Contact mechanics
Contents 1 History 2 Classical solutions for nonadhesive elastic contact 2.1 Contact between a sphere and a halfspace 2.2 Contact between two spheres 2.3 Contact between two crossed cylinders of equal radius R displaystyle R 2.4 Contact between a rigid cylinder with flatended and an elastic halfspace 2.5 Contact between a rigid conical indenter and an elastic halfspace 2.6 Contact between two cylinders with parallel axes 2.7 Bearing contact 2.8 The Method of Dimensionality Reduction 3 Hertzian theory of nonadhesive elastic contact 3.1 Assumptions in Hertzian theory 3.2 Analytical solution techniques 3.2.1 Point contact on a (2D) halfplane 3.2.2 Line contact on a (2D) halfplane 3.2.2.1 Normal loading over a region ( a , b ) displaystyle (a,b) 3.2.2.2 Shear loading over a region ( a , b ) displaystyle (a,b) 3.2.3 Point contact on a (3D) halfspace 3.3 Numerical solution techniques 4 Contact between rough surfaces
5
Adhesive
5.1 Bradley model of rigid contact 5.2 JohnsonKendallRoberts (JKR) model of elastic contact 5.3 DerjaguinMullerToporov (DMT) model of elastic contact 5.4 Tabor parameter 5.5 MaugisDugdale model of elastic contact 5.6 CarpickOgletreeSalmeron (COS) model 5.7 Influence of contact shape 6 See also 7 References 8 External links History[edit] When a sphere is pressed against an elastic material, the contact area increases. Classical contact mechanics is most notably associated with Heinrich Hertz.[4] In 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces. This stillrelevant classical solution provides a foundation for modern problems in contact mechanics. For example, in mechanical engineering and tribology, Hertzian contact stress is a description of the stress within mating parts. The Hertzian contact stress usually refers to the stress close to the area of contact between two spheres of different radii. It was not until nearly one hundred years later that Johnson, Kendall, and Roberts found a similar solution for the case of adhesive contact.[5] This theory was rejected by Boris Derjaguin and coworkers[6] who proposed a different theory of adhesion[7] in the 1970s. The Derjaguin model came to be known as the DMT (after Derjaguin, Muller and Toporov) model,[7] and the Johnson et al. model came to be known as the JKR (after Johnson, Kendall and Roberts) model for adhesive elastic contact. This rejection proved to be instrumental in the development of the Tabor[8] and later Maugis[6][9] parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials. Further advancement in the field of contact mechanics in the midtwentieth century may be attributed to names such as Bowden and Tabor. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact.[10][11] Through investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area. Such understanding also drastically changed the direction of undertakings in tribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces. The contributions of Archard (1957)[12] must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the normal force. Further important insights along these lines were provided by Greenwood and Williamson (1966),[13] Bush (1975),[14] and Persson (2002).[15] The main findings of these works were that the true contact surface in rough materials is generally proportional to the normal force, while the parameters of individual microcontacts (i.e., pressure, size of the microcontact) are only weakly dependent upon the load. Classical solutions for nonadhesive elastic contact[edit] The theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries. Some commonly used solutions are listed below. The theory used to compute these solutions is discussed later in the article. Contact between a sphere and a halfspace[edit] Contact of an elastic sphere with an elastic halfspace An elastic sphere of radius R displaystyle R indents an elastic halfspace to depth d displaystyle d , and thus creates a contact area of radius a = R d displaystyle a= sqrt Rd The applied force F displaystyle F is related to the displacement d displaystyle d by[16] F = 4 3 E ∗ R 1 / 2 d 3 / 2 displaystyle F= tfrac 4 3 E^ * R^ 1/2 d^ 3/2 where 1 E ∗ = 1 − ν 1 2 E 1 + 1 − ν 2 2 E 2 displaystyle frac 1 E^ * = frac 1nu _ 1 ^ 2 E_ 1 + frac 1nu _ 2 ^ 2 E_ 2 and E 1 displaystyle E_ 1 , E 2 displaystyle E_ 2 are the elastic moduli and ν 1 displaystyle nu _ 1 , ν 2 displaystyle nu _ 2 the Poisson's ratios associated with each body. The distribution of normal pressure in the contact area as a function of distance from the center of the circle is[1] p ( r ) = p 0 ( 1 − r 2 a 2 ) 1 / 2 displaystyle p(r)=p_ 0 left(1 frac r^ 2 a^ 2 right)^ 1/2 where p 0 displaystyle p_ 0 is the maximum contact pressure given by p 0 = 3 F 2 π a 2 = 1 π ( 6 F E ∗ 2 R 2 ) 1 / 3 displaystyle p_ 0 = cfrac 3F 2pi a^ 2 = cfrac 1 pi left( cfrac 6F E^ * ^ 2 R^ 2 right)^ 1/3 The radius of the circle is related to the applied load F displaystyle F by the equation a 3 = 3 F R 4 E ∗ displaystyle a^ 3 = cfrac 3FR 4E^ * The depth of indentation d displaystyle d is related to the maximum contact pressure by d = a 2 R = ( 9 F 2 16 E ∗ 2 R ) 1 / 3 displaystyle d= cfrac a^ 2 R =left( cfrac 9F^ 2 16 E^ * ^ 2 R right)^ 1/3 The maximum shear stress occurs in the interior at z ≈ 0.49 a displaystyle zapprox 0.49a for ν = 0.33 displaystyle nu =0.33 . Contact between two spheres[edit] Contact between two spheres. Contact between two crossed cylinders of equal radius. For contact between two spheres of radii R 1 displaystyle R_ 1 and R 2 displaystyle R_ 2 , the area of contact is a circle of radius a displaystyle a . The equations are the same as for a sphere in contact with a half plane except that the effective radius R displaystyle R is defined as[16] 1 R = 1 R 1 + 1 R 2 displaystyle frac 1 R = frac 1 R_ 1 + frac 1 R_ 2 Contact between two crossed cylinders of equal radius R displaystyle R [edit] This is equivalent to contact between a sphere of radius R displaystyle R and a plane. Contact between a rigid cylinder with flatended and an elastic halfspace[edit] Contact between a rigid cylindrical indenter and an elastic halfspace. If a rigid cylinder is pressed into an elastic halfspace, it creates a pressure distribution described by[17] p ( r ) = p 0 ( 1 − r 2 a 2 ) − 1 / 2 displaystyle p(r)=p_ 