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  • Kang, JiJun (2013) Determination of elastic-plastic and visco-plastic material properties from instrumented indentation curves. PhD thesis, University of Nottingham.

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  • Determination of Elastic-Plastic and Visco-Plastic

    Material Properties from Instrumented

    Indentation Curves

    JiJun Kang

    MEng

    Thesis submitted to The University of Nottingham

    for the degree of Doctor of Philosophy

    July 2013

  • i | P a g e

    For Dad and Mum

  • ii | P a g e

    Abstract

    Instrumented indentation techniques at micro or nano-scales have become more popular for

    determining mechanical properties from small samples of material. These techniques can be

    used not only to obtain and to interpret the hardness of the material but also to provide

    information about the near surface mechanical properties and deformation behaviour of bulk

    solids and/or coating films. In particular, various approaches have been proposed to evaluate

    the elastic-plastic properties of power-law materials from the experimental loading-unloading

    curves. In order to obtain a unique set of elastic-plastic properties, many researchers have

    proposed to use more than one set of loading-unloading curves obtained from different

    indenter geometries.

    A combined Finite Element (FE) analysis and optimisation approach has been developed,

    using three types of indenters (namely, conical, Berkovich and Vickers), for determining the

    elastic-plastic material properties, using one set of ‘simulated’ target FE loading-unloading

    curves and one set of real-life experimental loading-unloading curves. The results obtained

    have demonstrated that excellent convergence can be achieved with the ‘simulated’ target FE

    loading-unloading curve, but less accurate results have been obtained with the real-life

    experimental loading-unloading curve. This combined technique has been extended to

    determine the elastic and visco-plastic material properties using only a single indentation

    ‘simulated’ loading-unloading curve based on a two-layer viscoplasticity model.

    A combined dimensional analysis and optimisation approach has also been developed and

    used to determine the elastic-plastic material properties from loading-unloading curves with

    single and dual indenters. The dimensional functions have been established based on a

    parametric study using FE analyses and the loading and linearised unloading portions of the

    indentation curves. It has been demonstrated that the elastic-plastic material properties cannot

    be uniquely determined by the test curves of a single indenter, but the unique or more

    accurate results can be obtained using the test curves from dual indenters.

    Since the characteristic loading-unloading responses of indenters can be approximated by the

    results of dimensional analysis, a simplified approach has been used to obtain the elastic-

    plastic mechanical properties from loading-unloading curves, using a similar optimisation

  • iii | P a g e

    procedure. It is assumed that the loading-unloading portions of the curves are empirically

    related to some of the material properties, which avoids the need for time consuming FE

    analysis in evaluating the load-deformation relationship in the optimisation process. This

    approach shows that issues of uniqueness may arise when using a single indenter and more

    accurate estimation of material properties with dual indenters can be obtained by reducing the

    bounds of the mechanical parameters.

    This thesis highlights the effects of using various indenter geometries with different face

    angles and tilted angles, which have not been covered previously. The elastic-plastic material

    parameters are estimated, for the first time, in a non-linear optimisation approach, fully

    integrated with FE analysis, using results from a single indentation curve. Furthermore, a

    linear and a power-law fitting scheme to obtain elastic-plastic material properties from

    loading-unloading indentation curves have been introduced based on dimensional analysis,

    since there are no mathematical formulas or functions that fit the unloading curve well. The

    optimisation techniques have been extended to cover time-dependent material properties

    based on a two-layer viscoplasticity model, has not been investigated before.

  • iv | P a g e

    Acknowledgement

    This thesis would not have been possible without the support of my supervisors at the

    University of Nottingham. I am greatly indebted to Professor Adib Becker and Wei Sun for

    their continuous support, guidance and encouragement during my PhD study.

    I would like to thank all my friend and colleagues in the Structural Integrity and Dynamics

    (SID) group, the Department of Mechanical Engineering, The University of Nottingham for

    sharing their experience and knowledge. They have also made my time as a student enjoyable

    and memorable.

    Thanks to my brother, JunHyuk Kang, for encouraging, understanding, patience and

    supporting me during this period of time to finish my Ph.D program.

    Finally yet most importantly, I would like to thank my parents, who unconditionally support

    me financially and mentally, so I could complete this long journey without concern.

  • v | P a g e

    Nomenclature

    Projected area of the hardness impression

    π

    Radius of the circle of contact

    Ratio of the contact radius

    Depth of penetration

    C Independent of initial plastic strain

    Compliance of the loading instrument

    Total compliance

    Compliance of indenter material

    D The diameter of indenter (mm)

    d The diameter of indenter (mm)

    ⁄ The initial slope of the unloading curve

    Reduced modulus

    F( ) Objective function

    Dimensional functions

    h Spherical indenter at any point with radius r from the centre of contact

    Circle of contact

    Final depth of the contact impression after indenter removed

    Maximum displacement of indenter

    The hardness of Meyer’s law

    H’ Power law hardening with work hardening exponent

    Distance between the surface of specimen and the edge of contact at

    full load or

    Contact depth

    Hardness

  • vi | P a g e

    A specific position

    K (Yield coefficient)

    Elastic modulus of the elastic plastic network

    Elastic modulus of the elastic-viscous network

    m Power law index or

    N Total number of points

    n Work-hardening exponent

    Work hardening exponent

    Norton creep parameters

    P Indenter load

    Hertzian pressure distribution

    Mean contact pressure

    Maximum indenter load

    Unloading Force

    The (experimental) force from target data

    The predicted total force

    R Relative radius of the two contacting bodies’ curvature

    Radius of a rigid indenter

    A vector in the n-dimensional space

    S Initial slope of unloading curve or

    ( )

    UTS Ultimate tensile strength

    Total work done

    Work done during unloading

    Y Initial yield stress

    α The angle of indenter

  • vii | P a g e

    β Correction factor 1.034 for a Berkovich indenter and 1.024 for a

    Vickers indenter

    Geometric constant: 0.727 for conical and 0.75 for Berkovich and Vickers

    Y/E ⁄ A representative flow stress

    Stress in the elastic-viscous network

    Initial yield stress

    Stress in the elastic-plastic network

    Initial plasti