Nacre properties in the elastic range: Influence of matrix incompressibility

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Computational Materials Science 41 (2007) 96–106 www.elsevier.com/locate/commatsci

Nacre properties in the elastic range: Influence of matrix incompressibility Sergey Dashkovskiy a, Bettina Suhr

a,*

, Kamen Tushtev b, Georg Grathwohl

b

a b

Centre for Industrial Mathematics, FB3, University of Bremen, Germany Ceramic Materials and Components, FB4, University of Bremen, Germany

Received 14 September 2006; received in revised form 5 March 2007; accepted 22 March 2007 Available online 31 May 2007

Abstract The elastic behavior of nacre as a two phase composite is modelled with help of finite elements. The study is focused on the influence of the incompressibility of the matrix on the elastic modulus of the composite. A 2D finite element model is used to simulate tension, shear and bending tests. The simulation shows that the organic phase plays an important role on the stress distribution in the composite. In particular the elastic modulus of the composite increases strongly when the matrix becomes incompressible.  2007 Elsevier B.V. All rights reserved. Pacs: 02.60.Cb; 0.70.Dh; 46.25.y; 87.15.Aa; 87.15.La Keywords: Nacre; Elasticity; FEM; Incompressibility

1. Introduction Nacre is a molluscan shell material, which can be considered as a two phase biocomposite with a special nanostructure. The matrix phase consists of organic biopolymers. The other phase is aragonite in the form of platelets. This composite has attracted the attention of many researchers due to its high strength and toughness in comparison to the components it is made of. To design a man-made nanocomposite with similar properties the understanding of the interactions between its nanocomponents is required. There were many attempts to investigate nacre experimentally and to model it, see a brief review below. One of the main problems is that the constitutive properties of its components are almost unknown. The micro and nan*

Corresponding author. Present address. Bibliothekstrasse 1, 28359 Bremen, Germany. Tel.:+49 421 2184816. E-mail addresses: [email protected] (S. Dashkovskiy), [email protected] (B. Suhr), [email protected] (K. Tushtev), [email protected] (G. Grathwohl). 0927-0256/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.03.015

odimensions of the components of nacre make their direct testing very difficult. Hence inverse methods for their identification are necessary. In this paper we consider nacre only in an elastic range. Our aim is to demonstrate the influence of the incompressibility of the matrix on the stress distribution and on the elastic modulus of the composite. It will be shown that the Young’s modulus of the composite increases drastically when the organic matrix approaches to be incompressible, i.e., when its Poisson’s ratio becomes close to 0.5. A similar effect was pointed out by [1]. To the best knowledge of the authors, there are only few papers which deal with analyzing stress state of nacre in tension, shearing or bending. Here, we investigate the distribution of stresses in the platelets and in the organic matrix in different stress states depending on the properties of the matrix. The paper is organized as follows. In the next section the microstructure of nacre and its mechanical properties are described. Section 3 collects several models for nacre from the literature. To motivate our investigation we consider a model for pull-out of a platelet clamped between two

