Effects of elastic indenter deformation on spherical instrumented indentation tests: the reduced elastic modulus

Although the Hertz theory is not applicable in the analysis of the indentation of elastic-plastic materials, it is common practice to incorporate the concept of indenter/specimen combined modulus to consider indenter deformation. The appropriateness was assessed of the use of reduced modulus to incorporate the effect of indenter deformation in the analysis of the indentation with spherical indenters. The analysis based on finite element simulations considered four values of the ratio of the indented material elastic modulus to that of the diamond indenter, E/E(i) (0, 0.04, 0.19, 0.39), four values of the ratio of the elastic reduced modulus to the initial yield strength, E(r)/Y (0, 10, 20, 100), and two values of the ratio of the indenter radius to maximum total displacement, R/delta(max) (3, 10). Indenter deformation effects are better accounted for by the reduced modulus if the indented material behaves entirely elastically. In this case, identical load-displacement (P - delta) curves are obtained with rigid and elastic spherical indenters for the same elastic reduced modulus. Changes in the ratio E/E(i), from 0 to 0.39, resulted in variations lower than 5% for the load dimensionless functions, lower than 3% in the contact area, A(c), and lower than 5% in the ratio H/E(r). However, deformations of the elastic indenter made the actual radius of contact change, even in the indentation of elastic materials. Even though the load dimensionless functions showed only a little increase with the ratio E/E(i), the hardening coefficient and the yield strength could be slightly overestimated when algorithms based on rigid indenters are used. For the unloading curves, the ratio delta(e)/delta(max), where delta(e) is the point corresponding to zero load of a straight line with slope S from the point (P(max), delta(max)), varied less than 5% with the ratio E/E(i). Similarly, the relationship between reduced modulus and the unloading indentation curve, expressed by Sneddon`s equation, did not reveal the necessity of correction with the ratio E/E(i). The most affected parameter in the indentation curve, as a consequence of the indentation deformation, was the ratio between the residual indentation depth after complete unloading and the maximum indenter displacement, delta(r)/delta(max) (up to 26%), but this variation did not significantly decrease the capability to estimate hardness and elastic modulus based on the ratio of the residual indentation depth to maximum indentation depth, h(r)/h(max). In general, the results confirm the convenience of the use of the reduced modulus in the spherical instrumented indentation tests.

The reduced modulus in the analysis of sharp instrumented indentation tests

In the analysis of instrumented indentation data, it is common practice to incorporate the combined moduli of the indenter (E-i) and the specimen (E) in the so-called reduced modulus (E-r) to account for indenter deformation. Although indenter systems with rigid or elastic tips are considered as equivalent if E-r is the same, the validity of this practice has been questioned over the years. The present work uses systematic finite element simulations to examine the role of the elastic deformation of the indenter tip in instrumented indentation measurements and the validity of the concept of the reduced modulus in conical and pyramidal (Berkovich) indentations. It is found that the apical angle increases as a result of the indenter deformation, which influences in the analysis of the results. Based upon the inaccuracies introduced by the reduced modulus approximation in the analysis of the unloading segment of instrumented indentation applied load (P)-penetration depth (delta) curves, a detailed examination is then conducted on the role of indenter deformation upon the dimensionless functions describing the loading stages of such curves. Consequences of the present results in the extraction of the uniaxial stress-strain characteristics of the indented material through such dimensional analyses are finally illustrated. It is found that large overestimations in the assessment of the strain hardening behavior result by neglecting tip compliance. Guidelines are given in the paper to reduce such overestimations.

