Rigid Punch Research Articles

Continuous and discontinuous contact problem of a functionally graded orthotropic layer indented by a rigid cylindrical punch: Analytical and finite element approaches

AbstractIn this study, the continuous and discontinuous contact problems of a functionally graded (FG) orthotropic layer lying on a homogeneous and isotropic layer were investigated. The orthotropic layer was indented by a rigid cylindrical punch which was subjected to a concentrated normal force, and a homogeneous isotropic layer was firmly attached to a rigid substrate. For the analytical solution, the general expressions for the stresses and displacements were obtained in the presence of body forces by using elasticity theory and Fourier integral transforms. The continuous and discontinuous contact problems were reduced to the singular integral equations by means of boundary conditions. The system of the singular integral equations was solved using the Gauss‐Chebyshev integration formula. Then, the finite element solution of the two cases was also performed. Analytical results and numerical results (FEM) for critical load, initial separation distance, separation region in discontinuous contact, contact lengths under the punch, and contact stress distributions between the layer‐punch and layer‐layer were obtained for the dimensionless quantities.

Transient response of a finite thickness strip with thermoelectric effects loaded by an electrode

Thermoelectric (TE) materials exhibiting energy converting and storing capabilities connected with an electrode are increasingly applied in engineering sciences. The contact interfaces of the TE devices are prone to gradual degradation or even failure under thermal shock loading. The investigation of the dynamic contact problem of the TE materials loaded by an electrode is significant for designing reliable TE devices. The transient frictionless contact model of the TE materials with a finite thickness indented by a rigid flat punch is proposed and investigated in this paper. The considered thermal-mechanical coupling problem is converted into the corresponding singular integral equations by the Laplace transform and the Fourier transform techniques, which are solved numerically by a collocation method. Then, the numerical inverse Laplace transform is employed to obtain the time-domain solutions. The time-dependent and thickness-dependent surface normal energy flux and surface normal contact stress, which are significant for the transient contact behavior, are computed and analyzed. The magnitudes of the transient normal energy flux and normal contact stress differ from that in the steady-state. Specifically, the normal contact stress in the transient state is higher than that in the steady-state.

Hot indentation of a FGM-coated thermoelastic half-space by a conical punch: Approximated analytical solution of the contact problem

Hot indentation of a thermoelastic coated half-space by a rigid punch is considered. The coating is assumed to be functionally-graded with all group of properties arbitrary varying with depth. The deformation of the half-space caused by the simultaneous mechanical and thermal loading is considered. The punch is subjected to a normal centrally applied force and preliminary heated to a uniformly distributed temperature. The contact problem is reduced to the solution of a system of dual integral equations over two unknown functions: Hankel transform of normal stresses and heat flux in the contact area. Heat flux is obtained using the bilateral asymptotic method. This method was sufficiently generalized to take into account additional function describing the displacements of the surface caused by thermal heating and obtain contact stresses. The solution is obtained in an analytical form which clearly demonstrates the dependence of the contact characteristics from the initial physical parameters of the problem. Numerical analysis is provided to illustrate the effect of the presence of the coating and its thickness on the results. Differences between the homogeneous and functionally-graded coatings is illustrated.

Axisymmetric frictionless contact between an elastic layer thickness and a circular base under a rigid punch

The study presented in this work deals with analytical methods for an axisymmetric problem of an elastic layer partially reposing on a rigid circular base, and is indented along the upper surface with a rigid punch. The contact between the medium and the base is smooth. This boundary value problem is transformed into a system of dual integral equations. In contrast to the classical approach consisting in resolving the corresponding Fredholm equation of the second kind, the latter equations are obtained from an infinite algebraic system of simultaneous equations, where the particular case [Formula: see text] is verified. The results of this system are also obtained numerically. The normal displacement, normal stress, and the stress singularity factor are given analytically and shown graphically with discussion. By comparison with those predicted by the finite element method, the accuracy of the numerical method is approved.

EVOLUTION OF THE FIELD OF DISTRIBUTED DEFECTS IN A CRYSTAL DURING CONTACT INTERACTION WITH A SYSTEM OF RIGID STAMPS

The article discusses the mathematical modeling for the evolution of the stress-strain state and fields of defects in crystals during their contact interaction with a system of rigid punches. From a macroscopic point of view, the redistribution of defects is characterized by inelastic (viscoplastic) deformation, and therefore the processes under study can be classified as elastic-viscoplastic. Elastic and inelastic deformations are assumed to be finite. To take into account inelastic deformations, it is proposed to use a differential-geometric approach, in which the evolution of the fields of distributed defects is completely characterized by measures of incompatible deformations and quantified by material connection invariants. This connection is generated by a non-Euclidean metric, which, in turn, is given by a field of symmetric linear mappings that define (inconsistent) deformations of the crystal. Since the development of local deformations depends both on thecontact interaction at the boundary and on the distribution of defects in the bulk of the crystal, the simulation problem turns out to be coupled. It is assumed that the local change in the defect density is determined by the first-order Alexander Haasen Sumino evolutionary law, which takes into account the deviatoric part of the stress field. An iterative algorithm has been developed to find coupled fields of local deformations and defects density. The numerical analysis for the model problem was provided for a silicon crystal in the form of a parallelepiped, one face of which is rigidly fixed, and a system of rigid stamps acts on the opposite face. The three-constant Mooney Rivlin potential was used to model the local elastic response.

