Bio-inspired Material Design and Optimization

6th World Congresses of Structural and Multidisciplinary Optimization
Rio de Janeiro, 30 May - 03 June 2005, Brazil
Bio-inspired Material Design and Optimization
Xu Guo and Huajian Gao*
Max-Planck Institute for Metals Research
Heisenbergstr.3 D-70569 Stuttgart, Germany
Abstract Natural materials such as bone, tooth, and nacre are nano-composites of proteins and minerals with superior stiffness and toughness. At the most elementary structure level, bio-composites exhibit a generic microstructure consisting of staggered mineral bricks wrapped by soft protein in nanoscale. Why does nature design building blocks of biological materials in this form? Can we reproduce this kind of structure from the structural optimization point of view? We believe that biological materials are designed with simultaneous optimization of stiffness and toughness for maximum structural support and flaw tolerance. With this philosophy, an optimization problem is formulated under the assumption of appropriate material constitutive models and failure criteria. It is shown that, within this optimization framework, the staggered microstructure of biological materials can be successfully reproduced at the nanometer length scale. This study may have at least partially provided an answer to the question whether the nanostructure of biological materials is an optimized structure and what is being optimized. The results suggest that we can draw lessons from the nature in designing nanoscale and hierarchically structured materials.
Keywords: Bio-inspired mimetic; Material design; Optimization; Flaw tolerance
1.Introduction
Natural materials such as bone, tooth, and nacre are nano-composites of proteins and minerals with superior stiffness and toughness (Jackson et al., 1988). Experimental results show that the while the stiffness of bio-composites is close to that of its mineral constituent, its fracture strength and toughness are significantly higher than those of the mineral. This outstanding performance of bio-composites comes from their highly complex hierarchical structures at different length scales (Weiner and Wanger, 1998). For instance, sea shells have 2 to 3 orders of lamellar structure (Currey, 1997; Menig et al., 1998; Weiner and Wanger, 1998) and bone has 7 orders of hierarchy (Currey, 1984; Rho et al., 1998; Weiner and Wanger, 1998). Fig.1 shows the nanostructure of some typical bio-composites.
While the full hierarchical structure of bio-materials such as bone is extremely complex and variable, it is most interesting to observe that its basic building blocks, the mineralized collagen fibril are rather universal. They are generally designed at the nanoscale with nanometer sized hard mineral inclusions embedded in the soft protein matrix (Gao et al., 2003). For instance, at the lowest level of hierarchy, the nanostructure of bone (mineralized fibrils) is consisting of mineral platelets with thickness around a few nanometers aligned in a stagger pattern in a collagen matrix (Landis, 1995; Rho et.al., 1998; Weiner and Wanger, 1998). Dentin is a calcified tissue somewhat similar to bone, where the collagen-rich matrix is reinforced by calcium phosphate crystals (Weiner et al., 1999; Tesch et al., 2001; Weiner and Wanger, 1998). Similarly, the cell walls of wood are made of cellulose fibrils embedded in a soft semicellulose-lignin matrix.
Fig.1a Fig.1b Fig.1c
Fig.1 The nanostructure of some typical hard biological tissues. (a) tooth, (b) bone and (c) shell Because mineralized fibrils are the elementary unit of many complex bio-composites, it is important to understand how their mechanical properties depend on the properties of their constituents and the style of arrangement of different materials at the level of individual fibrils. The components of mineral fibril have extremely different mechanical properties. The mineral is stiff and brittle while the (wet) protein is much softer but also much tougher than the mineral. Increasing the amount of mineral particles will always increase the stiffness but also the brittleness of the bone tissue at the same time. How to make the bio-composite hard enough with high toughness? Nature solves this problem in an elegant way by the smart design of the size, shape and material distribution of the nanostructures of bio-composites.
