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A New Surface Node Method to Accurately Model the Mechanical Behavior of the Boundary in 3D State-Based Peridynamics
Full text linkAccurate numerical integration in 3D meshless peridynamic models
No full textIn mathematical terms, physical problems are often described by equations involving the computation of integrals, either because they appear in the weak form of differential equations or for the nature itself of the strong formulation. The recent theory of Peridynamics expresses the internal forces of mechanical problems as the integral of the nonlocal interactions of the material points. The numerical evaluation of those integrals strongly affects the accuracy of the solution of discretized peridynamic problems. The most common discretization of a peridynamic body is a meshless regular grid of points such that a cube is the volume associated to every point of the grid. Since the neighborhood of every (source) node is a sphere containing a large number of (family) nodes the intersection between the neighborhood and some of the cubic volumes associated to the family nodes is partial, and it is a complex function of the ratio between the horizon of the neighborhood (δ) and the grid spacing (h). Such a problem has affected the numerical solution of peridynamic problems since the first time Peridynamics was proposed. The paper presents the following two main results: - the exact solution to the geometrical problem of computing the partial volumes generated by all possible combinations of δ/h;- the application of the above exact evaluation to the numerical integration of peridynamic problems and the evaluation of the impact of the exact integration on the solution of structural problems. Several examples will illustrate various aspects of the newly proposed numerical integration techniques
An improved coupling of 3D state‐based peridynamics with high‐order 1D finite elements to reduce spurious effects at interfaces
Full text linkPeridynamics (PD) is a nonlocal continuum theory capable of handling fracture mechanisms with ease. However, its use involves high computational costs. On the other hand, Carrera Unified Formulation (CUF) allows one to use one-dimensional high-order finite elements, resulting in excellent accuracy while improving computational efficiency. To address the high computational cost of solving fracture problems, a coupling technique between these two theories is necessary. Various approaches have been proposed to couple peridynamic grids with finite element meshes in the literature. However, most of these approaches are affected by arbitrary choices of blending functions and tuning parameters or exhibit spurious effects at the interfaces. To overcome these issues, we propose a simple coupling technique based on overlapping PD/CUF regions and continuity of the displacement field at the interfaces. This approach is verified through static analysis of classical beams and thin-walled structures with applications in the aerospace industry
A general ordinary state-based peridynamic formulation for anisotropic materials
Full text linkA general ordinary state -based peridynamic formulation to model anisotropic materials in 2D and 3D is proposed. The new peridynamic constitutive model introduces two bond stiffness functions depending on the bond orientations. These functions are defined such that the components of the elasticity tensor evaluated by using the new formulation exactly reproduce those of classical continuum mechanics in the case of homogeneous deformation. Several numerical examples in 2D and 3D illustrate the validity of the proposed formulation for fully anisotropic materials. This formulation is also suitable to model monoclinic, orthotropic, transversely isotropic, and isotropic materials
A new method based on Taylor expansion and nearest-node strategy to impose Dirichlet and Neumann boundary conditions in ordinary state-based Peridynamics
Full text linkPeridynamics is a non-local continuum theory which is able to model discontinuities in the displacement field, such as crack initiation and propagation in solid bodies. However, the non-local nature of the theory generates an undesired stiffness fluctuation near the boundary of the bodies, phenomenon known as “surface effect”. Moreover, a standard method to impose the boundary conditions in a non-local model is not currently available. We analyze the entity of the surface effect in ordinary state-based peridynamics by employing an innovative numerical algorithm to compute the peridynamic stress tensor. In order to mitigate the surface effect and impose Dirichlet and Neumann boundary conditions in a peridynamic way, we introduce a layer of fictitious nodes around the body, the displacements of which are determined by multiple Taylor series expansions based on the nearest-node strategy. Several numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method
Accurate computation of partial volumes in 3D peridynamics
Full text linkThe peridynamic theory is a nonlocal formulation of continuum mechanics based on integro-differential equations, devised to deal with fracture in solid bodies. In particular, the forces acting on each material point are evaluated as the integral of the nonlocal interactions with all the neighboring points within a spherical region, called "neighborhood". Peridynamic bodies are commonly discretized by means of a meshfree method into a uniform grid of cubic cells. The numerical integration of the nonlocal interactions over the neighborhood strongly affects the accuracy and the convergence behavior of the results. However, near the boundary of the neighborhood, some cells are only partially within the sphere. Therefore, the quadrature weights related to those cells are computed as the fraction of cell volume which actually lies inside the neighborhood. This leads to the complex computation of the volume of several cube-sphere intersections for different positions of the cells. We developed an innovative algorithm able to accurately compute the quadrature weights in 3D peridynamics for any value of the grid spacing (when considering fixed the radius of the neighborhood). Several examples have been presented to show the capabilities of the proposed algorithm. With respect to the most common algorithm to date, the new algorithm provides an evident improvement in the accuracy of the results and a smoother convergence behavior as the grid spacing decreases