0 left(1 frac r^ 2 a^ 2 right)^ 1/2 where a displaystyle a is the radius of the cylinder and p 0 = 1 π E ∗ d a displaystyle p_ 0 = frac 1 pi E^ * frac d a The relationship between the indentation depth and the normal force is given by F = 2 a E ∗ d displaystyle F=2aE^ * d, Contact between a rigid conical indenter and an elastic halfspace[edit] Contact between a rigid conical indenter and an elastic halfspace. In the case of indentation of an elastic halfspace of Young's modulus E displaystyle E using a rigid conical indenter, the depth of the contact region ϵ displaystyle epsilon and contact radius a displaystyle a are related by[17] ϵ = a tan θ displaystyle epsilon =atan theta with θ displaystyle theta defined as the angle between the plane and the side surface of the cone. The total indentation depth d displaystyle d is given by: d = π 2 ϵ displaystyle d= frac pi 2 epsilon The total force is F = π E 2 ( 1 − ν 2 ) a 2 tan θ = 2 E π ( 1 − ν 2 ) d 2 tan θ displaystyle F= frac pi E 2left(1nu ^ 2 right) a^ 2 tan theta = frac 2E pi left(1nu ^ 2 right) frac d^ 2 tan theta The pressure distribution is given by p ( r ) = E d π a ( 1 − ν 2 ) ln ( a r + ( a r ) 2 − 1 ) = E d π a ( 1 − ν 2 ) cosh − 1 ( a r ) displaystyle p left(rright) = frac Ed pi aleft(1nu ^ 2 right) ln left( frac a r + sqrt left( frac a r right)^ 2 1 right)= frac Ed pi aleft(1nu ^ 2 right) cosh ^ 1 left( frac a r right) The stress has a logarithmic singularity at the tip of the cone. Contact between two cylinders with parallel axes[edit] Contact between two cylinders with parallel axes In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d:[16] F = π 4 E ∗ L d displaystyle F= frac pi 4 E^ * Ld The radii of curvature are entirely absent from this relationship. The contact radius is described through the usual relationship a = R d displaystyle a= sqrt Rd with 1 R = 1 R 1 + 1 R 2 displaystyle frac 1 R = frac 1 R_ 1 + frac 1 R_ 2 as in contact between two spheres. The maximum pressure is equal to p 0 = ( E ∗ F π L R ) 1 / 2 displaystyle p_ 0 =left( frac E^ * F pi LR right)^ 1/2 Bearing contact[edit] Main article: Bearing pressure The contact in the case of bearings is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: bore or hemispherical cup). The Method of Dimensionality Reduction[edit] Contact between a sphere and an elastic halfspace and onedimensional replaced model. Some contact problems can be solved with the Method of Dimensionality Reduction (MDR). In this method, the initial threedimensional system is replaced with a contact of a body with a linear elastic or viscoelastic foundation (see Fig). The properties of onedimensional systems coincide exactly with those of the original threedimensional system, if the form of the bodies is modified and the elements of the foundation are defined according to the rules of the MDR. [18] [19] However, for exact analytical results, it is required that the contact problem is axisymmetric and the contacts are compact. Hertzian theory of nonadhesive elastic contact[edit] The classical theory of contact focused primarily on nonadhesive contact where no tension force is allowed to occur within the contact area, i.e., contacting bodies can be separated without adhesion forces. Several analytical and numerical approaches have been used to solve contact problems that satisfy the noadhesion condition. Complex forces and moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated. In addition, the contact stresses are usually a nonlinear function of the deformation. To simplify the solution procedure, a frame of reference is usually defined in which the objects (possibly in motion relative to one another) are static. They interact through surface tractions (or pressures/stresses) at their interface. As an example, consider two objects which meet at some surface S displaystyle S in the ( x displaystyle x , y displaystyle y )plane with the z displaystyle z axis assumed normal to the surface. One of the bodies will experience a normallydirected pressure distribution p z = p ( x , y ) = q z ( x , y ) displaystyle p_ z =p(x,y)=q_ z (x,y) and inplane surface traction distributions q x = q x ( x , y ) displaystyle q_ x =q_ x (x,y) and q y = q y ( x , y ) displaystyle q_ y =q_ y (x,y) over the region S displaystyle S . In terms of a Newtonian force balance, the forces: P z = ∫ S p ( x , y ) d A ; Q x = ∫ S q x ( x , y ) d A ; Q y = ∫ S q y ( x , y ) d A displaystyle P_ z =int _ S p(x,y)~mathrm d A~;~~Q_ x =int _ S q_ x (x,y)~mathrm d A~;~~Q_ y =int _ S q_ y (x,y)~mathrm d A must be equal and opposite to the forces established in the other body. The moments corresponding to these forces: M x = ∫ S y p ( x , y ) d A ; M y = ∫ S x p ( x , y ) d A ; M z = ∫ S [ x q y ( x , y ) − y q x ( x , y ) ] d A displaystyle M_ x =int _ S y~p(x,y)~mathrm d A~;~~M_ y =int _ S x~p(x,y)~mathrm d A~;~~M_ z =int _ S [x~q_ y (x,y)y~q_ x (x,y)]~mathrm d A are also required to cancel between bodies so that they are kinematically immobile. Assumptions in Hertzian theory[edit] The following assumptions are made in determining the solutions of Hertzian contact problems: The strains are small and within the elastic limit. The surfaces are continuous and nonconforming (implying that the area of contact is much smaller than the characteristic dimensions of the contacting bodies). Each body can be considered an elastic halfspace. The surfaces are frictionless. Additional complications arise when some or all these assumptions are violated and such contact problems are usually called nonHertzian. Analytical solution techniques[edit] Contact between two spheres. Analytical solution methods for nonadhesive contact problem can be
classified into two types based on the geometry of the area of
contact.[20] A conforming contact is one in which the two bodies touch
at multiple points before any deformation takes place (i.e., they just
"fit together"). A nonconforming contact is one in which the shapes
of the bodies are dissimilar enough that, under zero load, they only
touch at a point (or possibly along a line). In the nonconforming
case, the contact area is small compared to the sizes of the objects
and the stresses are highly concentrated in this area. Such a contact
is called concentrated, otherwise it is called diversified.