Previous studies show that (Gao and Ji 2003; Ji and Gao 2004) large aspect ratios and the pattern of a staggered alignment of mineral platelets are the key factors contributing to the large stiffness of biomaterials. On the other hand, proteins between staggered mineral platelets play the essential role of absorbing and dissipating a large amount of fracture energy. Furthermore, as shown by Gao and Ji (2003), at the nanoscale, brittle mineral becomes insensitive to flaws which makes it possible for it to sustain large stress without brittle fracture and in turn enhance the toughness of biomaterials. In summary, it is apparent that both the organic and mineral components as well as their arrangement contribute equally to the strength of biomaterials. The bio-composite combines the optimal properties of its both components, the stiffness and the toughness. This rather unusual combination of material properties provides both rigidity and resistance against fracture. From the viewpoint of materials science, a better understanding of the underlying
*
Corresponding author. E-mail: hjgao@mf.mpg.de
construction principles might help designing better composite materials.
Nature materials have been perfected by the evolution through millions of years. The essential idea of bio-inspired material design is to see how, in some cases at least, the forms of these “well designed” nature materials can be explained by physical and mathematical laws. Once we can abstract the principle and mechanism of good design from nature, we can therefore use them to make advanced synthetic materials. From example, if we know some of the methods that natural systems have evolved to increase toughness, we can realize that holes, carefully used, can improve strength while at the same time actually making the structure lighter.
It is generally believed that the complex hierarchical structure is optimized to achieve a remarkable mechanical performance (Weiner and Wanger; 1998; Aksay and Weiner, 2004; Fratzl, 2004). Among the key features of the biological systems, the organization of bio-structures at nano-scale is the prominent one. The amazing rationality of these biological constructions naturally excites the interest to analyze them by using mathematical tools developed in the theory of mechanics and structural optimization. Although the “result” of the “optimization problem” is known, until now, however it is still unclear what performance index is optimized when nature designs the building blocks of bio-composites, i.e. in what sense the structure is optimal.
In order to obtain more insights for bio-inspired material design from nanoscale and up, in the present study, an optimization model is proposed to explain why nature designs building blocks of biological materials in the present form. We believe that biological materials are designed with simultaneous optimization of stiffness and toughness for maximum structural support and flaw tolerance. With this philosophy, an optimization problem is formulated under the assumption of appropriate material constitutive models and failure criteria. It is shown that, within this optimization framework, the staggered microstructure of biological materials can be successfully reproduced at the nanometer length scale. This study may have at least partially provided an answer to the question whether the nanostructure of biological materials is an optimized structure and what is being optimized. The results suggest that we can draw lessons from the nature in designing hierarchically structured materials.
This paper is organized as follows. In section 2, the mechanical properties of mineral and protein as well as their basic failure mechanisms will be discussed. Then in section 3, an optimization model used for the design of nanostructure of bio-composites with appropriate objective and constraint functions is proposed. Optimization results obtained with the proposed optimization model will be given in section 4. It is shown that the staggered pattern of the material distribution in the nanostructure of bio-composites can be successfully reproduced via the present optimization model. Finally, we end the paper with some concluding remarks.
2. Mechanical properties and failure mechanisms of the basic components of bio-composite at nanoscale
2.1 Mechanical property of mineral and its flaw tolerance behaviour at nanoscale
The present study focuses on the basic building block, the collagen-mineral composite, containing nano-sized mineral platelets (essentially carbonated hydroxyapatite), protein and water. These components have extremely different mechanical properties. The mineral is stiff and brittle, which can be considered as linear elastic material during the process of deformation in physiological conditions . It has Young’s modulus as high as about 100GPa with failure strain about 1%-2%. The stiffness of the biomaterials, which provides skeletal rigidity, is mainly provided by the mineral crystals.
Although the theoretic strength of mineral material in bio-composites is about 1GPa, it is, however, as fragile as a classroom chalk. It is very sensitive to the flaw in it at macroscopic length scale. The cracks in the mineral can always propagate at stresses much lower than its theoretical strength, which often leads to the catastrophe failure of structures. The high toughness of biomaterials, however, requires that the mineral constituent should sustain large stresses without brittle fracture. How does nature solve this problem?
