On the indentation of an inhomogeneous plastic solid

Summary The problem considered here is that of the indentation of a semi infinite, inhomogeneous rigid-plastic solid by a smooth, flat ended punch under conditions of plane strain. It is assumed that the yield stress of the solid k(x, y) has the form k ⁰+k(x, y) where k ⁰ is a constant and is small. A perturbation method of solution developed by Spencer [1] is used, and general results are obtained for arbitrary values of k(x, y). Some particular cases are then considered.     

High speed indentation of an elastic half-space by conical or wedge-shaped indentors

Self-similar problems of indentation of an elastic half-space by rigid cones or wedges are solved, assuming perfect adhesion, when the velocity of indentation is large enough for the area of contact to spread faster than the speed of P-waves. In contrast to the earlier study of the wholly subsonic case [2], the present problems can be solved in closed form without approximation. It emerges, too, that the no slip condition would be satisfied for a range of values of a finite coefficient of friction, in contrast to the situation in [2], where any finite friction is bound to allow some slip. A variety of wave fronts exist in the present problems and all of their amplitudes are found explicitly and discussed.School of Mathematics, University of Bath     

Scaling relationships in conical indentation of elastic-perfectly plastic solids

Using dimensional analysis and finite element calculations we derive several scaling relationships for conical indentation into elastic-perfectly plastic solids. These scaling relationships provide new insights into the shape of indentation curves and form the basis for understanding indentation measurements, including nano- and micro-indentation techniques. They are also helpful as a guide to numerical and finite element calculations of conical indentation problems. Finally, the scaling relationships are used to reveal the general relationships between hardness, contact area, initial unloading slope, and mechanical properties of solids.     

Modeling elasto-plastic indentation on layered materials using the equivalent inclusion method

This paper develops a fast semi-analytical model for solving the three-dimensional elasto-plastic contact problems involving layered materials using the Equivalent Inclusion Method (EIM). The analytical elastic solutions of a half-space subjected to a unit surface pressure and a unit subsurface eigenstrain are employed in this model; the topmost layer is simulated by an equivalent inclusion with fictitious eigenstrain. Accumulative plastic deformation is determined by a procedure involving an iterative plasticity loop and an incremental loading process. Algorithms of the fast Fourier transform (FFT) and the Conjugate Gradient Method (CGM) are utilized to improve the computation efficiency. An analytical elastic solution of layered body contact (O’Sullivan and King, 1988) and an indentation experiment result involving a layered substrate (Michler et al., 1999) are used to examine the accuracy of this model. Comparisons between numerical results from the present model and a commercial FEM software (Abaqus) are also presented. Case studies of a rigid ball loaded against a layered elasto-plastic half-space are conducted to explore the effects of the modulus, yield strength, and thickness of the coating on the hardness, stiffness, and plastic deformation of the composite body.     

Quasi-static normal indentation of an elasto-plastic substrate by a periodic array of elastic strip punches

The plane-strain, elasto-plastic, contact problem described in the title is treated for a substrate that is perfectly plastic post yield, and so simulates compression molding of some metals at elevated temperatures. The analysis uses finite elements and is verified with test problems and convergence checks. The key finding is that, in what might reasonably be viewed as a fully-plastic state, the molding pressure normalized by the yield stress is equal to a constant plus a term that increases linearly with the depth of indentation. This is in contrast to Tabor’s classical result for hardness testing that has the normalized pressure solely equal to a constant when response is fully plastic. The additional linear stiffening term found with the finite element analysis of the present configuration is confirmed experimentally. An explanation of the source of this stiffening term even with a perfectly-plastic substrate is offered. Contact stresses are also tracked as indentation proceeds. These stresses initially have high stress concentrations near the edges of the strip punches. However, these peak stresses abate rapidly with plastic flow and approach a nearly uniform distribution within the fully-plastic state. Implications for compression molding are discussed.     

