Mechanical models friction element

In analogy to indentation experiments, measurements of the lateral contact stiffness were used for determining the contact radius [114]. For achieving this, the finite stiffness of tip and cantilever have to be taken into account, which imposes considerable calibration issues. The lateral stiffness of the tip was determined by means of a finite element simulation [143]. As noted by Dedkov [95], the agreement of the experimental friction-load curves of Carpick et al. [115] with the JKR model is rather unexpected when considering the low value of the transition parameter A(0.2Further work seems to be necessary in order to clarify the limits of validity of the particular contact mechanics models, especially with regard to nanoscale contacts. 

A plastic material is defined as one that does not undergo a permanent deformation until a certain yield stress has been exceeded. A perfectly plastic body showing no elasticity would have the stress-strain behavior depicted in Figure 8-15. Under influence of a small stress, no deformation occurs when the stress is increased, the material will suddenly start to flow at applied stress a(t (the yield stress). The material will then continue to flow at the same stress until this is removed the material retains its total deformation. In reality, few bodies are perfectly plastic rather, they are plasto-elastic or plasto-viscoelastic. The mechanical model used to represent a plastic body, also called a St. Venant body, is a friction element. The... 

Tanaka et al. (1971) have used a two-element mechanical model (Figure 8-32) to represent fats as viscoplastic materials. The model consists of a dashpot representing the viscous element in parallel with a friction element that represents the yield value. 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

Two separate approaches can be used to model lung tissue mechanics. The traditional approach places a hnear viscoelastic system in parahel with a plastoelastic system. A hnear viscoelastic system consists of ideal resistive and comphant elements and can exhibit the frequency dependence of respiratory tissue. A plastoelastic system consists of dry-friction elements and comphant elements and can exhibit the volume dependence of respiratory tissue [Hildebrandt, 1970]. An alternate approach is to utilize a nonhnear viscoelastic system that can characterize both the frequency dependence and the volume dependence of respiratory tissue [Suki and Bates, 1991],... 

Newton s law in hydrodynamics, or in flnid mechanics, models the viscous friction between fluid elements. It links what is classically called a gradient of velocity (in fact a lineic density) to the local pressure P through a coefficient 77 called dynamic viscosity. This coefficient is a scalar in isotropic media and a tensor otherwise (Phan-Tien 2002). 

To account for the nonideal nature of real soUds and liquids, the theory of Unear viscoelasticity provides a generaUzation of the two classical approaches to the mechanics of the continuum-that is, the theory of elasticity and the theory of hydromechanics of viscous Uquids. Simulation of the ideal boundary properties elastic and viscous requires mechanical models that contain a combination of the ideal element spring to describe the elastic behavior as expressed by Hooke s law, and the ideal element dash pot (damper) to simulate the viscosity of an ideal Newton Uquid, as expressed by the law of internal friction of a liquid. The former foUows the equation F = D -x (where F = force, x = extension, and D = directional force or spring constant). As D is time-invariant, the spring element stores mechanical energy without losses. The force F then corresponds to the stress a, while the extension x corresponds to the strain e to yield a = E - e. 

A mechanical model to represent a rigid-plastic material is shown in Fig. 11.9(a). The model is simply a friction element that moves or slides only when the frictional resistance is overcome. Thus the constitutive equation for the friction element is... 

Phenomenologically, the behavior of polycarbonate can be represented by a mechanical model containing a friction or stick-slip element to represent yielding. Such a model was first introduced by Bingham (See Bingham, (1922) and Reiner, (1971)) to explain the behavior of certain fluids such as paint and later adapted to explain yielding in various materials including polymers with various modifications as shown in Fig, 11.15. 

Fig. 11.15 Various mechanical models with a yielding (friction) element. (After Brinson and DasGupta, (1975)).

Attempts have often been made to represent the behaviour of solid colloid systems on deformation by mechanical models, which consist of a combination of elastic and viscous elements. The elastic elements arc represented by springs and the viscous by dashpots, the motion of which is retarded by a viscous liquid. Since in actual systems one is also always dealing with a combination of elastic elements (by which potential energy can be stored) and of internal frictional resistances (in which energy is dissipated) the analogy is more than a formal one. One can indeed often make a useful picture of the deformation mechanism by the construction of such models. 