This problem has been studied by Gao et al (2003). They pointed out that there is a critical length scale of mineral material below which the mineral is insensitive to existing flaws, which can be expressed as:
22th
M
cri E h σγα≈ (1)
where γ is the surface energy and M
E is the Young’s modulus of mineral. α is a crack geometry dependent parameter and th σ is the theoretical strength of mineral.
As the thickness of mineral crystal drops below this length scale, the strength of a perfect mineral platelet is maintained in spite of defects. On this circumstance, the failure criterion is governed by theoretical strength rather than by the Griffith criterion . For typical values of γ,M E and th σ, it can be estimated that cri h is in the nanometer scale. This explains why the basic building
blocks of bio-composites are always taken the nanometer sizes.
2.2 Mechanical property of protein and its failure mechanism
Protein plays very important role in biomaterials. Although protein may have less effect on biomaterials’ stiffness than does mineral, however, it has a profound effect on biomaterial’s toughness. While the toughness of bone results from its complex hierarchical microstructure with several contributing toughening mechanisms, at the level of mineralized collagen fibrils, the soft organic matrix between the hard but brittle mineral plays a crucial role because it is a primary arrestor of cracks and source of energy dissipation. Previous studies (Gao and Ji, 2003; Ji and Gao, 2004) show that the tensile stress on the bio-composite is transmitted through the organic matrix mainly by shear, therefore the shear stiffness, failure strength and post yielding deformation behaviour are the controlling factors for the toughness of the bio-composites. Furthermore, it has been shown that the geometric arrangement of the inorganic and organic materials in the nanostructure of bio-composites also plays an essential role for the toughening effect. This naturally leads to the question that whether the staggered pattern in which the mineral and the organic material are organized in bio-composites is good enough from optimization point of view? This issue will be addressed in the following sections.
Since the mechanical behaviour of protein is vital for the toughness of bio-composites, here we will discuss it in more details. Smith et.al (1999) used atomic force spectroscopy to test the axial force-extension behavior of organic matrix exposed on a fractured nacre surface. The axial force-extension behavior exhibited an irregular “saw-tooth” character, so named because of the repeating pattern of a nonlinear force increase with extension followed by abrupt load drops. This phenomenon is attributed to the unfolding of complicated domains along the bio-macromolecular chain and overcoming sacrificial bonds induced by Ca +2 ions in organic matrix sequentially. This behavior is speculated to play a significant role in the mechanical behavior of biomaterials. The protein molecules can undergo large deformation as the protein domains unfold, which in turn increases the amount of energy dissipated before fracture and results in a very long flat tail of the force-extension curve as shown in Fig.2. In this work, the mechanical behavior of protein is modeled as )1()(e f P f e e γεσεσ−−==, where 5=γ and 20=P f
σMPa. e σ and e ε are equivalent stress and equivalent strain, respectively.
Fig.2 A schematic illustration of the force-extension curve of protein
3. Optimization problem
The present work aims at understanding the rationality of the existing brick and mortar form of the building blocks of bio-composites from optimization point of view. Mathematically speaking, bio-mimicking or bio-inspired material design can be formulating as an inverse optimization problem, i.e. finding a goal functional of an optimization problem with the inspiration that obtained from the analysis of existing forms of biomaterials. If the solution of this optimization problem can be in reasonable agreement with the reality, then the design principles embedded in the proposed optimization problem can be used to design high-performance man-made materials. This is just the essential idea of bio-mimicking research.
Here we have assumed implicitly that the building block of bio-composites is the “optimization result” of natural evolution, which is consistent with Neo-Darwin’s theory of natural selection. As for an optimization problem, the design variables, the objective as well as constraint functions (or functionals) are its three basic ingredients. In the following, these issues will be addressed respectively.
3.1 Design variables
In this work, we will formulate the corresponding optimization problem in the framework of optimal topology design, i.e. finding the optimal distribution of hard and brittle inorganic as well as soft and ductile organic materials in a basic unit-cell (RVE ) of the basic building blocks of bio-composites. To this end, an indicator function )(x ρ defined on the unit cell Y is introduced, which can only take the values of 0 and 1. If 0)(=x ρ, it is indicated that the current point x is occupied by hard mineral material, otherwise it is occupied by soft protein like material, see Fig.3 for reference.
