in military and commercial aerospace structures.

An excellent treatise on the structure and properties of cellular solids has been written by Gibson and Ashby (1988). While analytical methods for predicting thermal and thermo-mechanical properties of cellular media are well documented, research on fracture behavior of various foams is still in its infancy. Gibson and Ashby (1988) have presented approximate formulas for Mode I fracture toughness of cellular solids in terms of their relative density and tensile strength of the strut or ligament material. These are limited to cracks parallel to the principal material direction. Moreover, fracture behavior under mixed mode was not studied. In order to estimate the fracture toughness, the stresses in the crack-tip strut (first unbroken cell edge) is calculated in terms of the stress intensity factor. Then the maximum stress in the strut due to the bending moment is equated to the tensile strength of the strut material. The stress intensity factor that would produce such a bending moment is taken as the fracture toughness of the foam.

The SEM micrograph of carbon foam is shown in Fig. 1. The open-cell foam has irregularly sized and spaced cells. For high-density carbon foam (300–800 kg/m3), the length of cell edges is found to be in the range of 1–2 mm. For low-density carbon foam (160–300 kg/m3), the cell length is in the range of 200–600 μm. Unit cells of foams have been modeled as tetrakaidecahedra (a polyhedron containing 14 faces, 36 edges and 24 vertices). Li et al. (2003) modeled the open-cell carbon foam as a space frame and calculated the homogeneous elastic constants analytically. They performed a parametric study to understand how the variation in ligament properties affected the elastic constants of the foam. Sihn and Roy (2003) used three-dimensional finite elements to model the unit-cell of the carbon foam and studied the effects of anisotropy in the ligament material on the overall properties of the foam. In the present study the unit cell of the cellular solid is assumed to be a rectangular prism (actually a cube with side c), and the edges or the struts are assumed to have a square cross-section (h × h) as shown in Fig. 2. It should be mentioned that the purpose of the present study is to understand the effects of cell length and strut size on the fracture toughness of a model-foam. The fracture toughness of the carbon foam shown in Fig. 1 was studied experimentally and analytically in Choi and Sankar (2003).

On the macroscale, the cellular solid is considered as a homogeneous orthotropic material. A crack parallel to one of the principal material directions is assumed to exist in the solid and a small region surrounding the crack tip is modeled using Euler–Bernoulli beam finite elements. The commercial finite element software ABAQUS® was used for this purpose. The strut material is assumed to be isotropic, linearly elastic and brittle, and its elastic constants and tensile strength are assumed known. The crack is modeled by breaking several struts along the line of the intended crack. The properties of the strut material in this study are close to that of carbon foam investigated in a previous experimental study (Choi and Sankar, 2003), and they are given in Table 1.

Section snippets

Elastic constants of the foam

As will be seen later, the fracture models require knowledge of the orthotropic properties of the foam. In this section, formulas based on mechanics of materials type calculations for Young’s modulus and shear modulus are presented and the results are verified by finite element models. The cellular medium is assumed to consist of struts of square cross-section in a rectangular array in the 1–2 plane. In the following, a superscript * denotes the macroscopic properties of the foam, whereas a

Finite element based micromechanics for fracture toughness

In this section, we describe a finite element based micromechanics model for estimating the fracture toughness of the cellular solid. The crack is assumed to be parallel to one of the principal material axes, and Mode I, Mode II and mixed mode fracture conditions are considered. To determine the fracture toughness, a small region around the crack tip is modeled using beam elements, and a constant mode mixity KI/KII is considered. The boundary of the cellular solid is subjected to displacement

Analytical model for mode I fracture toughness

In order to derive an analytical model for fracture toughness, the stress intensity factor of the homogeneous model should be related to the actual stresses in the crack tip ligament of the foam. This can be obtained by assuming that the internal forces and bending moment in the crack tip strut are caused by a portion of the crack tip stress field ahead of the crack tip in the homogeneous model as illustrated in Fig. 19.

Let us define a non-dimensional factor α that describes the effective

Mode I fracture toughness of inclined cracks

So far our attention has been focused on cracks parallel to the principal material direction. The next step will be to study cracks inclined at an angle to the principal material direction. The procedures for predicting the fracture toughness of angled cracks are very similar to those described in the preceding sections. The only change is in the material elastic constants, which have to be transformed from the material principal directions to the global xy coordinate system where x-axis is

Summary and conclusions

A finite element based micromechanics method has been developed to determine the fracture toughness of cellular materials. A portion of the cellular medium surrounding the crack tip is modeled using beam finite elements. Displacement boundary conditions are applied such that they correspond to a given value of stress intensity factor in a homogeneous solid that has the same elastic constants as the cellular medium. The stresses developed in the beam elements (struts) are used to determine if

Acknowledgement

Thanks are due to Dr. Ajit Roy of Wright-Patterson Air Force Base for his guidance and technical discussions during the course of this project. Support from the NASA URETI Grant NCC3-994 managed by Glenn Research Center is also acknowledged.

References (6)

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    Cellular Solids: Structure and Properties

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  • Li, K., Gao, X.-L., Roy, A.K., 2003. Micromechanical analysis of three-dimensional open-cell foams using the matrix...
  • Sihn, S., Roy, A.K., 2003. Parametric study on effective Young’s modulus and Poisson’s ratio of open-cell carbon foam....
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