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DoITPoMS Teaching & Learning Packages Deformation of Honeycombs and Foams Deformation of Honeycombs and Foams (all content)

# Deformation of Honeycombs and Foams (all content)

Note: DoITPoMS Teaching and Learning Packages are intended to be used interactively at a computer! This print-friendly version of the TLP is provided for convenience, but does not display all the content of the TLP. For example, any video clips and answers to questions are missing. The formatting (page breaks, etc) of the printed version is unpredictable and highly dependent on your browser.

## Aims

On completion of this TLP you should:

• Understand what processes determine the elastic behaviour of a honeycomb structure;
• Know what processes cause the onset of yielding;
• Understand how these ideas for simple structures might be extended to more complex ones, such as foams.

## Before you start

There are no special prerequisites for this TLP.

## Introduction

There is an important class of materials, many biological, that are highly porous and made by bonding rods, ribbons or fibres in both regular and irregular structures. These include paper, bone, wood, packaging foam and insulating fibre mats and they can be made of polymers, metals, ceramics and natural materials. Despite the different structures and materials, there are many similarities in how they behave. An important class of these materials is where the rods or ribbons form cells, co-called cellular structures. Here we explain how such structures deform in compression. To understand the deformation processes more easily the behaviour of a regular honeycomb structure is described before extending the ideas to structures such as foams.

## Compression of a honeycomb: Experimental

The honeycomb studied here is an array of regular hexagonal cells, with the cell walls made of thin strips of aluminium. The structure is not quite as simple as it first appears because of the way in which it is made (see details). This causes one in every three cell walls to consist of two layers of aluminium bonded with adhesive, making some walls stiffer than others. In the honeycomb used here the thickness of an individual sheet, t, was 0.09 mm, and the length of each of the cell faces, l, was 6.30 mm. The relative density, the measured density as a fraction of the density of the solid material, is 0.008.

There are two different directions in which a hexagonal honeycomb can be compressed in the plane of the honeycomb.

 Cell walls lie parallel to the loading axis Cell walls lie diagonal to the loading axis

Although quantitatively different, the basic deformation processes are similar in both cases so only the situation where some cell walls lie parallel to the loading axis is described here.

Squares of honeycomb with 6 cells along each side were cut from a sheet of material. The samples were then compressed between flat, parallel platens at a constant displacement rate of 1 mm min-1, giving the stress-strain curve below.

The stress-strain curve has 3 distinct regions:

• an initial elastic region;
• followed by the onset of irreversible leading to region where the stress does not change with increasing strain, known as the plateau region;
• and lastly a region where the stress again begins to rise rapidly with increasing strain, known as densification.

#### Elastic region

The elastic region is characterised by the effective Young modulus of the material, that is the Young modulus of a uniform material that for the same imposed stresses gives rise to the same strains. This is found by taking the slope of the unloading curve, which helps to reduce the effects of any local plastic flow.

The measured Young modulus was 435 kPa.

#### Plateau region

With continued loading the stresses in the faces increase. Eventually these reach the flow stress of the material and irreversible yielding begins. The stress at which this occurred was approximately 15 kPa. The structure then began to collapse at an approximately constant stress, σP.

#### Densification

This continued up to a strain of ~ 0.7, at which point the stress started to rise much more rapidly as the faces from opposite sides of the cells were pressed up against one another.

This three-stage deformation behaviour is typical of virtually all highly porous materials, even ones made of very brittle materials. The next step is to try and quantitatively describe the observed behaviour.

## Elastic behaviour (I)

To start let us assume that the predominant contribution to the elastic strain comes from the axial compression of the vertical struts, as shown below.

We can estimate the magnitude of this strain as the cross-sectional area of solid material, AV, in a cut across just the vertical faces is less than that if the material were completely solid by the ratio of the cell wall thickness, t, to the horizontal distance across each cell, 2 l cos θ.

