Slip Problem Research Articles

Wall-slip effects in SnAgCu solder pastes used in electronics assembly applications

Solder paste is the most important strategic bonding material used in the assembly of surface mount components in electronics manufacturing. As the trend towards miniaturisation of electronic products continues, there is an increasing demand for better understanding of the flow and deformation that is, the rheological behaviour of solder paste formulations. Wall slip plays an important role in characterising the flow behaviour of solder paste materials. The problem of wall slip arises due to the various attractive and repulsive forces acting between the solder particles and the walls of the measuring geometry. These interactions could lead to the presence of a thin solvent layer adjacent to the wall, which gives rise to slippage. In rheological measurements, slip effects can generally be avoided by using roughened surfaces for measuring geometries. In this paper, a novel technique is developed to study the effect of wall slip in the rheological measurements of lead-free solder paste. The viscosity and oscillatory data obtained for three different solder paste samples (from measuring geometries of different surface roughness) have been analysed and compared. In viscosity measurements, slip effects were dominant at low shear rates and the use of serrated surfaces was found to be quite effective in minimizing slip effects. Oscillatory measurements were also affected by roughening the surfaces of measuring geometries.

Multiple Contacts of Similar Elastic Materials

The contact problem for an elastic body indenting a similar half-space resulting in multiple contacts is important for various applications. In this paper an exact fast numerical method based on singular integral equations is developed to solve the normal contact (including applied moments), partial slip, and shear-reversal problems for such contacts. The contact patches are considered to be fully interacting, with no simplifying assumptions. A contact algorithm to automatically generate trial values based on an analysis of the profile and to subsequently guide the solver toward convergence is detailed. Some applications are discussed, including regular rough cylinders and a regularly rough flat punch with rounded edges. The examples involve between 3 and 29 contacts. The partial slip problems include demonstration of cases with multiple stick zones in some contact patches and complete sliding in others.

Bulk-stress effects in double contacts of similar elastic materials

The effects of a remote applied stress on double contacts of similar elastic materials are examined using a numerical method based on Cauchy singular integral equations (SIE). In contrast to single contacts, the space of possible shear configurations is very rich. In many cases, the partial slip problem does not have a contact equivalent. An equation based on displacement continuity is derived to relate the stick-zone extents for any shear configuration; this equation is essential for the solution of general partial slip problems involving two stick-zones. A shear configuration map that shows the types of partial slip solution that occur for various shear load-remote bulk-stress combinations is obtained for biquadratic indenters. Lastly, it is shown that it is possible to predict the shear configuration in some cases for double cylindrical indenters.

The Kramers problem with accommodative boundary conditions for quantum Fermi gases

The Kramers problem of isothermal slip of a quantum Fermi gas with Cercignani boundary conditions is solved analytically. The velocity of isothermal slip is obtained as a function of the accommodation coefficient and the reduced chemical potential—the ratio of the chemical potential to the product of Boltzmann’s constant and the absolute temperature. The distribution function of the molecules is presented in explicit form.

The velocity slip problem: Accurate solutions of the BGK model integral equation

The velocity slip problem (also known as the Kramers problem) in the kinetic theory has been solved by a variety of techniques, and some accurate numerical as well as general variational results are available for both simple gases and gas mixtures. For the model equations, reasonably accurate solutions (within about 1% of the exact results) have been obtained by solving the related integral equations, but such techniques have not previously been pushed far enough to explore the possibilities of obtaining benchmark solutions. Some of the reasons for not further pursuing these techniques were limitations on the efficiencies and accuracies of the computer routines that could be constructed for the needed Abramowitz functions and limitations in terms of computational resources with respect to available precision and storage capabilities. These limitations have now effectively been removed with modern advances in computer technology and the computational tools that have become available, and it is the purpose of the present paper to report on our explorations in this regard within the specific context of the velocity slip problem as based on the BGK model which we have approached using the singularity subtraction technique in conjunction with numerical quadratures.

