Contact Mechanics

If two bodies contact each other, they deform and a contact area is created under the influence of an external applied force and surface interactions. Accurate knowledge on contact parameters like contact areaa and deformation δ as a function of the applied force as well as on the resulting stress distribution is a crucial requirement to under-stand effects in nature and for applications. Examples are reversible adhesives in both biological and non-biological systems.

Several theories exist that describe the contact situation and connect molecular inter-actions and continuum models. The most important theories of contact mechanics for linear elastic bodies are introduced without derivations in the following, oriented on reviews [124, 24, 127, 128].

Material Parameter

First, several material parameters must be introduced. The stress acting on a material is defined as the force per cross sectional area A. For normal loads in z direction this is

σ_z =F_z/A. (2.50)

The resulting relative elongation of the body is quantified by the strain _z = ∆L_z

L_z , (2.51)

with the total length in z-direction L_z and the length change ∆L_z. The elastic defor-mation of the body can be related to its Young’s modulus E which is defined for an isotropic elastic material¹⁰ as the proportionality factor between the applied strain and the resulting stress

σ_z =E_z. (2.52)

Typical values forEare in the range of tens of kPa (e.g. gels), some MPa (e.g. rubbers), up to some GPa (e.g. metals) [24]. When a material is stretched in one direction, it will normally contract perpendicular to this direction. This effect is considered by the Poisson ratio ν

ν = _x,y z

(2.53) for a material stretched inz direction. The Poisson ratioνis in the range of 0 up to 0.5 (0.5 for perfectly incompressible materials). Rubbers reach values close to the upper

10In general this have to be defined using tensor analysis

limit of 0.5 [24], whereas glass has a Poisson ratio of around 0.1 for example.

Contact Mechanic Models

The first model to describe the contact between two non- adhering, isotropic, linear elastic spheres was given by H. Hertz in 1881[143]. H. Hertz calculated the deformation δand contact areaaas a function of applied loadF, geometrical terms (R =R_iR_j/(R_i+ R_j)), and material parameters as accounted for the reduced modulus K

K = 4 3

1−ν_i²

E_i + 1−ν_j² E_j

⁻¹

. (2.54)

The resulting contact parameters as well as the stress distribution σ(r) in the contact zone (as a function of distance from the axial center r) are summarized in Figure 2.17 and Tab. 2.2 [127, 128]. Applying some simple conversions, the force to achieve a certain deformation δ can be obtained by

F^Hertz(δ) =KR^1/2δ^3/2. (2.55)

Taking adhesion in the contact area into account, Johnson, Kendall and Roberts (JKR)

Figure 2.16: Contact parameters of a sphere pressed against a flat substrate developed a more realistic model for soft contacts [144]. By balancing surface energies and the elastic potential, they calculated the contact area and the deformation as a function of applied load and surface energy per unit area w, as well as the resulting stress distribution (Figure 2.17 and Tab. 2.2). The JKR theory is valid for soft samples

with a large reduced radius and large adhesion forces. In case of zero external load, finite contact radius, deformation, and stress are observed due to adhesion [127]

a₀^JKR=

Due to the strength of adhesion, the interacting bodies still adhere while pulling (neg-ative loads) until a critical force F_adh is reached (see Tab. 2.2). It should be noted that the expression for the adhesion force is independent of the elastic properties of the material. Furthermore, the aspect of experimentally observed adhesion hysteresis can be explained using the JKR theory. The contact area is larger when unloading than in the loading case until rupture at a critical contact radius ofa_c= 0.63a₀ and a (negative) deformation ofδc=−(πw²R/12K)^1/3.

At the same time Derjaguin, Muller, und Toporov (DMT) developed an alternative model where adhesion is present around the contact zone [145]. The DMT model as-sumes that the surface profile is the same as for Hertzian contacts. Adhesion is included by an additional load caused by the surface forces around the contact area. The DMT theory can be applied to contacts of stiff samples with small radii and small adhesion.