Mechanical characterization of anisotropic elasto-plastic materials by indentation curves only

Anisotropy, usually orthotropy, arises in structural materials, particularly metals, due to production processes like laminations and concerns primarily parameters which govern the plastic behavior. Identification of such parameters is investigated here by a novel approach with the following features: experimental data provided by indentation curves only (not by imprint geometry); indenter shape with elliptical cross-section derived from classical conical or spherical shape and optimized by sensitivity analyses; indentation test repeated in near places after indenter rotation; deterministic inverse analyses centered on discrepancy function minimization and made computationally economical by an ‘a priori’ model reduction procedure.     

Size effects in indentation response of thin films at the nanoscale: A molecular dynamics study

The indentation response of Ni thin films of thicknesses in the nanoscale was studied using molecular dynamics simulations with embedded atom method (EAM) interatomic potentials. A series of simulations were performed in films in the [1 1 1] orientation with thicknesses varying from 4 to 12.8 nm. The study included both single crystal films and films containing low angle grain boundaries perpendicular to the film surface. The simulation results for single crystal films show that as film thickness decreases larger forces are required for similar indentation depths but the contact stress necessary to emit the first dislocation under the indenter is nearly independent of film thickness. The low angle grain boundaries can act as dislocation sources under indentation. The mechanism of preferred dislocation emission from these boundaries operates at stresses that are lower as the film thickness increases and is not active for the thinnest films tested. These results are interpreted in terms of a simple model.     

Elastic indentation of a rough surface by a conical punch

In the contact of a cone with a rough plane the mean pressure in the contact area is constant. In particular, above a critical ratio of the opening angle of the cone with respect to the rms gradient of surface roughness, the mean pressure is the same of that for nominally flat contact, no matter how large is the normal load. We introduce a new variable, namely, the local density of contact area, whose integral over the smooth nominal contact domain gives the real contact area. The results given by the theoretical model agree with the numerical simulations of the same problem presented in the paper.     

Finite element study for conical indentation of elastoplastic micropolar material

Numerous experiments have repetitively shown that the material behavior presents effective size dependent mechanical properties at scales of microns or submicrons. In this paper, the size dependent behavior of micropolar theory under conical indentation is studied for different indentation depths and micropolar material parameters. To illustrate the effectiveness of the micropolar theory in predicting the indentation size effect (ISE), an axisymmetric finite element model has been developed for elastoplastic contact analysis of the micropolar materials based on the parametric virtual principle. It is shown that the micropolar parameters contribute to describe the characteristic of ISE at different scales, where the material length scale regulates the rate of hardness change at large indentation depth and the value of micropolar shear module restrains the upper limit of hardness at low indentation depth. The simulation results showed that the indentation loads increase as the result of increased material length scale at any indentation depth, however, the rate of increase is higher for lower indentation depth, relative to conventional continuum. The numerical results are presented for perfectly sharp and rounded tip conical indentations of magnesium oxide and compared with the experimental data for hardness coming from the open literature. It is shown that the satisfactory agreement between the experimental data and the numerical results is obtained, and the better correlation is achieved for the rounded tip indentation compared to the sharp indentation.     

The frictional sliding response of elasto-plastic materials in contact with a conical indenter

Over the past decade, many computational studies have explored the mechanics of normal indentation. Quantitative relationships have been well established between the load–displacement hysteresis response and material properties. By contrast, very few studies have investigated broad quantitative aspects of the effects of material properties, especially plastic deformation characteristics, on the frictional sliding response of metals and alloys. The response to instrumented, depth-sensing frictional sliding, hereafter referred to as a scratch test, could potentially be used for material characterization. In addition, it could reproduce a basic tribological event, such as asperity contact and deformation, at different length scales for the multi-scale modeling of wear processes. For these reasons, a comprehensive study was undertaken to investigate the effect of elasto-plastic properties, such as flow strength and strain hardening, on the response to steady-state frictional sliding. Dimensional analysis was used to define scaling variables and universal functions. The dependence of these functions on material properties was assessed through a detailed parametric study using the finite element method. The strain hardening exponent was found to have a greater influence on the scratch hardness and the pile-up height during frictional sliding than observed in frictionless normal indentation. When normalized by the penetration depth, the pile-up height can be up to three times larger in frictional sliding than in normal indentation. Furthermore, in contrast to normal indentation, sink-in is not observed during frictional sliding over the wide range of material properties examined. Finally, friction between indenter and indented material was introduced in the finite element model, and quantitative relationships were also established for the limited effects of plastic strain hardening and yield strength on the overall friction coefficient. Aspects of the predictions of computational simulations were compared with experiments on carefully selected metallic systems in which the plastic properties were systematically controlled. The level of accuracy of the predicted frictional response is also assessed by recourse to the finite element method and by comparison with experiment.     