A common approach in linear elasticity is to superpose a number of
solutions each of which corresponds to a point load acting over the
area of contact. For example, in the case of loading of a halfplane,
the
Flamant solution
Schematic of the loading on a plane by force P at a point (0,0). A starting point for solving contact problems is to understand the effect of a "pointload" applied to an isotropic, homogeneous, and linear elastic halfplane, shown in the figure to the right. The problem may be either plane stress or plane strain. This is a boundary value problem of linear elasticity subject to the traction boundary conditions: σ x z ( x , 0 ) = 0 ; σ z ( x , z ) = − P δ ( x , z ) displaystyle sigma _ xz (x,0)=0~;~~sigma _ z (x,z)=Pdelta (x,z) where δ ( x , z ) displaystyle delta (x,z) is the Dirac delta function. The boundary conditions state that there are no shear stresses on the surface and a singular normal force P is applied at (0,0). Applying these conditions to the governing equations of elasticity produces the result σ x x = − 2 P π x 2 z ( x 2 + z 2 ) 2 σ z z = − 2 P π z 3 ( x 2 + z 2 ) 2 σ x z = − 2 P π x z 2 ( x 2 + z 2 ) 2 displaystyle begin aligned sigma _ xx &= frac 2P pi frac x^ 2 z (x^ 2 +z^ 2 )^ 2 \sigma _ zz &= frac 2P pi frac z^ 3 (x^ 2 +z^ 2 )^ 2 \sigma _ xz &= frac 2P pi frac xz^ 2 (x^ 2 +z^ 2 )^ 2 end aligned for some point, ( x , y ) displaystyle (x,y) , in the halfplane. The circle shown in the figure indicates a surface on which the maximum shear stress is constant. From this stress field, the strain components and thus the displacements of all material points may be determined. Line contact on a (2D) halfplane[edit] Normal loading over a region ( a , b ) displaystyle (a,b) [edit] Suppose, rather than a point load P displaystyle P , a distributed load p ( x ) displaystyle p(x) is applied to the surface instead, over the range a < x < b displaystyle a<x<b . The principle of linear superposition can be applied to determine the resulting stress field as the solution to the integral equations: σ x x = − 2 z π ∫ a b p ( x ′ ) ( x − x ′ ) 2 d x ′ [ ( x − x ′ ) 2 + z 2 ] 2 ; σ z z = − 2 z 3 π ∫ a b p ( x ′ ) d x ′ [ ( x − x ′ ) 2 + z 2 ] 2 σ x z = − 2 z 2 π ∫ a b p ( x ′ ) ( x − x ′ ) d x ′ [ ( x − x ′ ) 2 + z 2 ] 2 displaystyle begin aligned sigma _ xx &= frac 2z pi int _ a ^ b frac p(x')(xx')^ 2 ,dx' [(xx')^ 2 +z^ 2 ]^ 2 ~;~~sigma _ zz = frac 2z^ 3 pi int _ a ^ b frac p(x'),dx' [(xx')^ 2 +z^ 2 ]^ 2 \sigma _ xz &= frac 2z^ 2 pi int _ a ^ b frac p(x')(xx'),dx' [(xx')^ 2 +z^ 2 ]^ 2 end aligned Shear loading over a region ( a , b ) displaystyle (a,b) [edit] The same principle applies for loading on the surface in the plane of the surface. These kinds of tractions would tend to arise as a result of friction. The solution is similar the above (for both singular loads Q displaystyle Q and distributed loads q ( x ) displaystyle q(x) ) but altered slightly: σ x x = − 2 π ∫ a b q ( x ′ ) ( x − x ′ ) 3 d x ′ [ ( x − x ′ ) 2 + z 2 ] 2 ; σ z z = − 2 z 2 π ∫ a b q ( x ′ ) ( x − x ′ ) d x ′ [ ( x − x ′ ) 2 + z 2 ] 2 σ x z = − 2 z π ∫ a b q ( x ′ ) ( x − x ′ ) 2 d x ′ [ ( x − x ′ ) 2 + z 2 ] 2 displaystyle begin aligned sigma _ xx &= frac 2 pi int _ a ^ b frac q(x')(xx')^ 3 ,dx' [(xx')^ 2 +z^ 2 ]^ 2 ~;~~sigma _ zz = frac 2z^ 2 pi int _ a ^ b frac q(x')(xx'),dx' [(xx')^ 2 +z^ 2 ]^ 2 \sigma _ xz &= frac 2z pi int _ a ^ b frac q(x')(xx')^ 2 ,dx' [(xx')^ 2 +z^ 2 ]^ 2 end aligned These results may themselves be superposed onto those given above for
normal loading to deal with more complex loads.
