Indentation Fatigue Mechanics

Abstract

Instrumented indentation has been widely used in the determination of mechanical properties of materials due to its fast, simple, precise, and nondestructive merits over the past few years. In this chapter, we will present an emerging indentation technique, referred to as indentation fatigue, where a fatigue load is applied on a sample via a flat punch indenter, and establish the framework of mechanics of indentation fatigue to extract fatigue properties of materials. Through extensive experimental, theoretical, and computational investigations, we demonstrate a similarity between the indentation fatigue depth propagation and the fatigue crack growth, and propose an indentation fatigue depth propagation law and indentation fatigue strength law to describe indentation fatigue-induced deformation and failure of materials, respectively. This study provides an alternative approach for determining fatigue properties, as well as for studying the fatigue mechanisms of materials, especially for materials that are not available or feasible for conventional fatigue tests.

Introduction

Fatigue is a process of accumulations of material deformation and damage due to a repeated loading and unloading and is considered as one of major threats to the mechanical integrity of materials and structures. As a technical problem, the study of fatigue can be traced to as early as the mid-nineteenth century with the data collection of failure of railroad system by Germany technologist A Wohler (Suresh 1998). Since then, it has attracted tremendous attention over the last century, and great achievements have been made from macroscaled phenomena to micro-/nanoscaled deformation mechanisms (Estrin and Vinogradov 2010; Connolley et al. 2005), from qualitative descriptions to quantitative predictions (Rao and Farris 2008; Fleck and Smith 1984), and from engineering metals to biological and soft materials (Teoh 2000; Dirks et al. 2013; Tang et al.). Meanwhile, with the ever-growing applications of small structures, including nano-/micro-electromechanical systems (N/MEMS) and thin films over the last decades and emerging low-dimensional nanomaterials such as nanowires, nanofibers, and graphene, their properties usually exhibit differences than their materials counterpart at the macroscale (Alsem et al. 2007, 2008; Höppel et al. 2009; Li et al. 2003, 2014; Luo et al. 2015; Lee et al. 2008; Zhu et al. 2007), and the characterization of their mechanical properties in particular fatigue properties is challenging the traditional uniaxial mechanical testing technique that is mainly based on standard “dog-bone” shaped specimens. The development of an alternative testing technique that could help to address these challenges is highly desired.

Indentation, which origins from the hardness measurement, provides a compelling solution to measure mechanical properties of materials down to the nanosize (Cheng and Cheng 2004; Fischer-Cripps 2000). The merits of indentation include minor requirements of testing materials, easy to conduct, and high resolutions of measurement. During the experiment, the indentation load and displacement can be recorded continuously with the perpetration of the indenter to samples, and even deformation of materials can also be observed when the testing platform is integrated with advanced observational facilities such as transmission electron microscopy (TEM) and scanning electron microscope (SEM), usually referred to as in situ TEM-indentation (Warren et al. 2007) and SEM-indentation (Nowak et al. 2010), respectively. Significant efforts have been made to extract mechanical properties of materials from indentation data including elastic properties (Oliver and Pharr 2004; Yang et al. 2016; Li and Bhushan 2002; Dao et al. 2001; Lan and Venkatesh 2007; Jiang et al. 2009), plastic properties (Cheng and Cheng 1998; Chen et al. 2007; Xu and Chen 2010; Lee et al. 2010; Bucaille et al. 2003), fracture properties (Lawn and Wilshaw 1975; Tang et al. 2008; Xia et al. 2004; Sakaguchi et al. 1992; Miranzo and Moya 1984; Quinn and Bradt 2007), creep properties (Xu et al. 2008a; Cheng and Cheng 2001; Li et al. 1991; Yang and Li 1995; Chen et al. 2010; Stone et al. 2010), and relaxation properties (Xu et al. 2008b; Baoxing et al. 2010; Chu and Li 1977, 1980a; Hu et al. 2011; Chan et al. 2012) with a broad range of materials from hard metals, to composites, to biological materials, to soft matters. Usually, these properties can be determined from the measured indentation load and displacement curves and deformation profiles of indented materials at monotonic loading conditions. However, fatigue properties such as fatigue strength and fatigue-induced crack growth that rely on fatigue/cyclic loading conditions usually cannot be measured.