In order to use gradient based numerical algorithms for the solution of the proposed discrete valued design optimization problems, a conventional approach is to replace the integer variables with continuous variables and then introduce some form of penalty that steers the solution to discrete 0-1 values. This approach is also adopted in the present paper.
Fig.3 Illustration of the design variable and design domain
u
unit cell 0u u
1
=
Another design variable is the displacement control factor u α, which controls the process of deformation. Here we assume that the unit cell is loaded by prescribed displacements on the boundary of the unite cell, which takes the form of 0u u u u α== on u Y ∂.
3.2 Objective function
The determination of a suitable objective functional is a crucial aspect for the solution of an inverse optimization problem. As for the present bio-mimicking problem, it should reflect the essential idea (if there is) that nature used to “design” the biomaterials. In the present work, we postulate that biological materials are designed with simultaneous optimization of stiffness and toughness for maximum structural support and flaw tolerance. With this philosophy, the following objective functional is proposed:
))((0
0ΓΓ=E E f eff (2)
We use f as a measure of the ductility of the material. Here only one dimensional loading case (uni-axial tensile loading along principle deformation direction) is considered and eff E is the effective stiffness along the loading direction. Γ is the energy dissipated during the process of deformation before any material point in the unit cell reaches its critical failure status. 0E and 0Γ are two parameters used for the dimensionless of the objective function.
3.3 Constraint functions
Constraint function is another very important ingredient for an optimization problem, since it has vital influence on the final optimal solution. In a bio-mimicking problem, just like that of objective functional, the constraint functions should also reflect the basic challenges nature faced when designing biomaterials. From mechanics point of view, these constraint functions are closely related to the basic failure mechanisms of different materials as shown by Gao and Ji (2003). In the proposed optimization model, it requires that during the process of deformation, the effective strain of any material point in the unit cell should not exceed its critical value. According to the discussions in section 2, in this work, the effective failure strain is set to be 1% for hard and brittle mineral material while 150% for soft and ductile organic material. This is consistent with the typical deformation and failure mechanisms of these materials as discussed thoroughly in the literatures.
It is worth noting that in the present work, we tacitly assumed that the interfaces between different materials are strong enough. This can be justified by the fact that special polymers such as proteoglycans at the interface of organic and inorganic materials in bio-composites are capable of making the binding between them tight enough (Fratzl et al, 2004). Another problem is that 1% is the theoretical failure strain of mineral material, but if we take the unavoidable defects into consideration, is it still reasonable to use this theoretical strength value as the control parameter in the optimization problem?
This problem can be solved by restricting the size of the unit cell below the critical size of mineral derived from the argument of flaw tolerance. As pointed in section 2, when the mineral size drops below this critical length scale, the theoretical strength of mineral platelet can be maintained in spite of defects. From optimization point of view, this dimension restriction of the unit cell can also be seen as a result of fracture strength optimization, which is an essential part of the multidisciplinary optimization process of biomaterial design.
In summary, the optimization problem of bio-material design can be formulated as follows:
Find )()(Y L ∞∈x ρ,u α )(
min 0
0ΓΓ−=E E f eff (3) S.t. ∫∫=Y dY 0))((:))(()(x v x u x C εε in Y for every ad U v ∈ (4)
∫∫=Y
V dY )(x ρ 1)(0≤≤x ρ (5) 0u u u u α== on u Y ∂ (6)
P f n M f
n eq ερερεεε)](1[)())((:))(()(x x x u x u x −+≤= in Y (7)
where P M C x C x x C )](1[)()(n n ρρ−+= is the fourth order elasticity tensor at material point x . It is obtained by interpolating
between the elasticity tensors M C of
mineral and P C of protein, respectively. 1>n is a integer used for the penalization of the intermediate values of ρ. )(x eq ε is the equivalent strain at x . M f
ε and P f ε denote the failure strain of mineral and protein, respectively. T )0,1(0=u is the displacement mode vector, which indicates that the structure is only loaded along 1x direction.
ad U is the space of admissible test functions appeared in the weak form of equilibrium equation. V is the given amount of the
available mineral material. For simplicity, only 2-D case is considered, extensions to 3-D case is straightforward.
