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  "body": "\n## Example\n\nPeridynamics is a nonlocal model of continuum mechanics.\nLet a body occupy a region $B \\subseteq \\R^3$ in its reference configuration, with mass density $\\rho(x)$.\nLet $u(x, t)$ be the displacement of the point $x$ at time $t$.\nFix a horizon $\\delta \u003e 0$ and let $H_x = \\set{x' \\in B : \\abs{x' - x} \\le \\delta}$.\nIn the bond-based theory the equation of motion is\n$$\\rho(x) \\, \\frac{\\partial^2 u}{\\partial t^2}(x, t) = \\int_{H_x} f\\big(u(x', t) - u(x, t), \\, x' - x\\big) \\, dx' + b(x, t).$$\nHere $b$ is a given body force density.\nThe pairwise force function $f(\\eta, \\xi)$ gives the force per unit volume squared that $x'$ exerts on $x$.\nIts arguments are the relative position $\\xi = x' - x$ and the relative displacement $\\eta$.\nIt satisfies $f(-\\eta, -\\xi) = -f(\\eta, \\xi)$, so that action equals reaction.\nThe equation contains a time derivative of $u$ and a space integral of $u$, but no space derivative: it is an [[integro-differential equation|YRKW2KX2CW]].\n\nCheck: 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 $c \u003e 0$ is a constant.\nA 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.\n\n## Remarks\n\nSince no spatial derivatives of $u$ appear, the equation makes sense for displacements with jumps, so cracks can form and grow without a separate rule for fracture.\nIn the model above a bond breaks for good once its stretch exceeds a critical value.\nWhen $\\delta \\to 0$, for smooth displacements and suitable force functions, the model approaches classical linear elasticity.\n\n## Explanations\n\n### L1 human\n\nIn classical elasticity a point of a solid feels only its immediate neighbours, through derivatives of the displacement.\nIn peridynamics each point is linked by springs, called bonds, to all points within a fixed distance.\nThe force on a point is then an integral over all its bonds, and cracks are simply regions where bonds have broken.\n",
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