S. Dashkovskiy et al. / Computational Materials Science 41 (2007) 96–106

others assumed to be rigid in Section 4. The description of the finite element model, the obtained results and their discussion are reported in Section 5. We conclude with Section 6. 2. Microstructure and properties of nacre The volume fraction of aragonite platelets layered in parallel planes is about 95%. The thickness of the platelets varies from 0.2 to 1.5 lm [2,3]. The platelets are separated from each other by thin 30 nm organic layers composed mainly of proteins and polysaccharides. However, mineral bridges connecting two neighboring platelets are also observed [3]. The organic constituent, represents the interconnected matrix phase throughout the composite and is assumed to play an essential role in strengthening and toughening of the composite [4–6]. There are two main types of nacre with respect to the ordering of the platelets, sheet and columnar. The sheets in columnar nacre are arranged in columns perpendicular to the planes of sheets and hence have a high overlapping of each other in the same column. In contrary, the platelets of the neighboring planes in a sheet nacre have a smaller overlapping approaching to a half of their linear dimension, and the divisions of the plates are more randomly arranged than in columnar nacre. Mineral bridges between the platelets of neighboring planes being responsible for the mechanism of nacre formation are described as roughly circular aragonite columns with diameter about 50 nm and the height equal to the thickness of the organic matrix layer. Their geometry and distribution in the matrix was studied [3] modelling also their effect on the properties of the composite [7,3]. Furthermore the platelet surfaces have plenty of nanoasperities allowing a strong interconnection between the biomatrix and mineral platelets, ensuring a strong mechanical end chemical coupling essential for achieving the superior properties of nacre [7]. The mechanical properties vary in different types of nacre. Some typical test data are reported here from the literature. The Young’s modulus of platelets is assumed to be in the range of 50–205 GPa [8,9,4,10–12]. However, it is known that the aragonite crystal has a significant anisotropy. We refer to [13] for an experimental approach to study the elastic properties of platelets of nacre. See also [14] for the inverse method to study the constituent properties of nacre components. For our simulation we assume the tablets of aragonite to be homogeneous with the isotropic Young’s modulus of 100 GPa. The Poisson’s ratio is taken to be 0.3. Little is known about the properties of the biopolymer bonding the platelets due to its nanoscale size. It is also difficult to extract it, moreover, being extracted it would be denatured. However, it is clear that this organic material is essentially softer than aragonite. In [12] it is assumed that the Young’s modulus of the matrix is 5000 times less than of the platelets, and is estimated to be similar to polymeric

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rubber, i.e., 1 MPa. Different authors use the values of 1– 20 MPa, [10,6,12] or 15–20 GPa in [15–17] for the Young’s modulus of the organic layer. The shear modulus is used with values between 1.4 and 4.6 GPa, [4,11]. For our simulation we assume the isotropic Young’s modulus of the matrix to be 0.1 GPa and the Poisson’s ratio is varied from 0.1 to 0.499. The Young’s modulus of the composite is reported to be of the order of 70 GPa (dry) and 60 GPa (wet), and the tensile strength is about 170 MPa (dry) and 140 MPa (wet) [4], with pull-out of the platelets as the main mode of failure. Currey reports [2] the tensile strength to be 35–110 MPa. He notes that the work of fracture is very different in different loading directions and depends on whether it is tested dry or wet. The strength and toughness of nacre are 20– 30 times higher than the corresponding data of monolithic aragonite [13]. The fracture processes of nacre are investigated and modelled in [18,19]. In particular crack morphologies of nacre are described in [20]. It was observed in experiments [2,21] that the deformation of nacre is elastic up to some limits. Beyond the yield stress the deformation can be considered as elastic–plastic [2] or visco-elastic [5]. 3. Models review The organic matrix plays the central role in most of the nacre models. In a nonlinear analytic model [11] assume Kotha et al. that the load transfer between the platelets is due to the shear stresses in the matrix only. See the comparison of this assumption with our results below. Nukala and Simunovic [22] replace the organic matrix by shear elements to perform their calculations based on a statistic method. In a tension-shear chain model of Ji and Gao [5] the mineral platelets and the matrix are replaced by one dimensional tension and shear springs. The tension load is again transferred via shearing of the matrix. The soft matrix is assumed to play a key role for the high toughness and strength of nacre. Ji and Gao [23] adapted the virtualinternal-bound-model to nanostructured biocomposites and found that the protein layer can effectively enhance the toughness of biocomposites through crack shielding and impact protection. Okumura and de Gennes [12] developed a model in the frame of fracture mechanics. With help of the analytic solution they show that the reason for the high fracture toughness is the stress reduction at the crack tip. The effect of the matrix’ Poisson’s ratio on the stiffness of the composite was studied by Lie et al. [1]. The effect of nanoscale processes in the organic layers on the nonlinear deformation of nacre was considered by Qi et al. [24]. A micro-mechanical model is developed there to simulate the nonlinear deformation of the composite via progressive unfolding of protein chains in the matrix. Furthermore, the role of the mineral bridges between aragonite platelets of neighboring planes for the mechanical properties of nacre was considered from some authors. Song et al. [3] estimated that the total cross-sectional area

