Impact problems

All this work is based on the non-inertial approximation. However the practical utility of this approximation for the corresponding elastic problem has been demonstrated by Lord Rayleigh [Strutt (1906)], Hunter (1957) and Tsai (1968, 1971). If anything, the theory should be more realistic in the viscoelastic case. On the other hand. Aboudi (1979) who has developed a completely numerical 

These theories have applications to methods for deducing information on the mechanical properties of materials from the results of impact tests [see Lifshitz and Kolsky (1964) an Calvit (1967b)]. 

We now write down the equations governing the impact of an axisymmetric rigid indentor on a viscoelastic half-space, in the non-inertial approximation, using the results of the last section, specifically (5.2.22, 29). These hold for t t, the time when the contact area is maximum. However, if we extend the definition of 6 (t) so that 01 (0 = if it can be shown, by reference to (5.2.5, 7), that they also hold for t t. This is apparent on noting that for t t, j(t, t ) reduces to k lit-t ).Let the mass of the indentor be M. Then, assuming that the impact begins at / = 0, (5.2.29) and (5.2.22) give 

Assuming that the punch shape is given by (5.2.33), relations (5.3.1) give, by virtue of (5.2.34, 35)  

Equations (5.3.2) are the fundamental dynamical equations governing the impact problem, to which must be added initial conditions. We have 

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With the advances in computing hardware that have occurred over the last decade, three-dimensional computational analyses of shock and impact problems have become relatively common. In Lagrangian calculations, element erosion schemes have provided a means for handling the large deformations and material failure that is often involved, and Fig. 9.28 shows results of a penetration calculation which makes use of this methodology [68]. 

This analysis provides a lower anchor point for curves of impaction efficiency as a function of Stokes number. It applies also to non-Stokesian particles, discussed in the next section, because the point of vanishing efficiency corresponds to zero relative velocity between particle and gas. Hence Stokes law can be used to approximate the particle motion near the stagnation point. This is one of the few impaction problems for which an analytical solution is possible. 

It is evident from the discussions in Chapters 2 and 3 that the various aspects of hydrodynamic lubrication problems can range from the classical isoviscous, isothermal solutions of the simple Reynolds equation to short-time squeeze film behavior on impact. In dealing with elastohydrodynamic and impact problems, viscosity can no longer be taken as constant but instead must be introduced in a manner which correctly accounts for its response to temperature and pressure. As background for the appreciation of the intricacies of such problems we shall examine the effects of temperature and pressure on the viscosity of liquids, particularly those which can be used as lubricants. 

In Chapter One, Mechanical Engineering Scenario 3 examined what Maria, a mechanical engineering graduate, may experience in the work environment in 2030. In this scenario, she was using her ethics competency. This included how she conceptualized the impact problem on stakeholders and the globe, scaled eco-efficient solutions as an ethical matter. Based on her experiences in the PBL projects and Lean action-research project, Maria reflects with confidence on her mastery of ethics competency in the workplace. 

The three-dimensional properties of a laminate given by Eqns (6.11), (6.12), and (6.32) are needed in situations where out-of-plane stresses develop. Besides the obvious case of out-of-plane loading such as the local indentation and the associated solution of contact stresses in an impact problem, out-of-plane stresses typically arise near free edges of laminates, in the immediate vicinity of plydrops and near matrix cracks or delaminations. Typical examples are shown in Figure 6.4. The red lines indicate regions in the vicinity of which out-of-plane stresses [Pg.132]

Impact problems are becoming increasingly important to industry, with respect to safety issues. The designer has to take into account accidental loads of the material caused by dropped objects, collisions or explosions. In particular with respect to aramid materials, ballistic protection applications are an important issue. Advantageously, the materials should have a large capacity to absorb kinetic energy. 

Lotus effect, fluid dynamics, micro fluidics, hydrodynamic slippage, impact problem... 

Developing the analogy in our impact problem with a hydrophobic surface, we identify the wetting phase as air, so that the static contact angle with respect to the wetting phase is defined as o = (rt — %) and the triple contact line becomes no longer stable above u a (rr — 0o) - Furthermore, v is fixed by a critical capillary number Ca = u 7l/tlv that evolves like Ca l/9 . 

The method of multibody systems allows the dynamical analysis of machines and structures, see e.g. (Schiehlen Eberhard 2004) and (Schiehlen, Guse 8c Seifried 2006). More recently contact and impact problems featuring unilateral constraints were considered too, see (Pfeiffer Glocker 1996). A multibody system is represented by its equations of motion as... 

Zhong, Z.-H. 1993. Finite Element Procedures for Contact-Impact Problems, New York Oxford University Press. 

When a detailed evaluation of local damage is necessary or when the dynamic interaction between missiles and target is expected to be significant, an impact problem should be explicitly solved. The full description of the missile should therefore be available, since the application of an equivalent load function is not representative of the physical phenomena. 

Analysis of Impact Problems," Ph.D. Dissertation, University of Arizona, Tucson, 1988. 

H. Lankarani (1988) Canonical Equations of Motion and Estimation of Parameters in the Analysis of Impact Problems, Ph. D. Dissertation, Dept, of Aerospace and Mechanical Engineering, University of Arizona. 

In the absence of tangential motion, the equivalent viscoelastic problem is solvable, just as in the plane case - discussed in Sect. 3.10 - and by essentially the same methods. However, the problem is worth considering in some detail because, in contrast with the plane case, the indentation is determinate. This enables one to discuss impact problems, which are of considerable interest. These topics are covered in Sects. 5.2 and 5.3. 

The simplest case of (5.2.15) and (5.2.17) will now be discussed in more detail, partly to provide a more explicit illustration of these results, and partly because it is required in the next section in the context of impact problems. We assume that C t) is decreasing, having previously passed through a single maximum. The quantity /7 (0 has the form [see (2.4.18) and (3.10.4)] ... 

Sabin, G.C.W. (1983) Efficient numerical solution of the viscoelastic impact problem. Utilities Mathematica 23, 323 - 346... 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

Finite element analysis may also be used to determine the response of adhesive joints under impact loading. Adhesives generally exhibit strain rate-dependent material properties under impact conditions. The solution of impact problems is difficult to obtain from closed form solutions and the FEM is an excellent tool to solve such problems. Explicit finite element solvers can easily solve large deformation problems and dynamic events where inertia of the structure is of significant importance. Chapter 29 treats this subject. 

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