Relative Tangential Displacement Research Articles

Interface mismatch eigenstrain of non-slipping contacts between dissimilar elastic solids

The problems of non-slipping contacts between dissimilar elastic solids are studied under the conditions of plane strain. When two dissimilar solids are incrementally pressed into contact, a relative tangential displacement along the contact interface emerges because of the material property mismatch. The spatial derivative of the relative tangential displacement is referred to as the interface mismatch eigenstrain, which is separately investigated subject to the non-adhesive and adhesive surface conditions. The explicit solutions of the interface mismatch eigenstrain for non-slipping contacts with symmetrical indenter profiles are obtained. These results provide the foundation for improved precision contact deformation modelling at different scales.

Optimization of the Installation Sequence for the Temporary Fasteners in the Aircraft Industry

Abstract Riveting and bolting are common assembly methods in aircraft production. The fasteners are installed immediately after hole drilling and fix the relative tangential displacements of the parts that took place. A proper fastener sequence installation is very important because a wrong one can lead to a “bubble-effect” when gap between parts after fastening becomes larger in some areas rather than being reduced. This circumstance affects the quality of the final assembly. For that reason, the efficient methods for determination of fastening sequence taking into account the specifics of the assembly process are needed. The problem is complicated by several aspects. First of all, it is a combinatorial problem with uncertain input data. Second, the assembly quality evaluation demands the time-consuming computations of the stress–strain state of the fastened parts caused by sequential installation of fasteners. Most commonly used strategies (heuristic methods and approximation algorithms) require a large number of computational iterations what dramatically complicates the problem. The paper presents the efficient methods of fastener sequence optimization based on greedy strategy and the specifics of the assembly process. Verification of the results by comparison to commonly used installation strategies shows its quality excellence.

Mode decomposition in DCB and ENF tests with geometric and material assymetry

Mode decomposition in interlaminar fracture tests initially designed for pure modes I and II as Double Cantilever Beam (DCB) and End Notcehd Flexure (ENF) is analyzed. In the initial proposal of those tests crack arms have geometric and material symmetry and the crack advance occurs through the middle plane of the laminate. In the current study, the possibilty of crack arms of different materials or with different thickness is included. Two mode decomposition criterions are considered after the analysis of a slice near the crack tip. Those criterions are based on the stress and displacement state in the non-cracked zone near the crack tip. Two points that before the crack advance were the same point are named homologous. When the cohesive zone is developing, the stresses in homologous points are the same due to the principle of action and reaction. Nevertheless, relative displacement can be normal, related to mode I, tangential, related to mode II, or a combiantion of both, related to mixed mode I-II. In the present work the relative tangential displacement related to mode III is not considered. The contributions to the energy release rate corresponding to bending and shear in each mode are analyzed. The results obtained are applied to asymmetric DCB and ENF tests.

Experimental and numerical investigations on the effects of radius of curvature and longitudinal slope on the responses of curved bridges subject to seismic pounding

Because of the irregular geometries, earthquake-induced adjacent curved bridge pounding may lead to more complex local damage or even collapse. The relevant research is mainly concentrated on the numerical analysis which lack experimental verification and discussion by changing of structural parameters. In this paper, a scaled three-dimensional numerical model of a curved bridge is established based on 3D contact friction theory for investigating the uneven distribution of pounding forces at the expansion joint of the bridge. Shaking table tests were carried out at first on a curved bridge to validate the numerical model. A series of parametric studies were then conducted to examine the impacts of the radius of curvature and longitudinal slope of the superstructure of the curved bridge on its seismic pounding response. The results show that the maximum pounding force first increases and then decreases as the radius of curvature increases, but that it decreases monotonically with the growth of the longitudinal slope. These results suggest that controlling the radius of curvature and the longitudinal slope of the superstructure of the bridge can reduce the localized high stress that is induced by seismic pounding. Also, the unevenly distributed pounding forces can significantly increase the relative radial displacement of the bridge’s deck corners, although the relative tangential displacement may decrease. It is thus necessary to adopt effective anti-pounding measures to prevent the superstructure of the bridge from being unseated.

