Peridynamics is a non-local continuum mechanics theory developed to address the limitations of classical continuum mechanics, particularly in modeling material failure such such as fracture and damage. Unlike classical theories that rely on spatial derivatives of the displacement field (which become undefined at discontinuities like cracks), peridynamics formulates the equations of motion in terms of integral equations, allowing for the natural representation of discontinuities.
Principles and Formulation
The fundamental idea behind peridynamics is that material points interact with other material points within a finite range, called the horizon (represented by a radius $\delta$). These interactions are characterized by "bonds" that connect pairs of material points. The force between two interacting points is a function of their relative displacement.
The peridynamic equation of motion for a material point $\mathbf{x}$ at time $t$ is given by:
$$ \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x}, t) = \int_{\mathcal{H}{\mathbf{x}}} (\mathbf{T}(\mathbf{u}(\mathbf{x}', t) - \mathbf{u}(\mathbf{x}, t), \mathbf{x}' - \mathbf{x}) - \mathbf{T}(\mathbf{u}(\mathbf{x}, t) - \mathbf{u}(\mathbf{x}', t), \mathbf{x} - \mathbf{x}')) dV{\mathbf{x}'} + \mathbf{b}(\mathbf{x}, t) $$
where:
- $\rho(\mathbf{x})$ is the mass density at point $\mathbf{x}$.
- $\ddot{\mathbf{u}}(\mathbf{x}, t)$ is the acceleration of point $\mathbf{x}$.
- $\mathbf{u}(\mathbf{x}, t)$ is the displacement field.
- $\mathcal{H}_{\mathbf{x}}$ is the horizon of point $\mathbf{x}$, encompassing all points $\mathbf{x}'$ such that $|\mathbf{x}' - \mathbf{x}| \le \delta$.
- $\mathbf{T}$ is a pairwise force function (or "force state") that represents the force density exerted by $\mathbf{x}'$ on $\mathbf{x}$ (and vice versa) due to their relative displacement.
- $\mathbf{b}(\mathbf{x}, t)$ is the body force density.
A key feature is that bonds can break when their elongation exceeds a critical threshold, effectively modeling crack initiation and propagation without requiring additional fracture criteria or remeshing algorithms.
Types of Peridynamic Formulations
Peridynamic models are typically categorized based on how the force interactions (force states) are defined:
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Bond-Based Peridynamics (BB-PD):
- This is the original and simplest formulation, introduced by S.A. Silling in 2000.
- Each bond acts independently, and the force density between two points depends only on the relative displacement and initial distance of those two points.
- A significant limitation is that the Poisson's ratio is fixed (e.g., 1/4 for 3D isotropic materials), which restricts its applicability.
-
Ordinary State-Based Peridynamics (OSB-PD):
- Developed to overcome the Poisson's ratio limitation of BB-PD.
- The force on a bond is not just a function of that bond's deformation but depends on the collective deformation of all bonds within the horizon of the interacting points. This collective information is called a "deformation state."
- Allows for arbitrary Poisson's ratios and more complex material responses.
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Non-Ordinary State-Based Peridynamics (NOSB-PD):
- The most general formulation.
- Allows for even more complex material models, where the force between two points can depend on the deformation states of both interacting points.
- Can directly recover the classical stress-strain constitutive laws under certain conditions.
Advantages
- Natural handling of discontinuities: Cracks and damage are intrinsic outcomes of the model, emerging naturally from bond breaking, without the need for external fracture criteria or crack tracking algorithms.
- Unified approach: Models both deformation and failure within a single mathematical framework.
- Computational simplicity for fracture: Avoids the complexities of remeshing or singular stress fields associated with classical fracture mechanics.
- Applicability to complex failure modes: Well-suited for problems involving multiple interacting cracks, branching cracks, spalling, fragmentation, and delamination.
- Suitable for highly dynamic events: Its non-local nature is advantageous for high-strain-rate phenomena like impact and blast.
Limitations
- Computational cost: The non-local nature means each material point interacts with many neighbors within its horizon, leading to a higher computational cost compared to local methods like FEM, especially for large horizons or finely discretized domains.
- Surface effects/Ghost forces: In some implementations, particularly BB-PD, points near free surfaces or boundaries can experience spurious forces (often called "ghost forces") due to incomplete horizons, requiring special treatment or regularization.
- Material parameter calibration: Establishing a clear connection between peridynamic material parameters and classical continuum mechanics parameters (Young's modulus, Poisson's ratio, toughness) can be challenging, especially for complex state-based models.
- Consistency with classical continuum mechanics: Ensuring that peridynamics converges to classical continuum mechanics for smooth deformations (i.e., without discontinuities) requires careful formulation and interpretation.
Applications
Peridynamics has found applications in a wide range of engineering and scientific problems, including:
- Fracture Mechanics: Brittle fracture, ductile fracture, fatigue, stress corrosion cracking.
- Impact and Penetration: Ballistic impacts, blast loads, spalling.
- Composite Materials: Delamination, fiber breakage, matrix cracking.
- Geomechanics: Rock fracture, seismic wave propagation, hydraulic fracturing.
- Additive Manufacturing: Modeling failure in 3D-printed parts.
- Biomaterials: Bone fracture, soft tissue damage.
- Material Design: Predicting the performance and failure of novel materials.
History
Peridynamics was originally proposed by Dr. Stewart Silling at Sandia National Laboratories in 2000. His seminal paper, "Reformulation of the field equations of continuum mechanics to include nonlocal effects," laid the foundation for the theory. Since then, the field has seen rapid development, with numerous researchers contributing to its theoretical advancements, computational implementations, and diverse applications.