Mechanics of unloading of a rough surfaces pre-loaded joint

In this paper we study the problem of the contact characteristics changing on a decrease of the load applied to the preloaded joint of roughness surfaces. The penetration of a rigid sphere (indenter) into the elastic hardenable half-space is considered originally. The elastic crater restoring by unloading is also considered. In elastic-plastic material’s describing, Hollomon’s pawer law is used. To describe a contact of a rigid rough surface with an elastic plastic half-space, the discrete model of a rough surface is used. Microasperities are represented as a set of identical spherical segments, the height distribution of which corresponds to the bearing profile curve of the surface. To describe the bearing profile curve, incomplete beta function ratio is used. The relations of relative contact areas η and e η and gap densities at the joint e Λ and ce Λ on dimensionless loading q F and qe F at loading and unloading for different values of y ε and n are given. The obtained results are of practical importance for the sealing ability prediction of fixed sealing joints at the design stage, in particular for tightness supply of flange couplings and high pressure vessels seals.


Introduction
Many operational properties of joints of machine parts and devices, including tightness, are defined by contact interaction of rough surfaces [1]. In most cases, by contacting metal rough surfaces, the contact is elasticplastic. In [2,3] for the description of an elastic-plastic contact, the authors use the kinetic sphere indentation's load-displacement diagram. As it follows from [1], the sealing ability of a sealing joint is defined by the relative contact area and the density of gaps in a joint. From this point of view, the loading of sealing joints at their assembly by the increased load and maintenance of their tightness by lower load to providing the minimum weight and size characteristics is of practical interest. Especially it is actual at the flange joints and high pressure vessels design.
To solve this problem it is necessary to know how the contact characteristics (the relative contact area and the density of gaps) change during unloading preloaded fixed joint.

Contact of a rigid sphere with the elastic-plastic half-space
Let us consider the indentation of the rigid sphere (indenter) into the elastic-plastic half-space and the elastic crater restoring. For the description of an elastic-plastic hardening material we use the Hollomon's power law where n is the strain hardening exponent, E y y σy is the yield strength, E is the elastic modulus. By using a spherical indenter, the degree of deformation changes at all stage of loading and the primary result of such test is the kinetic indentation's loaddisplacement diagram [4] (Fig. 1).  (2) and the unload-displacement branch can be written as where C1, C2 are constants; α and γ are exponents. The initial unloading stiffness is In [3] parameter γ is determined by means of a calculation , 2 , c h is the depth at which the contact of an indenter and a material under load m P takes place. The problem of determination of the contact area at elastic restoring was considered in [5], however the author did not take into account effects of "sink-in/pileup". Under loading to the contact depth * e h (Fig. 2), the subsequent elastic restoring will be equal to 3 Contact of a rigid rough surface with the elastic-plastic half-space

The discrete model of a rough surface
Let us use the discrete model of a rough surface, in which microasperities are presented by identical spherical segments with the distribution of segments' peaks on height corresponds to the bearing profile curve of the real surface [1]. To describe the distribution of the reference curve is used incomplete beta function are incomplete and complete beta functions; where Rp, Rq, Rmax are height roughness parameters according to ISO 4281/1-1997. In this case, the density of the asperities distribution on height function is where εs is determined from the condition [1]. Geometrical parameters of a spherical segment are: ; ac is the radius of the base of the spherical segment.
where . max R R >> At elastic contact, the relation between the relative size of indentation of the single asperity and the relative load is defined according to the Hertz theory. , 3 where * E is the reduced (contact) elastic modulus. For elastic-plastic contact we use the equation from [6] , 2 *

The relative contact area
This problem has been considered in detail by [7]. When using the Eq. (14) for asperities of a rough surface, it is necessary to take into account that where ε is the relative approach a rough surface and the half-space; u is the initial distance to the peak of i-th asperity.
When the indentation of a rigid rough surface is equal ε, the total load P is described by equation where εe is the relative boundary an elastic contact, dnr is the number of peaks in a layer du, where .
For the real contact area, similar to Eq. (17), we have Determining Ari, we take into account that According to [8], we have , 2 For the relative contact area c r A A = η , we will finally obtain It should be noted that under unloading of a rigid rough surface, the size ∆h is identical for all asperities.
Using data from [2], we determine w0i as , ) By taking into account Eq. (16), we have The fig. 3 shows the relations of loading Fq(ε) and unloading Fqe(ξ−∆ε) of a contact of a rigid rough surface with the elastic-plastic half-space for different values εy and n calculated on Eqs. (21) and (32).
The fig. 4 shows the relations the relative contact area η and ηe versus the dimensionless force elasticgeometric parameter Fq and Fqe under loading and unloading, respectively.

The density of gaps in the joint
During indentation of rough surface on the value εRmax, contact of every single asperity is followed by "pileup/sink-in" effects, i.e. by plastic extrusion of the halfspace's material and its elastic squeezing [9]. For gaps volume is followed that ( ) where Ve is the total volume increase due to the elastic material squeezing for all the contacting asperities, Ve is the total volume decrease due to the plastic material extrusion for all the contacting asperities. Correspondingly, the joint's gaps density is For a symmetric profile (when The total volume due to the elastic squeezing of all asperities is uze is the elastic displacement of surface points out of the contact area [10]. From the eq. for spherical segment volume with height hf and unloaded crater radius ρi, the volume of plastically displaced material, attributable to a single hole, is  For example, the sealing ability of the joint of an elastoplastic contact is considered [11]. The intensity of the volume rate (per unit length along the seal perimeter) of ideally compressible gas through a sealing joint with a uniform distribution of contact pressure qc is where Cu is the dimensionless permeability functional that characterizes the sealing ability of the joint where p1 is the environment pressure, p is the atmosphere pressure, μ is the dynamic viscosity of the environment, l is the width of the sealing area, νn is the rate of effective micro-channels. At pressures above 16 MPa at a leakage calculations, it should to use the model of a real gas [12].
The rate of effective micro-channels [11] is