Abstract
There is no such thing as a rigid body. When mechanics is applied to the stability and dynamics of buildings, machinery, vehicles, or parts thereof, one must be able to deal with deformation. In this chapter we focus on elastic deformations , i.e. deformations which disappear completely when the forces causing them are no longer present.
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Notes
- 1.
Even though some of the things we encounter every day do come very close to being ideally elastic, like car tires.
- 2.
This approach is statistical mechanics .
- 3.
Clausius, Rudolf Julius Emanuel, German physicist, *Köslin (now Koszalin) 2.1.1822, †Bonn 24.8.1888.
- 4.
Had we done the simulation based on a larger system, the fluctuations would have been much smaller.
- 5.
Note that \(\delta \) in the following just means a ‘small change’ and does not refer to the displacement not being a state function. In fact, here we assume that the work is reversible.
- 6.
Here we use the symmetry of the stress tensor, i.e. \(\sigma _{ik}=\sigma _{ki}\). This can be shown as follows: The torque on a particular volume of the body is given by
$$\begin{aligned} \nonumber{} & {} \int (f_i x_k - f_k x_i) dV = \int \left( \frac{\partial \sigma _{il}}{\partial x_l} x_k - \frac{\partial \sigma _{kl}}{\partial x_l} x_i \right) dV \\ \nonumber= & {} \int \frac{\partial (\sigma _{il} x_k - \sigma _{kl} x_i)}{\partial x_l} dV - \int (\sigma _{il} \delta _{kl} - \sigma _{kl} \delta _{il}) dV \\ \nonumber= & {} \oint (\sigma _{il} x_k - \sigma _{kl} x_i) dA_l - \int (\sigma _{ik} - \sigma _{ki}) dV \;. \end{aligned}$$The volume integral must vanish, implying that \(\sigma _{ik}=\sigma _{ki}\), because the torque must be due to the forces acting on the surface.
- 7.
Hooke, Robert, English scientist, *Freshwater (Isle of Wight) 18.7.1635, †London 3.3.1703.
- 8.
Notation: \(\sigma _{xx}\), \(\sigma _{xy}\), etc. are equivalent to \(\sigma _{11}\), \(\sigma _{12}\), etc. The same is true for \(u_{xx}\) etc., i.e. \(u_{xx}=u_{11}\) etc. In addition \(\delta _{ii}=3\) but \(\delta _{11}=1\).
- 9.
If this is difficult to understand, you should think of a tightrope walker. The tension on the rope must be large compared to the weight force exerted by the walker in order to prevent slack.
- 10.
The integration constant is zero, because \(u_x = u_y = 0 \) at \(z = 0\) (neutral surface!).
- 11.
In the following we omit the indices x and y as well as l and t.
- 12.
Show this (Fig. 10.18).
- 13.
The h-factors follow according to the chain rule, i.e. \(d \zeta (x) /d\xi =( d \zeta (x)/dx )(d x/d \xi ) =\zeta ^\prime h\).
- 14.
You may wonder why we use (10.130) instead of \(\sigma =\sigma _o \cos (\omega t)\) and \(u= u_o \cos (\omega t-\delta )\), which is closer to the solution we had obtained for the forced oscillator on p. 169. The present expressions for \(\sigma \) and u follow from a simple shift of the time origin. They are more convenient in the present context. The various results, however, do not depend on which form we choose.
- 15.
The strain occurs instantaneously at \(t=0\). The dashpot cannot follow as quickly, i.e. only the \(\mu \)-elements contribute to the answer.
- 16.
The two models can be converted into one another via \(\mu _1^R = \mu _1^K \mu _2^K/(\mu _1^K+\mu _2^K)\), \(\mu _2^R = {\mu _2^K}^2/(\mu _1^K+\mu _2^K)\) and \(\eta ^R = (\mu _2^K/(\mu _1^K+\mu _2^K))^2 \eta ^K\). The index R indicates the relaxation or Zener model, the index K indicates its dual partner.
- 17.
This also explains why none of the models in Fig. 10.31 describes the entire frequency range.
- 18.
Low temperatures corresponds to high frequencies and vice versa. Notice that \(\log \omega \sim 1/T\).
- 19.
Looking closely at the experimental data you may be able to spot a couple of ‘junctions’ between shifted data sets.
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Hentschke, R. (2017). Basic Equations of the Theory of Elasticity\({}^\heartsuit \) . In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-48710-6_10
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