Indentation that is conducted under a cyclic loading with cyclic numbers >1000 is referred to indentation fatigue here with an emphasis on measurement of fatigue properties of materials. An early study by Li and Chu found that the flat punch indenter could continue to penetrate into the β-tin single crystal with applied cycles, and the recorded curves of indentation displacement-cyclic numbers were similar to that of fatigue crack propagation with the absence of third stage (Li and Chu 1979). With the same β-tin single crystal, later on, Chu and Li changed the fatigue conditions to fatigue spectrum with interruption by a peak loading and further found the similarity with that of fatigue crack propagation with a delayed retardation of indentation depth propagation caused by a peak loading (Chu and Li 1980b). By employing a Vickers diamond indenter, Kaszynski et al. investigated the indentation fatigue response of 316 L stainless steel and found that the indentation depth showed an approximately linear increase with the logarithm of the number of cycles (Kaszynski et al. 1998). This difference is believed to be led by the geometric shape of indenters and associated change of contact areas between indenter and materials in experiments (Li 2002; Yang and Li 2013). The similarities between indentation fatigue and fatigue crack propagation provide a critical hint to probe the fatigue properties of materials through instrumented indentation technique (Yang and Li 2013). However, why do such similarities exist? What is the deformation mechanism of the evolution of indentation depth propagation under a fatigue loading? What fatigue properties can be determined and how? How to quantitatively describe the depth propagation of indentation fatigue? What is the quantitative relationship between the depth propagation of indentation fatigue and fatigue crack propagation of conventional fatigue test? Over the last 10 years, we have been actively working on the study of indentation fatigue by integrating tools of experiments, theories, and computations to address these questions (Xu et al. 2010, 2007a, b, 2009; Xu and Yue 2006, 2007).

In this book chapter, we will summarize several important results based on our progresses and present quantitative mechanics descriptions and validations of indentation fatigue from indentation fatigue depth propagation to indentation fatigue strength of materials. In section “Mechanics Theory of Indentation Fatigue,” an elastic indentation fatigue mechanics model will be first developed, and its reduction to and comparison with indentation mechanical model under a monotonic loading will be discussed. And then we will show theoretical analysis on a similarity of stress field between the rim of contact area between flat punch indenter and surface of a sample and crack tips. In section “Indentation Fatigue Deformation,” extensive experimental results on polycrystalline copper and finite element analysis will be performed to demonstrate similarities between indentation fatigue depth propagation and fatigue crack growth, and an indentation fatigue depth propagation law will be developed. In section “Indentation Fatigue Damage,” we will extend the indentation fatigue mechanics to indentation fatigue damage with a focus on indentation fatigue-induced failure of materials and predict the fatigue strength of materials from indentation fatigue. Concluding remarks will be given in section “Concluding Remarks.”

Mechanics Theory of Indentation Fatigue

Indentation Load-Depth Curve

Consider a semi-infinite homogeneous elastic solid indented by a rigid frictionless flat punch, as schematized in Fig. 1, when the indentation load P is applied to the indenter, the penetration of the indenter (i.e., indentation depth) to the solid is h, and based on Hertz contact theory, it can be written as

$$ h=\frac{\left(1-{\nu}^2\right)}{Ed}P $$

(1)

where d is the diameter of the indenter, and E and v are Young’s modulus and Poisson’s ratio of the solid, respectively. Equation 1 is also referred to as Sneddon’s solution and serves the fundamental of determining Young’s modulus of materials from a flat punch indentation (Sneddon 1965). Besides, because of the constant diameter of the punch indenter, Eq. 1 suggests a linear variation of indentation depth with indentation load at a small deformation, indicating one of the benefits of flat punch indenter in indentation technique in comparison with other indenter shapes such as sharp indenter, Vicker indenter, or spherical indenter that will result in increased contact area with the increase of the indentation depth. Obviously, when the indentation load P increases or decreases, Eq. 1 holds instantaneously for a semi-infinite homogeneous elastic solid, and thus it can be used for both loading and unloading conditions. Assume the indentation load P is applied with a sinusoidal loading spectrum, we will have