This model is based on quasimolecular dynamics, in which the medium is assumed to be composed of an assembly of meso-scale discrete particles (i.e., finite elements). Tlie movement and deformation of the material system and its evolution are described by the aggregate movements of these elements. Two types of basic characteristics, geometrical and physical, are considered. In tlie geometrical aspect, sliapes and sizes of elements and tlie manner of their initial aggregation and arrangement are the important factors. In the physical aspect, mechanical, physical, and chemical characteristics, such as the interaction potential, phase transition, and chemical reactivity may be tlie important ones. To construct this model, many physical factors, including interaction potential, friction of particles, shear resistance force, energy dissipation and temperature increase, stress and strain at the meso- and macro-levels, phase transition, and chemical reaction are considered. In fact, simulation of chemical reactions is one of the most difficult tasks, but it is the most important aspect in shock-wave chemistiy. 

Actually, the data to be described in this paper are not primarily designed to monitor surface damage directly rather the influence that this damage has on the static friction of monofilament contacts. The acquisition of these data are an intrinsic part of an attempt to model the mechanical properties of non-woven monofilament assemblies. An assembly of this type accommodates a macroscopic strain by two means which involve internal motion in an element of monofilament which is strained locally by at least two contacts with adjacent monofilaments. The element may distort between fixed contact points. At high strains, however, the stresses on the contact points induce relative motion between monofilaments the critical stress is produced by the friction at the point contact. 

As for many immobilised enz3nnes, the hydraulic behaviour Is not adequately described by classical fluid mechanics. It was, therefore, necessary to develop a detailed mathematical model of the column hydraulics which together with a laboratory test procedure, would provide data on the basic mechanical properties of the enzyme pellet. The model Is based on a force balance across a differential element of the enzyme bed. The primary forces involved are fluid friction, wall friction, solids cohesion, static weight and buoyancy. The force balance Is integrated to provide generating functions for fluid pressure drop and solid stress pressure down the length of the column under given conditions. 

The irreversible transformation of energy into heat, e.g. in an electrical resistor, or due to friction in mechanical and hydraulic systems, is often modelled as a loss of free energy. In bond graphs, it is represented by an R element (resistive element). If the production of entropy is taken into account, the two-port RS element introduced by Thoma [22] is used (cf. Fig. B.2). The character S (Source) indicates the thermal port and expresses the entropy production. 

The bond graph model of the nominal system in integral causality is given in Fig. 3.18. The mechanical part of the engine is characterized by the viscous friction fm and inertia Jm- Load part is characterized by friction fs and inertia Js. Reducer part is represented by TF, and the axles stiffness at the input and output of the reducer is represented by C 1/K element. Modulated effort sources d and (U are the disturbing torques caused by the presence of the backlash. Axle velocities are represented on the bond graph model of Fig. 3.18 by two flow sensors Df % and Df ... 

The pure soil sample is a quarter size of the other two S-RM samples, it is tested under 100 kPa, 200 kPa and 300 kPa confining pressure in the biaxial numerical simulation to verify the selection of soil particles calculation parameters in the discrete element model are corresponding with the macro-mechanical parameters of soil. The internal friction angle and cohesion force can be calculated through the three experiments. 

Wang and Sun (2001) developed another numerical method to simulate textile processes and to determine the micro-geometry of textile fabrics. They called it a digital-element model. It models yams by pin-connected digital-rod-element chains. As the element length approaches zero, the chain becomes fully flexible, imitating the physical behavior of the yams. The interactions of adjacent yarns are modeled by contact elements. If the distance between two nodes on different yarns approaches the yam diameter, contact occurs between them. The yarn microstructure inside the fabric is determined by process mechanics, such as yarn tension and interyam friction and compression. The textile process is modeled as a nonlinear solid mechanics problem with boundary displacement (or motion) conditions. This numerical approach was identified as digital-element simulation rather than as finite element simulation because of a special yam discretization process. With the conventional finite element method, the element preserves... 

---