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of the mineral bridges is approximately one-sixth of the area of each platelet. The excellent performances of nacre are established to be intimately associated with the mineral bridges within the organic interface. In contrast, Katti et al. [6] found on the basis of FEM-calculations that the mineral contacts between aragonite platelets have insignificant effect of the elastic properties of nacre. Following experimental studies, various models were proposed which connect the mechanical properties of nacre with the form and the surface of aragonite platelets. The inelastic deformation in nacre is attributed by Evans et al. [8] and Wang et al. [21] to the nanoscale asperities on the surface of the platelets and their ability to provide a resistance to inter-facial sliding. In contrast, series of FE-simulations (Katti et al. [15]) suggested that nanoasperities have marginal effect on the mechanical response of nacre. Using three-dimensional finite element modeling Katti et al. [6,10,7,15–17,25] and Ghosh et al. [26] showed that interlocks between aragonite platelets are the key mechanism for the high toughness and strength of nacre. Platelet waviness was found from Barthelat et al. [27,14,13] as a most important micro-structural feature responsible for relatively large inelastic deformations and strength of nacre. The next chapter gives a motivation for our approach.

nite, we consider the platelets to be undeformable. Consider a piece of nacre under a tensile stress with aragonite platelets in white and the organic matrix in gray as sketched in Fig. 1. The thickness of the matrix layers is exaggerated to visualize the structure. The platelet B is clamped between the platelets A and C, (left side of Fig. 1). It is pulled out with a force F. Under our assumption only the matrix deforms. On the right side of Fig. 1 we consider the corresponding part of the matrix with the following simplified boundary conditions. The sides S2, S3 and S4 are fixed, i.e., the displacement vector is zero on these surfaces. Further, S1 and S5 are taken to be stress free, which can only reduce the resulting pull-out force F. On S6, S7 and S8 the displacement vector is constant and has only one non-zero component in the direction of tension, which is set to be 1 nm. It is assumed that the deformation remains in the elastic state. The elastic constants of the matrix are set as Young’s modulus and Poisson’s ratio, i.e. Eorg = 0.1 GPa, morg = 0.1–0.499, respectively. With help of FEM the force F for different values of morg is calculated, see Fig. 2. From this figure it is evident that the pull-out force F increases drastically when morg approaches the value of 0.5, i.e., when the organic matrix becomes almost incompressible. This motivates us to assume, that the Poisson’s ratio of the matrix can play an essential role on the elastic proper-

4. Motivation There were many attempts to derive a model predicting the mechanical behavior of nacre. However, there is still no comprehensive model with a generalized accessibility to the relevant properties of such nanocomposites. One of the main problems is that the constitutive properties of the components of the composite are almost unknown. As it is commonly accepted that nacre behaves elastically up to a certain limit, we confine our study to this case. The Poisson’s ratio of the organic matrix is varied in this study keeping all other elastic constants fixed. This study is then focused on the question what is the effect of the compressibility of the matrix on the effective Young’s modulus of the composite. A simplified model of the pull-out of one platelet under the assumption of a plane strain is used. As the organic matrix is believed to be essentially softer than the arago-

–4

8

x 10

7 6

F [N]

5 4 3 2 1 0 0.1

0.2

0.3 ν

0.4

Fig. 2. Tensile force versus Poisson’s ratio.

S2 S1 S8 F

A S3

B

0.5

S7

F S6 S5

C S4

Fig. 1. Simplified pull-out model of nacre.

S. Dashkovskiy et al. / Computational Materials Science 41 (2007) 96–106

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5.2. General setting

ties of the composite and maybe also on its fracture mechanism. In the following the elastic properties of nacre are investigated in dependence on morg. The stress distributions in the composite and in particular its dependence on morg will be studied in detail. Our model is based on the finite elements. Different geometries for the representative unit cell of nacre will be considered. Our consideration is not restricted to the symmetric case of platelets overlapping to 50% as in [1].

