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We considermaterial bodies exhibiting a response function for free energy, which depends on both the strain and its gradient. Toupin-Mindlin's gradient elasticity is characterized by Cauchy stress tensors, which are given by space-like Euler-Lagrange derivative of the free energy with respect to the strain. The present paper aims at developing a first version of gradient elasticity of non-Toupin-Mindlin's type, i.e., a theory employing Cauchy stress tensors, which are not necessarily expressed as Euler-Lagrange derivatives. This is accomplished in the framework of non-conventional thermodynamics. A one-dimensional boundary value problem is solved in detail in order to illustrate the differences of the present theory with Toupin-Mindlin's gradient elasticity theory. [ABSTRACT FROM AUTHOR]