- Email: [email protected]
Two-dimensional modelling of solid-fluid composites
Composite Shuctures Vol. 38, No. l-4, pp. 499-507, 1997 0 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0263-8223/97/$17.00 + 0.00 PII:SO263-8223(97)00085-8
Itvo-dimensional modelling of solid-fluid composites Tsutao Katayama Department of Mechanical Engineering, Doshisha University, Tanabe-cho, Tuzuki-gun, Kyoto-fi 610-03, Japan
Hidetake Yamamoto Faculty of Mechanical Engineering, Kobe City College of Technology, Gakuenhigashi-cho, Nishi-ku, Kobe 651-21, Japan
&
Kazuo Nishitani Postgraduate Course, Doshisha University, Tanabe-cho, Tuzuki-gun, Kyoto-fu 610-03, Japan
The purpose of this paper is to apply biomimetic-designed composites to artificial structures. From the results of numeric modelling analysis in biomechanics, we have learned the bone structures optimized to lighten weight and understood that the solid-fluid composite structure of the cancellous bone at the joint part works to distribute the joint load perfectly. In this paper, the two-dimensional honeycomb structure filled with fluid was investigated by way of a simplified solid-fluid composite material model of the cancellous bone. Hybrid finite element analyses illustrated that the solid-fluid phase interaction is effective in dispersing compressive load. Inplane indentation tests were carried out and in-plane deformation distributions of the solid-fluid composite specimens were measured. Consequently, as for the solid-fluid composite specimens whose cells were filled up with glycerine, a good enough cell deformation mode was obtained. 0 1997 Elsevier Science Ltd.
INTRODUCTION
have considered applying the mechanical functions of the bone’s composite structure to new material designs. We have recognized the solidfluid composite structure of the cancellous bone in the bio-joints to be a material system and investigated these functions as an intelligent material; we will try to develop new materials given these functions artificially. The purpose of this study is to learn from the cancellous bone, which has a good load dispersive and a high shock absorptive and to develop an artificial bio-structure, material prepared from those properties. The solid-fluid structure of the cancellous bone consists of the trabecular bone as a threedimensional (3-D) open-cell structure and the marrow in the bones as a viscous fluid. But simplified modelling appears to be effective in
Fibre-reinforced composites have attracted attention by way of tailored materials and have been studied. But they are used in practice as quasi-isotropic materials and their functions are not utilized effectively. However, it has been shown in many reports that biomaterials have fibre-reinforced composite structures and are optimally designed to loadings. We have studied bone structures from the viewpoint of composite materials and learnt the mechanism of optimum structures of the bone [1,2]. It was found from biomechanical results that the natural structures are perfectly optimally designed. The aim of this research was to develop biomimetic-designed composites. From the results of our biomechanics research we 499
500
7: Katayama, H. Yamamoto, K. Nishitani
analysing the essential functions of the cancellous bone because 3-D analysis complicates the problem. In this paper, the two-dimensional (2-D) honeycomb structure filled with fluid is investigated by way of a simplified solid-fluid composite material model for the cancellous bone. To evaluate the load-support system under compressive loadings, a 2-D closed hexagonal cell filled with fluid is used. At first, hybrid finite element analyses were used to estimate the influence of the solid-fluid phase interaction on the compressive deformation of the material. To verify the numeric result, inplane indentation tests were carried out and the load-indentation depth curves are obtained. And the cell’s in-plane deformation distributions of the solid-fluid composite specimens were measured.
MODELLING ANALYSIS OF CANCELLOUS BONE STRUCTURE Finite element modelling analysis of the cancellous bone in the bio-joints was investigated in order to obtain a biomimetic design of the new composites. The hexagonal unit used to replicate the solid-fluid composition of the cancellous bone is shown in Fig. 1. It is possible to express the dynamic behaviour of the solidfluid composite systems in the hybrid finite
element model with the following two assumptions [I]: the solid-fluid composition of the cancellous bone will be reproduced two-dimensionally by combining the hexagonal frame, which has a framework of two-noded beam elements rigidly jointed to each other, with the core which has six three-noded triangular elements [2]. Each hexagonal core will have a compressive modulus alone corresponding to the resistance of the bone marrow to the dynamic compressive stress and will contribute to the stress transmission of the cancellous bone independently. By assuming geometrical symmetry of the human proximal tibia to simplify the treatment, a 2-D model was applied to the right half. Figure l(a) shows the hybrid finite element model of the tibia1 cancellous bone. The hybrid model consists of triangular elements for the bone marrow, beam elements for the trabecular bone and triangular elements for the cortical bone. The material properties given to the finite elements are listed in Table 1. Inplane elastic moduli of triangular elements for the bone marrow were calculated as an isotropic body. However, the Poisson ratio of an incompressive fluid is 0.5, so 0.49 was selected in order to enable the numeric analysis to be carried out. The left-hand sides of the models were fixed in the horizontal direction only, and the bases were fixed in both the vertical and horizontal directions. A large displacement
Beam element for trabecula
Beam element
Triangular bone marrow
( a ) Using hybrid element modelling
Fig. 1. Finite element
model for cancellous
( b ) Using mono element modelling
bone of the proximal tibia. (a) Using hybrid element mono-element modelling.