Investigation of multibody receding frictional indentation problems of unbonded elastic functionally graded layers

In multibody indentation problems of functionally graded materials, characteristics and gradation of material constituents are greatly influence the response of these problems. The objective of the present study is two folds. The first is to develop a comprehensive and efficient 2D nonlinear inequality constrained variational finite element model capable of analyzing multibody frictional indentation problems of FGMs considering a realistic friction model. The second fold is to investigate the effects of material properties of the constituents of functional graded materials and their gradation, coefficient of friction, and indenter size on the response of multibody frictional indentation systems. For multibody contact configuration, different types of contact at different contact interfaces are modeled in the framework of contact mechanics. To overcome the drawbacks of classical Coulomb's law of friction, friction at the indentation interfaces is modelled using the nonclassical nonlinear local friction model to realize the tangential frictional stress and relative tangential displacements. All frictional contact constraints at all contact interfaces are exactly fulfilled without any penalty parameters. The isoparametric graded finite element formulation is utilized to realize the material properties of FGMs. Application of the developed model is verified with analytical models available in the literature and very well agreement is found. A case study of an unboned elastically graded layered homogeneous elastic substrate indented by a flat rigid indenter subjected to simultaneous normal and tangential loadings is comprehensively investigated. A detailed parametric study is conducted to explore the effect of the layer gradation index, coefficient of friction, and indenter width on the indentation response, which including contact pressure, tangential frictional stress and displacement field at the indenter-graded layer interface and graded layer-elastic substrate interface. Results show that the receding frictional contact response of FGMs can be controlled by appropriate choice of the layer gradient index.

Simulation of the airframe assembly process with regard to friction in fastening elements

One of the operations during the airframe assembly is the installation of temporary fastening elements. Temporary fasteners are installed to join the assembled parts and prevent the gap opening while drilling and reaming. If the diameter of temporary fastener is less than the diameter of predrilled hole, the relative tangential displacement of assembled parts can occur during the temporary assembly process. The correct simulation of this phenomenon requires the account for friction in fastening elements while solving the contact problem in order to find the displacements of joined parts. Such numerical approach is presented and discussed in the paper.

The Influence of Infilling Conditions on Flaw Surface Relative Displacement Induced Cracking Behavior in Hard Rock

The relative tangential displacement (or slip) and relative normal displacement of flaw surfaces are two of the main factors driving crack initiation and propagation of rocks. In this study, uniaxial compression experiments are performed in flawed marble under different cases, i.e., open flaw and flaw filled by gypsum, cement and resin, and different flaw inclinations, to study the influence of infilling conditions on the flaw surface relative displacement (i.e., relative tangential and normal displacement) and the induced cracking behaviors, such as crack initiation, propagation and crack patterns. The digital image correlation (DIC) technique is used to trace the evolution of relative deformation across the flaw surface and cracking process of flawed specimens. The results reveal that the difference in flaw surface relative displacement, at a given load, is introduced by different infilling materials. Among the four types of flaws, the filled flaw with smaller relative displacement is less likely to initiate cracks than the unfilled flaw with larger relative displacement. Moreover, the crack initiation position, angle, and types are also dependent on the relative deformation between the flaw surfaces. With the increase of the stiffness of the filler or flaw inclination angle, the effect of relative normal displacement on the cracking behavior is weakened, while the effect of relative tangential displacement increasingly dominates. As a result, the filled flaw with smaller relative normal displacement makes wing crack initiation points closer to flaw tips. Moreover, the filler with higher stiffness makes the wing crack initiation angle smaller, improves the possibility of the quasi-coplanar crack initiation, and suppresses the initiation of anti-wing cracks from the flaw tips in hard rock.

Numerical study on dynamic mechanism of brain volume and shear deformation under blast loading