As t << l cos θ, this is given by $${A_{\rm{V}}} = t/2l\;\cos \theta$$

Using the measured values of t ( = 0.09 mm) and l ( = 6.30 mm), and taking θ = 30° as the cells are hexagonal gives AV as 0.008. Taking the Young modulus of aluminium as 70 GPa, this predicts the Young modulus of the honeycomb to be 560 MPa. This is greater than the observed value of 435 kPa by more than 3 orders of magnitude and shows that axial compression of the vertical faces makes a negligible contribution to the elastic strain.

## Elastic behaviour (II)

If the axially loaded faces do not deform, then clearly it must be those at an angle to the loading axis that do, the diagonal faces. And because they are at an angle to the loading axis they will bend.

The bending of each face beams must be symmetrical about the mid-point of the face and can be estimated using beam bending theory. To do this each face is described as two beams, cantilevered at the vertices of the hexagonal cell and loaded at the centre point. Note that one beam (i.e. a half cell wall) is pushed upward, the other downward.

This [derivation] gives the Young modulus of the honeycomb as

$E = \frac{4}{{\sqrt 3 }}{E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^3}$

Using the measured values of t( = 0.09 mm) and l ( = 6.30 mm) and taking ES as 70 GPa, the elastic modulus is predicted to be 471 kPa. This is within 10% of the measured value for the sample.

The predominant contribution to the Young modulus is therefore from the bending of the diagonal faces.

## Yielding and plateau behaviour

The aluminium honeycomb will start to plastically deform if the stress in the faces anywhere exceeds the flow stress, σY, of the aluminium cell wall. We have already shown that the predominant contribution to the elastic strain is the bending of the diagonal faces. (see here). Furthermore we could estimate this if each face were considered to be made up of two beams, each of length l/2, cantilevered at the end connected to the vertical cell wall and acted upon by a force of magnitude F cos θ, where F is the force applied at the ends of the sample and θ is the angle between the diagonal face and the horizontal. It is clear then that the stress will be a maximum where the moment is greatest, that is at the vertices of the hexagonal cells.

It can be shown that the applied stress, σ, when the maximum stress in each face reaches the flow stress, σY, of the material making up the cell walls is given by (derivation)

$\sigma = \frac{4}{9} \cdot {\left( {\frac{t}{l}} \right)^2}{\sigma _{\rm{Y}}}$

Using the measured values of t ( = 0.09 mm), l ( = 6.30 mm) and σY ( = 100 MPa), predicts the yield strength of the honeycomb, σ, to be 9 kPa, somewhat lower than the measured value of 15 kPa. The stress we have estimated is the stress at which plastic flow will start in the outer surfaces at the cantilevered point. To enable plastic flow to spread through the thickness of the cell face requires that the stress is increased further by a factor of 1.5, giving a macroscopic flow stress of 13.5 kPa, much closer to the measured value.

Once the material has started to yield the cell walls begin to collapse. This occurs at an approximately constant stress until the cell walls impinge on one another when the stress begins to rise more rapidly with increasing strain.

## Densification

As the honeycomb yields in the plateau region, the regular hexagonal cell with a height (l + 2l sin θ) changes shape with the protruding apices being pressed toward one another to give cells with the shape of a bow-tie and a height l.

If the cells deform uniformly then the strain at which this occurs, εD, is given by

${\varepsilon _{\rm{D}}} = \ln \left( {\frac{l}{{l + 2l\sin \theta }}} \right) = \ln \left( {\frac{1}{{1 + 2\sin \theta }}} \right)$

Note true strain is used because the strains are large and compressive. As θ = 30°, εD, is predicted to have a magnitude of 0.7. Further increases in strain cause opposing cell walls to be pressed against one another and the stress required for further deformation increases rapidly. As can be seen in the stress strain curve below this prediction gives good agreement with the observed stress-strain curve.

It can be seen that this collapse does not occur uniformly throughout the whole structure, but layer by layer of cells. This behaviour is rather dependent on the size of the cell compared to that of the sample. Increasing the number of cells in a cross-section causes the behaviour to become more uniform as might be expected.

