A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is used to simulate a multi-frequency resistivity logging-while-drilling (LWD) tool response in a borehole environment. Presented in this study are the vector expression of Maxwell’s equations, three kinds of boundary conditions, stability weak formulation of Maxwell’s equations, and automatic hp-adaptivity strategy. The new hp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models, which further confirm the accuracy of the results using the Hermes library (http://hpfem.org/hermes) with a multi-frequency resistivity LWD tool response in a borehole environment.

In our previous study, we modeled the indentation performed on an elastic–plastic solid with a rigid conical indenter by using finite element analysis, and established a relationship between a nominal hardness/reduced Young’s modulus (Hn/Er) and unloading work/total indentation work (We/Wt). The elasticity of the indenter was absorbed in Er ≡ 1/[(1 − ν2)/E + (1 − νi2)/Ei], where Ei and νi are the Young’s modulus and Poisson’s ratio of the indenter, and E and ν are those of the indented material. However, recalculation by directly introducing the elasticity of the indenter show that the use of Er alone cannot accurately reflect the combined elastic effect of the indenter and indented material, but the ratio η = [E/(1 − ν2)]/[Ei/(1 − νi2)] would influence the Hn/Er–We/Wt relationship. Thereby, we replaced Er with a combined Young’s modulus Ec ≡ 1/[(1 − ν2)/E + 1.32(1 − νi2)/Ei] = Er/[1 + 0.32η/(1 + η)], and found that the approximate Hn/Ec–We/Wt relationship is almost independent of selected η values over 0–0.3834, which can be used to give good estimates of E as verified by experimental results.

We previously proposed a method for estimating Young’s modulus from instrumented nanoindentation data based on a model assuming that the indenter had a spherical-capped Berkovich geometry to take account of the bluntness effect. The method is now further improved by releasing the constraint on the tip shape, allowing it to have a much broader arbitrariness to range from a conical-tipped shape to a flat-ended shape, whereas the spherical-capped shape is just a special case in between. This method requires two parameters to specify a tip geometry, namely, a volume bluntness ratio Vr and a height bluntness ratio hr. A set of functional relationships correlating nominal hardness/reduced elastic modulus ratio (Hn/Er) and elastic work/total work ratio (We/W) were established based on dimensional analysis and finite element simulations, with each relationship specified by a set of Vr and hr. Young’s modulus of an indented material can be estimated from these relationships. The method was shown to be valid when applied to S45C carbon steel and 6061 aluminum alloy.

Dimensional and finite element analyses were used to analyze the relationship between the mechanical properties and instrumented indentation response of materials. Results revealed the existence of a functional dependence of (engineering yield strength σE,y + engineering tensile strength σE,b)/Oliver & Pharr hardness on the ratio of reversible elastic work to total work obtained from an indentation test. The relationship links up the Oliver & Pharr hardness with the material strengths, although the Oliver & Pharr hardness may deviate from the true hardness when sinking in or piling up occurs. The functional relationship can further be used to estimate the sum σE,y + σE,b according to the data of an instrumented indentation test. The σE,y + σE,b value better reflects the strength of a material compared to the hardness value alone. The method was shown to be effective when applied to aluminum alloys. The relationship can further be used to estimate the fatigue limits, which are usually obtained from macroscopic fatigue tests in different modes.

The effectiveness of Oliver & Pharr's (O&P's) method, Cheng & Cheng's (C&C’s) method, and a new method developed by our group for estimating Young's modulus and hardness based on instrumented indentation was evaluated for the case of yield stress to reduced Young's modulus ratio (σy/Er) ≥ 4.55 × 10−4 and hardening coefficient (n) ≤ 0.45. Dimensional theorem and finite element simulations were applied to produce reference results for this purpose. Both O&P's and C&C's methods overestimated the Young's modulus under some conditions, whereas the error can be controlled within ±16% if the formulation was modified with appropriate correction functions. Similar modification was not introduced to our method for determining Young's modulus, while the maximum error of results was around ±13%. The errors of hardness values obtained from all the three methods could be even larger and were irreducible with any correction scheme. It is therefore suggested that when hardness values of different materials are concerned, relative comparison of the data obtained from a single standard measurement technique would be more practically useful. It is noted that the ranges of error derived from the analysis could be different if different ranges of material parameters σy/Er and n are considered.

In a previously developed method for estimating Young’s modulus E by depth-sensing indentation with spherical-tipped Berkovich indenter, the E value is deduced from several functional relationships (established by finite element analysis) relating nominal hardness/reduced elastic modulus ratio (Hn/Er) and elastic work/total work ratio (We/W). These relationships are specified for different absolute bluntness/maximum displacement ratios (Δh/hm). This paper reports the generalization of the method by proposing a function to replace all the above mentioned Hn/Er−We/W relationships. The function contains only a parameter Vr ≡ Videal/Vblunt instead of Δh/hm, where Videal is defined as the indented volume bounded by the cross-sectional areas measured at the maximum displacement hm for an ideally sharp indenter, and Vblunt is that of the real indenter. The use of Vr to replace Δh/hm is for the purpose of extending the application of the method for non-spherical tipped Berkovich indenters. The effectiveness of the method for materials of prominent plasticity was demonstrated by performing tests on carbon steel and aluminum alloy using three Berkovich indenters with different tip shapes.

Both analysis and numerical calculations have been carried out to investigate the relationship between Young’s modulus and nonideally sharp indentation parameters. The results confirm that there exists an approximate one-to-one correspondence between the ratio of nominal hardness/reduced Young’s modulus (Hn/Er) and the ratio of elastic work/total work (We/W) for any definite bluntness ratio (Δh/hm) of a nonideally sharp indenter. Based on this relationship, the Young’s modulus of the indented material can be determined just from the values of Hn, We, and W, which are directly measurable quantities in an indentation test.