Multibody contact of two-dimensional anisotropic elastic/piezoelectric/magneto-electro-elastic solids

A unified expression for the generalized two-dimensional linear anisotropic elastic, piezoelectric, and magneto-electro-elastic (MEE) solids is established to simplify the connection between these three different kinds of materials. Here, the two-dimensional assumption includes the most general cases that in-plane and anti-plane deformations are allowed to couple each other. Based upon this unified expression, in this paper a recently developed boundary element method (BEM) for contact of two anisotropic elastic bodies is extended to solve the problems of multibody contact in which the bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids. When a special case of contact between multiple rigid punches and an anisotropic elastic, piezoelectric or MEE half-plane is considered, further reduction can be made by introducing three degrees of freedom (two rigid body translations and one rigid body rotation) for each punch. With this consideration, no meshes are required on the rigid punches. The usual contact modes of separation, stick, and slip all include the situation of conductor and insulator if any one of the multibody is piezoelectric or MEE. In other words, in this paper we successfully develop two BEMs to solve the generalized two-dimensional problems of multibody contact. One is for the general cases whose bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids, and the other is for the special cases of multiple rigid punches indenting on an anisotropic elastic, piezoelectric or MEE half-plane.

Sliding moving contact problem between a rigid cylindrical punch and a functionally graded orthotropic layer bonded to an isotropic homogeneous layer

In this study, moving contact problem for a rigid punch and a functionally graded orthotropic layer bonded to an isotropic homogeneous layer is solved, semi-analytically. The cylindrical punch moves steadily with a constant subsonic velocity over the upper layer and transmits concentrated normal and tangential forces. Using Fourier transform and Galilean transformation, the contact problem is converted to a Cauchy-type singular integral equation of the second kind, in which the contact stress and contact width are the unknowns. The numerical solution of the singular integral equation is obtained by using Gauss–Jacobi integration formulas. Numerical results for the contact stress, in-plane stress, and the contact width are given.

Frictionless Multi-field Coupling Contact Problem for a Thermoelectric Layer Loaded by Two Rigid Punches

Frictionless Multi-field Coupling Contact Problem for a Thermoelectric Layer Loaded by Two Rigid Punches

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model.

The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori, whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton-Hoff and Ramberg-Osgood materials. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Adhesion of graded elastic materials: A full self-consistent model and its application

A full self-consistent model (FSCM) of the axisymmetric adhesive contact between a rigid punch with an arbitrary surface shape and a power-law graded elastic half-space is developed. The self-consistent equation between the surface gap and the surface interaction (e.g., the Lennard–Jones force law) involves a nonlinear singular integral, posing a great challenge to numerical calculations. By applying the properties of Gauss’s hypergeometric function, the integral singularity is eliminated in the numerical calculation through Riemann–Stieltjes integral. Case studies for power-law punch profiles are performed and the self-consistent equation can be expressed in a dimensionless form with three dimensionless parameters, namely a shape index, a gradient exponent, and a new generalized Tabor number. The FSCM results are obtained by solving the self-consistent equation through the surface central gap control method and Newton–Raphson iterative method. For large generalized Tabor numbers, the force–displacement curves are ‘S-shaped’ and condense to the extended JKR limit in the high-load branch. As the generalized Tabor number decreases, a continuous transition from the extended JKR model to the Bradley model for the adhesion of power-law graded materials is obtained. It is found that the pull-off force of a graded material usually depends on the three dimensionless parameters, but for some cases of the shape index, it is not sensitive to the gradient exponent when the generalized Tabor number is fixed. Asymptotic solutions are derived to predict the unstable jump points, which coincide well with the FSCM predictions. The FSCM is applied to validate the extended Maugis–Dugdale (M–D) model of graded materials and it is found that the accuracy of the original M–D-n-k model using the maximum strength condition to determine the cohesive stress is limited. By introducing the rigid-limit-consistency condition of the pull-off force to determine the cohesive stress, the M–D-n-k model is improved and its predictions show good consistency with the FSCM results.

