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On theory of scale dependent beams and plates

Lurie S.A., Lykosova E.D., Popova E.I.

#### Abstract:

The theory of scale dependent beams and plates is obtained on the basis of the gradient theory of elasticity. It is shown that the use of the direct procedure for the method of hypothesis leads to an incorrect theory according to which the effective stiffness of thin systems modified by the scale of the gradient parameter. It was found that the cause of impropriety is in violation of the boundary conditions for the moment of stress on the longitudinal surface of the beams (plate). A method of constructing a correct theory, based on the reduction of potential energy for the gradient theory and on kinematic Bernoulli hypotheses theory of beams (plates) is proposed. We give the correct equations of the theory of Bernoulli beams, considering scale effects in the volume and scale effects due to surface interactions. As an example, we found the dependences of the natural frequencies for the thin beams on the scale parameter of the body and surface scale parameter.

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