Composite Materials Design - Using Hybrid FibresThe design tool used in the previous section can be modified to work with fibre reinforcements that are made up from more than one type of fibre. In order that we are not limited to a specific number of fibre types, it is easiest to write down the various elastic and physicalproperties of the fibres in Mathcad in the form of a vector - a vector is just a single column matrix with the number of rows equal to the number of elements in the vector - in our case the number of distinct fibre type in the hybrid fibre.
Vectors have been created for the longitudinal and transverse modulii and the density of each of the fibre types. We have also created a vector n to describe the number of fibres. This vector hase values starting from 1 up to the number of rows in one of the property vectors - using the rows() function ensures that the vector n will always have the same number of elements as there are fibres in the hybrid. For the design tool it is important that the designer can work with either weight or volume fractions so we create two additional vectors that describe the fractions by volume (fh) and weight (fhw) of each of the fibres in the hybrid. The sum of the numbers in any one vector should add to unity.   This may look complex but it is just a concise way of repeating the same calculation using each element of a vector. The arrow tells mathcad to vectorize the division of the vector of fibre weight fractions by the vector of fibre densities, i.e. to do the division element by element - which results is a new vector. Each term of the resulting vector is then divided by the sum of the ratios of weight fraction to density of each of the fibres (a scalar). The result is a vector containing the volume fractions of each fibre in the hybrid. The density of the hybrid fibre is then  which is just the sum of the product of the density of each fibre and its volume fraction. We can use the same formulation as before to convert between weight fraction fibres and volume fraction fibres in the composite.  where the the density of the fibre is written in terms of the sum shown previously. The volume fractions of the individual fibres in the composite can be obtained by multiplying the vector containing the individual fibre volume fractions in the hybrid fibre(which add to unity) by the above fraction. The result is a vector of volume fractions of the individual fibres in the hybrid mixed in the weight proportions defined by the vector fhw.  The program block that defines the weight to volume conversion is simply two statements, the first converts the weight fractions of fibres in the hybrid to volume fractions, storing the result in a temporary variable ff. The second converts the weight fraction of hybrid fibre in the composite to a volume fraction making use of the volume fractions of fibres in the hybrid, ff, and the densities of the individual fibres and the matrix. The various elastic and physical properties can be written in terms of the sum of series of individual properties multipled by their respective volume fractions.  Each function takes as its input a vector of the volume fractions of each of the fibres in the composite as a whole. So if the manufacturing inputs are in the form of the total volume fraction of hybrid fibre, f, the actual input required in the functions is f.fh. If the manufacturing inputs are in the form of the total weight fraction of hybrid fibre in the composite, f, and the weight fractions of the individual fibres in the hybrid alone are predefined, fhw, then the input to the equations would be fhv(f) which would carry out all the necessary conversions from weight to volume fractions. 
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