Point contact on a (3D) halfspace[edit]
Analogously to the
Flamant solution
h displaystyle h between two bodies can only be positive or zero h ≥ 0 displaystyle hgeq 0 where g N = 0 displaystyle g_ N =0 denotes contact. The second assumption in contact mechanics is related to the fact, that no tension force is allowed to occur within the contact area (contacting bodies can be lifted up without adhesion forces). This leads to an inequality which the stresses have to obey at the contact interface. It is formulated for the normal stress σ n = t ⋅ n displaystyle sigma _ n =mathbf t cdot mathbf n . At locations where there is contact between the surfaces the gap is zero, i.e. h = 0 displaystyle h=0 , and there the normal stress is different than zero, indeed, σ n < 0 displaystyle sigma _ n <0 . At locations where the surfaces are not in contact the normal stress is identical to zero; σ n = 0 displaystyle sigma _ n =0 , while the gap is positive, i.e., h > 0 displaystyle h>0 . This type of complementarity formulation can be expressed in the socalled Kuhn–Tucker form, viz. h ≥ 0 , σ n ≤ 0 , σ n h = 0 . displaystyle hgeq 0,,quad sigma _ n leq 0,,quad sigma _ n ,h=0,. These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of the underlying theory of the solid (e.g., linear or nonlinear solid in two or three dimensions, beam or shell model). By restating the normal stress σ n displaystyle sigma _ n in terms of the contact pressure, p displaystyle p , i.e., p = − σ n displaystyle p=sigma _ n the KuhnTucker problem can be restated as in standard complementarity form, i.e. h ≥ 0 , p ≥ 0 , p h = 0 . displaystyle hgeq 0,,quad pgeq 0,,quad p,h=0,. In the linear elastic case the gap can be formulated as h = h 0 + g + u , displaystyle h =h_ 0 + g +u, where h 0 displaystyle h_ 0 is the rigid body separation, g displaystyle g is the geometry/topography of the contact (cylinder and roughness) and u displaystyle u is the elastic deformation/deflection. If the contacting bodies are approximated as linear elastic half spaces, the BoussinesqCerruti integral equation solution can be applied to express the deformation ( u displaystyle u ) as a function of the contact pressure ( p displaystyle p ), i.e., u = ∫ ∞ ∞ K ( x − s ) p ( s ) d s , displaystyle u=int _ infty ^ infty K(xs)p(s)ds, where K ( x − s ) = 2 π E ∗ ln
x − s
displaystyle K(xs)= frac 2 pi E^ * ln xs for line loading of an elastic half space and K ( x − s ) = 1 π E ∗ 1 ( x 1 − s 1 ) 2 + ( x 2 − s 2 ) 2 displaystyle K(xs)= frac 1 pi E^ * frac 1 sqrt (x_ 1 s_ 1 )^ 2 +(x_ 2 s_ 2 )^ 2 for point loading of an elastic halfspace.[1] After discretization the linear elastic contact mechanics problem can be stated in standard Linear Complementarity Problem (LCP) form.[26] h = h 0 + g + C p , h ⋅ p = 0 , p ≥ 0 , h ≥ 0 , displaystyle begin array c mathbf h =mathbf h_ 0 +g+Cp ,\mathbf hcdot p=0 ,,,,mathbf pgeq 0,,,,hgeq 0 ,\end array where C displaystyle mathbf C is a matrix, whose elements are so called influence coefficients relating the contact pressure and the deformation. The strict LCP formulation of the CM problem presented above, allows for direct application of wellestablished numerical solution techniques such as Lemke's pivoting algorithm. The Lemke algorithm has the advantage that it finds the numerically exact solution within a finite number of iterations. The MATLAB implementation presented by Almqvist et al. is one example that can be employed to solve the problem numerically. In addition, an example code for an LCP solution of a 2D linear elastic contact mechanics problem has also been made public at MATLAB file exchange by Almqvist et al. Contact between rough surfaces[edit] When two bodies with rough surfaces are pressed into each other, the true contact area A displaystyle A is much smaller than the apparent contact area A 0 displaystyle A_ 0 . The mechanics of contacting rough surfaces are discussed in terms of normal contact mechanics and static frictional interactions. Natural and engineering surfaces typically exhibit roughness features, known as asperities, across a broad range of length scales down to the molecular level, with surface structures exhibiting self affinity, also known as surface fractality. It is recognized that the self affine structure of surfaces is the origin of the linear scaling of true contact area with applied pressure [27]. Assuming a model of shearing welded contacts, this linearity between contact area and pressure can also be considered the origin of the linearity of the relationship between static friction and applied normal force. In contact between a "random rough" surface and an elastic halfspace, the true contact area is related to the normal force F displaystyle F by[1][27][28][29] A = κ E ∗ h ′ F displaystyle A= frac kappa E^ * h' F with h ′ displaystyle h' equal to the root mean square (also known as the quadratic mean) of the surface slope and κ ≈ 2 displaystyle kappa approx 2 . The median pressure in the true contact surface p a v = F A ≈ 1 2 E ∗ h ′ displaystyle p_ mathrm av = frac F A approx frac 1 2 E^ * h' can be reasonably estimated as half of the effective elastic modulus E ∗ displaystyle E^ * multiplied with the root mean square of the surface slope h ′ displaystyle h' . For the situation where the asperities on the two surfaces have a Gaussian height distribution and the peaks can be assumed to be spherical,[27] the average contact pressure is sufficient to cause yield when p a v = 1.1 σ y ≈ 0.39 σ 0 displaystyle p_ mathrm av =1.1sigma _ y approx 0.39sigma _ 0 where σ y displaystyle sigma _ y is the uniaxial yield stress and σ 0 displaystyle sigma _ 0 is the indentation hardness.[1] Greenwood and Williamson[27] defined a dimensionless parameter Ψ displaystyle Psi called the plasticity index that could be used to determine whether contact would be elastic or plastic. The GreenwoodWilliamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks. An alternative definition of the plasticity index has been given by Mikic.[28] Yield occurs when the pressure is greater than the uniaxial yield stress. Since the yield stress is proportional to the indentation hardness σ 0 displaystyle sigma _ 0 , Micic defined the plasticity index for elasticplastic contact to be Ψ = E ∗ h ′ σ 0 > 2 3 . displaystyle Psi = frac E^ * h' sigma _ 0 > tfrac 2 3 ~. In this definition Ψ displaystyle Psi represents the microroughness in a state of complete plasticity and only one statistical quantity, the rms slope, is needed which can be calculated from surface measurements. For Ψ < 2 3 displaystyle Psi < tfrac 2 3 , the surface behaves elastically during contact.
