The influence of architecture on the deformation localisation of cellular materials

Transcription

1 The influence of architecture on the deformation localisation of cellular materials Dunlop John 1, Weinkamer Richard 1, Fratzl Peter 1, and Bréchet Yves 2 (1) Max Planck Institute of Colloids and Interfaces, Department of Biomaterials Potsdam, Germany (2) GPM2, Domaine Universitaire de Grenoble, Saint Martin d Heres Cedex, France john.dunlop@mpikg.mpg.de Abtract : In general cellular materials fail by localisation of deformation. For random cellular materials, the localisation plane due to compressive deformation is found to be perpendicular to the applied stress. For periodic cellular materials, however the localisation occurs in particular directions that are related to the underlying crystallography of the cellular architecture. In this contribution we treat the deformation localisation like a martensitic instability, that is a deformation corresponding to a soft elastic deformation mode. The model of Khachaturyan is applied to different cellular architectures both in two and three dimensions. Résumé : En générale les materiaux cellulaires rompent par localisation de la déformation. Pour les matériaux poreux non-ordonnés, le plan de localisation se trouve perpendiculaire a la contrainte, par contre pour les matériaux périodiques la localisation suit des directions bien particulières liées à la «cristallographie» de l architecture. Dans cette contribution nous analysons la relation entre l architecture et l orientation spatiale des bandes de localisation/instabilité dans un materiau cellulaire périodique. Nous appliquons les méthodes développées par Khachaturyan et nous traitons la localisation de la déformation comme un instabilité de type martensitique, c'est-à-dire correspondant à un mode élastique «mou». Le modèle de Khachaturyan est appliqué sur les structures cellulaires pour les architectures différentes en deux et trois dimensions. Key-words : Cellular Materials; Deformation Localisation; Lattice Dynamics 1 Introduction The interest in cellular materials has developed strongly in the last decades principally due to the possibility of optimising material properties through control of the cellular architecture, that is the size, shape and spatial organisation of the pores. Many different optimisation problems can be posed, including multi-criterion optimisation, some examples being; the optimisation of mechanical strength for minimal mass, maximising vibration damping for minimal mass, maximising fluid flow for catalytic or heat transfer applications or in optimising impact energy absorption (Bendsoe and Triantafyllidis 1990; Evans et al. 2001; Neves et al. 2002; Torquato et al In order to optimise cellular materials design it is necessary to have a good model to describe materials properties as a function of the underlying architecture. These models can also be used to predict the effects of architectural modification due to disease (e.g. osteoporosis) of trabecular bone. The modelling of mechanical properties has focused mainly on the elastic properties and failure stress or strength of cellular materials through the analysis of the behaviour of an elementary cell with periodically boundary conditions (Gibson and Ashby 1997 These approaches use the hypothesis of a macroscopically homogeneous deformation. Other approaches have used finite element methods in order to analyse the post-yield behaviour and the effect of imperfections (e.g. Chen et al In general, cellular materials fail through deformation localisation. For random cellular materials such as metal or polymeric foams, the deformation localisation occurs in a direction that is perpendicular to the applied compressive load (Tu et al. 2001; Dillard et al

