Soft elastic substrate

The differential equation that governs the transverse deflection w x) of the film midplane is again (5.5), modified to account for the resistance to transverse deflection exerted by the substrate. This resistance is represented by a pressure p x) over the midplane of the film, directed toward the substrate after values of x for which w x) is positive. The result is the ordinary differential equation 

Only the question of initial bifurcation is considered here post-buckling behavior is not examined. The undeflected configuration with w x) = 0 and p x) = 0 is always a solution. The objective is to find a range of parameters for which a nontrivial solution can be found. A spatially periodic deflection 

46) and (5.47) are substituted into (5.45) and if the integral is evaluated, it is found that TrlTm/A = Pm/Pg- Then, substitution of this result and 

It is readily established that the smallest magnitude of tm for which a positive real solution of this cubic equation for wavelength can be found is 

This value is therefore the critical stress for onset of plane strain wrinkling of the film on the substrate surface. The corresponding wavelength is 

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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. 

The mechanics of surface wrinkling necessitate that there be a reasonable modulus difference between the film of interest and the soft substrate (Es 2MPa for PDMS). In order to probe softer materials, the wrinkling metrology can be inverted, such that a sensor film of known modulus is adhered to a soft substrate of unknown modulus. Rearrangement of (1) leads to the following expression for the modulus of the soft, elastic substrate ... 

Solon, J., I. Levental, K. Sengupta, P.C. Georges, and P.A. Janmey. 2007. Fibroblast adaptation and stiffness matching to soft elastic substrates. Biophysics Journal 93(12) 4453-61. 

The relaxation process of residual stresses can also be followed indirectly by studying dewetting of spin-cast thin viscoelastic polymer films on soft elastic substrates [33]. There, the transient contribution of residual stresses leads to a nonmonotonic, two-stage dewetting behavior (see Fig. 1.7). The magnitude of the... 

In the remainder of this section, we will first analyze the competition between local shear and long-range elastic interaction in the case of hard solids. One of the central questions is how one can determine the size of one BK block. Finally, in Section VII.B, the sliding motion of a soft elastic body with large internal friction on a corrugated substrate will be discussed. In the latter case, friction arises not only as a consequence of finite shear forces in the individual microcontacts but also because of the internal friction of the elastic medium. 

The finite size effects in the contact between a spherical lens of polyurethane and a soft flat sheet of crosslinked polyfdimethyl siloxane) (PDMS) has been addressed by Falsafi et al. [37]. They showed that for deformations corresponding to contact diameters larger than the sheet thickness, the compliance of the system was affected by the glass substrate supporting the soft sheet. In order to minimize the finite size effects in the adhesion measurement of small elastomeric lenses, Falsafi et al. [38] and Deruelle et al. [39] used relatively thick elastic sheets to support their samples. 

If an elastic or insufficiently hard primer or paint has been applied under a less elastic top coat, or if the first coat (or set of coats) of oil-base paint has been second-coated before it is completely dry, not only will the paintwork remain soft for an unduly fong period, but cracking will also follow, as the upper layer cannot follow the movement. If the last coat is very thick this fault will frequently manifest itself in the form of alligatoring, i.e. the formation of cracks which do not penetrate all the films down to the substrate, and which may be present in the top layer only. 

Fig. 2 Mesenchymal stem cell differentiation on substrates with different elasticity (a-c) neurogenic differentiation on soft matrices (d-f) myogenic matrices showing upregulated muscle transcription factor (g—i) osteogenic differentiation on stiffer matrices. Reproduced with permission from Engler et al. [35]...

Fig. 13 Velocity dependence of frictional stress for a soft gel sliding on a smooth adhesive solid substrate. The result is based on the molecular picture in Fig. 12, which considers the thermal fluctuation of adsorption and desorption of the polymer chain, (a) The elastic term of the frictional stress of a gel. See text for a description of parameter u. (b) Summation of the elastic term and the viscous term. When v -C Vf, the characteristic polymer adsorption velocity, the elastic term is dominant. At v 2> the viscose term is dominant. Therefore, transition from elastic friction to lubrication occurs at the sliding velocity characterized by the polymer chain dynamics. (Modified from figure 1 in [65])...

The applicability of the analysis for a plastically deformed solid film on an elastically deformed substrate is not restricted to films of soft metals. This effect applies also to burnished films of other kinds of solid substances which can act as lubricants. Some of the practical ways of utilizing solids as lubricants involve compounding them with a substance, often an organic plastic of low shear strength, which bonds them to the surface to be protected, and hence the mechanical properties... 

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