Axisymmetric indentation of curved elastic membranes by a convex rigid indenter

Motivated by applications to seed germination, we consider the transverse deflection that results from the axisymmetric indentation of an elastic membrane by a rigid body. The elastic membrane is fixed around its boundary, with or without an initial pre-stretch, and may be initially curved prior to indentation. General indenter shapes are considered, and the load-indentation curves that result for a range of spheroidal tips are obtained for both flat and curved membranes. Wrinkling may occur when the membrane is initially curved, and a relaxed strain-energy function is used to calculate the deformed profile in this case. Applications to experiments designed to measure the mechanical properties of seed endosperms are discussed.     

On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials

Indentation is widely used to extract material elastoplastic properties from the measured force-displacement curves. One of the most well-established indentation techniques utilizes dual (or plural) sharp indenters (which have different apex angles) to deduce key parameters such as the elastic modulus, yield stress, and work-hardening exponent for materials that obey the power-law constitutive relationship. However, the uniqueness of such analysis is not yet systematically studied or challenged. Here we show the existence of “mystical materials”, which have distinct elastoplastic properties yet they yield almost identical indentation behaviors, even when the indenter angle is varied in a large range. These mystical materials are, therefore, indistinguishable by many existing indentation analyses unless extreme (and often impractical) indenter angles are used. Explicit procedures of deriving these mystical materials are established, and the general characteristics of the mystical materials are discussed. In many cases, for a given indenter angle range, a material would have infinite numbers of mystical siblings, and the existence maps of the mystical materials are also obtained. Furthermore, we propose two alternative techniques to effectively distinguish these mystical materials. The study in this paper addresses the important question of the uniqueness of indentation test, as well as providing useful guidelines to properly use the indentation technique to measure material elastoplastic properties.     

Determination of mechanical properties by instrumented indentation

The paper reviews the current state of the depth-sensing indentation (sometimes called nanoindentation), where the information on material behaviour and properties is obtained from the indenter load and depth, measured continuously during loading and unloading. It is shown how the contact parameters and principal characteristics are determined using pointed or spherical indenters. Indentation tests can be used for the measurement of hardness and elastic modulus, and also of the yield stress and for the construction of stress–strain diagrams, for the determination of the work of indentation and its components. Most devices use monotonic loading and unloading, but some also enable measurement under a small harmonic signal added to the basic monotonously increasing load. This makes possible continuous measurement of contact stiffness and the study of dynamic properties and the determination of properties of coatings. One section is devoted to the measurement on viscoelastic-plastic materials, where the delayed deforming must be considered during the measurement as well as in data evaluation. Instrumented indentation can also be used for the study of creep under high temperatures. The paper also discusses the errors arising in depth-sensing measurements and informs briefly about some other possibilities of the method.     

Discrete dislocation plasticity analysis of the wedge indentation of films

The plane strain indentation of single crystal films on a rigid substrate by a rigid wedge indenter is analyzed using discrete dislocation plasticity. The crystals have three slip systems at ±35.3^° and 90^° with respect to the indentation direction. The analyses are carried out for three values of the film thickness, 2, 10 and , and with the dislocations all of edge character modeled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Over the range of indentation depths considered, the indentation pressure for the 10 and thick films decreases with increasing contact size and attains a contact size-independent value for contact lengths . On the other hand, for the films, the indentation pressure first decreases with increasing contact size and subsequently increases as the plastic zone reaches the rigid substrate. For the 10 and thick films sink-in occurs around the indenter, while pile-up occurs in the film when the plastic zone reaches the substrate. Comparisons are made with predictions obtained from other formulations: (i) the contact size-independent indentation pressure is compared with that given by continuum crystal plasticity; (ii) the scaling of the indentation pressure with indentation depth is compared with the relation proposed by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 43, 411-423]; and (iii) the computed contact area is compared with that obtained from the estimation procedure of Oliver and Pharr [1992. An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564-1583].     

The correlation of the indentation size effect measured with indenters of various shapes

Experimental results are presented which show that the indentation size effect for pyramidal and spherical indenters can be correlated. For a pyramidal indenter, the hardness measured in crystalline materials usually increases with decreasing depth of penetration, which is known as the indentation size effect. Spherical indentation also shows an indentation size effect. However, for a spherical indenter, hardness is not affected by depth, but increases with decreasing sphere radius. The correlation for pyramidal and spherical indenter shapes is based on geometrically necessary dislocations and work-hardening. The Nix and Gao indentation size effect model (J. Mech. Phys. Solids 46 (1998) 411) for conical indenters is extended to indenters of various shapes and compared to the experimental results.     