2 Natural cellular materials, such as trabecular bone, also localise under compressive deformation however the direct link to architecture or fabric is still not clear (Nazarian et al For periodic cellular solids however the deformation localisation occurs along well defined crystallographic directions that depend on the microstructure (e.g. Côté et al This is a very similar to the deformation of single crystals, where deformation is localised along particular slip-planes. In metal physics the lattice dynamics approach developed by Born and Huang has been used to investigate the orientation dependence of the critical ideal shear stress of single crystals (Born and Huang 1954; Cook and De Fontaine 1969; Khachaturyan 1983; Wang et al. 1993; Fratzl and Penrose 1995 Born developed a set of three criteria for the elastic stability of a crystal (Born and Huang 1954) which for a cubic lattice are : C + C 0, C 0, C C 0, (1) > 44 > > where C ij are the elastic constants. These three criterion correspond respectively to; the existence of a Bulk modulus K = ( C C 12 )/ 3, the existence of a shear modulus C 44 and the existence of a shear modulus against tetragonal shear G' = ( C 11 C12 )/ 2 (Wang et al The third criterion is generally reached first, suggesting that as a first approximation we can just use just the difference between the Young s and shear modulus as a means of determining the onset of instability or shear localisation. In crystalline or polycrystalline materials such an instability is normally not reached under mechanical loading, as typically dislocation slip is activated, but these criteria have been applied successfully to the onset of melting in metals (Wang et al In some metals however such as in zinc or cadmium, with a large anisotropy in elastic constants a shear instability can be initiated. Porous materials with an underlying architecture can have elastic moduli which depend strongly on direction, meaning that shear instabilities can be initiated for particular architectures and cell geometries. What is required is a knowledge of the elastic stiffness matrix and how this varies for different deformation modes, macroscopic or global modes or localised perturbations. In this contribution we apply the mathematical tool developed for crystal lattices to the macroscopic crystal of a periodic cellular solid. We treat the deformation localisation much like a martensitic instability, that is a deformation corresponding to a soft elastic deformation mode (Khachaturyan 1983 A similar approach has been used by Triantafyllidis and coworkers, where they use a bifurcation analysis to determine the critical loading conditions for which a periodic cellular lattice loses stability (Triantafyllidis and Bardenhagen 1996; Triantafyllidis et al The model of Khachaturyan is applied to different cellular architectures both in two and three dimensions. The next section outlines the model development and is followed by initial results on simple structures in two and three dimensions. 2 Model x 2 u 3 (r) x 1 α r u 1 (r) u 2 (r ) u 1 (r ) r u 3 (r ) r u 2 (r) r Un-deformed Local Displacements Local Rotations Figure 1: Notation used after (Kumar and McDowell

3 The general approach of Khachaturyan involves calculating the energy of the lattice as a function of the pair-potential between atoms, and the underlying crystallography (Khachaturyan 1983; Fratzl and Penrose 1996 The pair potential gives the energy of a given pair of atoms as a function of the different degrees of freedom (translation and rotation) of each atom type. The calculation of the lattice energy is greatly simplified by the crystallography, as the underlying periodicity means it is only necessary to calculate the energy of each atom pair within the unit cell. The lattice energy is the sum of all the energy of all atom pairs in the unit cell as a function of deformation and rotation of each lattice point. This energy can then be minimised as a function of the lattice deformation modes in order to determine in which direction the lattice will deform or to look at which direction phase transformations will be preferred. In the following we describe our beam lattice as consisting of joints (atoms) connected by beams which have an associated pair potential related to the deformation energy of the beam. The energy of a single beam W as a function of displacements and rotations is given by the following equation: ( r, ) = u( r, K( r, u( r, r ) W, (2) r are the coordinates of the two ends of the beam, and ( ) where r and ' K r, is the stiffness matrix. u ( r, ) is the deformation/rotation vector of both beam ends, and in two dimensions is given by: u ( r ) ( u ( r) u ( r) u ( r) u ( ) u ( ) u ( ')), r = (3) u i for i = 1, 2 corresponds to the displacements of the beam ends and for i = 3 corresponds to the in-plane rotations of the beam ends (Figure 1 In three dimensions the deformation vector will have twelve components including the 6 degrees of freedom at each beam end. The stiffness matrices K ( r, ) of a beam in 2 and 3 dimensions can be found in standard mechanics textbooks (e.g. Bazant and Cedolin 2003 In general the total lattice energy can be written as: W Lattice u can be written as the sum of deformationwaves: u (, ) u( r, K ( r, u( r, ) = W r = (4) r, r, The heterogeneous deformation field ( r) () r = u( k) k 0 ~ T ikr e, (5) where the sum is over all k except k = 0 (which corresponds to the homogeneous displacement The vector u ~ ( k) is simply the Fourier transform of the deformation vector (including rotations) in real space. Equation (5) can then be substituted into equation (4) and then rearranged: Lattice T W = u ~ Lattice ~ (6) K Lattice k ( k) ( k). K ( k). u( k) is the reciprocal stiffness matrix corresponding to the entire lattice, and will be a 3 n by 3 n matrix in 2 dimensions (or 6 n by 6 n in 3 dimensions), where n is the number of joints (atoms) in the unit cell. Triantafyllidis and Schnaidt (1993) used a stiffness matrix in which the beam energy is written as a function of the applied compressive load in the beam. 3

