Asymptotic analysis of perforated plates and membranes. Part 2: Static and dynamic problems for large holes

• Published in: International journal of solids and structures Vol. 49; no. 2; pp. 311 - 317
• Main Authors: Andrianov, I.V., Danishevs’kyy, V.V., Kalamkarov, A.L.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 15.01.2012; Elsevier
• Subjects: Asymptotic homogenization; Asymptotic properties; Boundary value problems; Dynamics; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Homogenization; Homogenizing; Mathematical analysis; Perforated plate and membrane; Perforated plates; Physics; Singular perturbation; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Two-point Padé approximants; Unit cell; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Two-point Padé approximants; Asymptotic homogenization; Perforated plate and membrane; Singular perturbation; Elastic constant; Vibration; Perforated plate; Homogenization (mathematics); Multiscale method; Perturbation techniques; Size effect; Analytical method; Bending; Membrane; Asymptotic approximation; Padé approximation
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2011.10.004

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained.