Surface force elastic deformation

It has been also shown that when a thin polymer film is directly coated onto a substrate with a low modulus ( < 10 MPa), if the contact radius to layer thickness ratio is large (afh> 20), the surface layer will make a negligible contribution to the stiffness of the system and the layered solid system acts as a homogeneous half-space of substrate material while the surface and interfacial properties are governed by those of the layer [32,33]. The extension of the JKR theory to such layered bodies has two important implications. Firstly, hard and opaque materials can be coated on soft and clear substrates which deform more readily by small surface forces. Secondly, viscoelastic materials can be coated on soft elastic substrates, thereby reducing their time-dependent effects. 

Although the DMT theory attempts to incorporate distance-dependent surface interactions into the adhesion problem, it does not take into account the effect surface forces have on the elastic deformation. In other words, it does not predict the neck formation predicted by JKR. 

Erom the previous sections it is clear that there are a number of different possible models that can be applied to the contact of an elastic sphere and a flat surface. Depending on the scale of the objects, their elasticity and the load to which they are subjected, one particular model can be more suitably applied than the others. The evaluation of the combination of relevant parameters can be made via two nondimensional coordinates X and P [16]. The former can be interpreted as the ratio of elastic deformation resulting from adhesion to the effective range of the surface forces. The second parameter, P, is the load parameter and corresponds to the ratio of the applied load to the adhesive puU-off force. An adhesion map of model zones can be seen in Figure 2. 

The elasticity was quantitatively determined by analyzing the recorded force curves with the help of the Hertz model. The Hertz model describes the elastic deformation of two spherical surfaces touching imder the load, which was calculated theoretically in 1882 by Hertz. Other effects, such as adhesion or plastic deformation, were not included in this model. Sneddon extended the calculation to other geometries. For a cone pushing onto a flat sample, the relation between the indentation 5 and the loading force F can be expressed as ... 

Elastic deformation is a reversible process, whereby, if the applied load is released before the elastic yield value is reached, the particles will return to their original state. Plastic deformation and brittle fragmentation are non-reversible processes that occur as the force on the particles is increased beyond the elastic yield value of the materials. Brittle fragmentation describes the process where, as the force is increased, particles fracture into smaller particles, exposing new, clean surfaces at which bonding can occur. For plastically deforming materials, when the force is removed, the material stays deformed and does not return to its original state. Plastic materials are also known as time-dependent materials because they are sensitive to the rate of compaction. We can also speak of viscoelastic-type materials which stay deformed when the force is removed, but will expand slowly over time. 

How can the actual contact surface be measured One possibility is to measure the electrical resistance between two conductors and calculate the contact area from the measured resistance and the specific resistivity of the materials. Another possibility is to use an IR sensitive microscope to measure hot spots of a transparent solid that is in contact with a hot surface. With these methods it was found that the friction force is, in fact, proportional to the actual contact area. This implies that the true contact area must increase linearly with load. To illustrate how this is possible, we consider two extreme cases. In the first case, purely elastic deformation is considered. In the second case, we assume plastic deformation of the microcontacts. 

There are two major sources of the deformation in contact-mode SFM the elasticity of the cantilever and the adhesion between the tip and sample surface. For purely elastic deformation, a variety of models have been developed to calculate the contact area and sample indentation. The lower limit for the contact diameter and sample indentation can be determined based on the Hertz model without taking into account the surface interactions [79]. For two bodies, i.e. a spherical tip and an elastic half-space, pressed together by an external force F the contact radius a and the indentation depth 8 are given by the following equations ... 

In addition to the compression loading, uniaxial extension of entangled PDMS chains have been investigated by pulling a small portion of the material and measuring elastic response before the rupture happens [419]. The multiple ruptures observed in the force-distance curves (Fig. 43) have been interpreted as fractures of an entangled network of PDMS chains formed between the tip and the silica grafted surface. At small deformations, also the capillary forces were shown to contribute in the force. The elastic part of the curves was described us-... 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

Where one or both of the contacting surfaces becomes permanently deformed during sliding, the energy required to produce the deformation represents an additional component of the friction force. For engineering surfaces the amount of permanent deformation which can be tolerated is very limited so that the deformation friction is small in comparison with the adhesive friction. Elastic deformation only makes a significant contribution to the total friction when there is a high level of hysteresis in the elastic recovery, such as in vehicle tyres, and this is not normally a consideration when molybdenum disulphide is used. For practical purposes it can therefore be assumed that adhesive friction is the only type of friction which needs to be considered. 

Now consider the stresses arising inside a deformed elastic body. The forces acting on an elastic body fall into two types volume and surface forces. The volume forces act on the various elements of the body volume. For example, we assume that the force on an infinitesimal element of the volume dv is equal to Fdv, where F is the density of the volume force. 

Slip Region The slip region, as the name implies, is characterized by material slipping onto the roll surface with a corresponding elastic deformation of the particles. Frictional forces on the roll surface impart a forward motion to the bulk material and cause it to flow further into the region between the rollers. 

Near the contact, the vertical arrows at the dashed contour schematically represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from aH (Hertz) to aJKR (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius of the contacting sphere (or the reduced radii of the contacting spheres, if two spheres are in contact) and the elastic constant K (a combination of the Young s moduli and the Poisson s ratios of the contacting materials), defined as... 

For surfaces that deform plastically, the contact area A is proportional to the applied load. A single elastic contact deforms as as load increases, and would not be expected to follow this rule. However, when considering an exponential surface height distribution, which leads to a multiplicity of elastic asperity contacts, a linearity between load and contact area is recovered. Using instrumentation developed in the last 15yr, notably the atomic force microscope (AFM) and surface forces apparatus (SFA), researchers have explored the universality of friction-load proportionality over a much wider range of dimensions and surface characteristics. Indeed, SFA experiments have shown friction-load proportionality between atomically smooth mica surfaces in dry air over square micrometers of contact area. A contact mechanics expression for elastic contacts that incorporates the effects of adhesion was used. Similarly, AFM experiments of... 

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