$$ P(t)={P}_m+\Delta P\sin \left(\omega t\right) $$

(2)

where ΔP, P_m, and ω are the indentation load range, mean, and frequency, respectively. With Eqs. 1 and 2, when a sinusoidal loading is applied to a semi-infinite homogeneous elastic solid via a flat punch indenter, the indentation load-depth relationship will be

$$ h(t)=\frac{\left(1-{\nu}^2\right)}{Ed}\left({P}_m+\Delta P\sin \left(\omega t\right)\right) $$

(3)

Fig. 1

Schematic illustration of indentation fatigue on a semi-infinite solid via a flat punch indenter subjected to a fatigue loading P. d is the diameter of flat punch indenter, h is the indentation depth. The inset illustrates the cyclic fatigue loading

That is

$$ h(t)=\frac{\left(1-{\nu}^2\right)}{Ed}{P}_m\left(1+\frac{\Delta P}{P_m}\sin \left(\omega t\right)\right) $$

(4)

Apparently, the variation of indentation depth with experimental time is sinusoidal. Assume

$$ h(t)={h}_m+\Delta h\sin \left(\omega t\right) $$

(5)

where h_m and Δh are the mean and magnitude of the indentation depth. Thus, Eq. 4 can be rewritten as

$$ {h}_m=\frac{\left(1-{\nu}^2\right)}{Ed}{P}_m $$

(6)

$$ \Delta h=\frac{\left(1-{\nu}^2\right)}{Ed}\Delta P $$

(7)

Equation 6 is similar to Eq. 1 and indicates the indentation depth is induced by the mean load of the sinusoidal loading. Meanwhile, Eq. 7 indicates the dynamic response of indentation under the sinusoidal loading and can be written as

$$ \frac{E}{1-{\nu}^2}=\frac{1}{d}\frac{\Delta P}{\Delta h} $$

(8)

E/(1 − ν²)is usually referred to as the stiffness of materials. Apparently one can easily probe the variation of stiffness of materials by applying a dynamic loading. Equation 8 has served the theoretical model in the measurement of dynamic contact stiffness of materials and has been employed in the continuous stiffness measurement (CSM) technique in the instrumented indentation technique (Li and Bhushan 2002; Asif et al. 1999; Loubet et al. 2000).

Indentation Stress Intensity Factor

Consider the solid in Fig. 2a, the boundary condition of stress field beneath the flat punch indenter is (Sneddon 1965; Johnson 1985)

$$ {\sigma}_{zz}\left(x,0\right)=0,\, \mid x\mid >d/2 $$

(9a)

$$ {\tau}_{xz}\left(x,0\right)=0,{\displaystyle \begin{array}{cc}& \end{array}}0\le \mid x\mid \le d/2 $$

(9b)

Fig. 2

(a) Schematic representation of contact between the flat punch indenter and a semi-infinite solid. (b) Circumferentially cracked notch specimen subjected to far-field compressive load Xu et al. 2009 (reprinted with the permission from Elsevier)

In the cylindrical coordinate systemr, θ, z, the stress field can be written as (Sneddon 1965; Johnson 1985)