Blast-induced traumatic brain injury (b-TBI) is a kind of significant injury to soldiers in the current military conflicts. However, the mechanism of b-TBI has not been well understood, and even there are some contradictory conclusions. It is crucial to reveal the dynamic mechanism of brain volume and shear deformations under blast loading for better understanding of b-TBI. In this paper, the numerical simulation method is adopted to carry out comprehensive and in-depth researches on this issue for the first time. Based on the coupled Eulerian–Lagrangian method, the fluid–structure coupling model of the blast wave and human head is developed to simulate two situations, namely the head subjected to the frontal and lateral impacts. The simulation results are analyzed to obtain the underlying dynamic mechanisms of brain deformation. The brain volume deformation is dominated by the local bending vibration of the skull, and the corresponding frequency for the forehead skull under the frontal impact and the lateral skull faced to the lateral impact is as high as 8 kHz and 5 kHz, respectively. This leads to the high-frequency fluctuation of brain pressure and the large pressure gradient along the skull, totally different from the dynamic response of brain under head collisions. While the brain shear deformation mainly depends on the relative tangential displacement between the skull and brain and the anatomical structure of inner skull, being not related to the brain pressure and its gradient. By further comparing the medical statistics, it is inferred that diffuse axonal injury and brain contusion, the two most common types of b-TBI, are mainly attributed to brain shear deformations. And the von Mises stress can be adopted as the indicator for these two brain injuries. This study can provide theoretical guidance for the diagnosis of b-TBI and the development of protective equipment.

Simultaneous extraction of tensile and shear interactions at interfaces

A simple beam analysis is developed that accounts for the tensile and shear interactions between the surfaces of two beams that have been brought into contact. The generally coupled differential equations for the relative normal and tangential displacements between the interacting surfaces and the normal and shear components of the traction between the two surfaces are decoupled when a balance condition involving the bending stiffness and height of each beam is satisfied. This condition also allows the normal and shear components of the J-integral and the crack tip displacement to be obtained from measurements of the applied loads along with load point displacement and rotation of each beam. As a result, the derivative of each component of the J-integral with respect to its associated crack tip displacement simultaneously provides both the normal and shear traction–separation relations at any value of the mode-mix without invoking any assumption on the particular form. As an application of this method, the normal and shear traction–separation relations for a silicon/epoxy interface were determined over nominal mode-mixes ranging from −53° to 87.5° using non-symmetric end-loaded split (ELS) and end-notched flexure (ENF) specimens. It is found that many of the assumptions made previously in developing traction–separation relations based on potential or damage-based approaches do not apply to this particular interface, which should motivate future theoretical developments.

Closed-form solution of beam on Pasternak foundation under inclined dynamic load

The dynamic response of an infinite Euler–Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution. The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction. Thus, it is more realistic than the Winkler model, which only represents compressive soil resistance. However, the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation; hence, the beam under inclined loads cannot be considered in the model. In this study, a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model, which tends to disregard the tangential interaction between the beam and the foundation. The horizontal spring reaction is assumed to be proportional to the relative tangential displacement. The governing equation can be deduced by theory of elasticity and Newton's laws, combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition. Double Fourier transformation is used to simplify the geometric equation into an algebraic equation, thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam. Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution. The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS. Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam. Results are realistic and can be used as reference for future engineering designs.

Elastic Sinusoidal Wavy Surface Contact Under Full Stick Conditions

A purely normal contact problem of an elastic half-space with a three-dimensional periodic sinusoidal wavy surface and a rigid flat under the full stick condition is studied. The contacting points from mating surfaces have zero relative tangential displacement under the full stick condition. The scope of this study is restricted to a special case where the entire contact interface is in contact (referred to as complete contact) under the full stick condition. Complete contact is defined as when there are no gaps remaining between the surfaces. The corresponding state of stress of the half-space is derived analytically. According to the state of stress, we find (1) an analytical solution for the average pressure required to cause complete contact, (2) the location of the global maxima of the von Mises stress and (3) the critical magnitude of the waviness amplitude below which the plastic yielding of the half-space will never occur before the initiation of complete contact. The results are also compared with the solution under the perfect slip condition. We find that the location of the maximum von Mises stress may occur either on the contact interface or beneath it depending on the value of Poisson’s ratio.

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

• Analytical solution for dynamic response of infinite beam on foundation is developed. • The elastic foundation considered the horizontal tangential effect. • Double Fourier transformation is used for deduction and the residue theorem is employed to obtain the solution. • The analytical solution for the beam under inclined dynamic load is presented. The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.