Computer simulation of form-roll design

Changes in modern structural design have created a demand for products which are light but possess high strength. The objective is a reduction in fuel consumption and weight of materials to satisfy both economic and environmental criteria. Cold roll forming has the potential to fulfil this requirement. The bending process is controlled by the shape of the profile machined on the periphery of the rolls. A CNC lathe can machine complicated profiles to a high standard of precision, but the expertise of a numerical control programmer is required. A computer program was developed during this project, using the expert system concept, to calculate tool paths and consequently to expedite the procurement of the machine control tapes whilst removing the need for a skilled programmer. Codifying the expertise of a human and the encapsulation of knowledge within a computer memory, destroys the dependency on highly trained people whose services can be costly, inconsistent and unreliable. A successful cold roll forming operation, where the product is geometrically correct and free from visual defects, is not easy to attain. The geometry of the sheet after travelling through the rolling mill depends on the residual strains generated by the elastic-plastic deformation. Accurate evaluation of the residual strains can provide the basis for predicting the geometry of the section. A study of geometric and material non-linearity, yield criteria, material hardening and stress-strain relationships was undertaken in this research project. The finite element method was chosen to provide a mathematical model of the bending process and, to ensure an efficient manipulation of the large stiffness matrices, the frontal solution was applied. A series of experimental investigations provided data to compare with corresponding values obtained from the theoretical modelling. A computer simulation, capable of predicting that a design will be satisfactory prior to the manufacture of the rolls, would allow effort to be concentrated into devising an optimum design where costs are minimised.

A failure criterion for fillet welds

The work constitutes a study of the strength of mild steel fillet welds subject to static loading, and the behaviour of flange welded beam-column connections under combined bending and shear. Tests are conducted on short welds in the as-welded and stress relieved conditions, and also on full-size beam-column connections. It is shown that welds under compression have a lower strength than when under tension. Failure of the fillet weld is initiated at the weld root, the important factor controlling the initiation being weld ductility. The greater the residual stress, the lower the weld ductility and ultimate strength. Thermal stress relieving increases strength by as much as 30%. Weld failure plane is rarely at the throat and varies from 0° to 45° depending upon loading condition. Failure plane average stresses are related by a circular function which is expressed in terms of externally applied forces at limit state. The tension weld of a flange-welded beam-column connection always fails before the compression weld. The shear load sharing between the welds is a complex function of elastic compression of the web, elastic/plastic deformation of the flanges, load/deformation characteristics, and the type of load application. Bearing forces between the compression flange and column face produce low level bearing stresses and frictional forces which make a negligible contribution to shear load resistance. Three modes of connection failure are possible; 'end mode', 'bending mode' and 'shear mode', with a sudden change taking place between the two latter.

A critical reassessment of elastic unloading in sharp instrumented indentation experiments and the extraction of mechanical properties

This work examines the extraction of mechanical properties from instrumented indentation P-h(s) curves via extensive three-dimensional finite element analyses for pyramidal tips in a wide range of solids under frictional and frictionless contact conditions. Since the topography of the imprint changes with the level of pile-up or sink-in, a relationship is identified between correction factor beta in the elastic equation for the unloading indentation stage and the amount of surface deformation effects. It is shown that the presumption of a constant beta significantly affects mechanical property extractions. Consequently, a new best-fit function is found for the correlation between penetration depth ratios h(e)/h(max), h(r)/h(max) and n, circumventing the need for the assumption of a constant value for beta, made in our prior investigation [Acta Mater. 53 (2005) pp. 3545-3561]. Simulations under frictional contact conditions provide sensible boundaries for the influence of friction on both h(e)/h(max) and h(r)/h(max). Friction is essentially found to induce an overestimation in the inferred n. Instrumented indentation experiments are also performed in three archetypal metallic materials exhibiting distinctly different contact responses. Mechanical property extractions are finally demonstrated in each of these materials.

Elastic modelling of deformation twinning on the microscale

Magdeburg, Univ., Fak. für Maschinenbau, Diss., 2009

Tectonics of the Simplon massif and Lepontine gneiss dome: deformation structures due to collision between the underthrusting European plate and the Adriatic indenter