$$ \left\{\begin{array}{l}{\sigma}_{zz}=\frac{Eh}{2\left(1-{\nu}^2\right)}\underset{0}{\overset{\infty }{\int }}\left(1+ mz\right){e}^{- mz}{J}_o(mr)\sin \left(md/2\right) dm\\ {}{\sigma}_{rr}=\frac{Eh}{2\left(1-{\nu}^2\right)}\left[\underset{0}{\overset{\infty }{\int }}\left(1- mz\right){e}^{- mz}{J}_o(mr)\sin \left(md/2\right) dm\right]\\ \qquad \left. -\frac{1}{r}\underset{0}{\overset{\infty }{\int }}\frac{1}{m}\left(1-2\nu - mz\right){e}^{- mz}{J}_1(mr)\sin \left(md/2\right) dm\right]\\ {}{\sigma}_{\theta \theta}=\frac{Eh}{2\left(1-{\nu}^2\right)}\left[\underset{0}{\overset{\infty }{\int }}2\nu {m}^2{e}^{- mz}{J}_o(mr)\sin \left(md/2\right) dm\right]\\ \qquad\left.+\frac{1}{r}\underset{0}{\overset{\infty }{\int }}\frac{1}{m}\left(1-2\nu - mz\right){e}^{- mz}{J}_1(mr)\sin \left(md/2\right) dm\right]\\ {}{\sigma}_{zr}=\frac{Eh}{2\left(1-{\nu}^2\right)}\underset{0}{\overset{\infty }{\int }} mz{e}^{- mz}{J}_1(mr)\sin (mz) dm\end{array}\right. $$

(10)

where m is the variable function. J_α is the Bessel function of the first kind and is

$$ {J}_{\alpha }(x)=\sum \limits_{m=0}^{\infty}\frac{{\left(-1\right)}^m}{m!\Gamma \left(m+\alpha +1\right)}{\left(\frac{x}{2}\right)}^{2m+\alpha } $$

Within the contact range at z = 0, one can have

$$ {\left.{\sigma}_{zz}\right|}_{z=0}=\frac{Et_0}{\pi \left(1-{\nu}^2\right)}\underset{0}{\overset{\infty }{\int }}{J}_o(mr)\sin \left(md/2\right) dm $$

(11)

Because $\underset{0}{\overset{\infty }{\int }}{J}_o(mr)\sin \left(md/2\right) dm=\frac{1}{\sqrt{{\left(d/2\right)}^2-{r}^2}},\, r\le d/2$, one will have

$$ \begin{array}{l} {\left.{\sigma}_{zz}\right|}_{z=0}=\frac{Et_0}{\pi \left(1-{\nu}^2\right)}\underset{0}{\overset{\infty }{\int }}{J}_o(mr)\sin \left(md/2\right) dm\\ \qquad\quad =\frac{Et_0}{\pi \left(1-{\nu}^2\right)}\frac{1}{\sqrt{{\left(d/2\right)}^2-{r}^2}}\quad\, r\le d/2 \end{array} $$

(12)

That is

$$ {\sigma}_{zz}\left(r,0\right)=\frac{Eh}{\pi \left(1-{\nu}^2\right)}\frac{1}{\sqrt{{\left(d/2\right)}^2-{r}^2}},\, r\le d/2 $$

(13)

At the rim of indentation, ∣x∣ is approaching to d/2, one will have

$$ {\sigma}_{zz}\left(r,0\right)=\frac{Eh}{\pi \left(1-{\nu}^2\right)}\left(\frac{1}{\sqrt{dr}}+\cdots \right),\quad r\to d/2 $$

(14)

Equation 14 indicates there is a stress singularity at the edge of indentation with respect to $\sqrt{r}$.

Similarly, for a semi-infinite model-I crack subjected to a far-field compressive load P, as schematized in Fig. 2b, given the distanced/2from the origin of the coordinate system to the crack tip, the stress field at the crack tip is (Hertzberg 1995)

$$ \left(\begin{array}{c}{\sigma}_{zz}\\ {}\begin{array}{l}{\sigma}_{rz}\\ {}{\sigma}_{rr}\end{array}\\ {}{\sigma}_{\varphi \varphi}\end{array}\right)\to -\frac{2P}{\pi {d}^2\sqrt{s}}\cos \frac{\varphi }{2}\left(\begin{array}{ccc}1+\sin \dfrac{\varphi }{2}\sin \dfrac{3\varphi }{2}\\ {}\sin \dfrac{\varphi }{2}\cos \dfrac{3\varphi }{2}\\[8pt] {}1-\sin \dfrac{\varphi }{2}\sin \dfrac{3\varphi }{2}\\[8pt] {}2\nu \end{array}\right) $$

(15)

Comparison between Eqs. 14 and 15 implies that σ_zz in Eq. 15 is an asymptote of Eq. 13 at r → d/2, leading to an equivalence between Eqs. 14 and 15, and thus suggesting the same stress singularity with respect to √ at the rim of flat punch indentation and crack tip.