A novel and effective way to impose boundary conditions and to mitigate the surface effect in state-based Peridynamics
Full text linkPeridynamics is a nonlocal continuum theory capable of modeling effectively crack initiation and propagation in solid bodies. However, the nonlocal nature of this theory is the cause of two main problems near the boundary of the body: an undesired stiffness fluctuation, the so-called surface effect, and the difficulty of defining a rational method to properly impose the boundary conditions. The surface effect is analyzed analytically and numerically in the present paper in a state-based peridynamic model. The authors propose a modified fictitious node method based on an extrapolation with a truncated Taylor series expansion. Furthermore, a rational procedure to impose the boundary conditions is defined with the aid of the fictitious nodes. In particular, Neumann boundary conditions are implemented via the peridynamic concept of force flux. The accuracy of the proposed method is assessed by means of several numerical examples for a state-based peridynamic model: with respect to the peridynamic model adopting no corrections, the results are significantly improved even if low values of the truncation order for the Taylor expansion are chosen
Peridynamic simulation of elastic wave propagation by applying the boundary conditions with the surface node method
Full text linkPeridynamics is a novel nonlocal theory able to deal with discontinuities, such as crack initiation and propagation. Near the boundaries, due to the incomplete nonlocal region, the peridynamic surface effect is present, and its reduction relies on using a very small horizon, which ends up being expensive computationally. Furthermore, the imposition of nonlocal boundary conditions in a local way is often required. The surface node method has been proposed to solve both the aforementioned issues, providing enhanced accuracy near the boundaries of the body. This method has been verified in the cases of quasi-static elastic problems and diffusion problems evolving over time, but it has never been applied to a elastodynamic problems. In this work, we show the capabilities of the surface node method to solve a peridynamic problem of elastic wave propagation in a homogeneous body. The numerical results converge to the corresponding peridynamic analytical solution under grid refinement and exhibit no unphysical fluctuations near the boundaries throughout the whole timespan of the simulation
A peridynamic model for oxidation and damage in zirconium carbide ceramics
Full text linkZirconium carbide (ZrC) has potential to be applied in next-generation nuclear reactors for space missions and industrial applications. The mechanisms controlling ZrC oxidation dependence on temperature, material composition, pressure, porosity are not fully understood. In this work, we use a peridynamic modeling of diffusion/reaction across several regions observed in previous experiments to explain the oxygen diffusion mechanism and reaction kinetics. We emphasize the importance in the oxidation and damage process of a transition layer of partially-oxidized ZrC. The peridynamic model has an autonomously moving oxidation interface, and the delamination/detachment of oxide (induced by large volumetric expansion) is simulated here with an oxygen concentration-driven damage model. Once the diffusion properties are calibrated to match the measured oxygen concentration across the oxidation front, the speed of propagation of the oxidation front is predicted by a 1D peridynamic model in excellent agreement with experimental observations. An extension to 2D finds the shape of remaining unoxidized ZrC conforming to experimental observations
Peridynamic correspondence model for nearly-incompressible finite elasticity
No full textThis paper presents a correspondence model for use with peridynamic states in the context of nearly incompressible finite elasticity. An isochoric/volumetric decomposition is adopted, enabling the derivation of the peridynamic force state from a purely spherical, pointwise non-local deformation gradient and a deviatoric, bond-level non-local deformation gradient. This approach leads to a stable one-field, state-based peridynamic formulation that is free from zero-energy modes and capable of accurately capturing the mechanical behavior of elastic materials under large deformations, including those with low or negligible compressibility, typical of unfilled elastomers and isotropic soft biological tissues. Notably, the proposed correspondence model, based on a selective bond-associated deformation gradient, avoids the artificial stiffening commonly observed in standard displacement-based formulations near the incompressible limit. Moreover, its performance is shown to be independent of the specific compressibility ratio assumed in the hyperelastic constitutive law. The model has been successfully validated using classical polynomial strain energy functions through a series of illustrative examples involving both homogeneous and inhomogeneous finite deformations in isotropic hyperelastic solids
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