In both the GreenwoodWilliamson and Mikic models the load is assumed
to be proportional to the deformed area. Hence, whether the system
behaves plastically or elastically is independent of the applied
normal force.[1]
Adhesive
the area of contact was larger than that predicted by Hertz theory, the area of contact had a nonzero value even when the load was removed, and there was strong adhesion if the contacting surfaces were clean and dry. This indicated that adhesive forces were at work. The JohnsonKendallRoberts (JKR) model and the DerjaguinMullerToporov (DMT) models were the first to incorporate adhesion into Hertzian contact. Bradley model of rigid contact[edit] It is commonly assumed that the surface force between two atomic planes at a distance z displaystyle z from each other can be derived from the LennardJones potential. With this assumption F ( z ) = 16 γ 3 z 0 [ ( z z 0 ) − 9 − ( z z 0 ) − 3 ] displaystyle F(z)= cfrac 16gamma 3z_ 0 left[left( cfrac z z_ 0 right)^ 9 left( cfrac z z_ 0 right)^ 3 right] where F displaystyle F is the force (positive in compression), 2 γ displaystyle 2gamma is the total surface energy of both surfaces per unit area, and z 0 displaystyle z_ 0 is the equilibrium separation of the two atomic planes.
The Bradley model applied the
LennardJones potential
F a ( z ) = 16 γ π R 3 [ 1 4 ( z z 0 ) − 8 − ( z z 0 ) − 2 ] ; 1 R = 1 R 1 + 1 R 2 displaystyle F_ a (z)= cfrac 16gamma pi R 3 left[ cfrac 1 4 left( cfrac z z_ 0 right)^ 8 left( cfrac z z_ 0 right)^ 2 right]~;~~ frac 1 R = frac 1 R_ 1 + frac 1 R_ 2 where R 1 , R 2 displaystyle R_ 1 ,R_ 2 are the radii of the two spheres. The two spheres separate completely when the pulloff force is achieved at z = z 0 displaystyle z=z_ 0 at which point F a = F c = − 4 γ π R . displaystyle F_ a =F_ c =4gamma pi R. JohnsonKendallRoberts (JKR) model of elastic contact[edit] Schematic of contact area for the JKR model. JKR test with a rigid bead on a deformable planar material: complete cycle To incorporate the effect of adhesion in Hertzian contact, Johnson, Kendall, and Roberts[5] formulated the JKR theory of adhesive contact using a balance between the stored elastic energy and the loss in surface energy. The JKR model considers the effect of contact pressure and adhesion only inside the area of contact. The general solution for the pressure distribution in the contact area in the JKR model is p ( r ) = p 0 ( 1 − r 2 a 2 ) 1 / 2 + p 0 ′ ( 1 − r 2 a 2 ) − 1 / 2 displaystyle p(r)=p_ 0 left(1 cfrac r^ 2 a^ 2 right)^ 1/2 +p_ 0 'left(1 cfrac r^ 2 a^ 2 right)^ 1/2 Note that in the original Hertz theory, the term containing p 0 ′ displaystyle p_ 0 ' was neglected on the ground that tension could not be sustained in the contact zone. For contact between two spheres p 0 = 2 a E ∗ π R ; p 0 ′ = − ( 4 γ E ∗ π a ) 1 / 2 displaystyle p_ 0 = cfrac 2aE^ * pi R ~;~~p_ 0 '=left( cfrac 4gamma E^ * pi a right)^ 1/2 where a displaystyle a, is the radius of the area of contact, F displaystyle F is the applied force, 2 γ displaystyle 2gamma is the total surface energy of both surfaces per unit contact area, R i , E i , ν i , i = 1 , 2 displaystyle R_ i ,E_ i ,nu _ i ,~~i=1,2 are the radii, Young's moduli, and Poisson's ratios of the two spheres, and 1 R = 1 R 1 + 1 R 2 ; 1 E ∗ = 1 − ν 1 2 E 1 + 1 − ν 2 2 E 2 displaystyle frac 1 R = frac 1 R_ 1 + frac 1 R_ 2 ~;~~ frac 1 E^ * = frac 1nu _ 1 ^ 2 E_ 1 + frac 1nu _ 2 ^ 2 E_ 2 The approach distance between the two spheres is given by d = π a 2 E ∗ ( p 0 + 2 p 0 ′ ) = a 2 R displaystyle d= cfrac pi a 2E^ * (p_ 0 +2p_ 0 ')= cfrac a^ 2 R The Hertz equation for the area of contact between two spheres, modified to take into account the surface energy, has the form a 3 = 3 R 4 E ∗ ( F + 6 γ π R + 12 γ π R F + ( 6 γ π R ) 2 ) displaystyle a^ 3 = cfrac 3R 4E^ * left(F+6gamma pi R+ sqrt 12gamma pi RF+(6gamma pi R)^ 2 right) When the surface energy is zero, γ = 0 displaystyle gamma =0 , the Hertz equation for contact between two spheres is recovered. When the applied load is zero, the contact radius is a 3 = 9 R 2 γ π E ∗ displaystyle a^ 3 = cfrac 9R^ 2 gamma pi E^ * The tensile