Given Eq. 1, Eq. 13 can be rewritten as

$$ {\sigma}_{zz}\left(r,0\right)=-\frac{4P}{\pi {d}^2},\, r\le d/2 $$

(16)

According to the definition of stress intensity factor, we will have

$$ {\sigma}_{zz}=-\frac{K_I}{\sqrt{2\pi \left(d/2-r\right)}} $$

(17)

And thus the stress intensity factor near the contact rim of flat punch and solid can be expressed as

$$ {K}_I=\frac{F}{d\sqrt{\pi d/2}} $$

(18)

And Eq. 18 indicates the same stress intensity factor near the contact rim of flat punch and solid with that at the crack tip.

Under a fatigue loading condition, the stress intensity factor can be obtained similar to that of Eq. 2, and is

$$ {K}_{\mathrm{max}}=\frac{P_{\mathrm{max}}}{d\sqrt{\pi d/2}},\, \Delta K=\frac{\Delta P}{d\sqrt{\pi d/2}} $$

(19)

where P_max = ΔP/2 + P_m is the maximum indentation load.

Indentation Fatigue Deformation

Indentation Fatigue Depth Propagation Law

When a solid (engineering metals otherwise specified) is subjected to a fatigue loading, crack will initiate and propagate with the increase of cyclic number. The steady state (i.e., the 2nd stage) of the fatigue crack growth (da/dN)_s can be described by the well-known Paris equation (Ritchie 1977)

$$ {\left(\mathrm{d}a/\mathrm{d}N\right)}_s=C{\left(\Delta K\right)}^q $$

(20)

where ΔK = K_max − K_min is the nominal stress intensity factor range, and K_max and K_min are the maximum and minimum stress intensity factors, respectively. To highlight the competition between the intrinsic mechanism of crack tip growth and extrinsic mechanism of crack-tip shielding behind the tip, Eq. 20 can be modified to (Liu and Chen 1991; Dauskardt et al. 1992)

$$ {\left(\mathrm{d}a/\mathrm{d}N\right)}_s=C{\left({K}_{\mathrm{max}}\right)}^n{\left(\Delta K\right)}^m $$

(21)

where ΔK and K_max describe the intrinsic and extrinsic mechanisms of fatigue crack growth, respectively, and their dominance is reflected by the power indices m and n. m, n, and C are empirical constants and depend on material and microstructure, fatigue frequency, loading mode and environment, etc.

When an elastoplastic solid is subjected to a fatigue loading via a flat punch indenter, given the similarity of stress singularity in theory in section “Indentation Stress Intensity Factor” between the crack tip and the rim of indentation, in particular, the same stress intensity factor near the rim of indentation and crack tip, and inspired by Eq. 21, the indentation fatigue depth propagation at the steady state can be described quantitatively by using a power law

$$ {\left(\mathrm{d}h/\mathrm{d}N\right)}_s={C}_i{K_{\mathrm{max}}}^{n_i}\Delta {K}^{m_i} $$

(22)

where ΔK and K_max are obtained from Eq. 19. C_i, n_i, and m_i are constants of indentation fatigue and depend on material/microstructure and testing environments. Similar to the fatigue crack growth, the continuous sinking of the indenter into the solid suggests the accumulation of plastic deformation beneath the indenter and is driven by the stress concentration near the rim of contact between the indenter and solid surface. It reflects the intrinsic mechanism of indentation depth propagation with the cyclic number and is determined byΔK. In contrast, K_max indicates the elastic recovery during the indentation depth propagation and represents the extrinsic mechanism. We will validate Eq. 22 through both computations and experiments in the following.