For the calculation of eff E , a homogenization approach based on asymptotic expansion is used. eff E can be expressed as:
∫∫∂∂−=Y q p pq eff dY y E E Y E ])()([111111111x x χ (8)
where )(1111x χχ= is the characteristic function, which can be obtained by solving a boundary value problem defined on Y with periodic boundary conditions. Y denotes the area of the unit cell.
In the present work, the measure of the toughness of the bio-composites corresponding to the considered loading condition is defined as:
∫∫−=ΓY
dY )](:)())(1[(x x x εσρ (9)
where )(x σ and )(x ε are stress and strain at point x , respectively. An assumption is also made here that all of the work done on the soft material is dissipated as heat during the process of deformation.
3.4 Optimization algorithms
For the nonlinear constitutive model, although the sensitivities used for numerical optimization can be obtained analytically (for the limitation of space, the detailed derivations are omitted here), numerical experiments show that the behavior of the solution process is unstable and often lead to non-convergent results. This is can be partially attributed to the inaccurate numerical calculation of the sensitivities since at some stages of the solution process, the condition number of tangent stiffness matrix is large for the high contrast of the material stiffness properties of different materials. Taking this into consideration and in order for dealing with the 0-1constraints more directly, in the present work, Genetic Algorithm (GA) is used to solve the corresponding optimization problem.
A binary chain with 9+m bits is used to represent one point in the design space. The first m bits represent the material type in m elements and the last 9 bits represent the value of the displacement control factor. Its value is varied in the interval of [0.0, 5.1] with a resolution of 0.1.
4. Results and discussions
In this section, a numerical example will be given for illustration purpose. For the considered problem, the design domain is a rectangular one with an aspect ratio of 10:1. The length and width of the design domain are nm lenx 100= and nm leny 10=, respectively. The Young’s modulus of mineral and protein like material are =M E 100GPa and =P E 100MPa (corresponding to 5=γ), respectively. The effective failure strains for mineral and protein like material are 1% and 150%, respectively. The yield stress for protein like material is 20=P f
σMPa. 0E and 0Γ in objective function used for dimensionless are Y V E E M /0= and 4/0leny lenx p f P f
××=Γεσ, respectively. The available volume percentage of mineral in the design domain is chosen as 57%. The prescribed displacements are imposed on the lateral boundary of the unit cell in the horizontal direction. This loading condition is used to simulate the pull out process in the process zone in the front of the propagating crack, see Fig.4 for reference.
Fig.4 Loading condition in front of the tip of a propagating crack
We assume that the unit cell to be designed has one-quarter symmetry. Then only one quarter of the unit cell needs to be optimized. The design domain and the boundary conditions determined from both of the symmetry and periodic properties of the unit cell and loading condition are depicted in Fig.5. In Fig.5, the vertical displacement at the upper and lower boundaries and the horizontal displacements at the right boundary are set to zero values.
Fig.5 Design domain and boundary as well as loading conditions
14X3 and 7X6 meshes are first used to solve the same optimization. The individuals in the initial population for GA algorithm are generated randomly. The GA evolution histories are illustrated in Fig.6a-Fig.6l and Fig.7a-Fig.7l, respectively. The optimal values of the displacement control factor and objective function are 2.6 and 0386.0− for 14X3 mesh as well as 1.5 and 0946.0− for 7X6 mesh, respectively. It can be seen that for these meshes, the optimal distributions of the mineral and protein like material are also of staggered type.
