High-frequency scattering of elastic waves from cylindrical cavities

The scattering of time-harmonic plane longitudinal elastic waves by smooth convex cylindrical cavities is investigated. The exact solution for a circle is evaluated for wavelengths of the same order as the radius, and the geometrical and physical elastodynamics approximations are shown to be inadequate. The application of Watson's transformation exhibits the various diffraction effects and the relative importance of each is assessed. Excellent approximations for the scattered far-field are obtained with a hybrid method, in which an approximation for the surface field is constructed from the creeping wave contributions and this is then used in an integral representation. A generalization, based on the Geometrical Theory of Diffraction, of the hybrid method to cavities of smooth convex cross-section is presented and applied to the specific case of an ellipse. The predictions of the hybrid method compare well with numerical results obtained by an eigenfunction expansion method.     

Contact problem for an orthotropic half-space

Numerical and analytical solutions of the 3D contact problem of elasticity on the penetration of a rigid punch into an orthotropic half-space are obtained disregarding the friction forces.A numericalmethod ofHammerstein-type nonlinear boundary integral equations was used in the case of unknown contact region, which permits determining the contact region and the pressure in this region. The exact solution of the contact problem for a punch shaped as an elliptic paraboloid was used to debug the program of the numerical method. The structure of the exact solution of the problem of indentation of an elliptic punch with polynomial base was determined. The computations were performed for various materials in the case of the penetration of an elliptic or conical punch.     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

Determination of the mechanical constants of ZnS nanobelt by combining nanoindentation test and finite element method

The single nanobelt simplified as transversely isotropic is modeled by three dimension element during the modeling of finite element method (FEM), and the mechanical constants of ZnS nanobelt are obtained by combining nanoindentation test and FEM. In the forward analysis, the numerical loading curves at the appropriate penetration depth are simulated by using the purely mechanical indentation (PMI) and piezoelectric indentation (PI) modes to extract the numerical maximum indentation load and numerical loading curve exponent, and they are used to establish the dimensionless equations related with the mechanical constants of nanobelt by fitting the mechanical constants vs numerical maximum indentation load and numerical loading curve exponent curves. In the reverse analysis, the experimental indentation curve performed on ZnS nanobelt is fitted as the power function to obtain the maximum indentation load and the loading curve exponent and they are substituted into the dimensionless equations to solve the mechanical constants of the nanobelt. In order to verify the validity, the mechanical constants are inputted into ABAQUS software to obtain the computational loading curves under PMI and PI modes, and they are in good agreement with the experimental indentation curve of ZnS nanobelt. The combination solutions of mechanical constants under PMI mode is of larger total error than those under PI mode, and it indicates that the piezoelectric effect should be reasonably considered into the developed method, which is effective to determine the mechanical property of single nanobelt.     

An expanding cavity model incorporating strain-hardening and indentation size effects

An expanding cavity model (ECM) for determining indentation hardness of elastic strain-hardening plastic materials is developed. The derivation is based on a strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic power-law hardening material. Closed-form formulas are provided for both conical and spherical indentations. The indentation radius enters these formulas with its own dimensional identity, unlike that in classical plasticity based ECMs where indentation geometrical parameters appear only in non-dimensional forms. As a result, the newly developed ECM can capture the indentation size effect. The formulas explicitly show that indentation hardness depends on Young’s modulus, yield stress, strain-hardening exponent and strain gradient coefficient of the indented material as well as on the geometry of the indenter. The new model reduces to existing classical plasticity based ECMs (including Johnson’s ECM for elastic–perfectly plastic materials) when the strain gradient effect is not considered. The numerical results obtained using the newly developed model reveal that the hardness is indeed indentation size dependent when the indentation radius is very small: the smaller the indentation, the larger the hardness. Also, the indentation hardness is seen to increase with the Young’s modulus and strain-hardening level of the indented material for both conical and spherical indentations. The strain-hardening effect on the hardness is observed to be significant for materials having strong strain-hardening characteristics. In addition, it is found that the indentation hardness increases with decreasing cone angle of the conical indenter or decreasing radius of the spherical indenter. These trends agree with existing experimental observations and model predictions.     

