### Scientific Research and Essays

• Abbreviation: Sci. Res. Essays
• Language: English
• ISSN: 1992-2248
• DOI: 10.5897/SRE
• Start Year: 2006
• Published Articles: 2758

## A numerical method for second order strain gradient theory

##### Gerhard Silber1, Mansour Alizadeh1*, Mostafa Homayouni2 and S. Mohammad Ali Alvani2
1Institute for Materials Science, University of Applied Sciences, Frankfurt/Main, Germany. 2Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran.
Email: [email protected]

•  Accepted: 11 June 2012
•  Published: 05 November 2012

Abstract

In classical mechanics, based on the generalized Hook’s law, stress is only a function of strain. However, in the strain gradient theory, stress is not only a function of strain, but also a function of high-order gradients of strain. This relationship between stress and strain and its new boundary conditions is derived using the principle of minimum total potential energy. Meanwhile, a new stress tensor called total stress tensor is introduced, which can be used as the total stress tensor in the momentum equation. Also, a length scale parameter is introduced, which appears in nonlinear equations and large deformation conditions. In this article, total stress has been calculated using the high-order gradients of strain. To calculate the high-order gradient of strain tensor through numerical methods, first, each strain tensor element is estimated based on a quadratic function. Then, the high-order strain tensor gradient is calculated by the least square numerical method, and the total stress of each element is obtained by inserting these tensors in the total stress-strain equation. The results of the numerical solution are compared with those of the analytical solution. The proposed innovative method is very efficient despite its simplicity.

Key words: Strain gradient theory, principle of minimum total potential energy, total stress tensor, length scale parameter, least square method.