Fig.6a Step 1 Fig.6b Step 10
Fig.6c Step 30 Fig.6d Step 50
Fig.6e Step 80
Fig.6f Step 120
Fig.6g Step 150
Fig.6h Step 180
Fig.6i Step 220 Fig.6j Step 227
Fig.6k Step 228 Fig.6l Step 250
Fig.6 GA evolution history for 14*3 mesh (best individual in each generation)
5nm
Fig.7a Step 1 Fig.7b Step 11
Fig.7c Step 41 Fig.7d Step 51
Fig.7e Step 81 Fig.7f Step 91
Fig.7g Step 95 Fig.7h Step 120
Fig.7i Step 160 Fig.7j Step 200
Fig.7 GA evolution history for 7*6 mesh (best individual in each generation)
In order to further examine the optimality of the staggered type material arrangement, refined finite element mesh should be used. In order to enhance the computational efficiency of GA algorithm, under these circumstance, we insert the specific individuals constructed based on the inspirations obtained from the optimization results for coarse mesh. In this way, a lot of computational efforts can be saved. Fig.8a-Fig.8d show several such specific individuals. Optimal topologies obtained by GA algorithm for different meshes are shown in Fig.9. Both are of staggered type.
Fig.8a individual 1 for 14*6 mesh Fig.8b individual 2 for 14*6 mesh
Fig.8c individual 1 for 28*12 mesh Fig.8d individual 2 for 28*12 mesh
Fig.8 Specific individuals used for GA algorithm for different meshes
Fig.9a Optimal topology for 14*6 mesh Fig.9b Optimal topology for 28*12 mesh
Jager and Fratzl (2000) studied the mechanical properties of mineralized collagen and proposed a model with a staggered array of platelets that is in better agreement with results on molecular packing in collagen fibrils, see Fig.10 for reference. They showed this material arrangement leads to larger elastic modulus and fracture strain with the given amount of mineral in the fibril compared with a strictly parallel arrangement.
It can be seen that with the use of the proposed optimization model, the staggered arrangement of the hard and soft materials which is in reasonable agreement with that found in natural bio-composites can be reproduced (see Fig.1 and Fig.10). Then it seems that a plausible explanation for the convergent evolution in biology can be given (at least partially) from the optimization point of view. The obtained encouraging results confirm the belief that a staggered arrangement of mineral particles in the fibrils is mechanically superior to a strictly parallel arrangement from optimization point of view.
Fig.10a Fig.10b
Fig.10a Stagger model of mineralize collagen fiber (Jager and Fratzl, 2000) and Fig.10b the optimal material distribution
obtained by the proposed optimization model.
Fig.11 Stiffness and toughness of protein and mineral, as well as a few natural bio-composites. Taken from Fratzl (2004), which is based on a data compilation by Ashby et al. (Ashby, 1995)
At this point, it is interesting to have a look of the Ashby map shown in Fig.11. From this figure, it can be seen that proteins are tough but not very stiff. Mineral, on the contrary, is stiff but not very tough. Therefore the values of )/(00ΓΓ=E E f eff for these
materials are not large. But for bio-composites such as bone and detain, although the respective values of the stiffness and toughness of these materials cannot compare with those of mineral and protein, the product of their stiffness and toughness (ductility) is large. It 0.01
1.0101000.10.01 0.1 1.010
100
Stiffness (GPa)
Toughness (KJ/mm2)
means that these bio-composites combine the good properties of both its constituents. This also shows that the proposed objective function is reasonable from biology point of view.
5. Concluding remarks
The results of this study show that an optimization model with appropriate material constitutive models, failure criteria as well as objective and constraint functions can reproduce the realistic material distribution in the nanostructure of bio-composites. Although our approach to model the complex physical behaviour of biomaterials is certainly simplified, results obtained do show the independent roles played by mineral and soft protein like material and highlight the design principles used by the nature to produce the building blocks of bio-composites.
It seems that the maximization of ductility may be an objective of nature to design the basic building blocks of bio-composites. The insights gained from the present study are not only important for biological materials, but may also inspire novel ideas for the design of synthetic materials. From optimization point of view, bio-mimicking or bio-inspired material design can be formulated as an inverse optimization problem, i.e. finding a goal functional of an optimization problem under the condition that the solution to that problem is known. It is worth noting that the optimization problems in bio-inspired material design and engineering construction design are mutually inverse. In the former, the biological structure is known, but it is not clear what performance is optimized whereas the goal of the latter is to find a (unknown) optimal structure with the minimization (or maximization) of a given functional (LeeLavanichkul and Cherkaev, 2004). Using optimization principles to reveal the strategies and mechanisms used by nature to design biomaterials is a promising research field. The present work is only a preliminary step along this direction, more work needs to be done in this promising research field.