Computational Validation

Computational Method and Modeling

Modeling plastic behavior of materials under a fatigue loading has received considerable attention, and it requires the including of the Bauschinger effect, elastic shakedown, cyclic hardening or softening, ratcheting, and mean stress relaxation. A significant number of models have been developed over the last decades such as developed by Chaboche and Nouailhas (1989a, b), Ohno and Wang (1995), and Delobelle et al. (1995), and most of these models are based on the kinematic hardening model of Armstrong and Frederick (A-F model) (Jiang and Kurath 1996). A-F model has also been employed in the study of cyclic spherical indentation behavior by Huber and Tsakmakis (Moosbrugger and Morrison 1997; Abdel-Karim and Ohno 2000), but limited to a few number of cycles. In our simulations, two different polycrystalline copper alloys whose elastoplastic property proves to obey the classical kinematic hardening rule (namely the A-F model) will also be employed. The mechanical properties for a brittle copper alloy are 122.5 GPa for Young’s modulus, 0.35 for Poisson’s ratio, 33.32 MPa for initial yield stress, and 1.607 GPa for a linear hardening rate; the mechanical properties for a ductile copper alloy are 119.9 GPa for Young’s modulus, 0.35 for Poisson’s ratio, 73.50 MPa for initial yield stress, and 0.369 GPa for a linear hardening rate (Mclean 1965). We note that these two copper alloys can be distinguished based on measurements from the conventional uniaxial fatigue cracking test. For example, for fatigue cracking behavior of the brittle copper alloy, the index n is higher than m in Eq. 21, and the index n is less than m in Eq. 21 for fatigue cracking behavior of the ductile copper alloy. The plat punch indenter with radius d/2 = 1.0 mm is assumed to be rigid and frictionless. Figure 3a illustrates the history of fatigue loading with a sinusoidal manner. The application of a preloading to the mean fatigue load will help to minimize the effect of initial loadings. The simulations were conducted by the finite element software ABAQUS.

Fig. 3

(a) Schematic of the load history used in the numerical simulation of indentation fatigue. (b) The typical indentation load (P)-depth (h) curve. (c) The hysteresis loop of indentation load (P)-depth (h) curves under fatigue loading cycles 5th, 10th, 15th, 25th, 35th, and 50th. (d) Indentation depth (h)-time (t) curves. ΔP = 100 MPa and P_m = 100 MPa Xu et al. 2009 (reprinted with the permission from Cambridge University Press)

Results and Discussion

For the brittle copper as a representative, under ΔP=100 MPa and P_m = 100 MPa, the indentation load (P)-depth (h) curve is shown in Fig. 3b, and three stages are observed in response to the loading history in Fig. 3a. Stage II corresponds to the cyclic response of indentation to the fatigue loading, and is our focus. At the beginning of this stage, the indentation depth propagation rate (per cycle) is quite high, and then it decays with the increase of number of cyclic loading till to approximately reaching a constant. Overall, the indentation depth keeps increasing upon cyclic loading and shows a clear difference from the static elastoplastic indentation response. Besides, the propagation of indentation depth shows the same sinusoidal frequency as that of the applied indentation load. More importantly, there is a hysteresis loop in the indentation fatigue P-h curve at each cyclic loading, and several of them are magnified in Fig. 3c. As the number of cycles increases, the hysteresis loop tends to be more “closed” in comparison with the early ones, leading to a smaller enclosed area. Further, each hysteresis loop can be decomposed into two parts: the cyclic elastic indentation depth range ${h}_c^e$ and the cyclic plastic indentation depth range ${h}_c^p$. The former suggests the recovery of indentation depth during indentation fatigue, and the latter is associated with the continuous increase of indentation depth due to the propagation of plastic zone beneath the indenter. Both ${h}_c^e$ and ${h}_c^p$ decrease with the increase of indentation depth toward the overall closed hysteresis loop, and the decreasing rate for ${h}_c^p$ seems to be larger. Essentially, the area of hysteresis loop reflects the dissipated energy that is required to advance the indentation depth, and the dissipation rate will slow down as the indentation depth propagation increases (Xu et al. 2007b).

