Anssi T. Karttunen

Two-scale constitutive modeling of a lattice core sandwich beam

Abstract Constitutive equations are derived for a 1-D micropolar Timoshenko beam made of a web-core lattice material. First, a web-core unit cell is modeled by discrete classical constituents, i.e., the Euler–Bernoulli beam finite elements (FE). A discrete-to-continuum transformation is applied to the microscale unit cell and its strain energy density is expressed in terms of the macroscale 1-D beam kinematics. Then the constitutive equations for the micropolar web-core beam are derived assuming strain energy equivalence between the microscale unit cell and the macroscale beam. A micropolar beam FE model for static and dynamic problems is developed using a general solution of the beam equilibrium equations. A localization method for the calculation of periodic classical beam responses from micropolar results is given. The 1-D beam model is used in linear bending and vibration problems of 2-D web-core sandwich panels that have flexible joints. Localized 1-D results are shown to be in good agreement with experimental and 2-D FE beam frame results.

Micropolar modeling approach for periodic sandwich beams

Abstract A micropolar Timoshenko beam formulation is developed and used to model web-core sandwich beams. The beam theory is derived by a vector approach and the general solution to the governing sixth-order equations is given. A nodally-exact micropolar Timoshenko beam finite element is derived using the solution. Bending and shear stiffness coefficients for a web-core sandwich beam are determined through unit cell analysis, where the split of the shear forces into symmetric and antisymmetric parts plays a pivotal role. Static bending of web-core beams is studied using the micropolar model as well as modified couple-stress and classical Timoshenko beam models. The micropolar 1-D results are in best agreement with 2-D web-core beam frame results. This is because the micropolar beam allows antisymmetric shear deformation to emerge at locations where the 2-D web-core deformations cannot be reduced to 1-D by considering only symmetric shear behavior.

Closed-form solution for circular microstructure-dependent Mindlin plates

In this paper, we derive a general closed-form solution for the static, axisymmetric bending of solid and annular microstructure-dependent Mindlin plates based on the modified couple-stress theory. The solution is developed by employing a suitable change in displacement variables. The couple-stress solution contains modified Bessel functions which do not appear in the classical case. The stress distributions of the annular microstructure-dependent plate are obtained in terms of load resultants by using the general closed-form solution. Analytical and numerical examples considering solid and annular circular plates are presented. The examples include simply supported and clamped solid and annular plates subjected to a uniformly distributed load and annular plates attached to a point-loaded rigid shaft.