Acknowledgements
This work is supported by the Max Planck Society of Germany; China National Science Foundation under the grants of No.10225212, No.10472022; The Program for Changjiang Scholars and NCET Program provided by the Ministry of Education of China.
References
1.Jackson, A.P., Vincent, J.F.V., Turner, R.M., 1988. The mechanical design of nacre. Proc.R.Soc.London, Ser.B. 234, 415-440.
2.Currey, J.D., 1997. Mechanical properties of mother of pearl in tension. Proc.R.Soc.London, Ser.B.196, 443-46
3.
3.Mening, R., Meyers, M.H., Meyers, M.A., Vecchio, K.S., 2000. Quasi-static and dynamic mechanical response of Haliotis
rufescens (abalone) shell. Mater. Sci. Eng. A297, 203-211.
4.Weiner, S., Wanger H.D., 1998. The material bone: structure-mechanical function relations. Annu. Rev. Mater. Sci. 28,
271-298.
5.Currey, J.D., 1984. The mechanical adaptations of bone. Princeton University Press, Princeton, NJ, 24-37.
ndis, W.J., 1995. The strength of a calcified tissue depends in part on the molecular structure and organization of its
constituent mineral crystals in their organic matrix. Bone. 16, 533-544.
7.Weiner, S., Veis A., Beniash, E., Arad, T., Dilon, J.W., Sabsay, B., Siddiqui, F. 1999. Pertubular dentin formation: crystal
organization and the macromolecular constituent in human teeth. J. Struct. Biol. 126, 27-41.
8.Tesch, W., Eidelman, N., Roschger P., Goldenberg, F., Klaushofer, K., Fratzl, P., 2001. Graded microstructure and mechanical
properties of human crown dentin. Calcif. Tissue Int. 69, 147-157.
9.Gao, H., Ji, B., Jager, I.L., Artz, E., Fratzl, P., 2003. Materials become insensitive to flaws at nanoscale: lessons from nature.
Proc.Natl.Acad.Sci. USA. 100, 5597-5600.
10.Ji, B., Gao, H., 2004. Mechanical properties of nanostructure of biological materials. J.Mech.Phys.Solids. 52, 1963-1990.
11.Aksay, I.A., Weiner, S., 2004. Biomaterials. Is this really a field of research? J.Mater.Chem. 14, 2115-2123.
12.Fratzl, P., Burgert, I., Gupta, H.S., 2004. On the role of interface polymers for the mechanics of nature polymeric composites.
Phy.Chem.Chem.Phys. 6, 5575-5579.
13.Smith, B.L., Schaeffer, T.E., Viani, M., Thompson, J.B., Frederick, N.A., Kindt, J., Belcher, A., Stucky, G.D., Morse, D.E.,
Hansma, P.K., 1999. Molecular mechanistic origin of the toughness of natural adhesive, fibers and composites. Nature. 399, 761-763.
14.Jager, I.L., Fratzl, P., 2000. Mineralized collagen fibrils: A mechanical model with a staggered arrangement of mineral particles.
Biophysical Journal. 79, 1737-1746.
15.Fratzl, P., 2004: Hierarchical structural and mechanical adaptation of biological materials. R.L.Reis and S.Weiner (eds)
Learning from nature how to design new implantable biomaterials. Kluwer Academic Publishers. 15-34.
16.Ashby, M.F., Gibson, L.J., Wegst, U., Oliver, R., 1995: The Mechanical-properties of natural materials. 1. Material property
charts. Proc.R.Soc.London, Ser.A. 450, 123-140.
17.LeeLavanichkul, S., Cherkaev, A., 2004. Why the grain in tree trunk spirals: a mechanical perspective. Struct Multidis Optimi
28, 127-135.。