Computational Mechanics https://doi.org/10.1007/s00466-020-01816-2 ORIGINAL PAPER Strain gradient finite element model for finite deformation theory: size effects and shear bands Yooseob Song 1 · George Z. Voyiadjis 1 Received: 22 May 2019 / Accepted: 7 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract In this work, a thermodynamically consistent constitutive formulation for the coupled thermomechanical strain gradient plasticity theory is developed in the context of the finite deformation framework. A corresponding finite element solution is presented to investigate the microstructural features of metallic volumes. The developed model is established based on an extra Helmholtz-type partial differential equation, and the nonlocal quantity is calculated in a coupled method based on the equilibrium conditions. This approach is well known for its computational strength, however, it is also commonly accepted that it cannot capture the size effect phenomenon observed in the micro-/nanoscale experiments during hardening. In order to resolve this issue, a modified strain gradient approach which can capture the size effects under the finite deformation is constructed in this work. The shear problem is then solved to carry out the feasibility study of the developed model on the size effect phenomenon. Lastly, a plane strain problem under uniaxial tensile loading with shear bands is examined to perform the mesh sensitivity tests of the model during softening. Keywords Strain gradient plasticity · Nonlocal effective plastic strain · Finite deformation · Size effect · Shear band · Finite element implementation 1 Introduction A strain localization is generally shown as a narrow band of intense shearing strain. Needleman [1] showed that the conventional rate-independent plasticity model will cause an ill-posed problem during strain softening in finite element analysis. From a theoretical point of view, the strain local- ization is related to the change in the characteristics of the governing relations for the rate-independent materials. The forms of the partial differential equations vary from hyper- bolic to elliptic or vice versa. The finite element results of the localization problem, as a result, show the mesh depen- dent behavior. Not only the width of the shear bands, but also some characteristics of the materials such as the stiff- ness deduction are dependent on the mesh size or density, thus the conventional rate-independent models without any regularization cannot be used to solve the shear band prob- B George Z. Voyiadjis voyiadjis@eng.lsu.edu 1 Computational Solid Mechanics Laboratory, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA lems. One approach to resolve this problem is to employ the theory of viscoplasticity to provide the solution of the dynamic boundary-initial value problems for the regulariza- tion of localization of plastic deformation [2, 3]. Another promising approach is to bring in one or more intrinsic material length scales to the conventional plasticity theory. The earliest attempts of this approach are made by Aifantis [4, 5] who developed gradient-dependent plasticity theory by integrating the gradient terms into the conven- tional flow rule. Aifantis [4, 5] also shows that the width of the shear bands is strongly related to the coefficients of the gradient terms. Hereafter, the numerous gradient- enhanced plasticity theories have been proposed based on the work of Aifantis [4, 5] from a computational viewpoint, e.g. [6]. In these works, the higher-order strain gradient terms are directly incorporated to the constitutive relations, thus they have been termed as explicit gradient -enhanced theory (EGT) [7]. To implement the finite element method based on EGT, it is necessary to involve the nonstandard boundary conditions at the boundary of the plastic zone. Fur- thermore, in EGT, the use of C 1 -continuous interpolation functions is required, which in turn, makes the finite element implementation of EGT more challenging. To avoid the com- 123