Monday, June 20, 2011
Wednesday, June 1, 2011
Modelling Composite Material
As is the theme of the model we're developing, we are going to work with a model of composites that is widely used but probably popular primarily because of its simplicity. Models of composite behaviour are the subject of more than a lot of PhD's but for the narrow range of operating environments and the simple needs we have of the model the so called 'Slab Model' and the associated 'rules of mixtures' will serve more than adequately.
The Slab Model can be applied to any composite material that is made up of reinforcement ( such as fibreglass, carbon or steel mesh/ reo) that is encapsulated inside a Matrix material such as resin or concrete. The model consists of separating the reinforcement and the matrix into 2 separate slabs that are layered either one on top of the other or side by side and fixed together. Each component of the Slab occupies the same volume as it does in the actual material. The composite slab is then subject to the common strain constraint and the common stress constraint respectively.
The constant strain assumption is used when the load is in the same direction as the (long) fibres. This is because the fibres and the resin work in parallel to take the load and so must stretch by the same amount when loaded. The constant stress constrain is typically used when the loading is perpendicular to the direction the long fibres run as in this cast the resin and fibres are working in series ( cross-sections of the fibres joined with resin in between them) and so are both subject to the same stress ( i.e the same forces).
As the name suggests the common strain constraint means that under load (typically tensile load) both layers of the slab experience the same amount of strain. That is, the percentage elongation of the material is the same. This makes intuitive sense as the composite material will elongate as a whole. As we are only working with the linear region of elasticity we have the simple relationship between stress and strain where Stress/Strain = Elastic Modulus. As the reinforcements' elastic modulus much great than that of the resin ( c. 70GPa vs 2-3 GPa) the constant strain assumption implies that the stress in the higher modulus material ( the glass/carbon) will be much greater than that in the resin. That is, the reinforcement will do what reinforcement is supposes to do and that is take the lions share of the stresses the composite is exposed to.
Many physical parameters of the composite material can be estimated by taking into account the volume weighted contribution of each of the components of the slab. These are referred to as the 'Rules of Mixtures'. One simply takes the volume fractions of each component of the slab, multiplies the relevant physical parameter and sums them to get the approximate value of that parameter for the composite material.
When applying these you do need to think about the geometry of the situation you're working with. Unlike the Euler-Bernoulli Beam equation there is no term like the Area Moment of Inertia to take care of the geometrical considerations and you need to build up each case from 'first principals' and apply that appropriate assumptions for the arrangement.
For example, suppose that you wanted to estimate the elastic modulus under tension of unidirectional glass/ resin under axial load ( loaded along the long axis of the fibres) then you could just apply the rules of mixtures to calculate the volume weighted sum of the elastic modulus of each slab component. However, what about woven fabric where there fibres run in both directions? The load applied perpendicular to the load axis of the board which you would have used the constant stress assumption for now has fibres running across the width of board and so would be subject to the constraint strain assumption. Which is right? As the lateral distortion of the material will ultimately be constraint by the constant strain assumption governing the elongation of the width-wise fibres it is inconsistent with the constant strain assumptions as the 'series co-operation' that leads to the constants stress assumption would be subject to the constant strain assumptions being applied to the width-wise fibres. In this case you would use the constant strain assumption and the additional impact of the length-wise reinforcement would be taken into account by the volume fraction of the total volume of reinforcement (the sum of both direction-lying glass).
For our model the axial (long axis) elastic modulus (the elastic modulus in the direction of the boards-length) will be the most important parameter we will estimate this way as this is the value we need to plug into the Beam equation to determine the flex profile. However, many more can be estimated using the rules of mixtures.
Because the discussion of the various aspects of the model are limited to the requirements of our simple model, there are a number of effects that are being ignored. These effects are relatively small for the range operating parameters we are restricting ourselves to. In particular, we have assumed that the board will not be subject to strong shear forces that would distort the cross-sections (an requirement of the beam equation). This is reasonable in our case because the shear forces in the direction of the long axis of the board ( i.e perpendicular to the cross-section of the board are negligible and the board would snap long before the bending of the board could induce shear forces in this plane. The fact that the shear forces due to the lateral distortion that comes about when you strain a material in the normal direction ( the constant of proportionality is called Poisson's Ratio) is negligible compared to the strain due to axial loading.