Explicit approximations of the indentation modulus of elastically orthotropic solids for conical indenters

The elastic problem of the contact between an axisymmetric indenter and a general anisotropic (21 independent elastic constants) half space has not been solved explicitly in closed form. Implicit methods to determine the indentation modulus originate from the work of Willis [J. Mech. Phys. Solids 14 (1966) 163]; and are now available for conical, parabolic and spherical indenters [Philos. Mag. A 81 (2001) 447; J. Mech. Phys. Solids 51 (2003) 1701]. The particular case of orthotropy has also been investigated [ASME J. Tribol. 115 (1193) 650, 125 (2003) 223]. This paper proposes an explicit solution for the indentation moduli of a transversely isotropic medium and a general orthotropic medium under rigid conical indentation in the three principal material symmetry directions. The half-space Green’s functions are interpolated from their exact extreme values, then integrated and finally simplified. The proposed closed form expressions are in very good agreement with the implicit solution schemes of [Philos. Mag. A 81 (2001) 447; J. Mech. Phys. Solids 51 (2003) 1701].     

Determining plastic properties of a material with residual stress by using conical indentation

Instrumented indentation is a popular method for determining mechanical properties in engineering materials. However, there are several shortcomings and challenges involved with correctly interpreting the test results. We propose here a unified method for evaluating instrumented indentation testing conducted on a material that exhibits both strain hardening under yielding and which is subjected to uniform, equi-biaxial residual stresses. The proposed method is based on extensive finite element simulations that relate the parameter-space spanned by Young’s modulus, yield strength, strain hardening and residual stress, to the response from the indentation test. Based on reverse analysis, the proposed method can be used to determine two unknown quantities, such as yield strength and strain hardening. The technique involves utilizing the concept of representative strain and plural indenter-shapes.     

The dynamic indentation of an elastic half-space

A method is given for finding the distribution of traction beneath a conical or a wedge-shaped indentor, moving into an elastic half-space, for the two extreme cases of perfect adhesion between indentor and half-space, and perfectly lubricated indentors. Relations between indenting force, velocity of indentor and area of contact are investigated and found to be fairly insensitive to the friction, although in applications in which the shear traction is required a model with friction would be essential. Mathematically, the paper provides the first accurate solutions of any dynamic mixed boundary value problems with essentially coupled mixed conditions.

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Zusammenfassung Es wird eine Methode zur Bestimmung der Belastung eines keil-oder kegelförmigen Körpers angegeben, der in einen elastischen Halbraum eindringt, wobei die beiden Grenzfäll-vollständige Adhäsion zwischen eindringendem Körper und Halbraum sowie perfekte Schmierung-betrachtet werden. Beziehungen zwischen Eindrückkraft, Eindringgeschwindigkeit und Kontaktfläche werden untersucht. Dabei ergibt sich eine gewisse Unempfindlichkeit gegenüber Reibung, obschon ein Modell unter Berücksichtigung der Reibung dann erforderlich ist, wenn die Schubbeanspruchung benötigt wird. Mathematisch gesehen liefert die vorliegende Arbeit die erste exakte Lösung eines dynamischen gemischten Randwertproblems mit gekoppelten gemischten Bedingungen.

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School of Mathematics, Bath University     

Vibrations of the elasto-plastic pendulum

A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge-Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge-Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.     