The following simulations are conducted in two dimensions. Since the exact dimension of the aragonite platelets varies between different kinds of shells we chose values which we considered representative: width and height of the platelets 8190 nm and 420 nm, respectively. The width of the organic matrix is 30 nm. The overlap between the platelets was assumed to be 20–25%. In the simulations we investigated exclusively the elastic behavior of nacre. The necessary material parameters, Young’s modulus and Poisson’s ratio of the aragonite platelets and the organic matrix, were chosen as follows:

5. Description and simulation results of FEM-model 5.1. FEM software

Ep ¼ 100 GPa; mp ¼ 0:3; Eorg ¼ 0:1 GPa; morg ¼ 0:1–0:499

In the last years the research area ‘scientific computing’ located between applied mathematics and engineering gained strongly in importance. Nowadays not only model problems but real life applications with increasing complexity are simulated and demand an efficient programing of modern numerical algorithms. To name some examples: adaptive mesh refinement and time step control, discretizations with elements of higher order, fast iterative solvers for linear and non-linear systems of equations and multi-level algorithms. For the following simulations the open source toolbox ALBERTA was used (cf. [28]). ALBERTA was developed for the fast and flexible implementation of efficient software for complex applications based on the algorithms mentioned above. Moreover ALBERTA offers an environment in which existing algorithms can be improved and new algorithms can be developed. The toolbox makes possible the treatment of a big class of problems and the simple coupling of different solvers for systems of equations. One of the advantages of ALBERTA is its openness for expansion. Thereby one can on the one hand easily keep up with scientific progress and on the other hand minimize computational time for a specific problem by adjustment. This is exactly what was done in the following simulations where the PARDISO-solver of the University of Basel was used to solve the system of equations. The abbreviation stands for ‘Parallel Sparse Direct Linear Solver’: it is though a direct solver for sparse, big systems of equations for shared memory multiprocessors. During the implementation thread safety, velocity, robustness and memory efficiency played an important role, cf. [29–31].

We choose a formulation of the constitutive equation of elastic material behavior where the displacement is calculated. Using Hooke’s law the stress tensor is then to be computed. Some notations are introduced: We denote in a point (x, y) of the specimen the displacement u(x, y) = (ux,uy), the stress tensor r(x, y) and the strain tensor e(x, y). The equations of equilibrium are as follows: divðrÞ ¼ 0 and the Hooke’s law is r¼

E mE eðx; yÞ þ trðeðx; yÞÞI ð1 þ mÞ ð1 þ mÞð1  2mÞ

where I is the identity operator. A schematic drawing of the used geometry is presented in Fig. 3. We used a regular, triangular mesh which had a high resolution in the area of the polymer matrix and a low resolution in the area of the aragonite platelets. This was due to the fact that we expected the properties of the polymer matrix and the effects happening there to have a great influence on the composite’s mechanical behavior. 5.3. Tension test In this section a tension test is simulated where the specimen is elongated for 0.1% of its length. Under this condition, the specimen is still assumed to behave elastically. We denote in a point (x, y) of the specimen the outer normal by n and the length of the specimen by lx. With Γa

Γl

Γr

y Γb x Fig. 3. Schematic initial situation defining the names of the boundaries Ca, Cb, Cl, Cr.

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these notations the boundary conditions are as follows (compare Fig. 3): ux jCl ¼ 0

ryy jCl ¼ 0

ux jCr ¼ 0:001  lx ryy jCb ¼ 0

ryy jCr ¼ 0 uy jCb ¼ 0

rðx; yÞ  njCa ¼ 0 The effective Young’s modulus is calculated by Hooke’s law, whereas the stress equals force per unit length R rxx dy=ly E ¼ Cr ; Dlx =lx ly is the size of the specimen in y-direction. 5.3.1. Convergence studies and adaptive mesh refinement Before presenting our results of several tension tests some aspects on the numerical accuracy of the calculations have to be considered. In the following section we simulate a tension test and observe how much the degree of refinement of the triangulation and the polynomial degree of the test functions influence the calculated effective Young’s modulus, the maximal stress and the value of the error estimation. Below we use adaptively refined meshes based on residual error estimation. The concept of residual error estimation works roughly speaking as follows: The simulation yields an approximative solution which is piecewise polynomial, i.e. it is polynomial on every element of the mesh. The error is calculated element-wise and consists of several contributions. There is the error which arises because the approximative solution does not fulfill the constitutive equation. Furthermore, jumps in the gradient of the calculated solution as well as data approximation errors are taken into account. Based on these local errors one can refine elements of the mesh with a high error and calculate a total error for the whole specimen. In Table 1 there are the following values noted for the Poisson’s ratios morg = 0.2 and morg = 0.499 and for the polynomial degree of the Lagrange test functions n: # res. E krk1

total number of triangles in the mesh, residual error estimation, calculated effective Young’s modulus in GPa, the maximum of the Frobeniusnorm of the stress tensor in MPa.