modelling
and (b) using
501
Two-dimensional modelling of solid-fluid composites ‘lhble 1. Material properties for finite element models Material (finite element) Trabecular bone (beam) Bone marrow (triangular) Cortical bone (triangular)
Young’s modulus (MPa)
Shearing modulus (MPa)
5000 1000 14000
3500
corresponding to the dynamic compressive loading was applied, and the incremental finite element analyses were carried out. Figure 2 shows distributions of the maximum principal strain of the hexagonal units assuming that the cancellous bone is under compressive loading by the finite element method (FEM). Although the load is expected to be uniformly distributed on the articular surface in the ideal case, the stress propagation behaviour under one-point concentrated compressive loading was simulated in order to examine more clearly the pattern of stress dispersion under the great force assuming dynamic compressive loading. Figure 2(a) and (b) shows the results using hybrid element modelling and mono-element modelling for the cancellous bone, respectively. From Fig. 2(a), appropriate maximum principal strains were distributed sufficiently in the cancellous bone. It is expected that the applied compressive force on the articular surface is dispersed perfectly through the cancellous bone
Poisson’s ratio
0.49 0.35
and is transmitted to the cortical bone as a uniform compressive stress. However, large maximum principal strains will be distributed near the loading point because of the considerably high compressive stresses applied locally in the analysis. To emphaisize the effect of the solid-fluid interaction, the numeric results for the case of disregarding the solid-fluid interaction are shown in Fig. 2(b). The result indicates that the bone marrow only flows in the trabecular bone framework under compressive loading and does not contribute to the stress propagation through the cancellous bone. From the figure it was found that large maximum principal strains were distributed from the loading point through a part of the cancellous bone near the cortical bone and that maximum principal strains were not distributed sufficiently in the cancellous bone. Particularly, considerable large maximum principal strains were distributed under the loading point. Therefore, comparing Fig. 2(a) with (b), if the bone mar-
0.013
0.010
( a > Using hybrid element modelling Fig. 2. Distributions
of maximum
principal
( b ) Using mono element modelling
strain by FEM. (a) Using hybrid element element modelling.
modelling
and (b) using mono-
502
7: Katayama, H. Yamamoto, K. Nishitani
row was out of all relation to the stress propagation through the cancellous bone, ‘micro-buckling’ of the trabeculae would occur in the cancellous bone. Then, it is expected that the compressive loading is spent in the microfracture of the trabeculae and is poorly propagated to the cortical bone. The results coincident with the behaviour of the solid-fluid composite system under compressive loading were obtained from this analysis.
NUMERIC ANALYSIS OF 2-D SOLIDFLUID COMPOSITES From the numeric results of the cancellous bone we propose solid-fluid composites for the stable support of compressive loadings. Hybrid finite element analyses were done in order to estimate the influence of the solid-fluid phase interaction on the compressive deformation of the following material models. The deformation of the specimens was regarded as a plane strain condition due to the experimental conditions. Figure 3 shows the 2-D finite element models for the solid-fluid composites (following the type A specimen) and the solid material (following the type C specimen). The hybrid hexagonal model unit for the solid-fluid com-
posites consists of six beam elements with two rigid joints and six triangular elements with three nodes. The material properties given to the finite elements are listed in Table 2. The bulk modulus of the water under ordinary temperature and normal pressure was applied to the triangular elements. That is to say, the inplane elastic moduli of the triangular elements were calculated as an isotropic body. However, the Poisson ratio of an incompressive fluid is 0.5, and 0.49 was selected in order to enable the numeric analysis to be carried out. Considering the symmetry of the boundary conditions, the right-hand half of the specimen was analysed. The left-hand sides of the models were fixed in the horizontal direction only, and the bases were fixed in both vertical and horizontal directions. A displacement corresponding to the experimental results was applied, and the incremental finite element analyses were carried out. Figure 4 shows the deformation distributions of the solid-fluid composites and the solid material by FEM. As for the solid material, positive large deformations were distributed in the central part and along the diagonal line. Negative large deformations existed in the small-side part. The solid material supported the compressive load along the anisotropic axis of the honeycomb material. As for the solid-
Forced displacement
Forced displacemelit Beam element
Triangular element
( a ) For solid-fluid composites
Fig. 3. Finite element
models. (a) For solid-fluid
( b ) For solid materials
composites
and (b) for solid materials.