this study presents a review of published geological data, combined with original observations on the tectonics of the simplon massif and the Lepontine gneiss dome in the Western Alps. New observations concern the geometry of the Oligocene Vanzone back fold, formed under amphibolite facies conditions, and of its root between Domodossola and Locarno, which is cut at an acute angle by the Miocene, epi- to anchizonal, dextral centovalli strike-slip fault. the structures of the simplon massif result from collision over 50 Ma between two plate boundaries with a different geometry: the underthrusted European plate and the Adriatic indenter. Detailed mapping and analysis of a complex structural interference pattern, combined with observations on the metamorphic grade of the superimposed structures and radiometric data, allow a kinematic model to be developed for this zone of oblique continental collision. the following main Alpine tectonic phases and structures may be distinguished: 1. NW-directed nappe emplacement, starting in the Early Eocene (similar to 50 Ma); 2. W, SW and S- verging transverse folds; 3. transpressional movements on the dextral simplon ductile shear zone since similar to 32 Ma; 4. formation of the Bergell - Vanzone backfolds and of the southern steep belt during the Oligocene, emplacement of the mantle derived 31 - 29 Ma Bergell and Biella granodiorites and porphyritic andesites as well as intrusions of 29-25 Ma crustal aplites and pegmatites; 5. formation of the dextral discrete Rhone-Simplon line and the centovalli line during the Miocene, accompanied by the pull-apart development of the Lepontine gneiss dome - Dent blanche (Valpelline) depression. It is suggested that movements of shortening in fan shaped NW, W and sW directions accompanied the more regular NW- to WNW-directed displacement of the Adriatic indenter during continental collision.

Hydrodynamic induced deformation of and orientation of a microscopic elastic filament

We describe simulations of an elastic filament immersed in a fluid and subjected to a body force. The coupling between the fluid flow and the friction that the filament experiences induces bending and alignment perpendicular to the force. With increasing force there are four shape regimes, ranging from slight distortion to an unsteady tumbling motion. We also find marginally stable structures. The instability of these shapes and the alignment are explained by induced bending and nonlocal hydrodynamic interactions. These effects are experimentally relevant for stiff microfilaments.

Separating Growth from Elastic Deformation during Cell Enlargement1

Plants change size by deforming reversibly (elastically) whenever turgor pressure changes, and by growing. The elastic deformation is independent of growth because it occurs in nongrowing cells. Its occurrence with growth has prevented growth from being observed alone. We investigated whether the two processes could be separated in internode cells of Chara corallina Klien ex Willd., em R.D.W. by injecting or removing cell solution with a pressure probe to change turgor while the cell length was continuously measured. Cell size changed immediately when turgor changed, and growth rates appeared to be altered. Low temperature eliminated growth but did not alter the elastic effects. This allowed elastic deformation measured at low temperature to be subtracted from elongation at warm temperature in the same cell. After the subtraction, growth alone could be observed for the first time. Alterations in turgor caused growth to change rapidly to a new, steady rate with no evidence of rapid adjustments in wall properties. This turgor response, together with the marked sensitivity of growth to temperature, suggested that the growth rate was not controlled by inert polymer extension but rather by biochemical reactions that include a turgor-sensitive step.

Evolution of conducting channels in metallic atomic contacts under elastic deformation

We investigate both experimentally and theoretically the evolution of conductance in metallic one-atom contacts under elastic deformation. While simple metals like Au exhibit almost constant conductance plateaus, Al and Pb show inclined plateaus with positive and negative slopes. It is shown how these behaviors can be understood in terms of the orbital structure of the atoms forming the contact. This analysis provides further insight into the issue of conductance quantization in metallic contacts revealing important aspects of their atomic and electronic structures.

Mechanics of internal work (or work of deformation) in elastic bodies and systems in equilibrium, including the method of least work,

Mode of access: Internet.

Aplicação de indentação instrumentada na caracterização mecânica de poliuretana derivada de óleo de mamona

Neste trabalho são investigadas as propriedades mecânicas de poliuretana derivada do óleo de mamona, utilizando a técnica de indentação instrumentada com penetradores de geometrias piramidal e esférica. Foi analisada a influência da forma do penetrador utilizado nos ensaios de indentação instrumentada para se obter valores das propriedades mecânicas de polímero derivado de óleo de mamona. Os penetradores utilizados são de pontas piramidais dos tipos Berkovich e canto de cubo e esférico de raio igual a 150 μm em um Nanoindenter XP TM com cargas aplicadas entre 1 e 200 mN. As penetrações variam de acordo com o formato do penetrador, sendo maiores para pontas agudas. A dureza e o módulo de elasticidade foram determinados, utilizando o método de Oliver e Pharr. Verificou-se que os valores medidos para a dureza são maiores para penetradores mais agudos. Os valores obtidos com a ponta piramidal Berkovich foram de 0,14 GPa para pequenas penetrações e 0,12 GPa para maiores penetrações. Já os valores obtidos com ponta canto de cubo foram 25 a 30% maiores. Isso está relacionado com os volumes das regiões que apresentam deformações plásticas elevadas, no caso de penetradores agudos comparados com os volumes das regiões que sofrem deformações viscoelásticas. A viscosidade aparente determinada, utilizando penetrador esférico em testes de força aplicada constante, é igual a (22 ± 2) × 10(12) Pa.s.