load at which the spheres are separated, i.e., a = 0 displaystyle a=0 , is predicted to be F c = − 3 γ π R displaystyle F_ c =3gamma pi R, This force is also called the pulloff force. Note that this force is independent of the moduli of the two spheres. However, there is another possible solution for the value of a displaystyle a at this load. This is the critical contact area a c displaystyle a_ c , given by a c 3 = 9 R 2 γ π 4 E ∗ displaystyle a_ c ^ 3 = cfrac 9R^ 2 gamma pi 4E^ * If we define the work of adhesion as Δ γ = γ 1 + γ 2 − γ 12 displaystyle Delta gamma =gamma _ 1 +gamma _ 2 gamma _ 12 where γ 1 , γ 2 displaystyle gamma _ 1 ,gamma _ 2 are the adhesive energies of the two surfaces and γ 12 displaystyle gamma _ 12 is an interaction term, we can write the JKR contact radius as a 3 = 3 R 4 E ∗ ( F + 3 Δ γ π R + 6 Δ γ π R F + ( 3 Δ γ π R ) 2 ) displaystyle a^ 3 = cfrac 3R 4E^ * left(F+3Delta gamma pi R+ sqrt 6Delta gamma pi RF+(3Delta gamma pi R)^ 2 right) The tensile load at separation is F = − 3 2 Δ γ π R displaystyle F= cfrac 3 2 Delta gamma pi R, and the critical contact radius is given by a c 3 = 9 R 2 Δ γ π 8 E ∗ displaystyle a_ c ^ 3 = cfrac 9R^ 2 Delta gamma pi 8E^ * The critical depth of penetration is d c = a c 2 R = ( 9 4 ) 2 3 ( Δ γ ) 2 3 ( π 2 3
R 1 3 E ∗ 2 3 ) displaystyle d_ c = cfrac a_ c ^ 2 R =left( cfrac 9 4 right)^ tfrac 2 3 (Delta gamma )^ tfrac 2 3 left( cfrac pi ^ tfrac 2 3 ~R^ tfrac 1 3 E^ * ^ tfrac 2 3 right) DerjaguinMullerToporov (DMT) model of elastic contact[edit] The DerjaguinMullerToporov (DMT) model[7][31] is an alternative model for adhesive contact which assumes that the contact profile remains the same as in Hertzian contact but with additional attractive interactions outside the area of contact. The radius of contact between two spheres from DMT theory is a 3 = 3 R 4 E ∗ ( F + 4 γ π R ) displaystyle a^ 3 = cfrac 3R 4E^ * left(F+4gamma pi Rright) and the pulloff force is F c = − 4 γ π R displaystyle F_ c =4gamma pi R, When the pulloff force is achieved the contact area becomes zero and there is no singularity in the contact stresses at the edge of the contact area. In terms of the work of adhesion Δ γ displaystyle Delta gamma a 3 = 3 R 4 E ∗ ( F + 2 Δ γ π R ) displaystyle a^ 3 = cfrac 3R 4E^ * left(F+2Delta gamma pi Rright) and F c = − 2 Δ γ π R displaystyle F_ c =2Delta gamma pi R, Tabor parameter[edit] In 1977, Tabor[32] showed that the apparent contradiction between the JKR and DMT theories could be resolved by noting that the two theories were the extreme limits of a single theory parametrized by the Tabor parameter ( μ displaystyle mu ) defined as μ := d c z 0 ≈ [ R ( Δ γ ) 2 E ∗ 2 z 0 3 ] 1 / 3 displaystyle mu := cfrac d_ c z_ 0 approx left[ cfrac R(Delta gamma )^ 2 E^ * ^ 2 z_ 0 ^ 3 right]^ 1/3 where z 0 displaystyle z_ 0 is the equilibrium separation between the two surfaces in contact. The JKR theory applies to large, compliant spheres for which μ displaystyle mu is large. The DMT theory applies for small, stiff spheres with small values of μ displaystyle mu . Subsequently, Derjaguin and his collaborators[33] by applying Bradley's surface force law to an elastic half space, confirmed that as the Tabor parameter increases, the pulloff force falls from the Bradley value 2 π R Δ γ displaystyle 2pi RDelta gamma to the JKR value ( 3 / 2 ) π R Δ γ displaystyle (3/2)pi RDelta gamma . More detailed calculations were later done by Greenwood [34] revealing the Sshaped load/approach curve which explains the jumpingon effect. A more efficient method of doing the calculations and additional results were given by Feng [35] MaugisDugdale model of elastic contact[edit] Schematic of contact area for the MaugisDugdale model. Further improvement to the Tabor idea was provided by Maugis[9] who represented the surface force in terms of a Dugdale cohesive zone approximation such that the work of adhesion is given by Δ γ = σ 0
h 0 displaystyle Delta gamma =sigma _ 0 ~h_ 0 where σ 0 displaystyle sigma _ 0 is the maximum force predicted by the
LennardJones potential