Table 2. Material properties for finite element models Material (finite element) A5052H (BEAM) Glycerine (triangular)
Young’s modulus (MPa)
Bulk modulus (MPa)
Poisson’s ratio
7100 132
2200
0.49
Yield stress (MPa) 340 -
Two-dimensional modelling of solid-fluid composites
503
Aa [deg.] ( a ) For solid-fluid composites
Aa Ideg. Fig. 4.
( b ) For solid materials Distributions of a cell’s in-plane deformation by FEM. (a) For solid-fluid composites and (b) for solid materials.
fluid composites, positive large deformations were distributed in the central part, and negative large deformations were distributed in the part along the horizontal line. The solid-fluid composites supported the compressive load in the wide part of the surface not only in the solid material but also in the fluid material. Consequently, these results illustrated that the solid-fluid phase interactions worked well on the load-support system of the solid-fluid composites. EXPERIMENTAL SOLID-FLUID COMPOSITE MODELS Aluminium honeycomb sheet was used for the 2-D experimental model of the solid-fluid composite because aluminium alloy is easy to assemble. The honeycomb was made of alumin-
ium foil (A5052H) of 0.025 mm thickness and 23 mm width. The dimensions of the hexagonal cells are shown in Table 3. Honeycomb sheets were cut into pieces 85 mm high and 120 mm wide. The dimensions of the test piece are shown in Fig. 5. Four kinds of specimens were prepared to characterize the mechanical functions of the solid-fluid composite. Theses are listed in Table 4. Both type A and type B specimens are solid-fluid composite models, and both type C and type D specimens are solid material models. Type A and type B specimens are the solid-liquid composites and the solid-air composites, respectively. The hexagonal cells of the type A specimens and type B specimens were all filled with glycerine and air, respectively, under ordinary temperature and normal pressure to observe the effect of the compressibility. Both side surfaces of type A, type B and type C
‘Igble 3. Dimensions of hexagonal cells
Material A5052H
d (mm) 4.56
t (mm) 0.025
b (mm) 23.00
8 (degress) 101.0
7: Katayama, H. Yamamoto, K. Nishitani
504
Indenter
Enlarged figure of the loading point
Fig. 5. Experimental
conditions
specimens were all sealed with polyvinyl film using an epoxy elastic adhesive. But type C specimens were made of aluminium foil with holes of about 10 pm diameter. Type C specimens were used in order to determine the influence of the film, so the deformation of the specimens is not influenced by pressure in the hexagonal cells. Type D specimens have a honeycomb structure only.
LOAD-INDENTATION
DEPTH CURVES
An in-plane indentation test was proposed in order to evaluate the deformation in the specicompressive loading. The mens under experimental conditions of the in-plane indentation test are shown in Fig. 5. However, honeycomb sheets are stable to out-of-plane compression, the in-plane compressive deformations are anisotropic and complex because of the geometric deformations of the hexagonal cells. However, the in-plane deformations are evaluated easier when compressing honeycomb sheets, as shown in the figure. The indenter which has a flat head was set on the load cell of the universal testing machine. The specimens were set on the base to connect the indenter with the centre cell which was stiffened with polyester resin to obtain symmetric in-plane deformation and stable deformation. Indentation tests were carried out statically using the
of the in-plane
indentation
test.
testing machine at a cross-head speed of 5 mm/ min, and the load-indentation depth curves were obtained. Figure 6 shows typical loadindentation depth curves of four types of the specimen. The curve of the type D specimen is linear until the indentation depth reached 3 mm and shows temporary drops and rises in the load after the indentation depth went above 5 mm. The first knee point of the curve is the indentation depth of 3 mm because the initial deformation of the specimen shows simple elastic behaviour. When compressing cellular materials, the geometric rigidity decreases because of buckling. The curve becomes nonlinear over the knee point because of the buckling of the hexagonal cells. Structural condensation by the collapse of the hexagonal cells causes the load to rise. Although they have an energy absorption property, they do not suit the load-supporting material. The type C specimen shows itself to be a little more rigid than the type D specimen because both side surfaces of the type C specimen were sealed with polyvinyl film. But the type C specimen also shows linear behaviour until the indentation depth reached 3 mm and temporarily drops and rises with the load. The curve of the type C specimen is the same result as that for the type D specimen. That is to say, sealing with polyvinyl film did not change the in-plane compressive deformation pattern of the honeycomb sheet.
Table 4. List of specimens Specimen name
Base material condition
Fluid material
Side surfaces condition
Type Type Type Type
Without holes Without holes With holes Without holes
Glycerine Air None None
Film adhesion Film adhesion Film adhesion Free
A B C D
specimen specimen specimen specimen
Two-dimensional modelling of solid-fluid composites
505
100 A = Type A specimen 90
B = Type B specimen C = Type C specimen
80
D = Type D specimen 70
60 z 50 s 3 40
30
20
10
0
5 Indentation
10 depth
15
[mm]
Fig. 6. Measurement of a cell’s in-plane deformation.