For the polynomial degree n = 2 and n = 3 it was not possible to use refined meshes because of restrictions of memory capacity. Therefore the calculations are performed only on the macro-triangulation with 60,472 triangles. As we can see the calculated Young’s modulus varies only little with the number of triangles and the polynomial degree for Poisson’s ratio morg = 0.2 (less than 1% variation from mean value). For the higher Poisson’s ratio morg = 0.499 there are bigger differences notable. Although the variation from mean value is within less then 3% not too big, we see that the calculated Young’s modulus depends

Table 1 Dependency of the max. stress krk1, the Young’s modulus E and the estimated error res. on the number of triangles in the mesh #, the Poisson’s ratio of the organic matrix morg and the polynomial degree of the test functions n Res.

(a) morg = 0.2, n = 1 60,472 241,736 962,923

0.019 0.013 0.009

38.31 38.08 38.00

79.29 81.87 82.75

(b) morg = 0.499, n = 1 60,472 0.049 241,607 0.033 966,382 0.023

100.05 97.67 96.90

246.17 215.42 192.65

0.019

38.01

82.53

(d) morg = 0.499, n = 2 60,472 0.045

96.94

174.42

(e) morg = 0.2, n = 3 60,472

0.019

37.97

83.98

(f) morg = 0.499, n = 3 60,472

0.045

96.58

172.62

(c) morg = 0.2, n = 2 60,472

E

krk1

#

sensitively on the total number of triangles in the mesh and the polynomial degree of the test functions. On a coarse mesh with only 60,472 triangles and linear Lagrange test functions the effective Young’s modulus is calculated to be even bigger than the Young’s modulus of the aragonite. This numerical error which grows bigger when the Poisson’s ratio tends to 0.5 can be avoided when the mesh is fine enough. From these test we deduce that the numerical error of the simulation is small when the mesh is adaptively refined. As we are interested in the effective Young’s modulus it is sufficient to use linear Lagrange test functions although we can see in Table 1b that the stress krk1 is calculated too big. This is presumably an effect of locking, for further details see [32]. 5.3.2. Variation of the size of the specimen In this section we simulate tension tests with specimens of different size to find out whether this has a significant influence on the calculated effective Young’s modulus. The used geometries consist in x-direction of 2, 4 or 8 platelets (16.44 lm, 32.88 lm or 65.76 lm) and in y-direction of 6 or 12 layers (2.7 lm or 5.4 lm). For a visualization see Fig. 3 where a geometry consisting of four platelets in x-direction and 6 layers in y-direction (4 · 6) is presented. In Table 2 there the calculated effective Young’s moduli of the different sizes of the specimens dependent on the Poisson’s ratio of the organic matrix are presented. All values are given with two decimal places although for morg near 0.5 the accuracy of the calculations is not that high, compare Section 5.3.1. Note that the Young’s modulus of the specimens increases essentially, when morg approaches 0.5.

S. Dashkovskiy et al. / Computational Materials Science 41 (2007) 96–106 Table 2 Calculated Young’s modulus for differently sized specimen dependent on morg Tol. morg 0.10 0.20 0.30 0.40 0.45 0.48 0.49 0.499

2·6

2 · 12

4·6

4 · 12

8·6

0.13 E (GPa) 38.59 38.54 40.06 46.03 55.94 71.33 81.61 96.54

0.17 E (GPa) 38.38 38.34 39.91 46.16 56.23 71.76 82.26 97.18

0.17 E (GPa) 38.37 38.31 39.87 46.07 56.09 71.51 81.93 96.73

0.35 E (GPa) 38.09 38.04 39.63 46.12 56.62 72.11 82.31 97.59

0.35 E (GPa) 38.22 38.15 39.72 46.17 56.62 71.91 82.50 97.19

In the following simulations the meshes are adaptively refined until the residual error estimation is below the given tolerance. These tolerances are chosen differently for every specimen, as small as possible but big enough that the memory capacity is not exhausted. There are only small discrepancies between the calculated values; the relative error related to the mean value is about 1% for a fixed Poisson’s ratio. Hence a simulation of the tension test with the smallest specimen is sufficient. For the illustration of the stress state only a cutout of the middle part of the specimen is shown, as one can see in the schematic drawing of Fig. 4. In Figs. 5 and 6 the