It is now possible to quantitatively understand the entire stress-strain behaviour of a simple honeycomb. The next step is to extend these ideas to less regular structures, such as foams and fibrous structures.

## Other porous structures

Many other porous structures show the same type of stress-strain behaviour. The basic reasons are similar. The initial behaviour is elastic, until the stresses in the struts reach their flow or fracture stress. There is then a plateau region as the cells collapse, until the struts from opposite sides of the cells impinge on one another and the applied stress increases more rapidly. However the details can be very different. For instance the struts in ceramic foams tend to break, but a plateau region is still seen.

Many combinations of material and cell structure are possible. Seeing how the cells deform in a foam is more difficult than in the simple honeycomb. However this has been done using X-ray tomography as shown in the short video.

Deformation of cells in foam

Deformation of cells in foam

(For further details see J.A. Elliott et al, “In-situ deformation of an open-cell flexible polyurethane foam characterised by 3D computed microtomography”, J. Mater. Sci. 37 (2002) 1547-1555.)

Looking at the large cell on the right-hand side, it is clear that the deformation of the foam is similar to the honeycomb and the strain comes predominantly from the bending of the struts transverse to the loading axis.

For simplicity, consider the open-cell foam as having a cubic unit cell as shown below.

Note that each transverse strut has a vertical strut half-way along it, so that axial loading causes the struts transverse to the loading axis to bend, as shown in the diagram above. The Young modulus can now be estimated in a similar way to that for the honeycomb, except that the struts are assumed to have a square cross-section, rather than being rectangular as before and θ, the angle between the transverse strut and the horizontal is 0.

This gives an expression for the relative Young modulus, E/ES, as

$\frac{E}{{{E_{\rm{S}}}}} = k{\rm{ }}{\left( {\frac{\rho }{{{\rho _{\rm{S}}}}}} \right)^2}$

where E is the Young modulus of the porous material and ES that of the solid material and k is a numerical constant, experimentally found to approximately equal to 1 (derivation).

As can be seen in the graph above, experiments show this is correct for isotropic, open-cell foams and even appears to be obeyed where the struts are not slender beams and also, at least approximately, where the cells are closed rather than open. This is thought to arise because in most closed-cell foams most of the material is still along the edges of the cells, rather than being uniformly distributed across the faces. (The data is taken from various sources cited in L.J. Gibson and M.F. Ashby, "On the mechanics of three-dimensional cellular materials", Proc. Roy. Soc. A, 382[1782] (1982) 43-59.)

## Porous structures in bending

Porous structures are often used as a lightweight core separating two strong, stiff outer layers to form a sandwich panel. Like the I-beam, such structures have a greater resistance to bending per unit weight of material than a solid beam and so are useful where weight-saving and stiffness are important. Typical applications include flooring panels in aircraft or rotors in helicopter blades. Sandwich structures are also common in biological structures, such as leaves or spongy bone.

 From J. Banhart, Manufacture, Characterisation and Application of Cellular Metals and Metal Foams, Progress Mater. Sci., 2001, 46, pp.559-632. From Cell Biology by Thomas D. Pollard and William C. Earnshaw, Saunders 2004, pp.540 (Figure 34-4), courtesy of D.W. Fawcett, Harvard Medical School.

However some porous solids, such as wood are used without the stiff, strong outer layers. We might ask whether it is generally true, or under what conditions a porous rod will be stiffer in bending (i.e. give a smaller deflection for a given applied force) than a solid rod of the same length and overall mass.

Consider two rods one is porous and the other is solid. As each rod has the same length and mass and a circular cross-section, the porous one must have a larger radius.

The deflection, δ, of a cantilevered beam of length L under an imposed force W is given by

$\delta = \frac{1}{3}\frac{{W{L^3}}}{{EI}}$

For given values of W and L an increase in beam stiffness requires a higher value of the product EI. For a beam of circular cross-section <I = πr4/4. The porous beam has a larger radius and therefore a larger moment of area than the solid beam. However the porous beam also has a lower Young modulus. For the porous beam to be stiffer in bending, the rate at which the moment of area increases with radius must therefore be greater than the rate at which the Young modulus decreases.

