Understanding nanoindentation unloading curves

2002 ◽  
Vol 17 (10) ◽  
pp. 2660-2671 ◽  
Author(s):  
G. M. Pharr ◽  
A. Bolshakov
Keyword(s):  
Power Law ◽  
Material Properties ◽  
Elastic Contact ◽  
Physical Explanation ◽  
Material Constants ◽  
Plastic Materials ◽  
Power Law Exponent ◽  
Plastic Hardness ◽  
Elastic Plastic Materials ◽  
Indenter Shape

Experiments have shown that nanoindentation unloading curves obtained with Berkovich triangular pyramidal indenters are usually welldescribed by the power-law relation P = α(h − hf)m, where hf is the final depth after complete unloading and α and m are material constants. However, the power-law exponent is not fixed at an integral value, as would be the case for elastic contact by a conical indenter (m = 2) or a flat circular punch (m = 1), but varies from material to material in the range m = 1.2–1.6. A simple model is developed based on observations from finite element simulations of indentation of elastic–plastic materials by a rigid cone that provides a physical explanation for the behavior. The model, which is based on the concept of an indenter with an “effective shape” whose geometry is determined by the shape of the plastic hardness impression formed during indentation, provides a means by which the material constants in the power law relation can be related to more fundamental material properties such as the elastic modulus and hardness. Simple arguments are presented from which the effective indenter shape can be derived from the pressure distribution under the indenter.

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

2009 ◽  
Vol 24 (3) ◽  
pp. 1037-1044 ◽  
Author(s):  
Sara Aida Rodríguez Pulecio ◽  
María Cristina Moré Farias ◽  
Roberto Martins Souza
Keyword(s):  
Finite Element ◽  
Material Properties ◽  
Total Work ◽  
Plastic Materials ◽  
Maximum Penetration Depth ◽  
Finite Element Calculations ◽  
Maximum Penetration ◽  
Elastic Plastic Materials ◽  
Tip Radius ◽  
Hardness Values

In this work, the effects of indenter tip roundness on 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/hmax), which varied from 0.02 to 1. The proportional curvature constant (C), the exponent of depth during loading (α), the initial unloading slope (S), the correction factor (β), the level of piling-up or sinking-in (hc/hmax), and the ratio hmax/hf are shown to be strongly influenced by the ratio R/hmax. The hardness (H) was found to be independent of R/hmax in the range studied. The Oliver and Pharr method was successful in following the variation of hc/hmax with the ratio R/hmax through the variation of S with the ratio R/hmax. 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 (Wp/Wt) was found to be independent of the ratio R/hmax, 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.

scholarly journals Catalogue of the J-Q trajectories for CC(T) and SEN(T) specimens in tension

2011 ◽  
Vol 1 (3) ◽  
Author(s):  
Marcin Graba
Keyword(s):  
Fracture Toughness ◽  
Stress Distribution ◽  
Material Properties ◽  
Work Hardening ◽  
Theoretical Background ◽  
Engineering Analysis ◽  
Elastic Plastic ◽  
Plastic Materials ◽  
Elastic Plastic Materials ◽  
Work Hardening Exponent

AbstractIn this paper a short theoretical background about elastic-plastic fracture mechanics is presented and the O’Dowd-Shih theory is discussed. Using FEM, the values of the Q-stress determined for various elastic-plastic materials for two specimens in tension — SEN(T) specimen and CC(T) specimen are presented. The influence of geometry of the specimen, crack length and material properties (work-hardening exponent and yield stress) on the Q-parameter are tested. The numerical results were approximated by closed form formulas. The results are summarized in a catalogue of the Q-stress value, which may be used in engineering analysis for calculation of the real fracture toughness and the stress distribution near crack tip.

Regional variation of recession flow power-law exponent

2018 ◽  
Vol 32 (7) ◽  
pp. 866-872 ◽  
Author(s):  
Swagat Patnaik ◽  
Basudev Biswal ◽  
Dasika Nagesh Kumar ◽  
Bellie Sivakumar
Keyword(s):  
Power Law ◽  
Regional Variation ◽  
Power Law Exponent ◽  
Flow Power

scholarly journals Growth Rate Exponents of Richtmyer-Meshkov Mixing Layers

2005 ◽  
Vol 73 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Timothy T. Clark ◽  
Ye Zhou
Keyword(s):  
Flow Field ◽  
Power Law ◽  
Mixing Layer ◽  
Power Laws ◽  
Mixing Layers ◽  
Spatial Gradients ◽  
Power Law Exponent ◽  
Virtual Time ◽  
Time Origin ◽  
Density Ratios

The Richtmyer-Meshkov mixing layer is initiated by the passing of a shock over an interface between fluid of differing densities. The energy deposited during the shock passage undergoes a relaxation process during which the fluctuational energy in the flow field decays and the spatial gradients of the flow field decrease in time. This late stage of Richtmyer-Meshkov mixing layers is studied from the viewpoint of self-similarity. Analogies with weakly anisotropic turbulence suggest that both the bubble-side and spike-side widths of the mixing layer should evolve as power-laws in time, with the same power-law exponent and virtual time origin for both sides. The analogy also bounds the power-law exponent between 2∕7 and 1∕2. It is then shown that the assumption of identical power-law exponents for bubbles and spikes yields fits that are in good agreement with experiment at modest density ratios.

scholarly journals Free Convective MHD Flow Past a Vertical Cone with Variable Heat and Mass Flux

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
J. Prakash ◽  
S. Gouse Mohiddin ◽  
S. Vijaya Kumar Varma
Keyword(s):  
Boundary Layer ◽  
Power Law ◽  
Mass Flux ◽  
Numerical Study ◽  
Convection Boundary Layer ◽  
Vertical Cone ◽  
Local Skin ◽  
Surface Mass ◽  
Flow Past ◽  
Power Law Exponent

A numerical study of buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime with transverse magnetic field applied normal to the surface is considered. The heat and mass flux at the surface of the cone is modeled as a power law according to qwx=xm and qw*(x)=xm, respectively, where x denotes the coordinate along the slant face of the cone. Both Darcian drag and Forchheimer quadratic porous impedance are incorporated into the two-dimensional viscous flow model. The transient boundary layer equations are then nondimensionalized and solved by the Crank-Nicolson implicit difference method. The velocity, temperature, and concentration fields have been studied for the effect of Grashof number, Darcy number, Forchheimer number, Prandtl number, surface heat flux power-law exponent (m), surface mass flux power-law exponent (n), Schmidt number, buoyancy ratio parameter, and semivertical angle of the cone. Present results for selected variables for the purely fluid regime are compared with the published results and are found to be in excellent agreement. The local skin friction, Nusselt number, and Sherwood number are also analyzed graphically. The study finds important applications in geophysical heat transfer, industrial manufacturing processes, and hybrid solar energy systems.

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