Analysis of Plane Strain Deformations of Linearly Elastic Strain-Gradient Materials by the Finite Element Method

At small scales, numerous experimental studies have shown that material behavior strongly depends upon the specimen size. Classical theories are unable to explain this size dependence, whereas a strain gradient continuum theory has intrinsic length scales and may well describe mechanical deformations of small size bodies. In the current contribution, we develop a numerical software based on the finite element method (FEM) to analyze infinitesimal deformations of strain-gradient dependent materials by introducing auxiliary variables to enable the use of simple low order polynomials as basis functions. We use Lagrange multipliers to satisfy the non-classical boundary conditions pertinent to strain gradients. To verify the developed software, we analyze plane strain deformations of a clamped, transversely isotropic beam. The obtained stresses and displacements compare well with the analytical solutions available in the literature, thus verifying the numerical solution. The Method of Manufactured solutions (MMS) was used to further verify the developed code as the assumed displacements and the resulting stresses were successfully reproduced. To study the effect of the material characteristic length (lc) in isotropic and transversely isotropic materials, we numerically study some known problems of plane strain elasticity and compare the classical and strain-gradient solutions. As lc is increased, the beam becomes stiffer as evidenced by a decreased tip deflection under the same loads. This numerically predicted stiffening reflects the experimental findings. We also observe that as the beam thickness becomes much larger as compared to lc, the strain-gradient solution approaches the classical solution.