4 They then calculated the critical load at which the minimum eigenvalue of the stiffness matrix K Lattice ( k) tends to zero as a function of k. All the information concerning the energy of an arbitrary deformation mode is contained within the reciprocal lattice stiffness matrix. The k - dependence of the minimum eigenvalue gives information about what is the minimum energy deformation mode, and thus the potential direction dependence of localisation for arbitrary loading. Assuming shear localisation occurs in a plane, then this corresponds to looking at the eigenvalues along lines from the origin in reciprocal or k -space. The eigenvalue surface close to k = 0 gives information about long-wave deformation modes, and in the limit as k 0, the long wave limit, the slope of the eigenvalue surface gives the bulk elastic response of the lattice. A similar approach was used by Kumar and McDowell (2004) to calculate the elastic matrix for two dimensional periodic lattices. In the following we analyse minimum eigenvalue surfaces for different lattices, square, triangular, a bcc-square lattice in 2 dimensions and a cubic lattice in 3 dimensions. 3 Results for simple lattices in two and three dimensions Figure 2a illustrates the eigenvalue surfaces for a square lattice in two dimensions. In two dimensions three degrees of freedom are allowed, rotation of a beam end within the plane, and displacement in the x and y directions, meaning that there are three eigenvalues of the reciprocal stiffness matrix. We are interested in the minimum surface which will be associated with minimal energy deformation modes. Figures 2b and c shows the minimal eigenvalue surface, which appears to show a peak around the origin, however a zoom around the origin shows a finer structure consisting of valleys oriented along the 10 directions. This can also be more clearly seen in Figure 2d which shows the minimal eigenvalue as a function of the angle θ from 01 direction for two different radii. The eigenvalues of the reciprocal stiffness matrix not only give the minimum energies but also give information about the associated deformation mode; whether the mode is rotational, displacive in x or y or some combination of the three deformation modes. For example in the square lattice the lowest energy eigenmodes close to the origin correspond to pure shear modes along the 01 directions, modes further away from the origin are mixed displacement-rotational. Figure 2a: Surface of the three eigenvalues of the reciprocal stiffness matrix of a square lattice as a function of wave vector k = ( k 1, k 2 ) Figure 2b: Surface of the smallest eigenvalue of a square lattice as a function of wave vector k = ( k 1, k 2 ) For triangular lattices the minimal surface can also be calculated, as illustrated in Figure 3 The surface is peaked with a localised minimum around the origin with a circular symmetry which corresponds to the well known isotropy of the triangular lattice. Lattices containing more than one joint in the unit cell, (or a crystal consisting of interpenetrating Bravais lattices) can also be treated, however care must be taken to assemble the reciprocal stiffness matrix. As an example a 2D bcc lattice has been modelled, that is a 2D square lattice with an extra joint 4