Mechanics of indentation of plastically graded materials—II: Experiments on nanocrystalline alloys with grain size gradients

A systematic study of depth-sensing indentation was performed on nanocrystalline (nc) Ni-W alloys specially synthesized with controlled unidirectional gradients in plastic properties. A yield strength gradient and a roughly constant Young's modulus were achieved in the nc alloys, using electrodeposition techniques. The force vs. displacement response from instrumented indentation experiments matched very well with that predicted from the analysis of Part I of this paper. The experiments also revealed that the pile-up of the graded alloy around the indenter is noticeably higher than that for the two homogeneous reference alloys that constitute the bounding conditions for the graded material. These trends are also consistent with the predictions of the indentation analysis.     

Polymer indentation: Numerical analysis and comparison with a spherical cavity model

Three dimensional analyses of indentation of a polymer by a rigid indenter are carried out. The polymer is characterized by a finite strain elastic-viscoplastic constitutive relation and the calculations are carried out with a dynamic finite element program used to simulate quasi-static behavior. Two types of indenter are considered; a conical indenter and a pyramidal indenter. For each indenter type, different values of the sharpness of the indenter are considered and two rates of indentation are analyzed. Significant sink-in is found to occur in all cases considered. The amount of sink-in is found to be smaller for sharper indenters. The calculated values of both the nominal and true hardness do not differ significantly for the two indenter shapes. An expanding spherical cavity model is also considered and the predictions of this model are compared with the finite element results for various indenter shapes and indentation rates. The spherical cavity model is found to give fairly good agreement with the predictions of the finite element analyses for the nominal polymer hardness for both indenter shapes.     

Coupled thermo-mechanical modelling of bulk-metallic glasses: Theory, finite-element simulations and experimental verification

A three-dimensional, finite-deformation-based constitutive model to describe the behavior of metallic glasses in the supercooled liquid region has been developed. By formulating the theory using the principles of thermodynamics and the concept of micro-force balance [Gurtin, M., 2000. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989-1036], a kinetic equation for the free volume concentration is derived by augmenting the Helmholtz free energy used for a conventional metallic alloy with a flow-defect free energy which depends on the free volume concentration and its spatial gradient. The developed constitutive model has also been implemented in the commercially available finite-element program ABAQUS/Explicit (2005) by writing a user-material subroutine. The constitutive parameters/functions in the model were calibrated by fitting the constitutive model to the experimental simple compression stress-strain curves conducted under a variety of strain-rates at a temperature in the supercooled liquid region [Lu, J., Ravichandran, G., Johnson, W., 2003. Deformation behavior of the Zr-Ti-Cu-Ni-Be bulk metallic glass over a wide range of strain-rates and temperatures. Acta Mater. 51, 3429-3443].With the model calibrated, the constitutive model was able to reproduce the simple compression stress-strain curves for jump-in-strain-rate experiments to good accuracy. Furthermore stress-strain responses for simple compression experiments conducted at different ambient temperatures within the supercooled liquid region were also accurately reproduced by the constitutive model. Finally, shear localization studies also show that the constitutive model can reasonably well predict the orientation of shear bands for compression experiments conducted at temperatures within the supercooled liquid region [Wang, G., Shen, J., Sun, J., Lu, Z., Stachurski, Z., Zhou, B., 2005. Compressive fracture characteristics of a Zr-based bulk metallic glass at high test temperatures. Mater. Sci. Eng. A 398, 82-87].     

High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: Geometrically necessary dislocation densities

Experimental studies on indentation into face-centered cubic (FCC) single crystals such as copper and aluminum were performed to reveal the spatially resolved variation in crystal lattice rotation induced due to wedge indentation. The crystal lattice curvature tensors of the indented crystals were calculated from the in-plane lattice rotation results as measured by electron backscatter diffraction (EBSD). Nye's dislocation density tensors for plane strain deformation of both crystals were determined from the lattice curvature tensors. The least L²-norm solutions to the geometrically necessary dislocation densities for the case in which three effective in-plane slip systems were activated in the single crystals associated with the indentation were determined. Results show the formation of lattice rotation discontinuities along with a very high density of geometrically necessary dislocations.     

An indentation technique for estimating the energy density as fracture toughness with Berkovich indenter for ductile bulk materials

A technique is proposed to estimate the energy density as fracture toughness for ductile bulk materials with an indentation system equipped with a Berkovich indenter based on the theory of plastic deformation energy transforming into the indentation energy of fracture. With progressive increase of penetration loads, the material damage is exhibited on the effective elastic modulus. A quadratic polynomial relationship between the plastic penetration depth and penetration load, and an approximate linear relationship between logarithmic plastic penetration depth and logarithmic effective elastic modulus are exhibited by indentation investigation with Berkovich indenter. The parameter of damage variable is proposed to determine the critical effective elastic modulus at the fracture point. And the strain energy density factor is calculated according to the equations of penetration load, plastic penetration depth and effective elastic modulus. The fracture toughness of aluminum alloy and stainless steel are evaluated by both indentation tests and K_IC fracture toughness tests. The predicted S_cr values of indentation tests are in good agreement with experimental results of CT tests.