Adhesion and Friction Force Measurements Using an Optical Micro-Apparatus

An optical micro-apparatus was designed and developed to visualize and measure adhesion and friction forces for a steel ball in contact with a sapphire window. The apparatus allows for in situ optical investigation of the contact interface during the initiation, separation and simultaneous measurement of the normal force generated within the contact. A high precision z-stage was utilized to move the steel ball at low velocities (0.1 μm/s) to bring it into contact with the fixed sapphire window. In this study, the adhesion force between a 1–mm-diameter steel ball and sapphire window was measured and visualized when the surfaces approach and retract from each other. The experimental results demonstrated the presence of the adhesion force and stick area during the retraction of the contacting surfaces. The optical micro-apparatus is also equipped with a piezoelectric actuator to reciprocate the ball against the sapphire window to conduct sliding friction experiments at various normal loads. In this investigation, a new optical technique was also developed to measure the tangential displacement of the contacting interfaces during sliding friction tests. In this technique, the video of the contacting bodies during sliding friction cycle was processed to trace the center of the contact area and, consequently, measure the relative tangential displacement of the contact. The friction loops obtained using the measured tangential displacement exhibited a rectangular shape thus eliminating the system stiffness commonly observed in friction loop studies. The vertical and horizontal segments of the rectangular friction loops characterize the stick and slip regions of the contact, respectively.

Modeling of a creep process between rough surfaces under tangential loading

If two bodies with rough surfaces are brought into contact and an increasing tangential load is applied, there will be some relative tangential displacement of the bodies due to the contact stiffness and partial slip in the contact area. The sliding areas will increase with increasing load until the condition of complete sliding is fulfilled and macroscopic sliding starts. In this paper it is suggested that an essential part of the slow creep observed in many contact problems is due to this combined effect of elasticity and partial slip. To prove this hypothesis, tangential creep in contact of rough surfaces is simulated using the assumption of a constant coefficient of friction. The results of the simulation are compared with high resolution experimental measurements of creep between steel samples. In the most cases, the surface parameters can be adjusted to achieve a good description of the experimental data.

Crack Onset and Propagation in Composite Materials Using Finite Fracture Mechanics on Elastic Interfaces

A method for the prediction of crack onset and propagation along an interface in a fibre-reinforced composite system is devel- oped. The method is based on the Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM) and a coupled stress and energy criterion of the Finite Fracture Mechanics (FFM). Models similar to LEBIM, also called weak or imperfect interface models, are frequently applied to describe the behaviour of adhesively bonded joints. In the present paper the interface is modelled by a con- tinuous distribution of linear elastic-brittle springs where the normal and tangential stresses across the undamaged interface are, respectively, proportional to the relative normal and tangential displacements. The interface failure criterion applied takes into account the variation of the fracture toughness with the fracture mode mixity. In the present work a 2D collocational boundary element method (BEM) code is used to analyse debond onset and propagation at a fibre embedded in a matrix subjected to remote uniaxial transverse tension. One of the advantages of the method proposed here by applying the FFM criterion to the LEBIM is that we are able to make independent the two governing parameters of the model (critical stress and fracture toughness of the interface), which are related by an equation in the original LEBIM formulation. It seems that the predictions provided by FFM with LEBIM are quite similar to those obtained by the well-known Cohesive Crack Model (CCM). The advantage of the present method with respect to the CCM is evident, since a troublesome nonlinear analysis due to the softening can be avoided.

Complex variable formulation for non-slipping plane strain contact of two elastic solids in the presence of interface mismatch eigenstrain

In this paper, the problems of non-slipping contact, non-slipping adhesive contact, and non-slipping adhesive contact with a stretched substrate are sequentially studied under the plane strain theory. The main results are obtained as follows:(i) The explicit solutions for a kind of singular integrals frequently encountered in contact mechanics (and fracture mechanics) are derived, which enables a comprehensive analysis of non-slipping contacts. (ii) The non-slipping contact problems are formulated in terms of the Kolosov–Muskhelishvili complex potential formulae and their exact solutions are obtained in closed or explicit forms. The relative tangential displacement within a non-slipping contact is found in a compact form. (iii) The spatial derivative of this relative displacement will be referred to in this study as the interface mismatch eigenstrain. Taking into account the interfacemismatch eigenstrain, a new non-slipping adhesive contact model is proposed and its solution is obtained. It is shown that the pull-off force and the half-width of the non-slipping adhesive contact are smaller than the corresponding solutions of the JKR model (Johnson et al., 1971). The maximum difference can reach 9% for pull-off force and 17% for pull-off width, respectively. In contrast, the new model may be more accurate in modeling the non-slipping adhesion. (iv) The non-slipping adhesions with a stretch strain (S-strain) imposed to one of contact counterparts are re-examined and the analytical solutions are obtained. The accurate analysis shows that under small values of the S-strain both the natural adhesive contact half-width and the pull-off force may be augmented, but for the larger S-strain values they are always reduced. It is also found that Dundurs’ parameter β may exert a considerable effect on the solution of the pull-off problem under the S-strain.These solutions may be used to study contacts at macro-, micro-, and nano-scales.

