Elasticity and Geometry: From hair curls to the nonlinear by Basile Audoly, Yves Pomeau

By Basile Audoly, Yves Pomeau

We adventure elasticity all over in everyday life: within the straightening or curling of hairs, the irreversible deformations of auto our bodies after a crash, or the bouncing of elastic balls in ping-pong or football. the idea of elasticity is key to the hot advancements of utilized and basic technology, akin to the bio-mechanics of DNA filaments and different macro-molecules, and the animation of digital characters in special effects and fabrics technological know-how. during this e-book, the emphasis is at the elasticity of skinny our bodies (plates, shells, rods) in reference to geometry. It covers such themes because the mechanics of hairs (curled and straight), the buckling instabilities of under pressure plates, together with folds and conical issues showing at higher stresses, the geometric tension of elastic shells, and the delamination of skinny compressed motion pictures. It applies normal equipment of classical research, together with complicated nonlinear points (bifurcation conception, boundary layer analysis), to derive distinctive, totally specific options to express difficulties. those theoretical innovations are mentioned in reference to experiments. The ebook is self-contained. Mathematical necessities are vector research and differential equations. The publication can function a concrete creation to nonlinear tools in research.

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We can gain further insight into these approximations by analysing what happens when the strain is small but the approximation of small displacement does not hold: we assume λx ≈ 1 but let φ take an arbitrary value. Then, we find: lin xx ≈ cos φ − 1, nl xx ≈ 1 − cos φ. 9) is very small, in the right-hand side the linear terms and the non-linear terms are not small when evaluated separately; a cancellation takes place between them. Therefore, in the presence of finite rotations (here, when cos φ is not close to unity), it makes no sense to use the linearized definition of the strain tensor even though the physical strain may be very small : by doing so, one would discard non-linear terms that are of the same order of magnitude as the linear ones.

X For a linear spring (such that F (x) = k x) or even a non-linear one (F (x) is a non-linear function), this equation establishes a connection between the elastic response, namely its force–displacement curve F (x), and the elastic energy: the elastic energy can be determined by integration of the force with respect to displacement, while the force is conversely given as the gradient of the elastic energy. 55) extends the notion of energy to the case of a continuous elastic medium. 57) 30 We are implicitly assuming that all external forces are conservative, something that is indeed required for the equilibrium to be achieved at a minimum of some energy.

39) allows elimination of the displacement components ux and uy ; this provides an elementary proof of this equation. 3 Stress Unlike fluids in hydrostatic equilibrium, 20 small volume elements in an elastic solid are not only capable of transmitting forces normal to their surface, but also tangential forces. The magnitude of these normal forces may also explicitly depend on the direction of space, as happens for instance when a body is stretched along one particular direction (uniaxial stress).

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