The Slab Model can be applied to any composite material that is made up of reinforcement ( such as fibreglass, carbon or steel mesh/ reo) that is encapsulated inside a Matrix material such as resin or concrete. The model consists of separating the reinforcement and the matrix into 2 separate slabs that are layered either one on top of the other or side by side and fixed together. Each component of the Slab occupies the same volume as it does in the actual material. The composite slab is then subject to the common strain constraint and the common stress constraint respectively.
The constant strain assumption is used when the load is in the same direction as the (long) fibres. This is because the fibres and the resin work in parallel to take the load and so must stretch by the same amount when loaded. The constant stress constrain is typically used when the loading is perpendicular to the direction the long fibres run as in this cast the resin and fibres are working in series ( cross-sections of the fibres joined with resin in between them) and so are both subject to the same stress ( i.e the same forces).
As the name suggests the common strain constraint means that under load (typically tensile load) both layers of the slab experience the same amount of strain. That is, the percentage elongation of the material is the same. This makes intuitive sense as the composite material will elongate as a whole. As we are only working with the linear region of elasticity we have the simple relationship between stress and strain where Stress/Strain = Elastic Modulus. As the reinforcements' elastic modulus much great than that of the resin ( c. 70GPa vs 2-3 GPa) the constant strain assumption implies that the stress in the higher modulus material ( the glass/carbon) will be much greater than that in the resin. That is, the reinforcement will do what reinforcement is supposes to do and that is take the lions share of the stresses the composite is exposed to.
Many physical parameters of the composite material can be estimated by taking into account the volume weighted contribution of each of the components of the slab. These are referred to as the 'Rules of Mixtures'. One simply takes the volume fractions of each component of the slab, multiplies the relevant physical parameter and sums them to get the approximate value of that parameter for the composite material.
When applying these you do need to think about the geometry of the situation you're working with. Unlike the Euler-Bernoulli Beam equation there is no term like the Area Moment of Inertia to take care of the geometrical considerations and you need to build up each case from 'first principals' and apply that appropriate assumptions for the arrangement.
For example, suppose that you wanted to estimate the elastic modulus under tension of unidirectional glass/ resin under axial load ( loaded along the long axis of the fibres) then you could just apply the rules of mixtures to calculate the volume weighted sum of the elastic modulus of each slab component. However, what about woven fabric where there fibres run in both directions? The load applied perpendicular to the load axis of the board which you would have used the constant stress assumption for now has fibres running across the width of board and so would be subject to the constraint strain assumption. Which is right? As the lateral distortion of the material will ultimately be constraint by the constant strain assumption governing the elongation of the width-wise fibres it is inconsistent with the constant strain assumptions as the 'series co-operation' that leads to the constants stress assumption would be subject to the constant strain assumptions being applied to the width-wise fibres. In this case you would use the constant strain assumption and the additional impact of the length-wise reinforcement would be taken into account by the volume fraction of the total volume of reinforcement (the sum of both direction-lying glass).
For our model the axial (long axis) elastic modulus (the elastic modulus in the direction of the boards-length) will be the most important parameter we will estimate this way as this is the value we need to plug into the Beam equation to determine the flex profile. However, many more can be estimated using the rules of mixtures.
Because the discussion of the various aspects of the model are limited to the requirements of our simple model, there are a number of effects that are being ignored. These effects are relatively small for the range operating parameters we are restricting ourselves to. In particular, we have assumed that the board will not be subject to strong shear forces that would distort the cross-sections (an requirement of the beam equation). This is reasonable in our case because the shear forces in the direction of the long axis of the board ( i.e perpendicular to the cross-section of the board are negligible and the board would snap long before the bending of the board could induce shear forces in this plane. The fact that the shear forces due to the lateral distortion that comes about when you strain a material in the normal direction ( the constant of proportionality is called Poisson's Ratio) is negligible compared to the strain due to axial loading.
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