If the density of the porous beam is ρ and solid beam is ρS and both have the same length and mass, then the ratio of the radius of the porous beam, r, to that of the solid beam, rS, is

$\left( {\frac{{{\rho _{\rm{S}}}}}{\rho }} \right) = {\left( {\frac{r}{{{r_{\rm{S}}}}}} \right)^2}$

As I ∝ r4 the ratio of the second moments of area of the porous and solid beams, I and IS respectively, is

$\frac{I}{{{I_{\rm{S}}}}} = {\left( {\frac{{{\rho _{\rm{S}}}}}{\rho }} \right)^2}$

In other words I/IS increases as the inverse square of the relative density, ρ/ρS.

Now the expression derived above for the elastic modulus of an open-cell porous body was

$\frac{E}{{{E_{\rm{S}}}}} = k{\rm{ }}{\left( {\frac{\rho }{{{\rho _{\rm{S}}}}}} \right)^2}$

That is E/ES decreases as the square of the relative density. In other words although I is increasing with decreasing density, E is decreasing at the same rate. In this case there would be no advantage in using such a material in bending compared with the solid material.

As nothing can be done about the change in radius, and hence I, with relative density, a higher bending stiffness can only be obtained by ensuring that E/ES varies with ρ/ρS by a power less than 2. This is the case for the axial Young modulus of wood where the exponent lies closer to 1 rather than 2, as shown below.

(The data is taken from K.E. Easterling et al, “On the mechanics of balsa and other woods”, Proc. Roy. Soc. A, 383[1784] (1982) 31-41.) Such changes can be brought about by varying the cell structure, for instance by elongating the cells, as occurs in wood. However in the transverse (radial and tangential) directions E/ES for wood decreases much more rapidly with decreasing ρ/ρS. Here the exponent lies between 2 and 3.

## Summary

In this TLP, the elastic, yielding and densification behaviour of a simple honeycomb structure has been studied experimentally. It is shown that the deformation of a honeycomb structure is made up of 3 main regions: an elastic region, which ends when the maximum stress in the cell faces becomes equal to the flow stress of the material, allowing the cells compact at a constant stress, followed by a region in which the load rises rapidly with increasing strain, as the honeycomb is compacted.

Quantitative descriptions of the behaviour have been derived and compared with the experimental measurements. These show that the deformation behaviour of a honeycomb is determined not by the axial compression of those faces parallel to the loading axis, but by the bending of faces lying at some angle to the loading axis.

It has been shown that these ideas can be extended to describe the deformation behaviour of more irregular structures, such as foams. For foams that are isotropic and have open cells, it is predicted that the relative elastic modulus varies with the square of the relative density, consistent with observations in the literature.

The uses of such structures are described and it is shown that in isotropic open-cell foams, sandwich structures are required to obtain improved specific stiffness in bending. The enhanced stiffness of cellular structures such as wood arises from modifications to the structure that gives a different dependence of the relative stiffness on the elastic modulus.

## Questions

### Quick questions

You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!

1. How does elastic deformation of a honeycomb with hexagonal cells with some faces aligned parallel to the loading axis take place?

 a By elastic compression of the vertical faces b By elastic buckling of the vertical faces c By elastic bending of the diagonal faces

### Deeper questions

The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.

1. How might elongating the cells of a hexagonal honeycomb in the direction of loading change E/ES for a given relative density?

 a Decrease it b Have no effect c Increase it

### Quick questions

You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!

1. What is the criterion for the onset of yielding in a honeycomb?

 a That somewhere the stress should exceed the flow stress through the thickness of the film b That the maximum stress in the face exceeds the material flow stress c That the stress at the centre of the face should exceed the material flow stress

### Deeper questions

The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.