In case of zero external load, finite contact radius, deformation, and stress occur a^DMT₀ =

Due to adhesion, the interacting bodies adhere while pulling (negative loads) until a critical force is reached. The deformation parameters are summarized in Figure 2.17 and Tab. 2.2.

If the interaction energy is negligible or for very high loads (F > 10³πwR), the results of JKR and DMT models reduce to the equations given by the Hertz model.

As mentioned above, both the JKR and the DMT model are valid for different limits of material parameters. For quantification of the validity of the particular model, Tabor introduced ”the Tabor parameter” which is defined by the ratio between neck height

Table 2.2: Expressions for contact parameters of different contact mechanic models.

at critical deformation due to adhesion and the range of surface forces z₀ [146]

µ_T =

Maugis introduced a more general theory, which describes the transition range between the JKR and the DMT model and applies to all materials from large hard spheres with high surface energy to small soft bodies with low surface energies. The Maugis theory describes interactions by a Dugdale model and results in expressions for the contact parameters as a function of the so-called ”Maugis parameter” (µM ≈ 1.16µT). For analytic expressions of the contact parameters, the reader is referred to Ref. [127, 147].

Figure 2.18 shows an overview for the availability of the presented contact mechanic models.

A general expression for the contact parameters of two axisymmetric elastic bodies (i.e.

sphere, parabola, or cone) was given by Sneddon’s solution [127]. The reader should notice that the contact radius must be known for these expressions. Also, one can show for any punch that the load displacement can be written in the form

F(δ) =αδⁿ, (2.61)

with αincluding material parameters and ndependent on the geometry (n= 1 for flat cylinders, n= 2 for cones,n = 1.5 for paraboloids i.e. spheres) [128].

The expressions of the stress distributions presented above (see Tab. 2.2) have limita-tions in their phyical meanings, since the stresses in the descriplimita-tions are infinite at the edge of contact. This unphysical situation can be resolved using additional parameters.

One can show that the peak stress can only be in the order of the elastic modulus for

A

B

C

Figure 2.17: Contact mechanics of elastic bodies: 2.17A Contact radius as a function of applied load, 2.17B Deformation as a function of applied load for a soft sphere (R = 10µm, E = 1 MPa, w = 10 mJ/m²) pressed against a hard substrate (dotted lines show the hysteresis character of the JKR theory). 2.17C Stress in the contact zone of a soft sphere for an applied load of F = 1µN (dotted lines show the contact radius).

Figure 2.18: Availability of contact mechanic models (reproduced from [148]

Elsevier)c

soft materials or the predicted VdW stress of the interface.

Above, only total linear elastic deformations were treated. For indentations where plas-tic deformation or viscoelasplas-tic phenomena occur the situation is even more complex due to nonlinear behavior. Approaches to model these contact problems were devel-oped for example by Oliver and Pharr [149]. Also, heterogeneities and roughness are neglected in the continuum elastic theories described above and should be taken into account for a full realistic description [150, 151].

Table 2.3: Experimental methods to study responsive layers, FTIR: Fourier trans-form infrared spectroscopy, QMB: Quartz crystal micro balance, SFA: Surface force apparatus, AFM: Atomic force microscope, JKR: JKR apparatus

properties methods

chemical structure FTIR, QMB, spectroscopy

thickness and density scattering techniques, ellipsometry, reflectometry spectroscopy, microscopy, SFA, AFM surface interactions and mechanics SFA, AFM, JKR, microscopy

2.4 Experimental Methods: Atomic Force and Opti-cal Microscopy

For characterization and understanding of responsive surfaces experimental techniques are necessary. Novel scanning probe microscopy approaches and optical methods can give insights into the behavior of surfaces and interfaces [125]. Exemplary experimental approaches to study the chemical structure, the thickness and density of responsive PE surface, as well as surface interactions and mechanical properties are summarized in Table 2.3 [1, 17]. These methods are accomplished by new theoretical approaches and modeling techniques. This section focus on the used techniques in this thesis, in particular atomic force and optical microscopy. The discussion is oriented on [124, 125, 24, 126] and the cited literature.