The peridynamic formulation, recently introduced by Silling [1], is a nonlocal continuum theory for deformable bodies that does not use the spatial derivatives of the displacement field. Interactions between material particles are characterized by a pairwise force field that acts across a finite horizon. The same equations of motion are applicable over the entire body and no special treatment is required near or at defects. These properties make it a powerful tool to model problems that involve cracks, interfaces, and other defects. In the peridynamic theory, the time evolution of the displacement vector field u is given by the partial integro-differential equation
| ρ(x) ∂tt u(x,t) = | ∫ |
| f(u(x′,t)−u(x,t), x′−x,x) dx′ + b(x,t), (x,t)∈Ω×(0,T) (1) |
where f denotes a pairwise force field, ρ is the mass density, b is a prescribed body force density field, and Ω is a bounded set in R3. Here δ denotes the peridynamic horizon and Hδ(x)⊂Ω is a ball centered at x with radius δ. Equation (2) is supplemented by initial conditions for u(x,0) and ∂t u(x,0).
This project focuses on multiscale analysis of heterogeneous media using the peridynamic formulation. The objective is to resolve the deformation and the damage evolution in composites at both the structural scale and the micro scale.
We consider a peridynamic model of fiber-reinforced material. The associated initial value problem is a partial integro-differential equation with rapidly-oscillating coefficients
| ρє(x) ∂tt uє(x,t) = | ∫ |
| fє(uє(x′,t)−uє(x,t), x′−x,x) dx′ + bє(x,t). (2) |
The particular form of the pairwise force field fє is based on the bond stretch model, proposed in [2].
The macroscopic or homogenized equation is identified, utilizing Two-Scale Convergence and Semigroup Theory. A down-scaling method, justified by a strong convergence result, is provided which captures the local field fluctuations about the macroscopic displacement field. Finally, the down-scaling step is complemented with error estimates for sufficiently regular initial data.
This multiscale analysis is shown to provide the theoretical framework for a multiscale numerical method for computing the deformation of fiber-reinforced composites in the presence of residual forces.
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