Analysis of the tip roundness effects on the micro- and macroindentation response of elastic-plastic materials

In this work, the effects of indenter tip roundness oil the load-depth indentation curves were analyzed using finite element modeling. The tip roundness level was Studied based on the ratio between tip radius and maximum penetration depth (R/h(max)), which varied from 0.02 to 1. The proportional Curvature constant (C), the exponent of depth during loading (alpha), the initial unloading slope (S), the correction factor (beta), the level of piling-up or sinking-in (h(c)/h(max)), and the ratio h(max)/h(f) are shown to be strongly influenced by the ratio R/h(max). The hardness (H) was found to be independent of R/h(max) in the range studied. The Oliver and Pharr method was successful in following the variation of h(c)/h(max) with the ratio R/h(max) through the variation of S with the ratio R/h(max). However, this work confirmed the differences between the hardness values calculated using the Oliver-Pharr method and those obtained directly from finite element calculations; differences which derive from the error in area calculation that Occurs when given combinations of indented material properties are present. The ratio of plastic work to total work (W(p)/W(t)) was found to be independent of the ratio R/h(max), which demonstrates that the methods for the Calculation of mechanical properties based on the *indentation energy are potentially not Susceptible to errors caused by tip roundness.

Analysis of the effects of conical indentation variables on the indentation response of elastic-plastic materials through factorial design of experiment

In this work, the effects of conical indentation variables on the load-depth indentation curves were analyzed using finite element modeling and dimensional analysis. A factorial design 2(6) was used with the aim of quantifying the effects of the mechanical properties of the indented material and of the indenter geometry. Analysis was based on the input variables Y/E, R/h(max), n, theta, E, and h(max). The dimensional variables E and h(max) were used such that each value of dimensionless Y/E was obtained with two different values of E and each value of dimensionless R/h(max) was obtained with two different h(max) values. A set of dimensionless functions was defined to analyze the effect of the input variables: Pi(1) = P(1)/Eh(2), Pi(2) = h(c)/h, Pi(3) = H/Y, Pi(4) = S/Eh(max), Pi(6) = h(max)/h(f) and Pi(7) = W(P)/W(T). These six functions were found to depend only on the dimensionless variables studied (Y/E, R/h(max), n, theta). Another dimension less function, Pi(5) = beta, was not well defined for most of the dimensionless variables and the only variable that provided a significant effect on beta was theta. However, beta showed a strong dependence on the fraction of the data selected to fit the unloading curve, which means that beta is especially Susceptible to the error in the Calculation of the initial unloading slope.

Consolidation of elastic-plastic saturated porous media by the boundary element method

This paper presents a domain boundary element formulation for inelastic saturated porous media with rate-independent behavior for the solid skeleton. The formulation is then applied to elastic-plastic behavior for the solid. Biot`s consolidation theory, extended to include irreversible phenomena is considered and the direct boundary element technique is used for the numerical solution after time discretization by the implicit Euler backward algorithm. The associated nonlinear algebraic problem is solved by the Newton-Raphson procedure whereby the loading/unloading conditions are fully taken into account and the consistent tangent operator defined. Only domain nodes (nodes defined inside the domain) are used to represent all domain values and the corresponding integrals are computed by using an accurate sub-elementation scheme. The developments are illustrated through the Drucker-Prager elastic-plastic model for the solid skeleton and various examples are analyzed with the proposed algorithms. (c) 2008 Elsevier B.V. All rights reserved.