Size effects in the constrained deformation of metallic foams

The constrained deformation of an aluminium alloy foam sandwiched between steel substrates has been investigated. The sandwich plates are subjected to through-thickness shear and normal loading, and it is found that the face sheets constrain the foam against plastic deformation and result in a size effect: the yield strength increases with diminishing thickness of foam layer. The strain distribution across the foam core has been measured by a visual strain mapping technique, and a boundary layer of reduced straining was observed adjacent to the face sheets. The deformation response of the aluminium foam layer was modelled by the elastic-plastic finite element analysis of regular and irregular two dimensional honeycombs, bonded to rigid face sheets; in the simulations, the rotation of the boundary nodes of the cell-wall beam elements was set to zero to simulate full constraint from the rigid face sheets. It is found that the regular honeycomb under-estimates the size effect whereas the irregular honeycomb provides a faithful representation of both the observed size effect and the observed strain profile through the foam layer. Additionally, a compressible version of the Fleck-Hutchinson strain gradient theory was used to predict the size effect; by identifying the cell edge length as the relevant microstructural length scale the strain gradient model is able to reproduce the observed strain profiles across the layer and the thickness dependence of strength.     

Thermal stresses in an elastic sphere containing conical cuts

An axisymmetric thermoelastic problem for a sphere with conical cuts in the poles under the action of a quasi-stationary temperature field depending on the meridional angle and on the sphere radius is solved by a method based on the Castigliano variational principle. Orthonormalized systems of polynomials are used as the coordinate functions. Results of numerical calculations of the stress state of a spherical solid are presented.     

Thermal and inertia effects in externally pressurized conical thrust oil bearings

This paper presents theoretical and experimental investigations of thermal and inertia effects on the performance of externally pressurized conical thrust bearings. The analysis, as well as the experimental results, revealed that the increase in oil temperature due to pad rotation has a detrimental effect on the load carrying capacity, while it increases the flow rate. Increasing the speed of rotation, will increase or decrease the bearing load carrying capacity depending on the recess dimensions.Nomenclature c lubricant specific heat - F frictional torque - h film thickness - L load carrying capacity - P pressure - P pressure ratio (P/P ₁) - P ₁ inlet pressure - Q volume flow rate - r radius measured on cone surface - r radius ratio (R/R ₃) - R ₁ supply hole radius - R ₂ recess radius - R ₃ outside radius of bearing - S inertia parameter (0.15 ² R ₃ ² /P ₁) - T temperature - u, v, w velocity components (see Fig. 2) - z coordinate normal to cone surface - lubricant density - lubricant viscosity - 2 cone apex angle - rotational speed - recess depth     

结构性软土弹塑性模型的隐式算法实现

对于考虑软土结构性的高度非线性弹塑性本构模型,在采用Newton-CPPM隐式算法对模型进行数值实现的过程中容易出现Jacobian矩阵奇异和不收敛问题。为此,本文提出了两种改进隐式算法。考虑到Newton-CPPM隐式算法是局部收敛性算法,因此引入大范围收敛的同伦延拓算法对Newton-CPPM算法的迭代初值进行改进,形成了同伦-Newton-CPPM算法。考虑到Newton-CPPM隐式算法单个迭代步的计算量过大,因此借鉴显式算法的思想提出一种两阶段迭代算法,第一阶段先求出一致性参数,第二阶段采用类似于显示算法的方法进行回代得出状态变量的值。然后,以考虑软土结构性的SANICLAY模型为例,从弹塑性本构模型的组成和算法的特点两个角度分析了引起Jacobian矩阵奇异和不收敛问题的原因,并且在单单元计算的基础上,对全显式算法、传统隐式算法和两种改进隐式算法在计算收敛性、计算精度和计算效率方面进行了对比。最后,将同伦-Newton-CPPM算法和传统隐式算法用于地基承载力多单元计算中,结果表明该算法能够有效地解决Jacobian矩阵奇异和不收敛问题。