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stress state of the 4 · 6 specimen is presented. There we can see the three elements of the stress tensor rxx, rxy and ryy for the Poisson’s ratios morg = 0.2 and morg = 0.499. 5.3.3. Analysis of the stress state The simulated values of the stress tensor components are presented in Figs. 5 and 6. In the case of morg = 0.2 (Fig. 5) the distribution of the tensile stress component rxx in a platelet reaches its minimum at the ends of the platelet and in the juncture. Within the platelets it grows nearly linear from the ends. The maximum is reached in the place of juncture of the platelets of the next layers from above and below. This confirms the assumption in [4, p. 425] that the tensile stress in platelets builds up linearly from each end. However, there is assumed that the maximum of rxx is reached in the middle. This difference is probably due to the fact that the authors of [4] assume a higher overlapping of platelets than in our simulation. The tension stress in the horizontal layers of the organic matrix is zero, only where the horizontal layers meet the junctures the tension stress is bigger than zero (but still very small). For the shear stress component rxy we see two kinds of columnar domains with strongly different stress situations. One domain is the region of platelets junctures and the other is complementary. These domains are certainly

Fig. 4. Schematic drawing of the cutout presented.

Fig. 5. Tension test of the 4 · 6 specimen with morg = 0.2: elements of the stress tensor rxx, rxy, ryy (MPa) from up to down.

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Fig. 6. Tension test of the 4 · 6 specimen with morg = 0.499: elements of the stress tensor rxx, rxy, ryy (MPa) from up to down.

periodically repeated. In first approximation the stress in each of the domains can be assumed constant in horizontal direction. This is in agreement with the corresponding assumption 5 in [11] that the shear stresses in the organic on the sides of the platelets is constant. In the area where the junctures of platelets takes place the shear stress is essentially higher then in the rest of the specimen, where the stress is close to zero. This shows how the tensile load is transferred between the platelets. ryy is nearly constant in most points of the specimen and is close to zero. Only at the faces of juncture it is positive (maximal) in the matrix and negative at the ends of platelets. The platelets above and below the juncture show also local areas where ryy is bigger than zero. Consider what happens if the matrix becomes nearly incompressible, i.e., with morg = 0.499. As one can see in Fig. 6 the tension stress rxx is now homogeneously distributed within the platelets. At the junctures the tension stress is even 20% bigger than in the platelets (contrary to the case of morg = 0.2 where the platelets showed a higher tension stress than the organic matrix). So we can see, that the applied load is transferred by the matrix at the ends of the platelets. The tension stress in the horizontal layers of the organic matrix is bigger than zero near the junctures and almost zero everywhere else. Note that the arising stress is now higher then for morg = 0.2. Because the matrix is under a high hydrostatic pressure, the assumption that it does not yield is reasonable. The shear stress rxy is close to zero almost everywhere, only in the domains of junctions it becomes high and reaches its maxima and minima. The extremal values are higher then in case of morg = 0.2. The question arises, whether the soft organic can bear this load. In [11], e.g., it is experimentally determined that the inter-facial shear

stress at which the matrix yields is in the range of 37– 46 MPa, which is a bit higher as in our simulation. Here we note that an extension of our model to the case, when the matrix yields is necessary. This extension is planed for the nearest future. The component ryy has a similar distribution as for smaller Poisson’s ratios but it becomes also more homogeneous in the platelets. Again the maximal stress arises in the junctures. Comparing Figs. 6 and 5 we conclude that the stress distribution is different. The increase of the Poisson’s ratio leads to higher stresses in the platelets. Extremely high stresses in the matrix appear at the junction of platelets. Now let us consider what happens in the case of shearing of nacre.

