Strength of Composites in Compression

The strength of a fibre reinforced composite in compression is considerably lower than tension, the long thin fibres buckling easily under a compressive load - like a rope, fibres do not work well in compression. However, a particulate composite will have the same behaviour in tension as it does in compression.
The mathematical analysis of the strength of continuous aligned fibre composites in compression is complex so if you can either (a) skip straight to the answer, (b) follow through a simplified analysis or (c) jump into the rigorous analysis which requires a basic understanding of the calculus of Fourier series.

The Full Monty...

Timoshenko and Gere1 examined the compression problem from an energy standpoint, equating the work done buckling the fibres and shearing the mtrix with that done by the applied stress and minimising the latter value to obtain the strength of a fibre composite in compression. (in order to follow their method we need to understand (calculus of) fourier series). A simple first approximation would be to consider a failure criteria based on (Euler) buckling of the fibres





Composite will break at the lesser of matrix collapse or fibre buckling.






This is a little awkward since the strength is dependent on the aspect ratio of the fibres and decreases rapidly the fibe length increases.
In this section we will consider the effect of the matrix material surrounding the fibres acting to prevent fibre buckling. Using an energy balance approach we will consider the additional work done by the external load P on the fibres and the strain energy stroed in the fibres as they buckle and the induced strain energy in the matrix as it deforms to accommodate the buckling. We will ignore the work done by the external load (Umacro) in general deformation (compression) of the sample (fibres+matrix, Uc) ie.





Assume unit thickness and that the fibres are actually thin plates rather than fibres thus reducing the problem to two dimensions and making the analysis tractable.. In addition we assume that buckling only occurs due to loading in the x-direction. The deflection (d) of the fibre in the y-direction at any point along its length is given by





The deflection at this point is unknown because we have no idea what the buckling looks like; whether it is a simple sine wave with just a single mode and a single amplitude or whether the buckling is a more complex pattern that is better represented by the sum of multiple modes each with different amplitude, i.e. as a Fourier series. In the expression above, an is the amplitude of the sine wave whose mode is n.





We now have to consider two possible buckling modes; one where adjacent fibres buckle out of phase and one where the fibres buckle in phase. The former process cause either an extension or compression of the matrix, the latter results in only shear deformation of the matrix. The two possibilities are know as extension mode and shear mode respectively.


NOTE:



First we shall estimate the work done bending the fibre of length L. This is simply





The fibre occupies a volume of unit width by thickness, h, by length, L ie. hL. The moment of inertia, I, of a simple thin rectangular plate of thickness h, unit width is, so the total work done on 1 unit of fibre of length L is






The work done, W, by the external load P, acting onthe fibres is simply the product of load and deflection





Extension Mode

If the deflections of the individual fibres (plates) are out of phase then the deflection in the matrix will be 2d and the strain in the matrix will be ey=2d/2c. The additional elastic strain energy (per unit volume) induced in the (linear elastic ) matrix by fibre buckling is given by






which is clearly a function of position in the matrix. To get the average strain energy per unit volume we will integrate this expression over the length of the composite, ie.






The additional work done on matrix (per unit volume) as a consequence of the micro buckling of the fibres (plates). For the composite we can consider the matrix to be a unit cell volume of length L, width 2c and thickness, t = unit thickness. Thus the total work done in 1 unit of the matrix is





Now since W = Uf + Um, we can combine the equations and re-arrange






Clearly we wish to find the lowest load at which microbuckling will occur. The function above will have the smallest value when all the terms in each summation are zero except one. Thus we need only know the amplitude of the n’th mode of buckling, an. ie.





Now the minumum load is easy to find since we need only set the derivative of P with respect to n to zero and solve






Now we recognize that the volume fraction of fibres, f, is just or , from which it follows that





The composite stress is simply P/A = P/(h+2c) for unit thickness, thus the compressive strength of the composite assuming extension mode microbuckling is





Shearing Mode





In the shearing mode the microbuckling of the fibres occurs in-phase and it is assumed that the only strains in the matrix generated by the buckling are shear strains ie.





where u and v are the displacements in the x and y directions respectively. From the red line crossing the matrix in the figure below we can obtain the first term






The strain energy in the matrix is then the strain energy per unit volume 1/2Gg multiplied by the volume of the matrix unit cell (2cL, for unit thickness) ie.





The energy balance is then the same as in the extension mode where the additional strain energy in the matrix and fibres due to microbuckling is balanced by the work done by the external load buckling the fibres. Again to complete the energy minimisation we assume that amplitude of all buckling modes except one, the n’th, which has an amplitude an , are zero.


1 S.P.Timoshenko and J.M.Gere, Theory of Elastic Stability, McGraw-Hill, NY (1961).