Biomechanics of cellular solids
Introduction
In nature, materials with a cellular structure are widespread. Examples of natural materials with prismatic, honeycomb-like cells include wood and cork (Fig. 1a,b), while those with polyhedral cells include the inner core in plant stems and trabecular bone (Fig. 1c,d). They also appear as the cores in natural sandwich structures: in long, narrow plant leaves, such as the iris, and in shell-like bones, such as the skull (Fig. 1e,f). Natural tubular structures often have a honeycomb-like or foam-like core supporting a denser outer cylindrical shell, increasing the resistance of the shell to kinking or local buckling failure (e.g. plant stems and animal quills, Fig. 1g,h). Engineers design biomaterials with a cellular structure to replace or regenerate tissues in the body. For example, titanium foams are being considered as substitute materials for trabecular bone (Fig. 2a). And porous scaffolds used in tissue engineering are designed to mimic the body's extracellular matrix, allowing the attachment, migration, proliferation and functioning of cells (Fig. 2b).
The microstructural features of cellular solids affecting their mechanical response are most easily observed in engineering honeycombs and foams (Fig. 3). Honeycombs, with their prismatic cells, are referred to as two-dimensional cellular solids while foams, with their polyhedral cells, are three-dimensional cellular solids. The relative density is the density of the cellular solid divided by that of the solid it is made from, and is equivalent to the volume fraction of solid. Foams may be either open (with solid only at the edges of the polyhedra) or closed (with solid membranes over the faces of the polyhedra). The properties of the foam depend on those of the solid making up the cellular material; in materials such as wood, the cell wall is itself a multilayered composite material.
The stress–strain curve for a cellular solid in compression is characterized by three regimes (Fig. 4a): a linear elastic regime, corresponding to cell edge bending or face stretching; a stress plateau, corresponding to progressive cell collapse by elastic buckling, plastic yielding or brittle crushing, depending on the nature of the solid from which the material is made; and densification, corresponding to collapse of the cells throughout the material and subsequent loading of the cell edges and faces against one another. Many cellular materials have low relative densities (∼10–20%) so that they can be deformed up to large strains (∼70–80%) before densification occurs. In tension, the linear elastic response is the same as in compression, at least at small strains (Fig. 4b). As the strain increases, the cells become more oriented with the loading direction, increasing the stiffness of the material until tensile failure occurs.
The mechanical response of cellular solids has been modeled by representing the cellular structure in several ways. Initial models performed a structural analysis of a unit cell such as a hexagon (in two dimensions) or a dodecahedron (a 12-faced polyhedron) or tetrakaidecahedron (a 14-faced polyhedron) in three dimensions (Fig. 5) (Ko, 1965; Patel and Finnie, 1970; Gibson and Ashby, 1982a; Warren and Kraynik, 1997; Zhu et al., 1997a, Zhu et al., 1997b). The geometry of the unit cell makes the analysis tractable but may not give an exact representation of the real material (e.g. foams). An even simpler approach is to use dimensional analysis to model the mechanisms of deformation and failure observed in the cellular material without specifying the exact cell geometry (Gibson and Ashby, 1982b, Gibson and Ashby, 1997). This approach assumes that the cell geometry is similar in foams of different relative densities. It gives the dependence of the properties on the relative density and the solid properties, but requires experiments to determine the constants related to the cell geometry. A third approach is to use finite element analysis of either regular or random cellular structures (Silva et al., 1995, van der Burg et al., 1997). Finite element analysis allows local effects, such as imperfections, to be studied (Silva and Gibson, 1997a; Chen et al., 2001). It can also be used in conjunction with imaging techniques such as micro-computed tomography to model the exact geometry of a particular sample (e.g. trabecular bone), although this is computationally intensive (van Reitbergen, 1995).
