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A note on the ashby-gibson relation
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TECHNICAL NOTE A NOTE ON THE ASHBY-GIBSON K. D.
RELATION
PITHIA
South Bank University, Wandsworth Road, London SW8 212, U.K. (Received 8 February 1993)
Abstract-The Ashby-Gibson model of cellular materials is modified to take into account the space filling constraint. It is shown that the Ashby-Gibson relation is modified by a factor (b-* where I$ is the volume gas fraction.
INTRODUCTION The analysis by Ashby and Gibson of solid cellular materials has shown that the mechanical properties of foamed materials can be calculated from the relative density of the foamed material (11. This derivation however neglects the space filling constraint [2]. This constraint is important on two accounts: (i) the space filling constraint is important in the evolution of liquid foam [2]; this may arise when considering the manufacture of solid foams [3] and (ii) when considering building structures as cellular materials [4] in this model the cells correspond to macroscopic structures which can be changed in the limit of high gas fractions.
The strain is defined as s/l. Using eqn (3), we have
However, stress over the cell, given by F/l*. The elastic modulus of the foam Er is given by stress/strain. Therefore
where C, is a constant of proportionality. For a square cross-section of side f the second moment of inertia i is proportional to t4, i.e.
CELLULAR STRUCTURES WITH SPACE FILLING CONSTRAINT Consider an ideal cellular material, Fig. 1. We shall consider a cubic cell of length I. The material is also cubic of length L and contains N cells. The cross-section of the struts is square and of length 1. On the application of a force F, the deflection 6 of the struts is then given as
fCrt4 and from the definition of volume fractions Nil2
s=gl
5
(1)
where .E, is the elastic modulus of the strut. This relation applies provided there is a reasonable degree of cellular structure, i.e. in the limit of the dimensions considered the characteristics of the cell size are large in comparison with the strut dimensions, I > t.7 But I3 is the volume of one cell. Thus if the total volume is V and if d, is the volume gas fraction, then the volume of the cells is I’#. Therefore we have,
xcc t2 FE-.
1-4 4 l-4 9
(2)
E,
Nt4
E,”
zj?
(8)
(9) and using
(2) and(8)
where N is the number of cells. This is the space filling constraint provided I$ is constant. Therefore t This point emphasizes that the dimensions of the cells must be larger than the dimensions of the struts for bending to occur. The situation where the cell dimension is very much less than the strut dimension will give rise to a behaviour of the material which will take on the properties of the strut material
(7)
Then using (6) and substituting in (5)
where C is another constant of proportionality, NI’ = v4,
(6)
WV CONCLUSION The Ashby-Gibson model for solid foams is modified. The modification of the AshbyGibson relation results directly from the definition of the volume gas fraction. Not only does this model allow for the space filling 365
366
Technical Note
A
L
Fig. 1.
constraint but provides for a new model of the behaviour of materials approaching a relative density of zero. In the limit of zero relative density the Ashby~ibson relation is obtained. REFERENCES 1.
L. J. Gibson and M. F. Ashby, Celluar Solidr: Strucrure and Properties. Pergamon Press, Oxford (1988).
5. L. Weaire and N. Rivier, Soap cells and statisticsrandom patterns in two dimensions. Contemp. Pbyx 25, 59 (1984). S. C. Warburton, Ph.D. thesis, Cavendish Laboratory (1990). S. Chen and K. D. Pithia, South Bank University Project Report, On treating buildings as cellular materials (1991).
TECHNICAL NOTE A NOTE ON THE ASHBY-GIBSON K. D.
RELATION
PITHIA
South Bank University, Wandsworth Road, London SW8 212, U.K. (Received 8 February 1993)
Abstract-The Ashby-Gibson model of cellular materials is modified to take into account the space filling constraint. It is shown that the Ashby-Gibson relation is modified by a factor (b-* where I$ is the volume gas fraction.
INTRODUCTION The analysis by Ashby and Gibson of solid cellular materials has shown that the mechanical properties of foamed materials can be calculated from the relative density of the foamed material (11. This derivation however neglects the space filling constraint [2]. This constraint is important on two accounts: (i) the space filling constraint is important in the evolution of liquid foam [2]; this may arise when considering the manufacture of solid foams [3] and (ii) when considering building structures as cellular materials [4] in this model the cells correspond to macroscopic structures which can be changed in the limit of high gas fractions.
The strain is defined as s/l. Using eqn (3), we have
However, stress over the cell, given by F/l*. The elastic modulus of the foam Er is given by stress/strain. Therefore
where C, is a constant of proportionality. For a square cross-section of side f the second moment of inertia i is proportional to t4, i.e.
CELLULAR STRUCTURES WITH SPACE FILLING CONSTRAINT Consider an ideal cellular material, Fig. 1. We shall consider a cubic cell of length I. The material is also cubic of length L and contains N cells. The cross-section of the struts is square and of length 1. On the application of a force F, the deflection 6 of the struts is then given as
fCrt4 and from the definition of volume fractions Nil2
s=gl
5
(1)
where .E, is the elastic modulus of the strut. This relation applies provided there is a reasonable degree of cellular structure, i.e. in the limit of the dimensions considered the characteristics of the cell size are large in comparison with the strut dimensions, I > t.7 But I3 is the volume of one cell. Thus if the total volume is V and if d, is the volume gas fraction, then the volume of the cells is I’#. Therefore we have,
xcc t2 FE-.
1-4 4 l-4 9
(2)
E,
Nt4
E,”
zj?
(8)
(9) and using
(2) and(8)
where N is the number of cells. This is the space filling constraint provided I$ is constant. Therefore t This point emphasizes that the dimensions of the cells must be larger than the dimensions of the struts for bending to occur. The situation where the cell dimension is very much less than the strut dimension will give rise to a behaviour of the material which will take on the properties of the strut material
(7)
Then using (6) and substituting in (5)
where C is another constant of proportionality, NI’ = v4,
(6)
WV CONCLUSION The Ashby-Gibson model for solid foams is modified. The modification of the AshbyGibson relation results directly from the definition of the volume gas fraction. Not only does this model allow for the space filling 365
366
Technical Note
A
L
Fig. 1.
constraint but provides for a new model of the behaviour of materials approaching a relative density of zero. In the limit of zero relative density the Ashby~ibson relation is obtained. REFERENCES 1.
L. J. Gibson and M. F. Ashby, Celluar Solidr: Strucrure and Properties. Pergamon Press, Oxford (1988).
5. L. Weaire and N. Rivier, Soap cells and statisticsrandom patterns in two dimensions. Contemp. Pbyx 25, 59 (1984). S. C. Warburton, Ph.D. thesis, Cavendish Laboratory (1990). S. Chen and K. D. Pithia, South Bank University Project Report, On treating buildings as cellular materials (1991).