A General-Purpose, Inelastic, Rotation-Free Kirchhoff-Love Shell Formulation for Peridynamics

Abstract

We present a comprehensive rotation-free Kirchhoff-Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.

IGA-PD Penalty-Based Coupling for Immersed Air-Blast Fluid-Structure Interaction: A Simple and Effective Solution for Fracture and Fragmentation

“…At this stage, Y h is used to evaluate the discrete residual vector (also often called the nodal force vector) for the background domain, while Ỹh is used to evaluate the discrete residual vector for the foreground domain. The foreground-domain residual vector is then distributed to the background-domain DoFs using a linear transformation induced by Equation (19). The reader is referred to [7] for the details of this transformation.…”

“…The resulting strong coupling methodology resembles the classical Immersed Boundary Method [47] and a more recent Immersed Finite Element Method [48], but without the use of ad hoc smoothed delta functions to distribute the foreground-domain residual vector to the background DoFs. The foreground-domain residual vector distribution on the background DoFs is defined to be consistent with the test-function constraints given by Equation (19). This presents a clear benefit of using a variational formulation in the background domain, which is associated with the fluid mechanics part of the problem.…”

“…Constraining the foreground solid to the background fluid kinematics as in [7] gives the desired modularity together with strong coupling, however, the modeling of fracture and fragmentation in the immersed FSI simulations remains a challenge. While the foreground discretization such as PD can easily support discontinuous kinematic fields by locally breaking bonds between material points [11][12][13][14][15][16][17][18][19], the smooth background discretization of IGA [20][21][22][23][24][25][26][27][28] is not designed to excel in approximating discontinuous kinematics. Thus, constraining the foreground solution to its background counterpart results in an overly smooth foreground solution and, when coupled with continuum-damage (or phasefield [26,27]) approaches to model fracture and fragmentation, results in the size of damage zones that scales with that of the background mesh.…”

Coupling of IGA and Peridynamics for Air-Blast Fluid-Structure Interaction Using an Immersed Approach

“…In this section we focus on the PD formulation of the solid. The rate form of the energy balance law on Ω s may be expressed as [10,41]:…”