Discrete physically-based models in solid mechanics

The computational methods commonly used in solid mechanics are based on the spatial discretization of field equations of classical continuum. The governing equations are formulated in terms of partial derivatives of the displacement components which are not valid in the presence of cracks and other material discontinuities. As a consequence, they require special treatment of mathematical singularities and the definition of specific crack growth criteria. Moreover, an internal length parameter usually cannot be defined in these equations. Discrete approaches and lattice models, instead, avoiding any differential formulation of the elastic problem, result to be particularly suitable for problems involving discontinuities, because cracks are not viewed as a pathology of the displacement field but as a normal solution of the problem itself. Furthermore, microstructural effects can be modeled adopting quite simple strategies. We implemented in two and three dimensions, a mesh-free lagrangian micropolar lattice model arising from bond-based peridynamics, a new promising non-local theory of solid mechanics inspired by the atomistic struc- ture of matter and based on integral field equations. The model developed has specific and distinctive features and is based on an original implicit formulation derived analytically. By providing an appropriate mathematical and computational framework, the theoretical aspects of the conceived model and the software implementation strategy are discussed, then innovative applications are proposed. In particular, a quasi-static peridynamic for- mulation is applied to the study of the transition from local to nonlocal behavior of the stress and displacement fields in the vicinity of a crack front and other sources of stress concentration. Moreover a specific stochastic peridynamic model for glass in which the bond strengths are explicitly related with the size and orientation of the defects in the structural element is proposed. An original generalized micropolar peridynamic formulation is derived starting from the definition of a microelastic energy function which depends on three deformation parameters: the bond stretch, the bond shear deformation accounting for the rotational degree of freedom, and the particles relative rotation. Hence three different stiffness parameters for each peridynamic bond are defined and calibrated separately, leading to a more general PD model with arbitrary Poissons ratio and ap- plicable to a wide variety of mechanical problems. Two novel deformation-based failure criteria based on the definition of a bond shearing deformation limit are also introduced. Finally we developed an original 2D full orthotropic model in the micropolar peridynamic analysis framework which is characterized by four indepen- dent peridynamic microelastic moduli. An important and distinctive feature of the model is that the bond properties, i.e. the stiffness constants and critical deformation parameters, are continuous functions of bond orientation in the principal material axes. The introduction of the bond shear stiffness and the definition of a bond shearing deformation measure which accounts for particles rotations, eliminates the restrictions of the only two independent constants which affects other orthotropic peridynamic formulations and enables the model to predict the mechanical behavior of a wide variety of Cauchy orthotropic materials undergoing homogeneous and non-homogeneous deformations. Numerical visualizations and examples show the applicability of this discrete formulation in modeling a wide variety of practical engineering problems involving cracks, multiscale modelling, anisotropy and complex nonlinear behavior.