The interaction between the internal length scale and the spatial dimensions of a structure has implications on the observable mechanical behavior. Experimental studies that address this physical phenomenon, which is commonly referred to as size effect, reveal a stiffening tendency of the load-displacement behavior with decreasing specimen size. On the other hand, miniaturization plays a vital role in several sectors of research and industry. Against this background, the modeling and simulation-driven predictive analysis of the mechanical size effect is of growing importance. Since the standard continuum model is not able to capture this phenomenon, alternative so-called microcontinuum theories, in particular the micropolar or Cosserat model, have moved into a broader research focus.
This thesis contributes to the field of computational solid mechanics and presents novel results in the simulation of the size effect by means of micropolar elastoplasticity. It describes the development of a sufficiently general, numerically efficient and robust computational framework and its application.
A concise review of the basic concepts of non-linear micropolar continuum mechanics is given at the beginning of the thesis. The kinematical setting as initially developed by Steinmann is outlined, and thermodynamic consistent constitutive relations are derived. A novelJ2-based multi-criterion plasticity model is proposed. The micropolar initial boundary value problem is formulated as the basis of the following numerical developments.
The major part of this work is dedicated to the formulation of a finite element based solution procedure for the aforementioned micropolar initial value problem. The development of an appropriate solution approach and its linearization are spelled out to a great level of detail. Challenging aspects that arise in this context as a particular consequence of the relatively complex nature of the micropolar model are addressed. As a major novelty within this field, the implementation is hyperelastic-based and accounts for the kinematics of finite strain and finite curvature within the three-dimensional setting. Moreover, it is consistently linearized and incorporates the so-called F-bar methodology in order to account for quasi-isochoric deformations. A brief discussion of potential alternative methodologies completes the presentation.
The resulting computational model was implemented into an existing implicit finite element solver and is used to solve a comprehensive set of numerical examples. In this connection, the emphasis is put on the validation of important properties of the numerical solution framework, such as, e.g., the aimed-at quadratic convergence rates, and the general ability to capture quasi-isochoric deformations. For the sake of clarity, these properties are dissociated by presenting their validation in a step-by-step manner at problems with increasing complexity. Moreover, the model’s ability to predict the size effect within the infinitesimal and finite elastic and elastoplastic regime, respectively, is demonstrated. A comparative study against experimental data reveals certain advantages of the proposed multi-criterion plasticity model over the generalized formulation frequently employed in this field. Finally, a benchmark study quantifies the relative computational efficiency and robustness of the proposed numerical solution framework. In relation to a comparable computational model, the implementation allows for a drastically lowered consumption of CPU time and significantly increased time step sizes.
The computational framework developed in this thesis enables an efficient, robust and realistic simulation of non-linear elastoplastic size effects that may appear in the industrial practice of miniaturization.