A significant difference is seen between the curves of the type B and type C specimens. The curve of the type B specimen stayed linear until the indentation depth reached 3 mm, but did not have temporary drops and rises in load above 5 mm indentation depth. The curve had only one knee point because it was almost linear above 3 mm in the indentation depth. The initial deformation of the type B specimen was the same mechanism as with the type C specimen. It was considered that the difference between the curves of type B and type C specimens after the knee point was caused by increasing the air pressure in the hexagonal cells. When the curve is over the knee point, the geometric rigidity of the specimen decreased, and then the compressive deformations of the hexagonal cells raise the air pressure in the cells locally, which as a result, contributes to support the load. For the type A specimen the curve was linear until the indentation depth reached 8 mm. It was observed that polyvinyl film rose from the side surface of the hexagonal cells under the indenter because of the high liquid pressure of the glycerine.
IN-PLANE
DEFORMATIONS
OF THE CELL
The load-support system of the solid-fluid composites was examined by measuring the side surface deformation of the specimens. The side surface pictures were taken using a camera during the indentation tests. In order to minimize an error of measurement, the in-plane deformation of the specimens was carried out using the shifts of the joints between the hexagonal cells. The shifts of the joints were measured by the difference between the pictures under the indentation tests and the initial condition. From the side surface pictures of the specimens, three in-plane deformation modes of the hexagonal cells were found around the loading point, as shown in Fig. 7. Mode I deformations came from vertical compression of the rigid cell walls and were distributed in the central part of Mode II deformations came the specimens. from the horizontal compression at the joints between the thin cell walls on both sides of the mode I deformations and mode II deformations were distributed in both sides of the specimens.
T. Katayama, H. Yamamoto, K. Nishitani
506
Thin cell wall Initial condition
Moile I in-plane deformation
Mode III
Mode II modes of hexagonal cells
Fig. 7. Typic load-indentation
Mode III deformations came from the diagonal compression of joints between the rigid and the thin cell walls under both mode I and mode II deformations and were distributed in the parts between the mode I and mode II deformations. The in-plane deformation distribution including these three modes can be evaluated by measuring the change of angle between the thin cell walls as a parameter. Then the type A specimen is expected to have an good load dispersion property because the sufficient liquid pressure of the glycerin in each cell acts effectively on the load transmission. Mode I and mode III deformations are distributed where the changes of the angle are positive, and mode II deformations are distributed where the changes of the angle are negative. Figure 8 shows the in-plane deformation distributions of type A, type B and type C specimens at an indentation depth of 5.0 mm. As for the type C specimen, positive large deformations were distributed in the central part and along the diagonal line. The maximum deformations under the loading point are mode I deformations, and the large deformations in the part along the diagonal line are mode I or mode III deformations. Negative large deformations existed in the small side part. The type C specimen supports the compressive load along the anisotropic axis of the honeycomb material
depth curves.
because only the solid material supports the load. When the hexagonal cells collapse by mode I deformation under the loading point, large buckling comes out of mode I or mode III deformations in the part along the diagonal line. As for the type A specimen, positive large deformations were distributed in the central part, and negative large deformations were distributed along the horizontal line. The positive large deformations in the central part are mode I deformations, and the negative large deformations along the horizontal line are mode II deformations. The type A specimen supports the compressive load in the wide part, because there not only solid material but also liquid material supports the load. Considering the experimental conditions large bucklings never came out of mode II deformation along the horizontal line. As for the type B specimen, the cell’s in-plane deformation distribution displayed a result in between the type A and type C specimens.
CONCLUSION From the viewpoint of the biomimetic design, the 2-D honeycomb structure filled with fluid was investigated by way of a simplified solidfluid composite material model from a
Two-dimensional modeling of solid-fluid composites
507
-6
Aa Ideg. ( a ) Type A specimen
-6
Au [dc
Aa Ideg. ( b ) Type B specimen
6
-6 . .. na jcieg.1
ha 1deg.l
( c ) Type C specimen
Fig. 8. Distributions
of a cell’s in-plane
deformation.
(a) Type A specimen,
cancellous bone at the joint. The results of a hybrid finite element analysis pointed out the efficiency of the solid-fluid phase interaction of the solid-fluid composites on the load dispersion. In-plane indentation tests were carried out and the load-indentation depth curve of the solid-fluid composites showed a good load support. The cell’s in-plane deformation distribution of the solid-fluid composites displayed a high load dispersion property.
(b) type B specimen
and (c) type C specimen.
REFERENCES Katayama, T., Yamamoto, H., Ishiyama, H., Hirasawa, Y., Inoue, N. and Watanabe, Y., A study on initial of prosthesis based on biomechanical fixation behaviour of subchondral bone. In Proceedings of the 2nd Worid Congress of Biomechanics, Amsterdam, Vol. 1,1994, p. 106A. Katayama, T., Yamamoto, H. and Inoue, N., Optimum design of artificial joints considering initial fixation of prosthesis. Composite Struct., 1995, 32, 427-433.
