Abstract
The first part of this chapter is dedicated to the analysis of viscoelastic and viscoelasto-plastic rheological contact models (linear and nonlinear parallel spring-dashpot assemblies). The linear spring-dashpot model is studied, and a detailed survey of nonlinear models (like Simon-Hunt-Crossley model and its many variations) as well as other types of assemblies with dry friction elements is made. Emphasis is put on the model’s well-posedness, where complementarity systems may be used as a nice mathematical framework. The second part presents some well-posedness results (existence and uniqueness of solutions) of Lagrange dynamics with unilateral constraints and impacts, considering them as the limit of compliant systems when contact stiffness grows unbounded.
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Notes
- 1.
However, in many practical cases it is very short: \(4.10^{-4}\) s for a shock between a golf ball and a flat-nosed wooden projectile with a relative speed of 5.334 m/s [196], other authors report values between \(7.10^{-4}\) and \(5.10^{-4}\) s for pre-impact velocities between 10 and 60 m/s [55]; see other values of the same order in Sect. 4.3.10 for slender rods against a massive steel table.
- 2.
Although, as we shall see, this still requires advanced mathematical studies.
- 3.
Constant positions is a common assumption in impact mechanics.
- 4.
This model is often called a linear spring-dashpot model , or the Kelvin-Voigt model .
- 5.
Let us note that the following relationship means that the dam** coefficient is taken to be proportional to the square root of the stiffness coefficient.
- 6.
We will see in Sect. 2.1.3.4 that this approach is to be embedded into a complementarity model.
- 7.
This restitution coefficient will be denoted as \(e_{\mathrm{n}}\) in the rest of the book, where the subscript \(\mathrm{n}\) is for “normal”.
- 8.
Strictly speaking, this fact has to be proved. In the simple examples we have treated, we have been able to integrate the equations and to calculate the functions \(F_{n}(\cdot )\). Obviously, in slightly more complex cases this would not be possible.
- 9.
Here \(\dot{w}(t)=\nabla w(\eta )^{T}\dot{\eta }(t)\).
- 10.
This property, which shall be used elsewhere in this book, is implied by the dissipativity of the system, as a consequence of the passivity Linear Matrix Inequality [218, 254].
- 11.
Steel used for journal bearing, Japanese Industrial Standards.
- 12.
In fact, Hodgkinson considered sphere/sphere impacts under Hertz’ elasticity, which we simplify here to particle/particle impacts.
- 13.
Thus we assume implicitly that \(x(t)^{p}\) exists, or we could just write \(|x(t)|^{p}\).
- 14.
Often called the hysteresis factor [782] . The spring assures the compression/expansion, while the dam** creates dissipation and the hysteresis shape.
- 15.
Notice that contrary to what is written just above [542, Eq. (1)] and could be misleading, the contact model in (2.24) with \(p=\frac{3}{2}\) is not at all introduced in [1203].
- 16.
Such values should be checked and are given here just for the sake of providing an order of magnitude.
- 17.
It is unclear how the models which include the pre-impact velocity \(\dot{x}(t_{0})\), may be used in the context of multiple impacts, where some of the contact points are lasting before the collision.
- 18.
This is true if no external force acts on the body. As shown in [802], when the objects are separated by an external force, then Simon-Hunt-Crossley model may yield sticky contact forces, as illustrated in Fig. 2.4a.
- 19.
Indeed and most importantly, using such compliant models during persistent contact phases may produce spurious, unphysical oscillations of the contact force and acceleration during numerical simulations. This is visible on many numerical results presented in the literarure, e.g., systems with clearances [403, 489, 679, 941, 1179, 1226]. It may be preferable to switch to other contact models and numerical integrators outside collisions.
- 20.
\({\mathscr {B}}[p,q]=2 \int _{0}^{\frac{\pi }{2}}\cos ^{2p-1}(x) \sin ^{2q-1}(x) dx\).
- 21.
For instance, polymers or metals with sufficiently high temperature are known to exhibit viscoelastic behaviors.
- 22.
