Scaling in Cavity—Expansion Equations using the Isovector Method (original) (raw)
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New expanding spherical cavity model (ECM) for conical indentation is proposed. For polymeric materials description, the model incorporates isotropic non-monotonic strain hardening. For capturing the indentation size effect (ISE), the model incorporates the strain gradient dependence in yield strength based on lower-order strain gradient plasticity assumptions. Specifically, the forward gradient of the equivalent (accumulated) plastic strain is utilized as a non-local part of the yield strength. To predict the indentation depth-dependent hardness based on the proposed model, it is sufficient to numerically integrate one nonlinear ODE of the first order, and then calculate the definite integral. For the local perfect plasticity model, the hardness is obtained as an analytical expression that differs from known ECMs. The hardness estimate obtained numerically using the proposed model is compared with the experimental ISE data for polycarbonate (PC) and polymethyl methacrylate (PMMA). For the local perfect plasticity model, the formula obtained in the study is compared with the experimental data on the hardness of preliminary work-hardened materials. In both cases, the model shows good agreement with the experimental data. Fitting the experimental data on ISE, we found that intrinsic length scale of PMMA should be near 3 microns and near 9 microns for PC.
An expanding cavity model incorporating pile-up and sink-in effects
Journal of Materials Research, 2011
A new expanding cavity model (ECM) for describing conical indentation of elastic ideally-plastic material is developed. For the proposed ECM, it is assumed that the volume of material displaced by the indenter is equal to the volume loss, due to elastic deformation, in the material and depends on the pileup or sink-in. It was shown that the proposed ECM matches very well numerical data in the final portion of the transition regime for which the contact pressure lies between approximately 2.5Y and 3Y. For material of large E/Y ratio, the new ECM also matches very well numerical data in the plastic-similarity regime. For material of smaller E/Y ratio, the proposed ECM gives better results than the Johnson's ECM because pileup or sink-in is taken into account. I. INTRODUCTION Elastic, elastic-plastic and fully plastic regimes were observed for conical indentation. 1-4 The deformation process produced during conical indentation is well described by Love 1 when the regime is elastic. When plasticity occurs, it is more complex to describe indentation responses because the material of the sample exhibits multiaxial stress conditions with high gradients and large elastic-plastic strains in the indentation region. Strong non-linearities are induced by the unilateral contact and the involved large
Journal of Materials Research, 2009
The present paper aims to develop a robust spherical indentation-based method to extract material plastic properties. For this purpose, a new consideration of piling-up effect is incorporated into the expanding cavity model; an extensive numerical study on the similarity solution has also been performed. As a consequence, two semi-theoretical relations between the indentation response and material plastic properties are derived, with which plastic properties of materials can be identified from a single instrumented spherical indentation curve, the advantage being that this approach no longer needs estimations of contact radius with given elastic modulus. Moreover, the inconvenience in using multiple indenters with different tip angles can be avoided. Comprehensive sensitivity analyses show that the present algorithm is reliable. Also, by experimental verification performed on three typical materials, good agreement of the material properties between those obtained from the reverse a...
Expansion of cavities in brittle media
International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1967
A solution is presented of the problem of quasi-static expansion of a spherical or a cylindrical cavity located in an infinite medium composed of an idealized brittle material with the following properties: (i) Before failure it behaves as a Hooke solid; (ii) In intact state it fails according to a generalized form of the Griffith failure criterion; (iii) In a crushed state it obeys the Mohr-Coulomb failure criterion. The solution covers two different modes of failure of the material around the cavity: with a radially cracked zone for low values of ambient pressure, and without the zone for higher values of the ambient pressure. A tentat ire applicat ion of the theory in the field of indentation hardness testing is shown.
Elastic-plastic contact mechanics of indentations accounting for phase transformations
Experimental Mechanics, 2003
A contact mechanics model is developed which takes into account possible phase transformations in materials induced by hydrostatic and shear stresses associated with indentation. The proposed model allows prediction of the average thickness and approximate shape of the phase transformation zone in semiconductors and ceramics under various types of diamond indenters. The results of theoretical calculation are in good agreement with the available experimental data.
Plane-strain deformation of an elastic material compressed in a rough rectangular cavity
International Journal of Engineering Science, 2002
Deformation and stress distributions in a linear elastic solid, confined to a rigid cavity with rough walls and subjected to uniform compression from one end, are examined. Wall roughness is modeled by Coulomb friction. At the rigid walls, one boundary condition involves deformation and the other stresses, and this renders the problem non-standard. A Laplace transformation solution is constructed for a semiinfinite cavity, and a computational solution for a cavity with finite length. Agreement between the two solutions is good, and improves with increasing cavity lengths and higher coefficients of friction. There exists a critical value of the coefficient of friction below which the axial displacements decay monotonically with distance from the loaded end and the material points stay in contact with the rough walls. For supercritical values of the coefficient of friction, displacements and stresses on the rough walls exhibit oscillatory behavior in the axial direction. The material loses contact with the walls, and the analytical solution presented here loses validity.
Size effects in the conical indentation of an elasto-plastic solid
2012
The size effect in conical indentation of an elasto-plastic solid is predicted via the Fleck and Willis formulation of strain gradient plasticity (Fleck, N.A. and Willis, J.R., 2009, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045-1057). The rate-dependent formulation is implemented numerically and the full-field indentation problem is analyzed via finite element calculations, for both ideally plastic behavior and dissipative hardening. The isotropic strain-gradient theory involves three material length scales, and the relative significance of these length scales upon the degree of size effect is assessed. Indentation maps are generated to summarize the sensitivity of indentation hardness to indent size, indenter geometry and material properties (such as yield strain and strain hardening index). The finite element model is also used to evaluate the pertinence of the Johnson cavity expansion model and of the Nix-Gao model, which have been extensively used to predict size effects in indentation hardness.
Effect of friction and cavity depth on the elastic indentation of a cylindrical rod
Mechanics of Materials, 1997
Using finite element analysis, the elastic displacement of a rigid cylindrical rod pushed by an axial load into an existing cavity of the same diameter as the rod located in the surface of a half space was studied. It was found that such displacement is proportional to the applied load and decreases with increasing depth of the cavity. It is the largest for the slip condition over all the contact surfaces and the smallest for the stick condition over the same surfaces. Also for the same load, the maximum von Mises stress under the punch decreases with increasing depth of the cavity. If not glued to the punch, both the bottom and the lateral surfaces of the cavity are in contact with the punch during deformation.
Rapid Indentation of Transversely Isotropic or Orthotropic Half-Spaces
Journal of Applied Mechanics, 2000
The canonical problems of rapid indentation by, respectively, a rigid smooth wedge and a rigid smooth cylinder, are examined for a transversely isotropic or orthotropic half-space in plane strain. An exact transient analysis based on integral transforms is carried out for the case of contact zone expansion at a constant subcritical rate. Certain functions in the transform space can be factored in such a manner that the resulting solutions, despite anisotropy, have rather simple forms. This factorization is also exploited to obtain a compact exact formula for the Rayleigh wave speed, which serves as the critical contact zone expansion rate. Aspects of contact zone behavior for the two problems are illustrated for five specific materials.