Itvo-dimensional modelling of solid-fluid composites Tsutao Katayama Department of Mechanical Engineering, Doshisha University, Tanabe-cho, Tuzuki-gun, Kyoto-fi 610-03, Japan
Hidetake Yamamoto Faculty of Mechanical Engineering, Kobe City College of Technology, Gakuenhigashi-cho, Nishi-ku, Kobe 651-21, Japan
&
Kazuo Nishitani Postgraduate Course, Doshisha University, Tanabe-cho, Tuzuki-gun, Kyoto-fu 610-03, Japan
The purpose of this paper is to apply biomimetic-designed composites to artificial structures. From the results of numeric modelling analysis in biomechanics, we have learned the bone structures optimized to lighten weight and understood that the solid-fluid composite structure of the cancellous bone at the joint part works to distribute the joint load perfectly. In this paper, the two-dimensional honeycomb structure filled with fluid was investigated by way of a simplified solid-fluid composite material model of the cancellous bone. Hybrid finite element analyses illustrated that the solid-fluid phase interaction is effective in dispersing compressive load. Inplane indentation tests were carried out and in-plane deformation distributions of the solid-fluid composite specimens were measured. Consequently, as for the solid-fluid composite specimens whose cells were filled up with glycerine, a good enough cell deformation mode was obtained. 0 1997 Elsevier Science Ltd.
INTRODUCTION
have considered applying the mechanical functions of the bone’s composite structure to new material designs. We have recognized the solidfluid composite structure of the cancellous bone in the bio-joints to be a material system and investigated these functions as an intelligent material; we will try to develop new materials given these functions artificially. The purpose of this study is to learn from the cancellous bone, which has a good load dispersive and a high shock absorptive and to develop an artificial bio-structure, material prepared from those properties. The solid-fluid structure of the cancellous bone consists of the trabecular bone as a threedimensional (3-D) open-cell structure and the marrow in the bones as a viscous fluid. But simplified modelling appears to be effective in
Fibre-reinforced composites have attracted attention by way of tailored materials and have been studied. But they are used in practice as quasi-isotropic materials and their functions are not utilized effectively. However, it has been shown in many reports that biomaterials have fibre-reinforced composite structures and are optimally designed to loadings. We have studied bone structures from the viewpoint of composite materials and learnt the mechanism of optimum structures of the bone [1,2]. It was found from biomechanical results that the natural structures are perfectly optimally designed. The aim of this research was to develop biomimetic-designed composites. From the results of our biomechanics research we 499
500
7: Katayama, H. Yamamoto, K. Nishitani
analysing the essential functions of the cancellous bone because 3-D analysis complicates the problem. In this paper, the two-dimensional (2-D) honeycomb structure filled with fluid is investigated by way of a simplified solid-fluid composite material model for the cancellous bone. To evaluate the load-support system under compressive loadings, a 2-D closed hexagonal cell filled with fluid is used. At first, hybrid finite element analyses were used to estimate the influence of the solid-fluid phase interaction on the compressive deformation of the material. To verify the numeric result, inplane indentation tests were carried out and the load-indentation depth curves are obtained. And the cell’s in-plane deformation distributions of the solid-fluid composite specimens were measured.
MODELLING ANALYSIS OF CANCELLOUS BONE STRUCTURE Finite element modelling analysis of the cancellous bone in the bio-joints was investigated in order to obtain a biomimetic design of the new composites. The hexagonal unit used to replicate the solid-fluid composition of the cancellous bone is shown in Fig. 1. It is possible to express the dynamic behaviour of the solidfluid composite systems in the hybrid finite
element model with the following two assumptions [I]: the solid-fluid composition of the cancellous bone will be reproduced two-dimensionally by combining the hexagonal frame, which has a framework of two-noded beam elements rigidly jointed to each other, with the core which has six three-noded triangular elements [2]. Each hexagonal core will have a compressive modulus alone corresponding to the resistance of the bone marrow to the dynamic compressive stress and will contribute to the stress transmission of the cancellous bone independently. By assuming geometrical symmetry of the human proximal tibia to simplify the treatment, a 2-D model was applied to the right half. Figure l(a) shows the hybrid finite element model of the tibia1 cancellous bone. The hybrid model consists of triangular elements for the bone marrow, beam elements for the trabecular bone and triangular elements for the cortical bone. The material properties given to the finite elements are listed in Table 1. Inplane elastic moduli of triangular elements for the bone marrow were calculated as an isotropic body. However, the Poisson ratio of an incompressive fluid is 0.5, so 0.49 was selected in order to enable the numeric analysis to be carried out. The left-hand sides of the models were fixed in the horizontal direction only, and the bases were fixed in both the vertical and horizontal directions. A large displacement
Beam element for trabecula
Beam element
Triangular bone marrow
( a ) Using hybrid element modelling
Fig. 1. Finite element
model for cancellous
( b ) Using mono element modelling
bone of the proximal tibia. (a) Using hybrid element mono-element modelling.
