An immersed peridynamics method for fluid-driven damage and failure of anisotropic materials

Abstract

The immersed peridynamics (IPD) method is a fluid-structure interaction (FSI) model to simulate fluid-driven material damage and failure of an immersed structure, in which a peridynamic (PD) constitutive correspondence model is employed within a classical immersed boundary (IB)-type framework to describe stresses, forces, and structural deformations of a structural body, instead of classical continuum mechanics. This paper introduces an extension of the IPD method to simulate fluid-driven structural deformation, damage, and failure of anisotropic materials with complex geometries. We use quadrature rules attached to finite element (FE) meshes to generate both the PD points and their associated weights, which are used to approximate the PD integrals. We demonstrate that non-uniform discretizations improve both accuracy and volume conservation of hyperelastic materials along with accurately represented boundaries. To capture realistic biomaterial behaviors, we incorporate hyperelastic constitutive models including both isotropy and anisotropy into the proposed IPD method. In addition, a ductile failure model is adopted to simulate realistic failure processes of anisotropic materials. For non-failure cases, our numerical simulations demonstrate that the extended IPD method yields comparable accuracy with similar numbers of structural degrees of freedom for different choices of peridynamic horizon sizes. For failure tests, we investigate the effect of a fiber orientation on deformations and failure processes using realistic biomaterial models with varying fiber directions. We further demonstrate that the developed method generates grid-converged simulations of damage growth, crack formation and propagation, and rupture under large deformations, including purely fluid-driven failure processes.

Figures (31)

• Figure 1: Schematics of the computational domain $\Omega$ and the time-dependent Lagrangian and Eulerian coordinate systems in the IPD formulation along with PD material points, bonds, and horizons. The PD node $\bm{\mathrm{X}} \in \Omega_0^{\text{s}}$ interacts with its neighborhoods within a finite range called horizon denoted by $\bm{\mathrm{\mathcal{H}_X}} \subset \Omega_0^{\text{s}}$.
• Figure 2: Three different material discretizations of Cook's membrane. The uniform discretization imposes uniformly distributed material volumes along the material body. On the other hand, the material volumes in the rescaled and irregular representations require exact volume computations to avoid nonphysical behaviors. A solid line indicates the boundary represented by each discretization and black circles are PD structural nodes.
• Figure 3: An example of non-uniform discretization with FE quadrilateral meshes and corresponding nodes. The blue points are FE and PD nodes that share the same locations and the green shaded regions are areas (i.e., volumes) computed by FE meshes. The nodal volumes achieved by the FE elements on the boundary need to be corrected for accurate PD volumetric forces to avoid insufficient PD volumetric forces in the IPD formulation.
• Figure 4: Schematic diagram for the Cook's membrane benchmark. The initial configurations of the immersed structure and a fluid are denoted by $\Omega_0^{\text{s}}$ and $\Omega_0^{\text{f}}$, respectively. The entire computational domain is $\Omega = \Omega_0^{\text{s}} \cup \Omega_0^{\text{f}}$. Zero fluid velocity is enforced on the other boundaries of the computational domain.
• Figure 5: Deformations of two-dimensional Cook's membrane with the values of $J$ at material points using the neo-Hookean material model with $G = 83.3333 \, \frac{\text{dyn}}{\text{cm}^2}$. The left panel shows uniformly distributed volumes, the center panel shows non-uniformly distributed volumes, and the right panel shows irregularly distributed volumes. The deformations are represented using $381$ solid DoF for the uniform discretization and $575$ solid degrees of freedom (DoF) for the rescaled and irregular discretizations. We set $\epsilon = 2.015 \Delta X$. The numerical Poisson's ratio is fixed at $\nu_{\mathrm{stab}} = 0.4$.