5.4. Shearing test In this section we simulate a shearing test with differently sized specimens. We investigate the dependency of the calculated effective shear modulus l on the Poisson’s ratio of the organic matrix and the size of the specimen. To our knowledge in the literature there are no commonly accepted boundary conditions for shearing tests of composite materials. We decided to define the following boundary conditions, for which homogeneous materials show the desired behavior: rxx  0, rxy = const., ryy  0 and the shear modulus was calculated correctly. ux jCa ¼ 0:001  lx ujCl ¼ ujCr ujCb ¼ 0

uy jCa ¼ const:

S. Dashkovskiy et al. / Computational Materials Science 41 (2007) 96–106 Table 3 Calculated effective shear modulus dependent on the size of the specimen and morg Tol. morg 0.10 0.20 0.30 0.40 0.45 0.48 0.49 0.499

2·6

2 · 12

4·6

4 · 12

6 · 104 l (MPa) 661.41 607.74 562.70 524.46 507.79 498.71 496.07 493.83

6 · 104 l (MPa) 660.49 607.05 562.10 524.04 507.51 498.14 495.29 493.33

2 · 103 l (MPa) 661.44 607.84 562.76 524.46 507.79 498.76 495.97 493.43

2 · 103 l (MPa) 660.36 607.02 562.15 524.11 507.17 498.25 495.52 493.22

The effective R shear modulus l is calculated by applied stress rapp :¼ l1x Ca rxy divided by the resulting shear angle. In Table 3 the calculated effective shear moduli for the differently sized specimens and different Poisson’s ratios of the polymer matrix are shown. The notation is the same as in 5.3.2. In contrary to the Young’s modulus E, the shear modulus l decays with increasing Poisson’s ratio. But the influence of morg on l is not that essential as on E. The size of the specimen has a negligible influence on the shear modulus; the relative error related to the mean value is about 1% for a fixed Poisson’s ratio. For the illustration of the stress state only a cutout of the middle part of the specimen is shown, as one can see in the schematic drawing of Fig. 7. In Figs. 8 and 9 the

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stress state of the 4 · 6 specimen is shown for the Poisson’s ratios morg = 0.2 and morg = 0.499. Only a cutout of the middle part of the specimen is shown to avoid boundary effects. 5.4.1. Analysis of the stress state The stress distributions in the shearing test are presented on Figs. 8 and 9 for morg = 0.2 and morg = 0.499, respectively. We see that the tensile stress rxx is nearly zero far from the domains of platelets junctures, where it reaches its maximum and minimum values. In the case of almost incompressible matrix, i.e., with morg = 0.499, the stress free part in the middle of the platelets are considerably larger then for morg = 0.2. For both Poisson’s ratios the same effect is observed: The left end of a platelet bends down, the middle is stress free and the right end bends up. Because of the brick and mortar structure of nacre there are also stress contributions of the bending ends of over- and underlying platelets. The authors cannot explain this effect. The shear stress rxy can be considered as nearly constant far from the domains of platelets juncture, where it reaches its extremal values. At some few points it becomes slightly negative, which is probably due to numerical errors and therefore not displayed in the figures. The situation is similar for both morg = 0.2 and morg = 0.499. The organic matrix at the junctures is stress free. The maximum value is reached above and below the junctures (for morg = 0.499 additionally left and right from the junctures) which happens due to the bending of the platelets.

Fig. 7. Schematic drawing of the cutout presented.

Fig. 8. Shearing test of the 4 · 6 specimen with morg = 0.2: elements of the stress tensor rxx, rxy, ryy (MPa) from up to down.

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Fig. 9. Shearing test of the 4 · 6 specimen with morg = 0.499: elements of the stress tensor rxx, rxy, ryy (MPa) from up to down.

ryy can be considered as zero in the most points of the specimen in both cases. It becomes large in the organic at the platelets’ junctures, especially for morg = 0.499. It is worthy to note that for all stress components there are only small differences between their values in the matrix and in the platelets at places far from the platelets junctures, i.e., stress is rather high in the matrix. Together with the high stresses of ryy in the junctures, this raises the question whether the soft matrix can hold such a load? Probably it yields. This shows that an elastic model for the shearing of nacre should be extended to viscoelastic or plastic one or a model with platelets pullout and friction. In the following we briefly consider the bending test. 5.5. Bending test In this section we simulate a bending test to evaluate the influence of the size of the specimen and the Poisson’s ratio of the organic matrix on the Young’s modulus (See Fig. 10). In the simulation the stress S = S(y) is determined (corresponding to the moment) which belongs to a force F of 10 N applied to the middle of the top side surface. To gain a unique solution and to enforce a symmetric bending of