modelling
and (b) using
501
Two-dimensional modelling of solid-fluid composites ‘lhble 1. Material properties for finite element models Material (finite element) Trabecular bone (beam) Bone marrow (triangular) Cortical bone (triangular)
Young’s modulus (MPa)
Shearing modulus (MPa)
5000 1000 14000
3500
corresponding to the dynamic compressive loading was applied, and the incremental finite element analyses were carried out. Figure 2 shows distributions of the maximum principal strain of the hexagonal units assuming that the cancellous bone is under compressive loading by the finite element method (FEM). Although the load is expected to be uniformly distributed on the articular surface in the ideal case, the stress propagation behaviour under one-point concentrated compressive loading was simulated in order to examine more clearly the pattern of stress dispersion under the great force assuming dynamic compressive loading. Figure 2(a) and (b) shows the results using hybrid element modelling and mono-element modelling for the cancellous bone, respectively. From Fig. 2(a), appropriate maximum principal strains were distributed sufficiently in the cancellous bone. It is expected that the applied compressive force on the articular surface is dispersed perfectly through the cancellous bone
Poisson’s ratio
0.49 0.35
and is transmitted to the cortical bone as a uniform compressive stress. However, large maximum principal strains will be distributed near the loading point because of the considerably high compressive stresses applied locally in the analysis. To emphaisize the effect of the solid-fluid interaction, the numeric results for the case of disregarding the solid-fluid interaction are shown in Fig. 2(b). The result indicates that the bone marrow only flows in the trabecular bone framework under compressive loading and does not contribute to the stress propagation through the cancellous bone. From the figure it was found that large maximum principal strains were distributed from the loading point through a part of the cancellous bone near the cortical bone and that maximum principal strains were not distributed sufficiently in the cancellous bone. Particularly, considerable large maximum principal strains were distributed under the loading point. Therefore, comparing Fig. 2(a) with (b), if the bone mar-
0.013
0.010
( a > Using hybrid element modelling Fig. 2. Distributions
of maximum
principal
( b ) Using mono element modelling
strain by FEM. (a) Using hybrid element element modelling.
modelling
and (b) using mono-
502
7: Katayama, H. Yamamoto, K. Nishitani
row was out of all relation to the stress propagation through the cancellous bone, ‘micro-buckling’ of the trabeculae would occur in the cancellous bone. Then, it is expected that the compressive loading is spent in the microfracture of the trabeculae and is poorly propagated to the cortical bone. The results coincident with the behaviour of the solid-fluid composite system under compressive loading were obtained from this analysis.
NUMERIC ANALYSIS OF 2-D SOLIDFLUID COMPOSITES From the numeric results of the cancellous bone we propose solid-fluid composites for the stable support of compressive loadings. Hybrid finite element analyses were done in order to estimate the influence of the solid-fluid phase interaction on the compressive deformation of the following material models. The deformation of the specimens was regarded as a plane strain condition due to the experimental conditions. Figure 3 shows the 2-D finite element models for the solid-fluid composites (following the type A specimen) and the solid material (following the type C specimen). The hybrid hexagonal model unit for the solid-fluid com-
posites consists of six beam elements with two rigid joints and six triangular elements with three nodes. The material properties given to the finite elements are listed in Table 2. The bulk modulus of the water under ordinary temperature and normal pressure was applied to the triangular elements. That is to say, the inplane elastic moduli of the triangular elements were calculated as an isotropic body. However, the Poisson ratio of an incompressive fluid is 0.5, and 0.49 was selected in order to enable the numeric analysis to be carried out. Considering the symmetry of the boundary conditions, the right-hand half of the specimen was analysed. The left-hand sides of the models were fixed in the horizontal direction only, and the bases were fixed in both vertical and horizontal directions. A displacement corresponding to the experimental results was applied, and the incremental finite element analyses were carried out. Figure 4 shows the deformation distributions of the solid-fluid composites and the solid material by FEM. As for the solid material, positive large deformations were distributed in the central part and along the diagonal line. Negative large deformations existed in the small-side part. The solid material supported the compressive load along the anisotropic axis of the honeycomb material. As for the solid-
Forced displacement
Forced displacemelit Beam element
Triangular element
( a ) For solid-fluid composites
Fig. 3. Finite element
models. (a) For solid-fluid
( b ) For solid materials
composites
and (b) for solid materials.
Table 2. Material properties for finite element models Material (finite element) A5052H (BEAM) Glycerine (triangular)
Young’s modulus (MPa)
Bulk modulus (MPa)
Poisson’s ratio
7100 132
2200
0.49
Yield stress (MPa) 340 -
Two-dimensional modelling of solid-fluid composites
503
Aa [deg.] ( a ) For solid-fluid composites
Aa Ideg. Fig. 4.