Indentation responses of piezoelectric films

Frictionless indentation responses of transversely isotropic piezoelectric film/rigid substrate systems under circular cylindrical indenter (i.e., punch), conical indenter (i.e., cone), and spherical indenter (i.e., sphere) are investigated. Both insulating and conducting indenters are considered. The technique of Hankel transformation is employed to derive the corresponding dual integral equations for the mixed boundary value indentation problems. For the two limiting cases of infinitely thick and infinitely thin piezoelectric films, closed-form solutions are obtained. For piezoelectric films of finite thickness, a numerical method is constructed to solve the dual integral equations and semi-empirical models having only two unknown parameters are proposed for the responses of indentation force, electric charge and electric potential, and contact radius. With the two parameters inferred from the numerical results, the semi-empirical formulae are found to provide good estimates of the indentation responses for the two limiting cases of infinitely thick and thin piezoelectric films, as well as those in between. The inferred parameters in the proposed semi-empirical formulae for normalized indentation force and electric charge are checked against four different piezoelectric materials and are found to be insensitive to the selection of piezoelectric materials. It is believed that the proposed semi-empirical indentation formulae are useful in developing experimental indentation techniques to extract the material properties of piezoelectric films.     

Characterisation of non-linear viscoelastic foods by the indentation technique

A method to determine the non-linear viscoelastic constitutive constants from indentation force–displacement data corresponding to different indentation speeds has been developed. The method consists of two parts. In the first part, the force–displacement data is expressed as two functions which represent the strain and the time-dependent responses, respectively. From these functions, the time-dependent constants and the instantaneous force–displacement response are obtained. In the second part, the strain-dependent variables are determined from the instantaneous force–displacement response through an inverse analysis based on the Levenberg–Marquardt method. The method was verified by numerical experiments using the properties of cheese as examples.     

Stresses in a surface obstacle undercut due to rapid indentation

Surface obstacle undercuts such as tunnels act as guides for waves produced by rapid indentation of the surface. The diffraction of the waves can produce severe dynamic stresses near the undercut termnini. Utilizing the semi-infinite crack as a short-time model, this article examines these stresses for the case of indentation by a rigid smooth wedge. Although the distance between the terminus and wedge apex gives the problem a characteristic length, the solution can be obtained by summing a dynamically similar solution with respect to a speed parameter. The results show that if fracture occurs at the terminus prior to the arrival of wave reflections, it may proceed at a noticeable angle to the undercut plane.     

Modeling wheel-induced rutting in soils: Indentation

The analysis of indentation of rigid cylindrical wheels into frictional/cohesive soils is presented. Three- and two-dimensional numerical simulations were performed using the finite element code ABAQUS to assess the influence of soil strength parameters, dilatancy, and wheel geometry on the relationship between the indentation force and wheel sinkage. The effect of three-dimensionality in the indentation process is studied in detail. Three-dimensional effects were found to be minor for clays though significant for sands. An approximate analytic approach is also presented, which relates indentation force and wheel sinkage for given wheel geometry and material parameters. Theoretical results are compared with preliminary experimental data obtained from small-scale indentation tests, and satisfactory qualitative agreement is shown. The results described in the paper are regarded as reference for numerical and analytic modeling of wheel rolling, to be presented in a separate paper.     

Two contact problems in anisotropic thermoelasticity

Some basic equations recently derived by Clements are used to consider two contact problems in anisotropic thermoelasticity. The first problem concerns the determination of the thermal stress in an anisotropic half-space due to a heated load over a section of the boundary. The second problem concerns the indentation of a half-space by a heated rigid punch. In particular, indentation by a cylindrical punch is considered and some numerical results obtained.

---

Zusammenfassung Einige Grundgleichungen, die Clements neulich abgeleitet hat, dienen als Grundlage, um zwei Kontaktprobleme in der anisotropischen Thermoelastizität zu betrachten. Im ersten Problem geht es darum, die Wärmespannung in einem anisotropischen Halbraum zu bestimmen, die von einer erhitzten Last über einem Teil der Grenzlinie hervorgebracht wird.Im zweiten Problem geht es darum, einen Halbraum durch eine erhitzte starre Punze auszuzacken. Insbesondere wird das Auszacken durch eine kreisartize Punze betrachtet, und es ergeben sich numerische Angaben.

     

THE THREE-DIMENSIONAL UNSTEADY BOUNDARY LAYER OVER AN IMPULSIVELY STARTED ROTATING DISK

The unsteady boundary layer over an impulsively started rotating disk isstudied,a complete solution describing the smooth transition from vortex diffusionat ωt=0 to Kármán’s steady solution is obtained by series expansion and itsnumerical continuation. The angle of body streamlines,together withexperimental values, are given as the function of time t as well as the momentcoefficient C_M and on-coming velocity w(∞).     

Stress measurement of SS400 steel beam using the continuous indentation technique

An analytical model is presented for determining surface residual stress using continuous indentation. The elastic residual stress is assumed to have no influence on contact area or hardness and to be uniform over a volume that is several times larger than the indentation mark. A step-by-step analysis for the residual-stress-induced load difference at a given depth is outlined here and such concepts as stress interaction, stress-sensitive contact morphology, and reversible contact recoveries during a stress relaxation are described. Finally, the proposed method is applied to the interpretation of the continuous indentation results obtained from an SS400 steel beam in which controlled bending stresses are generated. The stress estimated, however, showed a high scatter due to plastic pile-up deformation. When the optically measured contact area is used as an alternative of the contact area calculated from the unloading curve, the re-evaluated stress agrees well with the already known applied stress.     