Wave time-domain hypersingular integral applied to co-seismic slip of P- and S-waves from crack propagation

This work presents a wave time-domain hypersingular integral equation (WTD-HIE) method for modeling the 3D crack propagation problem of co-seismic slip under fully coupled electromagnetothermoelastic P- and S-wave fields through theoretical analysis and numerical simulations. First, the general extended displacement wave solutions are obtained by analyzing wave time-domains using the Green’s function method. Then, based on the wave time-domain boundary element method, the problem is reduced to solving a set of WTD-HIEs coupled with nonlinear boundary integral equations, in which the unknown functions are the general extended displacement discontinuity waves. The behavior of the general extended singular stress indices around the crack front terminating at the slip surface is analyzed by time-domain main-part analysis. The general extended singular stress waves and the extended dynamic stress intensity factors are obtained from closed-form solutions. In addition, a numerical method for the problem is proposed, in which the extended displacement discontinuity waves are approximated by the product of time-domain density functions and polynomials. Finally, the radiation distribution of near-field extended dynamic stress intensity factors for P- and S-waves at the crack surface are calculated, and the results are presented toward demonstrating the applicability of the proposed method.

Direct GPS P-Code Acquisition Method Based on FFT

Recently, direct acquisition of GPS P-code has received considerable attention to enhance the anti-jamming and anti-spoofing capabilities of GPS receivers. This paper describes a P-code acquisition method that uses block searches with large-scale FFT to search code phases and carrier frequency offsets in parallel. To limit memory use, especially when implemented in hardware, only the largest correlation result with its position information was preserved after searching a block of resolution cells in both the time and frequency domains. A second search was used to solve the code phase slip problem induced by the code frequency offset. Simulation results demonstrate that the probability of detection is above 0.99 for carrier-to-noise density ratios in excess of 40 dB·Hz when the predetection integration time is 0.8 ms and 6 non-coherent integrations are used in the analysis.

The screw dislocation in a three-quarter plane

The state of stress in a three-quarter-plane undergoing antiplane shear deformation is studied, due to the presence of a screw dislocation along one of the projection lines extended from the free surfaces. A simple, accurate formula for the state of stress along the line is found, providing a useful kernel for the solution of crack and contact edge slip problems.

Quantum phase slips in a confined geometry

We consider tunneling of vortices across a superconducting film that is both narrow and short (and connected to bulk superconducting leads at the ends). We find that in the superconducting state the resistance, at low values of the temperature $(T)$ and current, does not follow the power-law dependence on $T$ characteristic of longer samples but is exponential in $1∕T$. The coefficient of $1∕T$ in the exponent depends on the length or, equivalently, the total normal-state resistance of the sample. These conclusions persist in the one-dimensional limit, which is similar to the problem of quantum phase slips in an ultranarrow short wire.

The influence of layering and pre-existing joints on the development of internal structure in normal fault zones: the Lodève basin, France

Abstract This paper examines the role of mechanical stratigraphy on the evolution of normal fault geometry and fault zone internal structure, using a well-exposed normal fault system from the Permian Lodève Basin, southern France. Faults formed early during the syn-deformation tilting history of the basin tend to have steeper segments in the competent sandstone layers due to refraction, assisted by pre-existing early bedding-perpendicular joints, where displacement remained on the order of bed thickness. Faults which continued to slip during tilting have a more complex structure of splays due both to the space incompatability problem of slip at fault bends of this irregular geometry, and because tilting favours the generation of new splays at a different angle to the earlier faults experiencing rotation. Continued deformation between faults and their splays often causes both distributed deformation in between the two, and reconnection of splays to the main fault forming isolated lenses. Thus, fault zone complexity increases greatly as slip exceeds competent bed thickness, owing both to the presence of the mechanical layering, and the fact that this layering is being tilted.

Torsion of rough elastic half-space by rigid punch

Torsion of an elastic half-space by a rigid punch is investigated. The boundary of the half-space is assumed to be rough. Two geometries of the punch-parabolic and flat end are considered. It is shown that the contact area consists of stick and slip zones. This fact, which is well-known in the classical torsional contact of the elastic half-space with the smooth surface and the parabolic punch, also holds true for the flat-ended punch if the boundary roughness is involved. The partial slip problems are reduced to the integral equations, which are solved numerically. The presented results show the effects of boundary roughness on the shear stresses, size of the stick area and the relation between the twisting moment and the angle of twist.

