Localization of deformation is associated with a softening behaviour of the analyzed structure or specimen. In the classical (homogenized) continuum framework a negative slope of the stress-strain relation (strain softening) introduces a material instability, which may result in the loss of well-posedness of the partial differential equations governing the boundary value problem. Localization then occurs in a set of measure zero and a pathological discretization sensitivity is observed in numerical simulations. To remedy this difficulty the continuum theory must be enhanced to become nonlocal or dependent on higher-order deformation gradients. Such continua include a characteristic length scale, which defines the width of localization bands (Chapter 2).
In this thesis we consider a gradient plasticity theory, in which the yield function depends on the Laplacian of an invariant plastic strain measure. The gradient term acts as a stabilizer and allows to preserve the well-posedness of the rate boundary value problem in the softening regime. The essential consequence of the gradient-dependence is that, unlike in the classical theory, the plastic consistency condition becomes a partial differential equation. Weak satisfaction of the equilibrium and plastic consistency conditions is assumed to solve the boundary value problem of nonlinear statics with small deformations (Chapter 3).
An incremental-iterative algorithm with a consistent tangent operator is derived. Independent finite element discretization of the displacements and plastic multipliers is used. The presence of the Laplacian sets the requirement of C1-continuity of the plastic multiplier field. When the first derivatives of this quantity are interpolated as well, and connected to the plastic multiplier field by a penalty constraint, an alternative C0-element formulation is obtained. Several one-dimensional, quadrilateral and triangular elements with mixed interpolation have been implemented. The solution procedure is compared with the classical return mapping algorithm and the implications of the gradient dependence of the yield strength are described (Chapter 4).
The Huber-Mises, Drucker-Prager and Rankine gradient-dependent yield/fracture functions are formulated and applied in several mesh-sensitivity studies. It is numerically verified that the internal length incorporated in the theory sets the size of the localization zone and that the post-peak results do not suffer from the spurious sensitivity to mesh refinement or alignment. The regularization is equally effective for the description of localized failure under shear, tension and mixed-mode conditions.
The performance of the implemented elements is compared using the Huber-Mises gradient plasticity for a one-dimensional tension test and a biaxial compression test. It is shown that the rectangular element with serendipity interpolation of displacements and hermitian interpolation of plastic strains is the most reliable. The influence of imperfections and the uniqueness of the numerical solution is also investigated (Chapter 5).
Two geotechnical problems are analyzed using the (non)associated Drucker-Prager gradient plasticity theory with cohesion softening. Physically meaningful results are obtained in the post-critical regime. Shear banding and bulging localization modes are predicted in the plane strain biaxial compression test. The stability of a slope and a vertical embankment is also examined (Chapter 6).
The vertex-enhanced Rankine gradient plasticity is applied to Mode-I and mixed-mode concrete fracture problems. Several plane stress configurations (a beam in four-point-bending, a bar in direct tension, SEN beam and DEN specimen) are analyzed and the results are compared with experimental findings. The fracture energy and the internal length scale (which determines the crack band width) are the material constants and an exponential softening diagram is used. The results of gradient plasticity simulations show a good agreement with the experiments and the size effect is correctly reproduced (Chapter 7).
In the conclusions some robustness requirements for the developed mixed elements and the observations from the numerical simulations are summarized. Some prospects of the gradient plasticity development are suggested.