Peridynamics

Example

Peridynamics is a nonlocal model of continuum mechanics. Let a body occupy a region $B \subseteq \R^3$ in its reference configuration, with mass density $\rho(x)$. Let $u(x, t)$ be the displacement of the point $x$ at time $t$. Fix a horizon $\delta > 0$ and let $H_x = \set{x' \in B : \abs{x' - x} \le \delta}$. In the bond-based theory the equation of motion is $$\rho(x) \, \frac{\partial^2 u}{\partial t^2}(x, t) = \int_{H_x} f\big(u(x', t) - u(x, t), \, x' - x\big) \, dx' + b(x, t).$$ Here $b$ is a given body force density. The pairwise force function $f(\eta, \xi)$ gives the force per unit volume squared that $x'$ exerts on $x$. Its arguments are the relative position $\xi = x' - x$ and the relative displacement $\eta$. It satisfies $f(-\eta, -\xi) = -f(\eta, \xi)$, so that action equals reaction. The equation contains a time derivative of $u$ and a space integral of $u$, but no space derivative: it is an integro-differential equation.

Check: in the prototype microelastic brittle model, $f(\eta, \xi) = c \, s \, \frac{\xi + \eta}{\abs{\xi + \eta}}$ for $\abs{\xi} \le \delta$, where $s = \frac{\abs{\xi + \eta} - \abs{\xi}}{\abs{\xi}}$ is the bond stretch and $c > 0$ is a constant. A rigid translation $u(x, t) = a$ gives $\eta = 0$ and $s = 0$, so the integral vanishes, and with $b = 0$ the equation reduces to $\ddot u = 0$, which a constant $u$ satisfies.