· Home · Table of Contents · Fundamental & Applied Research | ## Stress-strain relationship evaluated by load-depth curve obtained from indentation techniqueM. Futakawa, I. IokaJapan Atomic Energy Research Institute Tokai-mura, Naka-gun, Ibaraki-ken, 319-1195, Japan T. Wakui, Y Tanabe University of Niigata 8050, Ikarashi-Ninocho, 275-8588, Japan Niigata-shi, 950-2181, Japan Contact |

It is difficult for conventional hardness measuring to get quantitative information in terms of the mechanical properties to describe material deformation. Indentation technique being measurable on load-depth curve is very promising technology for the evaluation of mechanical properties, in particular for micro- and nano-meter zone of materials. The novel technology used the indentation technique combined with numerical calculation is developed in order to quantitatively evaluate the mechanical properties. Inverse analyses on the loading and unloading curves measured by the instrumented indentation machine with a hemispherical indenter were carried out using Kalman filter.

This technique was applied to evaluate mechanical properties on two kinds of materials; aluminum alloy A5056 and nickel based alloy Inconel 600. The constitutive equations with the estimated material constants were used to the simulation for the uniaxial tensile tests on them. The simulated engineering stress-strain curves were compared with the experimental results. As a result, it was confirmed that the stress-strain relationship evaluated by the presented technique could describe appropriately the deformation under uniaxial tensile loading. The technique is very promising as small specimen tests to study the properties of irradiated materials because of the available irradiation volume.

Hardness has not been sufficiently interpreted from the viewpoint of physical meaning yet, although it is broadly used to qualitatively evaluate the change of mechanical properties as a sort of non-distractive technique. For example, the embrittleness by neutron irradiation was examined systematically for various kinds of steel materials. The identification of material properties had been tried from the indentation depth profile and pill-up profile around indents. Indentation technique with a depth sensing measuring system is very promising and convenient for evaluating mechanical properties of materials at micro- or nano-meter scale level: thin surface layers, coatings, ion implanted layer, corroded surface layer, irradiated materials, etc. However, the stress distribution under the indenter becomes too complicated to be directly related with the stress-strain relationship that we can evaluate from uniaxial tensile tests, because of 3-dimensional stress distribution under the indenter. In particular, a lot of efforts have been made so far to examine the relationship between the indentation load vs depth curve behavior and the various fundamental mechanical properties of very thin surface area and/or films.

Generally, the stress analyses using finite element method (FEM) get to be mainstream as the stress and/or deformation evaluation for structural components. The constitutive equations applicable to FEM are useful from the viewpoint of engineering. Some analyses using FEM, which can treat elastic and plastic deformations, have been carried out to examine effects of interaction between the thin film and the substrate on the load/depth curve. Bhattacharya, et al. have developed semi-analytically functional equations of the thin film on the substrate through systematic FEM simulations to predict the hardness variation with depth under various conditions and to separate the elastic properties of the film from those of the substrate [1]. Likewise, the present authors have applied the depth-sensing indentation technique to investigate the degradation of mechanical properties of corroded surface layers of some ceramic materials [2]. Inverse analyses using FEM on the load/depth curve have been carried out to quantitatively evaluate the variation of mechanical properties of the corroded surface layers. Giannakopoulos, et al. investigated in detail the influence of mechanical properties, Young's modulus, yield stress, and hardening coefficient, on the deformation of materials under a Vickers indenter by using FEM analyses [3]. Field and Swain focused on the behavior of pilling up and sinking in to characterize mechanical properties [4]. Based on the FEM results, Cheng, *et al*. addressed the limitation for the determination of the stress-strain relationship from the load-depth curve measured using the depth sensing indentation technique with conical and pyramidal indenters [5].

In this paper, as taking account of the above background on the indentation technique with a depth sensing measuring system, the authors present the novel technique used the indentation technique combined with numerical calculation to identify the material constants describable the stress-strain relationship. Inverse analyses on the loading and unloading curves measured by the instrumented indentation machine with the hemispherical indenter were carried out using Kalman filter. The technique was applied to aluminum alloy A5056 and nickel based alloy Inconel 600. The simulation with the estimated material constants was carried out to compare with the deformation under uniaxial tensile loading.