In this review, we first summarize the results of micromechanical models developed using dimensional analysis and compare them to selected results from the analysis of unit cells and finite element modeling. We then show how the models can be applied to honeycomb- and foam-like natural cellular solids (wood and trabecular bone). The role that cellular materials play in increasing the efficiency of natural sandwich structures (e.g. iris leaves) and tubular structures (e.g. plant stems, animal quills) is also described. Cellular materials are increasingly used as biomaterials to replace or regenerate tissue in the body. Here we describe titanium foams that are being considered for bone replacement and porous scaffolds for the regeneration of a wide variety of tissues, including skin, nerve, liver, bone and cartilage.
Section snippets
Two-dimensional cellular solids (honeycombs), in-plane loading
When loaded uniaxially in the plane of the hexagonal cells, the cell walls of a honeycomb initially deform by bending (so long as the wall thickness, t, is small compared with the wall length, l) (Fig. 6a,b). The Young's modulus, E* can be related to the relative density, , the modulus of the solid, Es, and the cell geometry (h/l, ) using structural mechanics. A stress acting in the x1 direction induces a load P on the end of the inclined cell wall:where b is the wall
Wood: a two-dimensional cellular solid
The structure of two species of wood, cedar and oak, are shown in Fig. 8. In both softwoods and hardwoods, the bulk of the cells are long prismatic cells (tracheids in softwoods and fibres in hardwoods). The annual growth rings are distinguished by alternating bands of thin- and thick-walled cells, corresponding to spring and summer, respectively. The cells making up the rays are squatter and more box-like. Hardwoods also have vessels, relatively large diameter channels through which fluids are
Trabecular bone: a three-dimensional cellular solid
Trabecular bone exists at the ends of the long bones, within the vertebral body, and in the core of shell-like bones such as the skull. It has a cellular structure, with a relative density typically between about 0.05 and 0.3. Bone grows in response to load, so that the density of trabecular bone depends on the magnitude of the loads and the orientation of the trabeculae depends on the direction of the loading (Fig. 12). Low-density trabecular bone resembles an open-cell foam. High-density
Natural sandwich structures: plant leaves, skull
Engineers sometimes make use of cellular solids as the cores of sandwich structures: components with two stiff, strong outer faces separated by a lightweight honeycomb or foam core. The separation of the faces by the core increases the moment of inertia of the section with little increase in weight, making sandwich structures attractive for resisting bending and buckling. Examples of engineering sandwich panels include aircraft flooring panels and downhill skis. Sandwich structures also occur
Natural tubular structures: plant stems, animal quills
Tubular structures in nature often have a dense outer cylindrical shell with an inner honeycomb- or foam-like core. Examples include plant stems, animal quills and bird feather quills (known as the rachis) (Fig. 17). In plant stems, this is known as the “core-rind” structure (Niklas, 1992). Quills from porcupines, hedgehogs, echidnas and spiny rats all have a similar structure (Vincent and Owers, 1986; Karam and Gibson, 1994). In nature, these structures are loaded primarily in bending and/or
Cellular solids as biomaterials
Cellular solids are increasingly used as biomaterials to replace or regenerate damaged or diseased tissue. Here we consider two examples: open-cell titanium foams for bone implants and porous scaffolds for tissue engineering, for the regeneration of a wide range of tissues (e.g. skin, nerve, bone, cartilage).
Recently, there have been a number of novel processes developed for making metallic foams (Ashby et al., 2000). Open-cell foams are of the most interest as biomaterials, as the
Summary
The cellular structure of wood resembles that of an engineering honeycomb. The stiffness and strength of different species of woods depend on the density of the species and the direction of loading: woods are much stiffer and stronger when loaded along the grain than across it. Models based on dimensional analysis of the mechanisms of deformation in honeycomb models (uniaxial compression of the cell wall for loading along the grain and bending of the cell wall for loading across the grain) give
Acknowledgements
I am grateful for a number of stimulating and productive collaborations with colleagues over many years: Prof. MF Ashby of Cambridge University Engineering Department (cellular solids, including wood); Prof. TA McMahon (deceased), formerly of Harvard University Division of Applied Sciences (trabecular bone); Prof. WC Hayes, formerly of the Orthopaedic Biomechanics Laboratory, Beth Israel Deaconess Medical Center, Harvard Medical School (trabecular bone); and Prof. IV Yannas of the Department of
References (86)
- et al.