5 located in the centre of the unit cell giving diagonal bracing. The minimal eigenvalue surface of this system is illustrated in Figures 4a and 4b. What is interesting is that although there is local minimum at the origin (Figure 4b) there are global minima in the 01 directions (Figure 4a), suggesting that localised or short-wave deformation modes may be activated in the system. Figure 2c: Zoom of the fine structure of the minimal eigenvalue surface around the origin, showing the low energy valleys in the 10 directions Figure 2d: Radial distribution of minimum eigenvalues, normalised by the Young S modulus and length around k = 0 from Figure 2a. Dotted line is for a radius of 10 6 wave vectors from the origin, and filled line is _ wave vectors to the origin Figure 3: The fine structure of the minimal eigenvalue surface of a triangular lattice, showing the circularly symmetric low eigenvalue valley around the origin Figure 4a: Surface of the minimal eigenvalue of a bcc-lattice (two joints per unit cell) as a function of wave vector k Figure 4b: Zoom of the centre of the eigenvalue surface, showing the low energy minimum valley close to the origin Figure 5: Iso-contour surface of the minimum eigenvalue of a cubic lattice as a function of the wave vector k = ( k1, k2, k3 ). Note the direction dependence of the surface in the 100 directions. In the last example the procedure developed above is applied to a three dimensional cubic beam lattice. Figure 5 illustrates an iso-contour surface of the minimum eigenvalues of the reciprocal stiffness matrix for a cubic lattice. The surface is highly oriented having wings 5

6 extending in the 100 directions, corresponding to the directions of localisation in a cubic lattice. 4 Conclusions This paper has briefly outlined the development of a model to investigate the localisation behaviour of periodic cellular materials. The underlying periodicity allows the direct transfer of techniques developed in solid state physics to describe macroscopic lattice properties. The lattice dynamics approach allows the calculation of eigenvalue surfaces or iso-surfaces which is in turn linked to the minimal energy of the system. This has been applied to several different two and three dimensional lattices and can in principal be applied to arbitrary periodic cellular solids, although the calculations will be limited by the number of degrees of freedom. The full analysis of such surfaces will need to be compared with experiments on real porous solids and future work will be done to do mechanical testing of porous solids with different architectures produced using rapid prototyping, as well as other computational experiments such as finite element analysis. References Bazant, Z. P. and L. Cedolin (2003 Stability of Structures. Mineola, Dover. Bendsoe, M. P. and N. Triantafyllidis (1990 International Journal of Solids and Structures 26(7): Born, M. and K. Huang (1954 Dynamical Theory of Crystal Lattices. Oxford, Oxford University Press. Chen, C., T. J. Lu, et al. (2001 International Journal of Mechanical Sciences 43: Cook, H. E. and D. De Fontaine (1969 Acta Metallurgica 17: Côté, F., V. Deshpande, et al. (2006 International Journal of Solids and Structures 43: Dillard, T., F. N'Guyen, et al. (2005 Philosophical Magazine 85(19): Evans, A. G., J. W. Hutchinson, et al. (2001 Progress in Materials Science 46(3-4): Fratzl, P. and O. Penrose (1995 Acta Metallurgica et Materialia 43(8): Fratzl, P. and O. Penrose (1996 Acta Materialia 44: Gibson, L. J. and M. Ashby (1997 Cellular Solids: Structure and Properties. Cambridge, Cambridge University Press. Khachaturyan, A. G. (1983 Theory of Structural Transformations in Solids. New York, John Wiley and Sons. Kumar, R. S. and D. L. McDowell (2004 International Journal of Solids and Structures 41: Nazarian, A., M. Stauber, et al. (2006 Bone 39: Neves, M. M., O. Sigmund, et al. (2002 International Journal for Numerical Methods in Engineering 54: Torquato, S., S. Hyun, et al. (2003 Journal of Applied Physics 94(9): Triantafyllidis, N. and S. Bardenhagen (1996 Journal Of The Mechanics And Physics Of Solids 44(11): Triantafyllidis, N., M. D. Nestorovic, et al. (2006 Journal of Applied Mechanics 73: Triantafyllidis, N. and W. C. Schnaidt (1993 Journal of the Mechanics and Physics of Solids 41(9): Tu, Z. H., V. P. W. Shim, et al. (2001 International Journal of Solids and Structures 38: Wang, J., S. Yip, et al. (1993 Physical Review Letters 71(25):