( b ) For solid materials Distributions of a cell’s in-plane deformation by FEM. (a) For solid-fluid composites and (b) for solid materials.
fluid composites, positive large deformations were distributed in the central part, and negative large deformations were distributed in the part along the horizontal line. The solid-fluid composites supported the compressive load in the wide part of the surface not only in the solid material but also in the fluid material. Consequently, these results illustrated that the solid-fluid phase interactions worked well on the load-support system of the solid-fluid composites. EXPERIMENTAL SOLID-FLUID COMPOSITE MODELS Aluminium honeycomb sheet was used for the 2-D experimental model of the solid-fluid composite because aluminium alloy is easy to assemble. The honeycomb was made of alumin-
ium foil (A5052H) of 0.025 mm thickness and 23 mm width. The dimensions of the hexagonal cells are shown in Table 3. Honeycomb sheets were cut into pieces 85 mm high and 120 mm wide. The dimensions of the test piece are shown in Fig. 5. Four kinds of specimens were prepared to characterize the mechanical functions of the solid-fluid composite. Theses are listed in Table 4. Both type A and type B specimens are solid-fluid composite models, and both type C and type D specimens are solid material models. Type A and type B specimens are the solid-liquid composites and the solid-air composites, respectively. The hexagonal cells of the type A specimens and type B specimens were all filled with glycerine and air, respectively, under ordinary temperature and normal pressure to observe the effect of the compressibility. Both side surfaces of type A, type B and type C
‘Igble 3. Dimensions of hexagonal cells
Material A5052H
d (mm) 4.56
t (mm) 0.025
b (mm) 23.00
8 (degress) 101.0
7: Katayama, H. Yamamoto, K. Nishitani
504
Indenter
Enlarged figure of the loading point
Fig. 5. Experimental
conditions
specimens were all sealed with polyvinyl film using an epoxy elastic adhesive. But type C specimens were made of aluminium foil with holes of about 10 pm diameter. Type C specimens were used in order to determine the influence of the film, so the deformation of the specimens is not influenced by pressure in the hexagonal cells. Type D specimens have a honeycomb structure only.
LOAD-INDENTATION
DEPTH CURVES
An in-plane indentation test was proposed in order to evaluate the deformation in the specicompressive loading. The mens under experimental conditions of the in-plane indentation test are shown in Fig. 5. However, honeycomb sheets are stable to out-of-plane compression, the in-plane compressive deformations are anisotropic and complex because of the geometric deformations of the hexagonal cells. However, the in-plane deformations are evaluated easier when compressing honeycomb sheets, as shown in the figure. The indenter which has a flat head was set on the load cell of the universal testing machine. The specimens were set on the base to connect the indenter with the centre cell which was stiffened with polyester resin to obtain symmetric in-plane deformation and stable deformation. Indentation tests were carried out statically using the
of the in-plane
indentation
test.
testing machine at a cross-head speed of 5 mm/ min, and the load-indentation depth curves were obtained. Figure 6 shows typical loadindentation depth curves of four types of the specimen. The curve of the type D specimen is linear until the indentation depth reached 3 mm and shows temporary drops and rises in the load after the indentation depth went above 5 mm. The first knee point of the curve is the indentation depth of 3 mm because the initial deformation of the specimen shows simple elastic behaviour. When compressing cellular materials, the geometric rigidity decreases because of buckling. The curve becomes nonlinear over the knee point because of the buckling of the hexagonal cells. Structural condensation by the collapse of the hexagonal cells causes the load to rise. Although they have an energy absorption property, they do not suit the load-supporting material. The type C specimen shows itself to be a little more rigid than the type D specimen because both side surfaces of the type C specimen were sealed with polyvinyl film. But the type C specimen also shows linear behaviour until the indentation depth reached 3 mm and temporarily drops and rises with the load. The curve of the type C specimen is the same result as that for the type D specimen. That is to say, sealing with polyvinyl film did not change the in-plane compressive deformation pattern of the honeycomb sheet.
Table 4. List of specimens Specimen name
Base material condition
Fluid material
Side surfaces condition
Type Type Type Type
Without holes Without holes With holes Without holes
Glycerine Air None None
Film adhesion Film adhesion Film adhesion Free
A B C D
specimen specimen specimen specimen
Two-dimensional modelling of solid-fluid composites
505
100 A = Type A specimen 90
B = Type B specimen C = Type C specimen
80
D = Type D specimen 70
60 z 50 s 3 40
30
20
10
0
5 Indentation
10 depth
15
[mm]
Fig. 6. Measurement of a cell’s in-plane deformation.