It is not mentioned in [1287] how this condition may be guaranteed.
- 23.
The shape in Fig. 2.5b for large \(\beta _{2}\) presents strong similarities with the experimental curves shown in [602].
- 24.
High-velocity impacts of tennis balls, which are not spheres but shells, involves some buckling effects and cannot be modeled with such simple equations.
- 25.
Coulomb’s friction is introduced in more detail in Sect. 5.3.
- 26.
The word gephyroidal comes from the Greek “bridge”.
- 27.
Such a definition is logical if one thinks of sgn\((\cdot )\) as the subdifferential of \(f(u)=|u_{1}|+|u_{2}|+\ldots +|u_{n}|\).
- 28.
Compare the value of the dam** in this sequence of approximating problems with the value of the dam** in (2.9). It is a common calculation to compute \(e_{\mathrm{n}}\) for the spring-dashpot model, see [175, Eq. (3.44)].
- 29.
\(W^{1,p}\), \(1 \le p \le \infty \), denotes Sobolev spaces [191] .
Definition 2.2 Let \(1 \le p \le +\infty \). The Sobolev space \(W^{1,p}(I)\), where \(I \subset \mathbb {R}\) is an open interval (bounded or not), is the set of functions \(f(\cdot )\) such that
(i) \(f \in L^{p}(I)\).
(ii) There exists a function \(g \in L^{p}(I)\) such that \(\int _{I}f \dot{\varphi }=-\int _{I}g\varphi \) for all \(\varphi \in {\mathscr {D}}\) whose support is contained in I.
Any function \(f \in L_{p}\) possesses a distributional derivative that belongs to \({\mathscr {D}}^{\star }\) (see definitions in Appendices A.1 and A.2). Then \(f \in W^{1,p}\) if this distributional -or generalized- derivative coincides in \({\mathscr {D}}^{\star }\) with a function in \(L_{p}\). See also Sect. A.1.3 for basic facts about strong and weak\(\star \) convergence.
- 30.
This why \(\varPhi \) is assumed to be convex: this secures a unique projection.
- 31.
The use of the kinetic metric to analyze impact dynamics in Lagrangian systems, may be traced back to [581, 589, 683], and in the first edition of this book [202]. It has been deeply used in [209, 210, 228].
- 32.
Note that closed is to be taken here in the physical or real-world meaning, whereas closed in the Paoli-Schatzman’s problem is to be taken in the topological sense, i.e., the whole space itself is in fact closed.
- 33.
See [533, p. 154]: a property is generic in E if the set G of elements of E which possess it, contains a dense (in E) open set. In a sense, one deduces from the property of density of a set in another one that there are elements of G “almost everywhere” in E.
- 34.
i.e., Q([0, T]) is finite, i.e., it consists of a finite set of numbers \(c_{1},\cdots ,c_{n}\). In other words, the external action is piecewise-constant, with a finite number of values.
- 35.
Let a surface S in \(\mathbb {R}^{3}\) be given by \(q_{3}=f(q_{1},q_{2})\), with \(\frac{\partial f}{\partial q_{1}}(q_{10},q_{20})=\frac{\partial f}{\partial q_{2}}(q_{10},q_{20})\not = 0\) (these two vectors span the tangent plane to S at P) and the \(q_{3}\)-axis is normal to S at \(P=(q_{10},q_{20},q_{30})\). Then the Gauss or total curvature of S at P is equal to the determinant of the Hessian of \(f(q_{1},q_{2})\) at P, i.e., the matrix \(\frac{\partial ^{2} f}{\partial q_{1}\partial q_{2}} \in \mathbb {R}^{2 \times 2}\). It is for instance easy to verify that a plane given by \(q_{3}=aq_{1}+bq_{2}\) has zero total curvature at any of its points. The ideas generalize for higher dimensions.
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Brogliato, B. (2016). Viscoelastic Contact/Impact Rheological Models. In: Nonsmooth Mechanics. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-28664-8_2
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DOI: https://doi.org/10.1007/978-3-319-28664-8_2
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