Swelling of a single foam film under slipping

Aqueous foams are often used under flow, and one of the biggest challenges is to create predictive models of their rheology. Our investigations deal with the problem of slip at surfaces, and more precisely sliding bubbles inside a narrow horizontal tube. Using a new experimental procedure based on a Fabry–Perot interferometer, our first results underline some thickness variations for a vertical film separating two bubbles, and moving horizontally. The velocity-dependent thickness is quantitatively described by a power law. We discuss the possible origins of this effect, and the consistency with available models of bubble slipping. Our observations confirm that, under dynamical conditions, there can be complex liquid re-distribution between the different elements of the foam structure.

Is the constitutive relation for entangled polymers monotonic?

The dynamic and steady state behavior of highly entangled polybutadiene solutions has been studied in shearing flow using high-spatial-resolution particle tracking velocimetry. In Couette geometry, the velocity profile is approximately linear upon the imposition of a step shear rate, but becomes significantly curved at later times, and attains the steady shape only after more than 100 strain units. In steady state, local shear rate varies smoothly across the gap and there is no shear banding. The constitutive relation is constructed from local stress and local shear rate extracted from velocity profiles. This procedure is not affected by the common problems of slip and edge instabilities as encountered with cone and plate geometry. The constructed constitutive curve is clearly monotonic for all tested solutions with entanglement numbers up to 47, notwithstanding a local power law index in the stress plateau as small as 0.01. The slope of the stress plateau is sensitive to both the chain polydispersity and entanglement number. Transient shear banding was observed under imposition of step shear in cone and plate geometry. This banding, however, disappears at longer times, which again confirms a monotonic steady state constitutive relation.

A numerical analysis of partial slip problems under Hertzian contacts

In this study the tangential partial slip problems in Hertzian contact regions are treated by a numerical technique. The tangential loading may include tangential forces in the contact plane and a twisting moment normal to the contact plane. The Coulomb’s law of friction and the property that the direction of friction must oppose the relative motion lead to nonlinear equations. The Newton-Raphson method is utilized to solve these nonlinear equations. Numerical results for tangential tractions and sizes of stick and slip zones may be determined, and they agree with existing analytical results for circular contacts.

An approximate method in transport theory. II

Maxwell had constructed first approximate solutions of the basic slip problems in the rarefied gas flows by distinguishing between incident and outgoing distributions at a surface. He used a single parameter approximation to the incident distribution and a conservation relation. The method has been improved in the recent past through the use of two conservation relations and two-parameter approximations to the incident distribution. We have explored in this work a method to improve upon these latter approximations by using an arbitrary order parameter representation of the incident distribution. For a test problem, we have considered the Milne problem of one speed transport [S. K. Loyalka, Phys. Fluids 24, 1912 (1982)]. We have found that a three-parameter representation yields a result for extrapolation distance that is accurate to four figures. We have explored approximations up to order 512, and we have found that the accuracy of results is improved systematically with the order of the approximation, and results of any desired accuracy can be obtained.

Thermal slip of Fermi gases

A kinetic equation which describes the behavior of Fermi gases in a half-spatial problem of thermal slip has been deduced for the first time. For the problem, an analytic solution has been found and the function of distribution of particles flying to a wall has been obtained in explicite form. It has been analyzed how the slip equation and velocity depend on the parameter that is defined as the ratio of the chemical potential to the product of Boltzmann’s constant by the absolute temperature.

Effect of Temperature on Slip Coefficients of Molecular Gases

Results obtained by accurate analytical methods applied to the problem of molecular-gas slip over a rigid spherical surface are reported. The Boltzmann equation is modified to take into account rotational degrees of freedom in the BGK model is used as a master kinetic equation. The calculated slip coefficients are shown to depend on the Prandtl number and on the gas temperature. Slip coefficients for several molecular gases are plotted as functions of temperature.

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).

Analytical Solution of the Problem of Second-Order Thermal Slip with Allowance for the Rotational Degrees of Freedom of Gas Molecules

Exact analytical methods are applied to description of second-order thermal slip for a polyatomic gas (with the number of atoms in the molecule N ≥ 2) with allowance for the rotational degrees of freedom of the gas molecules. The thermal slip coefficients are determined for some particular molecular gases. It is shown that the second-order thermal slip coefficients of polyatomic gases depend on the Prandtl number.

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1]. p]The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients Cm and KTS, respectively. p]In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10]. p]In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients Cm and KTS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.