Keywords: grammatical evolution, constitutive equation, Hooke's law.

Grammatical evolution (GE), which is a type of an evolutionary algorithms, that can determine an executable program or program fragment that will achieve a good fitness value for the given objective function to be minimized. The search algorithm of GE is very similar to the genetic algorithm (GA) except for the translation rules from the genotype (bit string) to the phenotype (function or program). The translation rule of GE is defined in Backus naur form (BNF).

After the original GE algorithm is described, three improved algorithms are explained for improving the convergence property of the original GE. The improved algorithms were named as scheme 1, scheme 2, and scheme 3, respectively. The scheme 1 is designed to overcome the difficulty related to the remainder selection which is too sensitive to the variation of the decimal number. The schemes 2 and 3 are designed to overcome the problems of the selection probability of the recursive (non-terminal) and terminal rules, respectively.

In the numerical example, GE is applied for determining the constitutive equation between stress and strain components in the plane strain state. The results show that GE can find the similar stiffness equation as Hooke's law, which is determined from experiments. Comparing the convergence properties of the schemes show that the scheme 1+2+3 is the fastest and the scheme 1+2 is the second-fastest. Therefore, we can conclude that the scheme 3 is effective for this problem.

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