A significant difference is seen between the curves of the type B and type C specimens. The curve of the type B specimen stayed linear until the indentation depth reached 3 mm, but did not have temporary drops and rises in load above 5 mm indentation depth. The curve had only one knee point because it was almost linear above 3 mm in the indentation depth. The initial deformation of the type B specimen was the same mechanism as with the type C specimen. It was considered that the difference between the curves of type B and type C specimens after the knee point was caused by increasing the air pressure in the hexagonal cells. When the curve is over the knee point, the geometric rigidity of the specimen decreased, and then the compressive deformations of the hexagonal cells raise the air pressure in the cells locally, which as a result, contributes to support the load. For the type A specimen the curve was linear until the indentation depth reached 8 mm. It was observed that polyvinyl film rose from the side surface of the hexagonal cells under the indenter because of the high liquid pressure of the glycerine.
IN-PLANE
DEFORMATIONS
OF THE CELL
The load-support system of the solid-fluid composites was examined by measuring the side surface deformation of the specimens. The side surface pictures were taken using a camera during the indentation tests. In order to minimize an error of measurement, the in-plane deformation of the specimens was carried out using the shifts of the joints between the hexagonal cells. The shifts of the joints were measured by the difference between the pictures under the indentation tests and the initial condition. From the side surface pictures of the specimens, three in-plane deformation modes of the hexagonal cells were found around the loading point, as shown in Fig. 7. Mode I deformations came from vertical compression of the rigid cell walls and were distributed in the central part of Mode II deformations came the specimens. from the horizontal compression at the joints between the thin cell walls on both sides of the mode I deformations and mode II deformations were distributed in both sides of the specimens.
T. Katayama, H. Yamamoto, K. Nishitani
506
Thin cell wall Initial condition
Moile I in-plane deformation
Mode III
Mode II modes of hexagonal cells
Fig. 7. Typic load-indentation
Mode III deformations came from the diagonal compression of joints between the rigid and the thin cell walls under both mode I and mode II deformations and were distributed in the parts between the mode I and mode II deformations. The in-plane deformation distribution including these three modes can be evaluated by measuring the change of angle between the thin cell walls as a parameter. Then the type A specimen is expected to have an good load dispersion property because the sufficient liquid pressure of the glycerin in each cell acts effectively on the load transmission. Mode I and mode III deformations are distributed where the changes of the angle are positive, and mode II deformations are distributed where the changes of the angle are negative. Figure 8 shows the in-plane deformation distributions of type A, type B and type C specimens at an indentation depth of 5.0 mm. As for the type C specimen, positive large deformations were distributed in the central part and along the diagonal line. The maximum deformations under the loading point are mode I deformations, and the large deformations in the part along the diagonal line are mode I or mode III deformations. Negative large deformations existed in the small side part. The type C specimen supports the compressive load along the anisotropic axis of the honeycomb material
depth curves.
because only the solid material supports the load. When the hexagonal cells collapse by mode I deformation under the loading point, large buckling comes out of mode I or mode III deformations in the part along the diagonal line. As for the type A specimen, positive large deformations were distributed in the central part, and negative large deformations were distributed along the horizontal line. The positive large deformations in the central part are mode I deformations, and the negative large deformations along the horizontal line are mode II deformations. The type A specimen supports the compressive load in the wide part, because there not only solid material but also liquid material supports the load. Considering the experimental conditions large bucklings never came out of mode II deformation along the horizontal line. As for the type B specimen, the cell’s in-plane deformation distribution displayed a result in between the type A and type C specimens.
CONCLUSION From the viewpoint of the biomimetic design, the 2-D honeycomb structure filled with fluid was investigated by way of a simplified solidfluid composite material model from a
Two-dimensional modeling of solid-fluid composites
507
-6
Aa Ideg. ( a ) Type A specimen
-6
Au [dc
Aa Ideg. ( b ) Type B specimen
6
-6 . .. na jcieg.1
ha 1deg.l
( c ) Type C specimen
Fig. 8. Distributions
of a cell’s in-plane
deformation.
(a) Type A specimen,
cancellous bone at the joint. The results of a hybrid finite element analysis pointed out the efficiency of the solid-fluid phase interaction of the solid-fluid composites on the load dispersion. In-plane indentation tests were carried out and the load-indentation depth curve of the solid-fluid composites showed a good load support. The cell’s in-plane deformation distribution of the solid-fluid composites displayed a high load dispersion property.
(b) type B specimen
and (c) type C specimen.
REFERENCES Katayama, T., Yamamoto, H., Ishiyama, H., Hirasawa, Y., Inoue, N. and Watanabe, Y., A study on initial of prosthesis based on biomechanical fixation behaviour of subchondral bone. In Proceedings of the 2nd Worid Congress of Biomechanics, Amsterdam, Vol. 1,1994, p. 106A. Katayama, T., Yamamoto, H. and Inoue, N., Optimum design of artificial joints considering initial fixation of prosthesis. Composite Struct., 1995, 32, 427-433.