**C**,_{0}**E**,_{0}**R**and S_{max}are input as initial values.**Z**of L and dL/dD at step S is input._{s}**Y**of L and dL/dD at step S is calculated by FEM code._{s}- The estimated values on
**C**and**E**are given by the following equation of Kalman
filter using each value, - Go back to (2) and repeat the process up to S
_{max}. Finally we can obtain the optimal values**C**_{Smax}estimated at final step S_{max}.

**FEM model**

The inverse analysis was carried out using an explicit FEM code, LS-DYNA[6], which enables us to robustly analyze a large deformation accompanying with contacting behavior. In the analysis, the indenter and specimen were treated as axisymmetric two-dimensional bodies to take calculative efficiency into account, as illustrated in Fig.1. The shapes of indenter apexes were varied to examine the shape effect on the estimated material constants: conical and hemispherical ones. The modeled indenters were perfectly rigid. The contact interface between the specimen and the indenter was assumed to be frictionless, because the frictional force induced with friction coefficient up to 0.3 had few effects on the load-depth relationship. The mesh size is given to be sufficiently fine to keep accuracy: the minimum element size around the apex contacting zone was 0.05mm. The total number of the elements used in the model was 1509. The loading rate in the calculation was small enough to neglect an inertia effect as a static condition.

Fig 1: FEM modal |

**Identification with inverse analysis**

The constitutive equation of the material installed into the model was assumed to be a simple power-law which is generally believed to be applicable to normal metallic materials as follows:

s = Ee s £ s_{y} | (1) |

s = A(e_{0}+e)^{n}
| (2) |

e_{0} = (s / _{y}A)^{1/n} - (s_{y} / E )
s >s_{y}
| (3) |

where, s is true stress, e true strain, E Young's modulus, s_{y} yield stress, A work hardening coefficient and n work hardening exponent. Hereafter, we have to identify the following material constants; E, s_{y}, A and n through the inverse analysis on the load-depth, L-D, curve.

Fig 2: Flow chart for identification of material constants . |

The flow chart of the identification with inverse analysis is illustrated in Fig. 2. Here, **C** and **E** are determinants of material constants and estimated errors of material constants. **Z** and **Y** are determinants of experimental and calculated values on L and dL/dD. **R** is determinant of error in measuring systems. S_{max} is the maximum number of steps in the divided L-D curve. **H** is d**Y**/d**C** The procedure of the inverse analysis is as follows.

C = _{s}C + _{s-1}E_{s}H_{s}R^{-1}(Z - _{s}Y
_{s}) | (4) |

E = _{s}(E_{s-1}^{-1} + H_{s}^{T}R^{-1}H_{s})^{-1}
| (5) |

**Indentation test**

The indentation tests were carried out on the polished surfaces of specimens of A5056 and Inconel 600 at room temperature using the conical indenter that has a hemispherical apex with radius of 1.2 mm. A testing machine, DUH-201 (Shimazu Co, Japan), was used for it. During loading and unloading, the load and indent depth were continuously measured with a resolution of 19.6 mN and 1 nm, respectively. A load was imposed through the indenter on the specimen's surface with loading rate of 2.7 mN/s, held for 1 s, and then removed. The maximum loads, 60 mN for A5056 and 150 mN for Inconel 600, were chosen not to be affected by the size effect manly due to the roughness of the indenter apex and the roughness of the specimen surface, as taking both the indentation depth and the radius of the hemispherical apex into account.

**Uniaxial tensile test**

The uniaxial tensile tests were carried out at room temperature to verify the estimated material constants. The specimen was made of aluminum alloy A5056 and nickel based alloy Inconel 600, whose dimensions are illustrated in Fig. 3. The elongation of the specimen was measured at the crosshead. The tensile loads were applied at the crosshead speeds of 0.2 mm/min for A5056 and 2.25 mm/min for Inconel 600, respectively.

Fig 3: Dimensions of specimens for uniaxial tensile test . |

**Effect of indenter shape
**

In order to investigate the effects of the shape of indenter on the estimated material constants, the numerical simulations were carried out systematically by using the FEM model illustrated in Fig. 2. Figure 4 shows that the stress-strain curves used in the simulation and the L-D curves obtained by the conical and the hemispherical indenters. The simulated materials have the material constants which are hardly distinguished through the L-D curve using the conical indenter among them as reported by Cheng, et al. [5], as seen in Fig. 4(b).