Contact Analysis of Elastic Layer Supported by a Wedge

In this paper the contact analysis of elastic layer supported by a wedge is considered in plane. The problem is formulated with closed formed integral equations. The length of the contact region, the pressure between the wedge and the layers, and the pressure between the layer and the punch are unknowns. Both the material and layer are elastic in this problem. This problem can be defined as a layer supported over two wedges with perpendicular angle. The upper surface of the layer is assumed as circle with a large radius. The thickness of the layer is finite. Singular integral equations are used in the formulation of the problem. The benefits of this formulation are the following: the problem can easily be generalized for the case of forces acting through many rigid punches. The solution gives the contact stress directly and the solution of the singular integral equations is an appropriate way in terms of numerical solution technique. The application of this problem is the train wheel that is contacted to the connection part of the rails. It is shown that the divergent terms at the kernels cancel each other by considering the equilibrium conditions. As a numerical example, the contact problem between the wheel and rails is investigated.

Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function

A solution of an orthotropic elastic plane problem is derived for a frictional rigid flat punch on a half plane with an oblique edge crack. A mapping function is used for arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped half plane problems can be solved by exchanging the mapping function. Therefore, once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, the stress distributions are shown for a frictional coefficient and an oblique edge crack angle. Four elastic constants are expressed by two independent parameters.

Indentation of geometrically exact beams

We investigate arbitrarily large indentations of beams in two dimensions by a frictionless and rigid cylindrical punch whose axis is perpendicular to the plane of the beam. The beam is modeled as a geometrically exact beam that admits large displacements and rotations. Absence of both friction and inter-penetration between the contacting surfaces allows the normal contact pressure and the separation between the surfaces to be related by the Karush–Kuhn–Tucker (KKT) conditions. This is then implemented in a nonlinear finite element framework through both penalty and augmented lagrangian methods. We solve for the contact domain and the forces and moments acting within it, and relate them to the total applied force and the extent of indentation. We report results for several punch radii and beam thicknesses and consider both pinned and clamped beams. In this, we discuss the importance of nonlinear effects such as wrapping of the beam around the punch and the stiffening of the beam due to axial stretch. We demonstrate the importance of utilizing a nonlinear beam formulation by comparing our results with the predictions of theories predicated on small deformations. For example, while the latter claim loss of contact with the beam in the vicinity of the punch’s apex as the indentation proceeds, we find that the punch and the beam remain in contact.

Analysis of Thermomechanical Cracks of Disc Brake Used in Heavy Trucks

In the present work, the study of fatigue failure of the disc brake of Volvo truck under thermomechanical effects across the disc thickness is reported. Both analytical and numerical approaches have been used. In the analytical approach, flat rigid asymmetric punch contact behavior was assumed at surface interaction, while in the numerical approach, the 3D finite element model of the vented disc type is subjected to thermomechanical cycles using ABAQUS software. Linear elastic material models were used to simulate the material's behavior under the applied pad pressure on the disc. Surface-to-surface type of contact was used for bodies that have arbitrary shapes with relatively large contact areas at disc pad interfaces. The crack initiated due to hydraulic pressure of 1.47 MPa on the disc was analyzed under mixed-mode (mode I and II) conditions. This analysis has shown that the crack propagated up to critical length of 6.02 mm and finally, failure of the disc brake was observed.

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

Effects of a tilted flat-ended punch on the receding contact between a graded and a homogeneous layer

This paper considers the receding contact problem between an exponentially graded layer and a homogeneous layer under the indentation of a tilted rigid flat-ended punch. In the presence of an offset load deviated from the symmetry axis of the rigid punch, the contact pressures both under the rigid punch and along the receding contact interface between the layers lose symmetry. Depending on the magnitude of eccentricity, either complete or incomplete contact may take place under the rigid flat-ended punch, whereas a receding type of contact always occurs along the interface separating the graded and the homogeneous layers. Fourier integral transforms help to convert the governing equations and mixed boundary conditions of the double-contact problem into dual Fredholm integral equations of the second kind with Cauchy-type singular kernels. Gauss–Chebyshev and Gauss–Jacobi numerical quadratures are subsequently employed to discretize and collocate the dual singular integral equations, together with the four force and moment equilibrium equations, for the complete and incomplete contact, as well as the transitional status corresponding to the critical load eccentricity. Extensive parametric studies indicate that the critical load eccentricity depends on the property gradation of the graded layer and the layer-thickness ratio, but not on the magnitude of the indentation load. The extent of contact both under the rigid flat-ended punch and along the receding contact interface is also independent of the magnitude of the indentation load. The results suggest reasonable departures from the contact properties of a homogeneous half-plane under a tilted rigid flat-ended punch, indicating the valuable information on optimizing the double-contact properties in terms of the property gradation and layer-thickness ratio under a given eccentricity.

