TAILIEUCHUNG - Second order homogenization of quasi periodic structures

The paper develops a general framework to derive the effective properties of quasi-periodic elastic medium. By using the asymptotic expansion method, the solution is expanded to the second order by solving a sequence of minimization problems. | Vietnam Journal of Mechanics, VAST, Vol. 40, No. 4 (2018), pp. 325 – 348 DOI: SECOND ORDER HOMOGENIZATION OF QUASI-PERIODIC STRUCTURES 1 Duc Trung Le1,∗ , Jean-Jacques Marigo2 Sorbonne Universit´es, UPMC Univ Paris 06, Institut d’Alembert, Paris, France 2 Laboratoire de M´ecanique du Solide, Ecole Polytechnique, Palaiseau, France ∗ E-mail: Received Frebuary 07, 2018 Abstract. The paper develops a general framework to derive the effective properties of quasi-periodic elastic medium. By using the asymptotic expansion method, the solution is expanded to the second order by solving a sequence of minimization problems. The effective stiffness tensors fields entering in the expression of the macroscopic energy are obtained by solving several families of microscopic problems posed on the unit cell and which bring into play only the microstructure. As an illustrative example, we consider an anti-plane elastic case of a heterogeneous cylinder made of a bi-layer laminate and submitted to the gravity. The unit cell being one-dimensional, all the associated elementary problems can be solved in a closed form and one shows that the effective energy of the medium expanded up to the second order depends not only on the strain gradient, but also on the gradient of the volume fraction θ characterizing the repartition of the two materials in the laminate. Keywords: homogenization, quasi periodic, strain gradient theories, asymptotic expansions. 1. INTRODUCTION The aim of the homogenization theory is to derive the macroscopic behavior of a system which is heterogeneous at the microscopic level. Homogenization has first been developed for periodic structures by using two-scale asymptotic expansions. The main assumption which justifies the scale separation is that the size of the period is small compared to the size of the medium, their ratio being denoted by e. From the mathematical foundations of the .

• Cơ khí - Chế tạo máy
• Second order homogenization
• Quasi periodic
• Strain gradient theories
• Asymptotic expansions
• Macroscopic energy
• Generalized continuum
• Second gradient medium
• Higher order homogenization
• Non homogeneous boundary conditions
• Representative volume element