On oblique contact of creeping solids

Oblique indentation of power-law creeping solids by a rigid die is analysed in three dimensions with perfectly plastic behaviour emerging as an asymptotic case. Indenter profiles are prescribed to be axisymmetric for simplicity but not by necessity. Invariance and generality is aimed at, as the problem is governed by only four essential parameters, i.e. the die profile, p, the indentation angle, γ, the power-law exponent, n, and the coefficient of friction, μ. The solution strategy is based on a self-similar transformation resulting in a reduced problem corresponding to flat die indentation of complete contact. The reduced auxiliary problem, being independent of loading, history and time, was solved by a three-dimensional finite element analysis characterized by high accuracy. Subsequently, cumulative superposition was used to resolve the original problem and global and invariant relations between force, depth and contact area were determined. Detailed results are given for the location and shape of the contact region and stick/slip contours as well as for local states of surface stresses and deformation at flat and spherical indenters. Due to the asymmetry prevailing, it was found that in the spherical case, contact contours proved to be oval and shifted, although with normal and tangential forces only weakly coupled. Finite friction as compared to full adhesion proved to have only a minor effect on global relations. The framework laid down may be applied to the contact of structural assemblies subjected especially to elevated temperatures and also to various issues such as compaction of powder aggregates, flattening of rough surfaces and plastic impact.     

Smooth static and dynamic indentation of a cantilever beam

The static and dynamic indentation of structural elements such as beams and plates continue to be intriguing problems, especially for scenarios where large area contacts are expected to occur. Standard methods of indentation analyses use a beam theory solution to obtain an overall load–displacement relationship and then a Hertzian contact solution to calculate local stresses under the indenter. However, these techniques are only applicable in a fairly limited class of problems: the stress distribution in the contact region will differ significantly from a Hertzian one when the contact length exceeds the thickness of the beam. The indentation models developed herein are improvements over existing GLOBAL/LOCAL models for static and dynamic indentation of cantilever beams. Maximum contact stresses, beam displacements, and contact force time histories are obtained and compared with the predictions of current static and dynamic indentation models. The validity of the solutions presented herein is further assessed by comparing the results obtained to the predictions of modified beam theory solutions.     

Oliver–Pharr indentation method in determining elastic moduli of shape memory alloys—A phase transformable material

Instrumented indentation test has been extensively applied to study the mechanical properties such as elastic modulus of different materials. The Oliver–Pharr method to measure the elastic modulus from an indentation test was originally developed for single phase materials. During a spherical indentation test on shape memory alloys (SMAs), both austenite and martensite phases exist and evolve in the specimen due to stress-induced phase transformation. The question, “What is the measured indentation modulus by using the Oliver–Pharr method from a spherical indentation test on SMAs?” is answered in this paper. The finite element method, combined with dimensional analysis, was applied to simulate a series of spherical indentation tests on SMAs. Our numerical results indicate that the measured indentation modulus strongly depends on the elastic moduli of the two phases, the indentation depth, the forward transformation stress, the transformation hardening coefficient and the maximum transformation strain. Furthermore, a method based on theoretical analysis and numerical simulation was established to determine the elastic moduli of austenite and martensite by using the spherical indentation test and the Oliver–Pharr method. Our numerical experiments confirmed that the proposed method can be applied in practice with satisfactory accuracy. The research approach and findings can also be applied to the indentation of other types of phase transformable materials.     

On Brinell and Boussinesq indentation of creeping solids

As an alternative to traditional tensile testing of materials subjected to creep, indentation testing is examined. Axisymmetric punches of shapes defined by smooth homogeneous functions are analysed in general at power law behaviour both from a theoretical and a computational point of view. It is first shown that by correspondence to nonlinear elasticity and self-similarity the problem to determine time-dependent properties admits reduction to a stationary one. Specifically it is proved that the creep rate problem posed depends only on the resulting contact area but not on specific punch profiles. As a consequence the relation between indentation depth and contact area is history independent. So interpreted, the solution for a flat circular cylinder (Boussinesq) is not only of intrinsic interest but serves as a reference solution to generate results for various punch profiles. This is conveniently carried out by cumulative superposition and in particular ball indentation (Brinell) is analysed in depth. A carefully designed finite element procedure based on a mixed variational principle is used to provide a variety of explicit results of high accuracy pertaining to stress and deformation fields. Universal relations for hardness at creep are proposed for Boussinesq and Brinell indentation in analogy with the celebrated formula by Tabor for indentation of strain-hardening plastic materials. Quantitative comparison is made with a diversity of experimental data attained by earlier writers and the relative merits of indentation strategies are discussed.     