h 0 displaystyle h_ 0 is the maximum separation obtained by matching the areas under the Dugdale and LennardJones curves (see adjacent figure). This means that the attractive force is constant for z 0 ≤ z ≤ z 0 + h 0 displaystyle z_ 0 leq zleq z_ 0 +h_ 0 . There is not further penetration in compression. Perfect contact occurs in an area of radius a displaystyle a and adhesive forces of magnitude σ 0 displaystyle sigma _ 0 extend to an area of radius c > a displaystyle c>a . In the region a < r < c displaystyle a<r<c , the two surfaces are separated by a distance h ( r ) displaystyle h(r) with h ( a ) = 0 displaystyle h(a)=0 and h ( c ) = h 0 displaystyle h(c)=h_ 0 . The ratio m displaystyle m is defined as m := c a displaystyle m:= cfrac c a . In the MaugisDugdale theory,[36] the surface traction distribution is divided into two parts  one due to the Hertz contact pressure and the other from the Dugdale adhesive stress. Hertz contact is assumed in the region − a < r < a displaystyle a<r<a . The contribution to the surface traction from the Hertz pressure is given by p H ( r ) = ( 3 F H 2 π a 2 ) ( 1 − r 2 a 2 ) 1 / 2 displaystyle p^ H (r)=left( cfrac 3F^ H 2pi a^ 2 right)left(1 cfrac r^ 2 a^ 2 right)^ 1/2 where the Hertz contact force F H displaystyle F^ H is given by F H = 4 E ∗ a 3 3 R displaystyle F^ H = cfrac 4E^ * a^ 3 3R The penetration due to elastic compression is d H = a 2 R displaystyle d^ H = cfrac a^ 2 R The vertical displacement at r = c displaystyle r=c is u H ( c ) = 1 π R [ a 2 ( 2 − m 2 ) sin − 1 ( 1 m ) + a 2 m 2 − 1 ] displaystyle u^ H (c)= cfrac 1 pi R left[a^ 2 (2m^ 2 )sin ^ 1 left( cfrac 1 m right)+a^ 2 sqrt m^ 2 1 right] and the separation between the two surfaces at r = c displaystyle r=c is h H ( c ) = c 2 2 R − d H + u H ( c ) displaystyle h^ H (c)= cfrac c^ 2 2R d^ H +u^ H (c) The surface traction distribution due to the adhesive Dugdale stress is p D ( r ) = − σ 0 π cos − 1 [ 2 − m 2 − r 2 a 2 m 2 ( 1 − r 2 m 2 a 2 ) ] f o r r ≤ a − σ 0 f o r a ≤ r ≤ c displaystyle p^ D (r)= begin cases  cfrac sigma _ 0 pi cos ^ 1 left[ cfrac 2m^ 2  cfrac r^ 2 a^ 2 m^ 2 left(1 cfrac r^ 2 m^ 2 a^ 2 right) right]&quad mathrm for quad rleq a\sigma _ 0 &quad mathrm for quad aleq rleq cend cases The total adhesive force is then given by F D = − 2 σ 0 m 2 a 2 [ cos − 1 ( 1 m ) + 1 m 2 m 2 − 1 ] displaystyle F^ D =2sigma _ 0 m^ 2 a^ 2 left[cos ^ 1 left( cfrac 1 m right)+ frac 1 m^ 2 sqrt m^ 2 1 right] The compression due to Dugdale adhesion is d D = − ( 2 σ 0 a E ∗ ) m 2 − 1 displaystyle d^ D =left( cfrac 2sigma _ 0 a E^ * right) sqrt m^ 2 1 and the gap at r = c displaystyle r=c is h D ( c ) = ( 4 σ 0 a π E ∗ ) [ m 2 − 1 cos − 1 ( 1 m ) + 1 − m ] displaystyle h^ D (c)=left( cfrac 4sigma _ 0 a pi E^ * right)left[ sqrt m^ 2 1 cos ^ 1 left( cfrac 1 m right)+1mright] The net traction on the contact area is then given by p ( r ) = p H ( r ) + p D ( r ) displaystyle p(r)=p^ H (r)+p^ D (r) and the net contact force is F = F H + F D displaystyle F=F^ H +F^ D . When h ( c ) = h H ( c ) + h D ( c ) = h 0 displaystyle h(c)=h^ H (c)+h^ D (c)=h_ 0 the adhesive traction drops to zero. Nondimensionalized values of a , c , F , d displaystyle a,c,F,d are introduced at this stage that are defied as a ¯ = α a ; c ¯ := α c ; d ¯ := α 2 R d ; α := ( 4 E ∗ 3 π Δ γ R 2 ) 1 / 3 ; A ¯ := π c 2 ; F ¯ = F π Δ γ R displaystyle bar a =alpha a~;~~ bar c :=alpha c~;~~ bar d :=alpha ^ 2 Rd~;~~alpha :=left( cfrac 4E^ * 3pi Delta gamma R^ 2 right)^ 1/3 ~;~~ bar A :=pi c^ 2 ~;~~ bar F = cfrac F pi Delta gamma R In addition, Maugis proposed a parameter λ displaystyle lambda which is equivalent to the Tabor parameter μ displaystyle mu . This parameter is defined as λ := σ 0 ( 9 R 2 π Δ γ E ∗ 2 ) 1 / 3 = 1.16 μ displaystyle lambda :=sigma _ 0 left( cfrac 9R 2pi Delta gamma E^ * ^ 2 right)^ 1/3 =1.16mu where the step cohesive stress σ 0 displaystyle sigma _ 0 equals to the theoretical stress of the LennardJones potential σ t h = 16 Δ γ 9 3 z 0 displaystyle sigma _ th = cfrac 16Delta gamma 9 sqrt 3 z_ 0 Zheng and Yu [37] suggested another value for the step cohesive stress σ 0 = exp ( − 223 420 ) ⋅ Δ γ z 0 ≈ 0.588 Δ γ z 0 displaystyle sigma _ 0 =exp