the specimen, the displacement in y-direction at the midpoints of the side surfaces and in x-direction at the midpoint of the bottom side is set to zero. The boundary conditions in a point (x, y) with the outer normal n are as follows: rðx; yÞ  njCl ¼ SðyÞ rðx; yÞ  njCr ¼ SðyÞ rðx; yÞ  njCa [Cb ¼ 0 uy ð0; ly =2Þ ¼ uy ðlx ; ly =2Þ ¼ ux ðlx =2; 0Þ ¼ 0 In this setting the upper part of the specimen is under compression and the lower part under tension. It is assumed that the Young’s moduli of tension and compression are equal. The Young’s modulus E is calculated by the following equations: (compare [33]) l

dlx 1 M 2y ; ¼ exx jlx ¼ 2 EJ lx where lx, ly are the measures of the specimen in x-, y-direction, dlx is the change of length in x-direction, e is the strain tensor, M = Flx is the applied momentum and R ly l3 J ¼ 2ly y 2 dy ¼ 12y . It follows: 2

S

S

y x Fig. 10. Schematic drawing of the bending test.

S. Dashkovskiy et al. / Computational Materials Science 41 (2007) 96–106 Table 4 Calculated effective Young’s modulus for differently sized specimen dependent on morg Tol. morg 0.10 0.20 0.30 0.40 0.45 0.48 0.49 0.499

E¼

10 · 22

12 · 6

5 · 11

– E (GPa) 37.40 37.39 39.04 45.60 56.10 72.60 83.49 99.65

– E (GPa) 37.66 37.74 39.46 46.04 56.42 72.65 83.40 99.67

0.016 E (GPa) 38.78 38.66 40.07 45.97 55.74 70.67 80.49 93.58

3Fl2x : dlx l3y

The size of the specimen is 80 lm · 10 lm (10 platelets in x-direction and 22 layers is y-direction). Moreover we simulate a specimen with half width and height (5 · 11) and a specimen with significantly different aspect ratio: 98.64 lm · 2.7 lm (12 · 6). In Table 4, the calculated effective Young’s modulus in dependency on the size of the specimen and morg is shown. Due to restrictions of the memory capacity we simulate the two larger specimen (10 · 22 and 12 · 6) on the macro-triangulation, i.e. the mesh is not refined and therefore in Table 4 there is no tolerance given. The smallest specimen (5 · 11) is simulated on a refined mesh. The comparison of these three specimens shows that neither the size of the specimen itself nor the aspect ratio have a great influence on the calculated Young’s modulus; the relative error related to the mean value is about 4%. The specimen simulated with a finer mesh has for increasing Poisson’s ratio a smaller numerical error which allows a more accurate calculation of the Young’s modulus. In a bending test the upper half of the specimen is under compression and the lower half is under tension. Corresponding to our expectations the calculated effective Young’s moduli are quite similar to those calculated in the tension test, with the same dependency on the Poisson’s ratio, i.e., E increases drastically when morg comes close to 0.5. 6. Conclusions We have simulated the tension, shear and bending tests of nacre with help of 2D finite element models. Only the elastic range of deformation has been simulated. The Young’s modulus of the composite has been calculated. We have considered the influence of incompressibility of the organic matrix on the Young’s modulus of the composite and on the stress distribution in a specimen. We have found, that the Young’s modulus becomes essentially higher for higher values of the Poisson’s ratio morg of the matrix and reaches its maximum when the matrix becomes

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incompressible, i.e., when morg becomes close to 0.5. We have also found that the stress distribution changes for different values of morg, especially in case of tension test. This knowledge can be used for the design of the artificial nanocomposites similar to nacre. Acknowledgements The authors are thankful for the fruitful conversations with Michael Murck engineer at the institute ‘‘Ceramic Materials and Components’’ at the University of Bremen. This research was partially supported by the Bremen Center for Scientific Computing and Applications (BreSCA).

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