Mixed Convection Adjacent to a Suddenly Heated Horizontal Circular Cylinder Embedded in a Porous Medium

The mixed convection caused when a horizontal circular cylinder is suddenly heated is investigated in the situation when the initial flow past the cylinder is uniform and its direction either upwards or downwards. An analytical series solution, which is valid at small times, is obtained using the matched asymptotic expansions technique. A numerical solution, which is valid at all times and for any values of the Rayleigh and Péclet numbers, is also obtained using a fully implicit finite-difference method. Three different regimes, when either the free or forced convection is dominant or when they have the same order of magnitude, are considered. In the free convection dominated regime, two vortices develop near the sides of the cylinder in both situations of an upward or downward external flow. Comparisons between the analytical and numerical results at small times, as well as a detailed discussion of the evolution of the numerical solution are presented. The numerical results obtained for large Rayleigh, Ra, and Péclet Pe, numbers show that a thermal boundary-layer forms adjacent to the cylinder for any value of the ratio Ra/e. The steady state boundary-layer analysis, similar to that performed by Cheng and Merkin, is analysed in comparison to the numerical solution obtained for large values of Ra and Pe at very large times.     

A new indentation model for snow

Plate indentation tests have been used widely to characterize the properties of terrains. In particular, pressure-sinkage curves obtained from these tests have been used for vehicle-terrain interaction predictions. However, there is a lack of physical basis to properly interpret the meaning of these empirical curves such that they cannot be related to fundamental material properties. Also, the relation between the plate indentation tests and static (non-rolling) pneumatic tire indentation is not clear. In this paper, we conducted finite element analysis of circular plate indentation and static tire indentation simulations for fresh snow of different depths. The results indicate that the pressure-sinkage relationship for the plate indentation test is qualitatively similar to that for static tire indentation. Three deformation zones have been identified for these tests using pressure-sinkage and density-sinkage data: a small elastic zone (Zone I), a propagating hardening plastic zone (Zone II) and a densification (finite depth) zone (Zone III). The onset of a finite-depth zone was identified where the pressure bulb beneath the plate/tire has reached the bottom of snow. It is shown that Zone I and Zone II correspond to a semi-infinite terrain typical of vehicle-soil interaction, whereas Zone III corresponds to a finite-depth domain for snow and other multilayered media. The plastic constraint underneath the indenters suggests a quasi-uniaxial stress state such that a simple 1-D indentation model was proposed for Zone I, a spherical cavity expansion solution was adapted for Zone II, and an upper bound solution was adapted for Zone III. The results of the prediction of the transition between Zone II and Zone III as well as the pressure-sinkage relationships compared well with finite element solutions of plate indentation and static tire indentation tests, and with field data.     

Analytical solution of the contact problem of a rigid indenter and an anisotropic linear elastic layer

We use the Stroh formalism to study analytically generalized plane strain deformations of a linear elastic anisotropic layer bonded to a rigid substrate, and indented by a rigid cylindrical indenter. The mixed boundary-value problem is challenging since the a priori unknown deformed indented surface of the layer contacting the rigid cylinder is to be determined as a part of the solution of the problem. For a rigid parabolic prismatic indenter contacting either an isotropic layer or an orthotropic layer and a flat rigid punch indenting a half space, the computed solutions are found to agree well with those available in the literature. Parametric studies have been conducted to delimit the length and the thickness of the layer for which the derived relation between the axial load and the indentation depth caused by the rigid cylinder is valid. The indentation of a face centered cubic crystal with the plane of indentation oriented differently from the principal planes of symmetry has also been studied to illustrate the applicability of the technique to general layers made of anisotropic materials. Results presented herein can serve as benchmarks with which to compare solutions obtained by other methods.     

A numerical approach to spherical indentation techniques for material property evaluation

In this work, some inaccuracies and limitations of prior indentation theories, which are based on experimental observations and the deformation theory of plasticity, are investigated. Effects of major material properties on the indentation load-deflection curve are examined via finite element (FE) analyses based on incremental plasticity theory. It is confirmed that subindenter deformation and stress-strain distribution from deformation plasticity theory are quite dissimilar to those obtained from incremental plasticity theory. We suggest an optimal data acquisition location, where the strain gradient is the least and the effect of friction is negligible. A new numerical approach to indentation techniques is then proposed by examining the FE solutions at the optimal point. Numerical regressions of obtained data exhibit that the strain-hardening exponent and yield strain are the two key parameters which govern the subindenter deformation characteristics. The new indentation theory successfully provides a stress-strain curve and material properties with an average error of less than 3%.