1. For a highly porous structure E/ES is proportional to (ρ/ρS)n. In which case will the structure show an increased bending stiffness? Hence explain why many porous materials are often used as the core in sandwich structures.

 a n > 2 b n = 2 c n < 2

2. Consider a honeycomb loaded with some faces parallel to the compressive loading direction, in which the shape of the hexagonal cell is such that θ < 0, how would the material deform elastically in the transverse direction?

 a contract inwards b no tranverse movement c expands outwards

## Going further

### Books

• L.J. Gibson and M.F. Ashby, Cellular solids: structure and properties, Cambridge University Press, 2nd edition (1997).
Covers honeycombs and foams, both open and close celled, as well as the effects of gases and liquids in the cells. It also discusses the properties of bone, wood and the iris leaf as highly porous solids.
• K.K. Chawla, Fibrous materials, Cambridge University Press, 2nd edition (1998).
Covers fibrous and some woven structures.
• D. Boal, Mechanics of the Cell, Cambridge University Press, 2002.
See chapter 3 on two-dimensional networks.

## How the honeycomb is made

The honeycomb is made by printing a pattern of parallel, thin stripes of adhesive onto thin sheets of aluminium. These sheets are then stacked in a heated press to cure the adhesive and slices cut through the thickness of the sheet. The slices, or block form, are then gently stretched and expanded to form a sheet of continuous hexagonal cell shapes.

## Derivation of Young modulus

Here we estimate the Young modulus when the load is applied to the honeycomb made of hexagonal cells with some of the faces parallel to the loading axis where the displacement results from bending of diagonal faces The thickness of the cell wall is t and its through-thickness width is w and the length of each of the hexagonal faces is l.

The displacements in the upper and lower halves of the diagonal faces must be symmetrical about the centre, with the lower half being bent downward and the upper half in the opposite direction. Each face can therefore be treated as if it were made up of two cantilevered beams. Each is length l/2, cantilevered at the end fixed to the vertical face and loaded at the other end.

From beam bending theory the displacement of the loaded end, δ, of a cantilevered beam is given by

$\delta = \frac{1}{3}\frac{{W{L^3}}}{{EI}}$

where W is the applied load, L is the length of the beam, E is the Young modulus and I is the second moment of area (definition).

Here the bending beam lies at an angle (90-θ)° to the axis of loading. The component of the applied force in the direction normal to the beam is therefore F cos θ. As the length of each beam is l/2, then the displacement from its original position (in the direction normal to the diagonal beam) is

$\delta = \frac{1}{{24}}\frac{{F{l^3}}}{{{E_{\rm{S}}}I}} \cdot \cos \theta$

ES is the modulus of the cell wall material from which the honeycomb is made and the second moment of area of the cell wall, I, is wt3/12.
Just as with the force, the direction of the displacement is also not in the direction of loading. The vertical component of the displacement (that is in the loading direction) due to the bending of the complete cell face, Δx, and taking the displacements produced by the two half-beams is

Δx = 2 δ cos θ

Substituting for δ and I gives Δx as

$\Delta x = \frac{F}{{{E_{\rm{S}}}w}} \cdot {\left( {\frac{l}{t}} \right)^3}{\cos ^2}\theta$

This can be expressed as a strain by dividing this downward displacement by the original vertical height of the honeycomb structure, that is two vertical half-faces and the vertical component of the diagonal face, giving the strain, ε, under an imposed load F as

$\varepsilon = \frac{F}{{{E_{\rm{S}}}w}} \cdot {\left( {\frac{l}{t}} \right)^3}\frac{{{{\cos }^2}\theta }}{{l{\rm{ }}(1 + \sin \theta )}}$
Now the applied force F acts over an area w l cosθ, so that the stress, σ, can be expressed as $\sigma = \frac{F}{{w{\rm{ }}l\cos \theta }}$

The Young modulus of the honeycomb in this direction, E (= σ/ε), is therefore $E = {E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^3}\frac{{(1 + \sin \theta )}}{{{{\cos }^3}\theta }}$
As the cells are regular hexagons θ = 30° and this becomes $E = \frac{4}{{\sqrt 3 }}{E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^3}$

## Derivation of yield stress in the honeycomb

Here we estimate the stress at which the honeycomb starts to deform irreversibly where some of the faces are aligned parallel to the loading axis and the predominant contribution to the strain comes from the bending of the diagonal faces. (see here) The thickness of the cell wall is t, its through-thickness width is w and the length of each of the hexagonal faces is l.