Variations in strength and structure of cancellous bone at the knee
Journal of Biomechanics
(1974) - et al.
Pore strain behavior of collagen-glycosaminoglycan analogues of extracellular matrix
Biomaterials
(1995) - et al.
Effect of inclusions and holes on the stiffness and strength of honeycombs
International Journal of Mechanical Sciences
(2001) The relationship between the elasticity tensor and the fabric tensor
Mechanics and Materials
(1985)- et al.
Micromechanics of fibroblast contraction of a collagen-GAG matrix
Experimental Cell Research
(2001) The mechanical behavior of cancellous bone
Journal of Biomechanics
(1985)- et al.
Mechanical consequence of trabecular bone loss and its treatmentA three-dimensional model simulation
Bone
(2002) - et al.
X-ray quantitative computed tomographythe relations to physical properties of proximal tibial trabecular bone specimens
Journal of Biomechanics
(1989) - et al.
A model of vertebral trabecular bone architecture and its mechanical properties
Bone
(1990) - et al.
Biomimicking of animal quills and plant stemsnatural cylindrical shells with foam cores
Materials Science and Engineering
(1994)
Elastic buckling of cylindrical shells with elastic cores—I Analysis
International Journal of Solids and Structures
Elastic buckling of cylindrical shells with elastic cores—II Experiments
International Journal of Solids and Structures
Differences between the tensile and compressive strengths of bovine tibial trabecular bone depend on modulus
Journal of Biomechanics
Yield strain behavior of trabecular bone
Journal of Biomechanics
Mechanical properties of trabecular bonedependency on strain rate
Journal of Biomechanics
Preparation and characterization of poly (L-lactic acid) foams
Polymer
Dependence of yield strain of human trabecular bone on anatomic site
Journal of Biomechanics
Time-lapsed microstructural imaging of bone failure behavior
Journal of Biomechanics
An expression relating breaking stress and density of trabecular bone
Journal of Biomechanics
The effects of non-periodic microstructure and defects on the compressive strength of two-dimensional cellular solids
International Journal of Mechanical Sciences
Modeling the mechanical behavior of vertebral trabecular boneeffects of age-related changes in microstructure
Bone
The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids
International Journal of Mechanical Sciences
The fabric dependence of the orthotropic elastic constants of cancellous bone
Journal of Biomechanics
A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models
Journal of Biomechanics
Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells
Journal of the Mechanics and Physics of Solids
Analysis and Design of Structural Sandwich Panels
Materials Selection in Mechanical Design
The mechanical properties of natural materials. I Material property charts
Proceedings of the Royal Society of London
Metal FoamsA Design Guide
Prediction of elastic parameters of wood
Wood Science
Mechanics of Wood and Wood Composites
Buckling of core-stabilized cylinders under axisymmetric external loads
Journal of Aerospace Science
The compressive behavior of bone as a two-phase porous structure
Journal of Bone and Joint Surgery
Acta Orthopaedica Scandinavica
The anisotropic elasticity of the plant cell wall
Wood Science and Technology
The longitudinal Young's modulus of Pinus radiata
Wood Science and Technology
Isotropy of uniaxial yield strains for bovine trabecular bone
Journal of Orthopaedic Research
TimberIts Nature and Behaviour
On the mechanics of balsa and other woods
Proceedings of the Royal Society of London
Physical properties of trabecular bone
Calcified Tissue Research
Cited by (808)
Compressive response and optimization design of a novel hierarchical re-entrant origami honeycomb metastructure
2024, Engineering StructuresEquivalent-oriented model for sandwich panels with ZPR accordion honeycomb
2024, International Journal of Mechanical SciencesMachine Learning approaches for the design of biomechanically compatible bone tissue engineering scaffolds
2024, Computer Methods in Applied Mechanics and EngineeringGaussian random field-based characterization and reconstruction of cancellous bone microstructure considering the constraint of correlation structure
2024, Journal of the Mechanical Behavior of Biomedical Materials