left( cfrac 223 420 right)cdot cfrac Delta gamma z_ 0 approx 0.588 cfrac Delta gamma z_ 0 to match the LennardJones potential, which leads to λ ≈ 0.663 μ displaystyle lambda approx 0.663mu Then the net contact force may be expressed as F ¯ = a ¯ 3 − λ a ¯ 2 [ m 2 − 1 + m 2 sec − 1 m ] displaystyle bar F = bar a ^ 3 lambda bar a ^ 2 left[ sqrt m^ 2 1 +m^ 2 sec ^ 1 mright] and the elastic compression as d ¯ = a ¯ 2 − 4 3 λ a ¯ m 2 − 1 displaystyle bar d = bar a ^ 2  cfrac 4 3 ~lambda bar a sqrt m^ 2 1 The equation for the cohesive gap between the two bodies takes the form λ a ¯ 2 2 [ ( m 2 − 2 ) sec − 1 m + m 2 − 1 ] + 4 λ a ¯ 3 [ m 2 − 1 sec − 1 m − m + 1 ] = 1 displaystyle cfrac lambda bar a ^ 2 2 left[(m^ 2 2)sec ^ 1 m+ sqrt m^ 2 1 right]+ cfrac 4lambda bar a 3 left[ sqrt m^ 2 1 sec ^ 1 mm+1right]=1 This equation can be solved to obtain values of c displaystyle c for various values of a displaystyle a and λ displaystyle lambda . For large values of λ displaystyle lambda , m → 1 displaystyle mrightarrow 1 and the JKR model is obtained. For small values of λ displaystyle lambda the DMT model is retrieved. CarpickOgletreeSalmeron (COS) model[edit] The MaugisDugdale model can only be solved iteratively if the value of λ displaystyle lambda is not known apriori. The CarpickOgletreeSalmeron approximate solution [38] simplifies the process by using the following relation to determine the contact radius a displaystyle a : a = a 0 ( β ) ( β + 1 − F / F c ( β ) 1 + β ) 2 / 3 displaystyle a=a_ 0 (beta )left( cfrac beta + sqrt 1F/F_ c (beta ) 1+beta right)^ 2/3 where a 0 displaystyle a_ 0 is the contact area at zero load, and β displaystyle beta is a transition parameter that is related to λ displaystyle lambda by λ = − 0.924 ln ( 1 − 1.02 β ) displaystyle lambda =0.924ln(11.02beta ) The case β = 1 displaystyle beta =1 corresponds exactly to JKR theory while β = 0 displaystyle beta =0 corresponds to DMT theory. For intermediate cases 0 < β < 1 displaystyle 0<beta <1 the COS model corresponds closely to the MaugisDugdale solution for 0.1 < λ < 5 displaystyle 0.1<lambda <5 . Influence of contact shape[edit] Even in the presence of perfectly smooth surfaces, geometry can come into play in form of the macroscopic shape of the contacting region. When a rigid punch with flat but oddy shaped face is carefully pulled off his soft counterpart, its detachment occurs not instantaneously but detachment fronts start at pointed corners and travel inwards, until the final configuration is reached which for macroscopically isotropic shapes is almost circular. The main parameter determining the adhesive strength of flat contacts occurs to be the maximum linear size of the contact[39]. The process of detachment can as observed experimentally can be seen in the film[40]. See also[edit] Adhesive
Adhesive
References[edit] ^ a b c d e f Johnson, K. L, 1985, Contact mechanics, Cambridge
University Press.
^ Popov, Valentin L., 2010, Contact Mechanics and Friction. Physical
Principles and Applications, SpringerVerlag, 362 p.,
ISBN 9783642108020.
^ H. Hertz, Über die berührung fester elastischer Körper (On the
contact of rigid elastic solids). In: Miscellaneous Papers. Jones and
Schott, Editors, J. reine und angewandte Mathematik 92, Macmillan,
London (1896), p. 156 English translation: Hertz, H.
^ Hertz, H. R., 1882, Ueber die Beruehrung elastischer Koerper (On
Contact Between Elastic Bodies), in Gesammelte Werke (Collected
Works), Vol. 1, Leipzig, Germany, 1895.
^ a b c K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy
and the contact of elastic solids, Proc. R. Soc. London A 324 (1971)
301313
^ a b D. Maugis, Contact,
Adhesion
External links[edit] [1]: More about contact stresses and the evolution of bearing stress
equations can be found in this publication by NASA Glenn Research
Center head the NASA Bearing, Gearing and Transmission Section, Erwin
Zaretsky.
[2]: A MATLAB routine to solve the linear elastic contact mechanics
problem entitled; "An LCP solution of the linear elastic contact
mechanics problem" is provided at the file exchange at MATLAB Central.
[3]:
Contact mechanics
