Improvement of indentation technique for measuring general biaxial residual stresses in austenitic steels
Introduction
The effects of residual stress and strain fields on materials may be important based on the material's function (e.g. fatigue, failure, corrosion, and wear). In order to consider these effects in engineering design, the amount of this residual stress/strain and its nature, being compressive or tensile, should be measured. There are several ways to measure residual stresses, which are often expensive, complex, and applicable only to certain materials [1]. Indentation is a nondestructive method which has widely been used in recent years because of its simplicity of use and applicability on nano-to macro-scales [2]. A major advantage of the indentation method is its in situ application, which can be very practical in oil and gas industries to examine the residual stress in pipelines and welded areas. Also, by this method, material properties such as Young's modulus and yield stress can be measured in addition to the residual stress. Suresh and Giannakopoulos [3] provided the first relationship between indentation parameters and residual stress based on Tsui et al.'s [4] works. For this purpose, they derived a relationship between the contact areas of indentation with and without the existence of residual stress. In their method, the residual stress field was considered equibiaxial. After that, many researchers attempted to develop their model in accordance with the general state of residual stress. For instance, Carlsson and Larsson [1] introduced a new parameter describing the relationship between contact areas and indentation while considering pile-up or sink-in effects using a conical indenter. Then, they developed their method to measure non-equibiaxial residual stresses in materials with an elastic-perfect plastic mechanical behavior.
Lee and Kwon [5,6] also developed Suresh et al.‘s idea and separated the equibiaxial surface residual stress into mean stress σm (hydrostatic stress) and plastic-deformation-sensitive shear deviator stress σD. Next, by measuring the difference of indentation force between stressed and stress-free specimens indented to a specified penetration depth, the equibiaxial residual stress was determined. They also developed their model to estimate the general non-equibiaxial stress state using a Knoop indenter to measure the stress ratio in the general stress state. Lee et al. [7,8] and Kwon et al. [[9], [10], [11]] are still developing their model to obtain the general stress field and the ratio of biaxial stresses.
Wang et al. [12] employed a Berkovich indenter and proposed a model for calculating the equibiaxial residual stress according to the indentation work during the indentation process. Their basic assumption was that during indentation, elastic responses are completely independent of residual stresses on the indented surface. Therefore, the elastic unloading part of load-depth curves is expected to be unaffected by residual stresses.
Huber and Heerens [13] and Heerens et al. [14] performed a pervasive analysis of the problem by examining how the mechanical properties of the surfaces of the material under investigation are influenced by a residual stress field. It should be noted that Huber and Heerens [13] and Heerens et al. [14] incorporated spherical indentation in place of the sharp indenters utilized by other methods. However, in most studies sharp indenters are of interest because the hardness and contact area measured by these indenters are independent of indentation depth, which is a particular advantage when explaining the results [15]. On the other hand, despite spherical indenters, sharp indenters are capable of measuring non-equibiaxial residual stress fields. Several case studies have also been conducted in order to investigate residual stress in particular materials, but their methods are not guaranteed for all materials and occasions. Pham and Kim [16] performed an extensive finite-element (FE) analysis to explore the effects of residual stress and plastic parameters on the indentation response in structural steel. They developed a reverse algorithm to predict unknown parameters (including residual stress, yield stress, and work-hardening exponent) from the indentation test. Their results were in agreement with those of experiments on SS400 and SM490 steels. Misra and his coworkers [[17], [18], [19], [20]] conducted extensive studies on nanograined/ultrafine-grained (NG/UFG) austenitic steels with nanoindentation to investigate the deformation mechanisms and phase revisions in this type of steels. To increase the accuracy of their studies, Misra et al. [20] considered the effect of loading rate or straining rate on the accuracy of nanoindentation results.
From among all the models reviewed above, Suresh's model, Wang's model and Lee and kwon's model are most reliable ones and can be applied for a wide range of materials with equibiaxial residual stresses. Lee and kwon's model also has the advantage of calculating non-equibiaxial residual stress fields.
The purpose of this study was to investigate the accuracy of the three considered models in estimating residual stresses in austenitic steels and then, to correct the error in the selected model. To this end, numerous two-dimensional (2D) and three-dimensional (3D) FE simulations were performed with equibiaxial and non-equibiaxial stress fields. Then, by studying and formulating the changes of error in measuring residual stress according to the stress ratio, a procedure was suggested to decrease the measurement inaccuracy of residual stresses in austenitic steels.
Section snippets
Theoretical background
The schematic diagram of the load-displacement (P-h) curve for a typical indenter is illustrated in Fig. 1. The most important data extracted from this chart are: dP/dh, the slope of the initial unloading portion of the P-h curve; hc, contact depth (considering pile-up or sink-in); hr, residual plastic displacement under the indenter after full unloading; and hmax and Pmax, the maximum indentation depth and maximum indentation load, respectively.
Fig. 2 depicts two kinds of material responses
Numerical analysis
Due to the large number of cases with various combinations of material properties and residual stresses, it is more economical to use numerical analysis instead of empirical experiments. The commonality of the three considered models is in measuring the equibiaxial stresses. Since in that case, the loading conditions are symmetric, it is more convenient to perform a series of two-dimensional (2D) simulations with equibiaxial stress fields in order to find out which one of the three mentioned
Results and discussions
As mentioned before, the accuracy of three reviewed model was obtained in this study by comparing the calculated residual stresses with the applied residual stresses in the FE simulation for austenitic stainless steels. The error of the calculated residual stress in each case was obtained by Equation (10):
The graphical view of the error in measuring equibiaxial residual stresses by the three mentioned models are presented in Fig. 7a–c. In these figures,
Conclusion
In this paper, three models, i.e. Suresh's model, Lee and Kwon's model, and Wang's model, were reviewed for residual stress calculation through indentation. Since Suresh's and Wang's model are only applicable in equibiaxial residual stress field, it was first assumed that the stress filed was equibiaxial. Using 2D finite-element simulations, it was found out that Wang's and Suresh's models have more than 70% error in residual stress measurement in the considered material, while the Lee and
Declaration of competing interest
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.
References (28)
- et al.
On the determination of residual stress and strain fields by sharp indentation testing.: Part I: theoretical and numerical analysis
Acta Mater
(2001) - et al.
A further study on representative models for calculating the residual stress based on the instrumented indentation technique
Comput Mater Sci
(2014) - et al.
A new method for estimating residual stresses by instrumented sharp indentation
Acta Mater
(1998) - et al.
Measurement of residual-stress effect by nanoindentation on elastically strained (1 0 0) W
Scripta Mater
(2003) - et al.
Estimation of biaxial surface stress by instrumented indentation with sharp indenters
Acta Mater
(2004) - et al.
A contact size-independent approach to the estimation of biaxial residual stresses by Knoop indentation
Mater Des
(2015) Correction of indentation load–depth curve based on elastic deformation of sharp indenter
Mech Mater
(2014)Indentation method to measure the residual stress induced by ion implantation
Nucl Instrum Methods Phys Res Sect B Beam Interact Mater Atoms
(2006)- et al.
On the effect of a general residual stress state on indentation and hardness testing
Acta Mater
(2008) - et al.
Determination of equi-biaxial residual stress and plastic properties in structural steel using instrumented indentation
Mater Sci Eng, A
(2017)