Estimation of tensile strengths of metals using spherical indentation test and database

Highlights

• A database containing the tensile strengths and corresponding indentation load-depth curves was established.

• With the assistance the database, the tensile strengths of tested materials can be acquired quickly.

• The accuracy of the database method in estimating strengths are verified with both virtual and experimental materials.

Abstract

The indentation technique provides a non-destructive characterization method for in-service equipment. Recent years have seen an increased interest in determination of tensile properties from spherical indentation test (SIT) through inverse optimization theory, while this approach is computational costing and time consuming. This paper aims to propose a novel method for estimation of strength properties based on the application of a database. Firstly, systematically varied Ludwik constitutive parameter vectors were used to establish a database that contains indentation force-depth curves (extracted from the simulated results of indentation test) and corresponding strength values (calculated with the parameter vectors). The indentation force-depth curve of a concerned material was then processed by the database, as a result of which the yield and ultimate strengths can be obtained. A series of virtual materials were used to validate the accuracy of this method. The results show that the database method provides an impressive accuracy and stability in estimating strengths. The database method was subsequently applied to eight real materials. It is found that the predicted strengths agree well with the values from uniaxial tests for most materials, while the database method may be not suitable for materials that show obvious tension-compression asymmetry.

Introduction

For many in-service pressure vessels and pipes, it is vital to reevaluation their mechanical properties in a non-destructive way after long-time service in harsh conditions [1]. The yield strength and ultimate tensile strength exhibit the loading-bearing ability of materials, and are therefore significant in safety assessment of structures. Traditionally, they are obtained from the results of from uniaxial tensile testing. While for in-service equipment, uniaxial tensile testing is not allowed because it requires specimen preparation with a certain volume of material and thus will destroy the structures. In recent decades, the spherical indentation test (SIT) has become a common technique to investigate the elastic and plastic properties of in-service structures due to its non-destructive nature and convenient operation [[2], [3], [4]].

The relationship between indentation and uniaxial tensile/compression stress field was firstly studied by Tabor [5] and Johnson [6]. Since that, many researchers focused on obtaining the tensile properties from the indentation response of materials. Based on Hertz contact theory, Young's module E can be determined using the initial slope of unloading curve and the contact area with a reliable accuracy by the widely accepted O&P method [7]. However, determination of strength values is still challenging since the stress states of specimens between indentation tests and uniaxial tensile/compression tests are fundamentally different. Traditional approaches which employ representative strains [[8], [9], [10], [11], [12], [13], [14]] or dimensional analysis [[15], [16], [17], [18], [19]] usually require several dimensionless functions and include many coefficients. These coefficients are calibrated by extensive FE simulations. Moreover, they are based on some specific assumptions, e.g. expanding cavity assumption [11], kick's law [15]. For this reason, the use of these approaches to determinate tensile properties are limited.

In addition to above approaches, numerical studies have been performed to examine tensile properties using indentation simulations integrated with optimization algorithms [[20], [21], [22], [23], [24], [25], [26], [27], [28]]. Tensile properties are determined by minimizing an objective function that decribes an average deviation between the experimental and predictive curves. Earlier studies [20,28] often designated the force-depth curve as the input data to define the objective function as it is easy to acquire and sensitive to mechanical properties of materials. The objective function is therefore defined as

$$f(\mathit{x})=\sum _{j=1}^M{(\frac{F_j^{\text{exp}}-F_j^{\text{sim}}(\mathit{x})}{F_j^{\text{exp}}})}^2$$

where $\mathit{x}$ is a vector of the elastoplastic parameters to be optimized, $F_j^{\text{exp}}$ is the force of the experimental curve at the depth of data point j, $F_j^{\text{sim}}(\mathit{x})$ is the force of predictive curve at the same depth, and M is the total number of force-depth points that are acquired by linear interpolation and evenly spaced in the indentation curve. A critical issue in the optimization theory is the uniqueness of the solution. This situation is particularly common for a sharp indenter showing geometrically self-similarity. Researches from Dao and Cao et al. [15,29] show that a sharp indenter with a fixed semi-vertical angle can only determine a single representative strain. Therefore, a sharp indentation test provides a unique solution of Young's module and representative stress, but cannot uniquely determine strain hardening exponent. While for a spherical indenter, different ratio of indentation depths to the diameter of indenter are related to different representative strain, which can better improve the uniqueness issues. Further studies [30] indicates that if the h_max/D is larger than 0.06, the uniqueness issues problem can be resolved. Another issues presented is the convergence of the algorithms. To date, various algorithms has been used to find the best fit between the experimental and predictive curve, such as genetic algorithm (GA) [23], swarm particle optimization (SPO) [24]. All the algorithms are reported to converge well with accurate results. However, the drawbacks are also evident. Firstly, for most optimization procedures, the convergent speed is strongly related to the initial guess of the parameters vector x and the selection of the algorithm. Both an improper initial guess of the parameters and an ill-posed algorithm will increase the uncertainty of the convergence. Moreover, the iterative FE simulations are required in every single calculation which yields a considerable computational cost.

In this paper, a database method is proposed to determine the yield and ultimate strengths from a spherical indentation force-depth curve. This method is able to solve the issues of computational cost and instability in optimization methods. Ludwik constitutive equation was used to describe the hardening behaviors of materials. Then systematically varied parameter combinations were used to establish a database that contain indentation force-depth curves (extracted from the simulated results of indentation test) and corresponding strength values (calculated with the parameter combinations). By analyzing the discrepancies of input curve and that from the database, the strength values can be acquired. This work is organized as following. In Section 2, procedures for determination of strength values using the database method was introduced. Section 3 presents the validation strategy using numerical experiments. The potentialities and limitations of the database method for real materials are investigated in Section 4. Section 5 gives conclusions of this work.