From beam bending theory, the maximum stress, σmax, in a cantilevered beam of length L due to a force, W, applied normal to the beam is

${\sigma _{{\rm{max}}}} = \frac{{6{\rm{ }}W{\rm{ }}L}}{{w{\rm{ }}{t^2}}}$

Now in our situation W = F cos θ and L = l/2, which gives smax as

${\sigma _{{\rm{max}}}} = \frac{{3{\rm{ }}F{\rm{ }}l}}{{w{\rm{ }}{t^2}}} \cdot \cos \theta$

We assume that yielding will start when σmax = σY, giving the applied force, F, at which yielding starts as

$F = \frac{{w{\rm{ }}{t^2}}}{{3{\rm{ }}l{\rm{ }}\cos \theta }} \cdot {\sigma _{\rm{Y}}}$

Now the applied force F acts over an area wl cos θ, so that the stress, σ, can be expressed as

$\sigma = \frac{F}{{w{\rm{ }}l\cos \theta }}$

Substituting for F gives the applied stress at which the honeycomb starts to yield in terms of the yield stress of the cell wall material, σY

$\sigma = \frac{1}{3} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot \frac{1}{{{{\cos }^2}\theta }} \cdot {\sigma _{\rm{Y}}}$

For a regular hexagon, where θ = 30°, this becomes

$\sigma = \frac{4}{9} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot {\sigma _{\rm{Y}}}$

At this value of σ, yielding begins at the surfaces of the cell wall, at the cantilevered end of the beam where the stresses will be greatest. For plastic yielding to spread through the thickness of the cell wall requires that the stress is increased by a factor of 1.5, giving a macroscopic flow stress of

$\sigma = \frac{2}{3} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot {\sigma _{\rm{Y}}}$

## Derivation of Young modulus in highly porous foam

Here we estimate the Young modulus of an open-cell foam, represented schematically as shown below, where each cubic cell has a side length l and is made of struts with a square cross-section of thickness t, whose Young modulus is ES. The derivation is similar to that used for the honeycomb. However because the cell is cubic rather than hexagonal, θ = 0, giving the deflection of a single half-beam, δ, as

$\delta = \frac{1}{{24}}\frac{{F{l^3}}}{{{E_{\rm{S}}}I}}$

where I is the second moment of area, where I = t4/12. This gives the deflection in the loading direction, Δx, as

$\Delta x = 2\delta = \frac{F}{{{E_{\rm{S}}}t}}{\rm{ }}{\left( {\frac{l}{t}} \right)^3}$

This gives the strain, ε, as

$\varepsilon = \frac{F}{{{E_{\rm{S}}}}}{\rm{ }} \cdot \frac{{{l^2}}}{{{t^4}}}$

The stress, σ, arising from the applied force F is

$\sigma = \frac{F}{{{l^2}}}$

This gives an expression for the Young modulus of the open-cell foam, E

$E = {E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^4}$

Now

$\frac{\rho }{{{\rho _{\rm{S}}}}} \propto {\left( {\frac{t}{l}} \right)^2}$

The relative elastic modulus of the open-cell foam, E/ES, is therefore related to the relative density, ρ/ρS, according to

$\frac{E}{{{E_{\rm{S}}}}} = k{\rm{ }}{\left( {\frac{\rho }{{{\rho _{\rm{S}}}}}} \right)^2}$

where k is a constant approximately equal to 1.

Academic consultant: Bill Clegg and Athina Markaki (University of Cambridge)
Content development: Duncan McNicholl and David Brook
Photography and video: Brian Barber and Carol Best
Web development: David Brook and Lianne Sallows

This DoITPoMS TLP was funded by the UK Centre for Materials Education, the Worshipful Company of Armourers and Brasiers', and the Department of Materials Science and Metallurgy, University of Cambridge.