On the competition between adhesive and surface effects in the nanocontact properties of an exponentially graded coating

This paper aims at investigating the adhesive nanocontact properties between a rigid cylindrical punch and an exponentially graded coating perfectly bonded with a homogeneous half-plane substrate. The interface of nanocontact is modeled with the Maugis–Dugdale adhesive contact model and the Steigmann–Ogden surface mechanical theory. Under plane strain assumption, governing equations and boundary conditions of the nanocontact problem are converted into triple integral equations. They are numerically solved for the determination of nanocontact length, pressure distribution and the application range of the adhesive zone, by the use of Gauss–Chebyshev quadrature, Gauss–Legendre quadrature and a self-designed iterative algorithm. The method of solution and numerical algorithm are first validated against literature results. Extensive parametric studies are then conducted with respect to the adhesive energy density, Tabor’s parameter, surface material constants and inhomogeneity index of the exponentially graded coating. Based on these results, the relationship between adhesive and surface effects can be identified as competitive. Both are of great importance in the determination of nanocontact behaviors of graded materials and structures. In addition, the Maugis–Dugdale adhesive contact model is found to converge to the Johnson–Kendall–Roberts model under large Tabor’s parameters. Large indenters, high adhesive energy densities, soft substrates and small adhesive cut-off distances are all helpful to the transition. The current work provides a generalized solution framework to the nanocontact problems of graded materials and structures in the simultaneous presence of adhesive and surface effects.

Контактная задача о взаимодействии штампа в форме параболоида вращения и пороупругого слоя, закрепленного на упругом основании

The contact problem of the introduction of a rigid punch in the form of a paraboloid of rotation into a poroelastic layer fixed on an elastic half-space is considered on the basis of the equations of the theory of poroelastic bodies of Cowin-Nunziato. It is assumed that there is no friction in the contact zone. With the help of the Hankel integral transformation, the problem posed is reduced to solving the integral equation for an unknown contact stress. The collocation method is used to solve the integral equation. The area of contact, the values of contact stresses are found, and the deformation of the surface outside the punch is investigated. The relationship between the force acting on the punch and its displacement has also been investigated. A comparative analysis of the studied quantities for different values of the parameters of the poroelastic layer and the elastic foundation has been carried out. Numerical results are presented in the form of graphs.

Effect of Pressurization and Annealing on Hole Expansion

Abstract Due to the continuous push toward environmental regulations to reduce the impact on the environment, more sustainable manufacturing is in demand. Easily available material like iron-carbon based alloys are denser but exhibit higher strength and good formability. Alternatively, low dense materials have the potential to reduce the weight, but they lose the “easy-to-deform” spot. In addition, the traditional method to characterize the material is not sufficient to capture the material model which would accurately predict the complex nature of deformation. Thus, the present study is focused on how to make the material more deformable in the process by evaluating the parameters in deformation through the hole expansion process. For this study, four tests were chosen, namely hemispherical dome test, cylindrical tool test, conical tool test, and biaxial test. Except the biaxial test, all tests use the rigid tool punch to deform the hole. The cruciform specimen with a center hole was used to make the sample, which fits in all the considered tests. Force–displacement curves were plotted and discussed. In addition, tests were also performed on annealed materials to understand the hole expansion in ductile material. Based on the results it was observed that biaxial tests do not provide any pressurization effect and all tests which include the rigid tool to deform the hole doe. Due to the pressurization effect, the hole was expanded more. It was also noted that the hole expansion was more in ductile material and pressurization effect increases with ductile material.

Dynamic frictional contact mechanics between a functionally graded orthotropic medium and a moving flat punch

The present study establishes a general theory for dynamic frictional contact mechanics between an orthotropic graded half-plane and a moving rigid punch of a flat profile. The rigid punch moves over the orthotropic graded half-plane at a constant subsonic speed. Analytical method is developed for the contact mechanics problem based on two-dimensional elastodynamics. Stresses in the orthotropic graded medium are determined considering plane elasticity. Governing partial differential equations are solved analytically by the use of Galilean and Fourier transformations. The unknown functions appearing in the displacement components are determined considering the surface boundary conditions. The mixed boundary value problem is then reduced to a singular integral equation of the second kind which is solved numerically using an appropriate expansion-collocation technique. The contact stress results of the present analytical study are compared to those generated through a finite element analysis regarding the elastostatic case. Moreover, the contact stress results obtained in the elastodynamic case are also verified using some analytical results available in the literature. Successful agreement is attained in all the comparative analyses. Normal contact stress, lateral contact stress and punch stress intensity factors are calculated for various values of punch speed, coefficient of friction, in-homogeneity constant and orthotropic elastic constants. Results of the present study clearly indicates that the utilization of elastodynamic theory is crucial to get more realizable and accurate estimations of contact stresses especially in high speed sliding conditions.