However, the L-D curves obtained by the hemispherical indenter reflect sufficiently the differences of their material constants. Figure 5 shows then plastic zone profiles developed under conical and hemispherical indenters. The plastic zone is described with the normalized z an r coordinates by the indentation depth h_{I}. The plastic zone develops analogously in the case of the conical indenter but increases unanalogously with the depth in the case of the hemispherical indenter. So that, the different stress distribution is formed with the indentation depth under the hemispherical indenter. The accuracies of the material constants estimated by using the conical and hemispherical indenters are shown in Fig. 6. The material is assumed to have s_{y} of 200 MPa, A of 1400 MPa, n of 0.4 and E of 200 GPa in Eqs (1) to (3). Poisson's ration is 0.3. In the inverse analysis the initial values were ranged from 100 to 300 MPa in s_{y}, from 1200 to 1600 MPa in A and from 0.3 to 0.5 in n. It is seen in Fig.6 that the accuracy of the material constants estimated by the hemispherical indenter is higher than that done by the conical one. It was confirmed from these results that the hemispherical indenter is much more suitable for the inverse analysis to identify the material constants than the conical indenter.

Fig 4: Simulation on the load and depth curves. |

Fig 5: Plastic zone profiles developed under conical and hemispherical indenters. |

Fig 6: Comparison of accuracies of estimated material constants between conical and hemispherical indenters . |

**Uniaxial tensile test
**

Figure 7 shows the L-D curves measured using the spherical indenter for the A5051 and Inconel 600, and the calculated results using material constants in Eqs (1)-(3) identified according to the presented method with the inverse analysis on the L-D curve. Regardless of the materials, the calculated L-D curves agree with the experimental ones well. The load and elongation curves were measured from the uniaxial tensile test using plate and rod specimens shown in Fig. 3 and were compared with the calculated results using the FEM model shown in Fig. 8 which has the estimated material constants. Figure 9 shows the load and elongation curves up to the onset of necking behavior. The calculated results shows give a good agreement with the experimental one. As a result, it can be said that the presented method is useful to identify the material constants.

Material |
E , GPa |
s |
A, MPa |
n |

A5051 |
73 |
77 |
322 |
0.126 |

Inconel 600 |
170 |
263 |
1425 |
0.293 |

Table 1: Material constants estimated by the inverse analyses on L-D curves of A5051 and Inconel 600. |

Fig 7: Comparison of L-D curves between experimental and calculated results.. |

Fig 8: FEM models for uniaxial tensile tests. |

Fig 9: Comparison of load-elongation curves between experimental and calculated results. |

Indentation technique being measurable on load-depth curve is very promising technology for the evaluation of mechanical properties, in particular for micro- and nano-meter scale zone of materials, including very thin layers. The novel technology used the indentation technique combined with numerical calculation is developed in order to quantitatively evaluate the mechanical properties. The inverse analyses with Kalman filter were carried out on the loading and unloading curves measured by the instrumented indentation machine with the hemispherical indenter. It was confirmed through the comparison between the experimental and the calculated results on the uniaxial tensile deformations of A5051 and Inconel 600 that the technique is useful to identify the material constants. The technique will be applicable to not only the quantitative evaluation of material constants in micro and nano-meter scale zones including very thin surface layer, i.e. corroded layer, ion implanted layer, but also the small specimen testing technique for irradiated materials.

- A. K. Bhattacharya and W.D. Nix, Analysis of elastic and plastic deformation associated with indentation testing of thin films on substrates, Int. J. Solids Structures, Vol. 24, No.12, pp.1287-1298, 1988.
- M. Futakawa, T. Wakui, I. Ioka and M. Eto, Mechanical-property evaluation of thin corroded surface layer of ceramic materials by microindentation technique, J. Euro. Ceram. Soci.,20,1135(2000)
- A.E. Giannakopoulos, P.L. Larsson and R. Vestergaard, Analysis of Vickers indentation, Int. J. Solids Structures, Vol. 31, No. 19, pp.2679-2708, 1994.
- J. S. Field and M. V. Swain, Determining the mechanical properties of small volumes of material from submicrometer spherical indentations, J. Mater. Res., Vol.10, No.1, pp.101-112, 1995.
- Y. Cheng and C. Cheng, Can stress-strain relationships be obtained from indentation curves using conical and pyramidal indenters ?, J. Mater. Res., Vol.14, No.9, pp.3493-3496, 1999.
- LS-DYNA USER' MANUAL, Livermore Software Tech. Co., 1995.

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