BEM analysis of crack onset and propagation along fiber–matrix interface under transverse tension using a linear elastic–brittle interface model

The behavior of the fiber–matrix interface under transverse tension is studied by means of a new linear elastic–brittle interface model. Similar models, also called weak or imperfect interface models, are frequently applied to describe the behavior of adhesively bonded joints. The interface is modeled by a continuous distribution of linear-elastic springs which simulates the presence of a thin adhesive layer (interphase). In the present work a new linear elastic–brittle constitutive law for the continuous distribution of springs is introduced. In this law the normal and tangential stresses across the undamaged interface are, respectively, proportional to the relative normal and tangential displacements. This model not only allows for the study of crack growth but also for the study of crack onset. An important feature of this law is that it takes into account the variation of the fracture toughness with the fracture mode mixity of a crack growing along the interface between bonded solids, in agreement with previous experimental results. The present linear elastic–brittle interface model is implemented in a 2D boundary element method (BEM) code to carry out micromechanical analysis of the fiber–matrix interface failure in fiber-reinforced composite materials. It is considered that the behavior of the fiber–matrix interphase can be modeled by the present model although, strictly speaking, there is usually no intermediate material between fiber and matrix. A linear-elastic isotropic behavior of both fiber and matrix is assumed, the fiber being stiffer than the matrix. The failure mechanism of an isolated fiber under transverse tension, i.e., the onset and growth of the fiber–matrix interface crack, is studied. The present model shows that failure along the interface initiates with an abrupt onset of a partial debonding between the fiber and the matrix, caused by presence of the maximum radial stress at the interface, and this debonding further develops as a crack growing along the interface.

On the feasibility of shear stress estimation at interfaces of block-structured medium

In the work, we study the stress state at interfaces of block-structured medium and its effect on the deformation response of the materials to dynamic loading by the movable cellular automaton method. The simulation results suggest several deformation response parameters qualitatively dependent on the character of the stress state. These parameters are the relative tangential shear displacements initiated under dynamic loading and certain characteristics of the frequency spectrum of their velocity in block-structured medium. The performed analysis allows us to put forward a complex approach to the stress state diagnostics of fragments of active interfaces in various block-structured medium, particularly in hierarchical geomedia.

Numerical prediction of crack propagation and crack widths in concrete structures

The paper deals with the numerical prediction of crack propagation and crack widths in concrete structures. The early stages of cracking, characterized by distributed micro-cracking, are described by a smeared crack model with rotating crack directions, whereas the subsequent formation of a macro-crack is described by embedding a discontinuity in the displacement field of the respective finite elements. This so-called crack model with delayed embedded discontinuities allows correcting initial mispredictions of the crack direction during crack evolution. It accounts for crack opening in the direction normal to the crack and for relative tangential displacements of the crack faces with transfer of shear forces across the crack faces. After validation of the model by a well-known test series on combined normal and shear loading of concrete specimens, it is applied to two problems of practical relevance in engineering practice. The first one deals with the load-carrying behavior of steel anchors embedded into concrete and the second with the structural behavior of pre-damaged concrete slabs strengthened by different types of concrete overlays.

A crack model with delayed embedded discontinuities

AbstractIn this contribution the combination of a smeared rotating crack model with a crack model based on the strong discontinuity approach and formulated within the framework of elements with embedded discontinuities is presented. This so–called crack model with delayed embedded discontinuities allows considering crack opening in the direction normal to the crack and relative tangential displacements of the crack faces with transfer of shear forces across the crack faces. The advantages of this approach are shown by the numerical simulation of an anchor pull out test. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)