Figure 3d presents the variation of indentation depth with the history of cyclic loadings. Similar to the indentation fatigue P-h curve in Fig. 3b, three stages that correspond to the loading history are also obtained, and the second stage under a cyclic loading will be investigated. It further confirms that an indentation depth rate (dh/dN) arrives in a stable state with the increase of cyclic numbers at the second loading stage, and this stable indentation depth propagation ratio (dh/dN)_s depends on the indentation load range and the maximum load, and radius of indenter, i.e., ΔK and K_max and d/2. With d/2 = 1.0 m, Fig. 4a gives the variation of the steady-state depth rate (dh/dN)_s with ΔK and K_max. A power-law function (i.e., Eq. 22) of (dh/dN)_sas both the stress intensity range ΔK and the maximum stress intensity K_max is obtained. When the indenter radius d/2 = 3.0 mm, the functional form of Eq. 22 holds. Further, the exponents n_i, m_i can be obtained, and n_i=5.2>m_i=1.5, suggesting that (dh/dN)_s is dominated by K_max in comparison with ΔK. Similar to the fatigue crack propagation, ΔK can be considered the driving force for propagation of plastic zone due to the stress concentration at the rim of contact (associated with the cyclic plasticity), which promotes indentation depth propagation and represents the intrinsic mechanism. Meanwhile, K_max is associated with the elastic recovery of indentation depth, which recovers elastic deformation during indentation fatigue and represents the extrinsic mechanism. The dominance of K_maxin the indentation fatigue depth propagation for the brittle copper alloy is qualitatively similar to the conventional fatigue cracking behavior of brittle materials (Liu and Chen 1991; Dauskardt et al. 1992; Ritchie 1999), where their crack growth rate is dominated by K_max in Eq. 21.

Fig. 4

Variation of the steady-state rate of indentation fatigue depth (dh/dN)_s with the stress intensity range ΔK and the maximum stress intensity K_max from the numerical simulation on (a) a brittle copper alloy and (b) a ductile copper alloy Xu et al. 2009 (reprinted with the permission from Elsevier)

Following the similar computational procedures, we performed the finite element analysis of indentation fatigue on the ductile copper alloys. As ΔK and K_max change with different indenter radii, Fig. 4b shows the corresponding obtained(dh/dN)_s, which confirms that the indentation fatigue depth propagation law (Eq. 22) holds. The exponents are n_i=4.2<m_i=5.6, which indicates that the dominance of ΔK over that of K_max for ductile copper alloys, also consistent well with the fatigue cracking behaviors of ductile materials (Liu and Chen 1991; Dauskardt et al. 1992).

In summary, these computational results validate the proposed power-law function, i.e., Eq. 22, and the steady-state indentation fatigue depth propagation rate is a power-law function of K_max and ΔK for both brittle and ductile metals, similar to that fatigue crack growth. Besides, the dominance of K_max and ΔK can be determined by comparing their exponents. In addition, note that the coefficients in Eq. 22C_i, n_i, and m_i depend on materials and testing conditions but are insensitive to indenter radii.

Experimental Validation

Material Choice and Testing Platform

99.9% polycrystalline copper (with 0.0262% Zn, 0.0145% P, 0.003% Pb, and 0.1266% Fe) with an average grain size 32μm was chosen, and the surface of each specimen (15 mm in diameter, by 15 mm in length) was polished to minimize the effect of surface roughness before each experiment. Besides, annealing procedures for specimens were made at temperatures 150 °C for 35 minutes to relieve the residual stress. The material of the flat punch indenter was made by high temperature alloys and its deformation was neglected in comparison with the indentation depth. The diameter of the indenter was 0.993 mm.

A sinusoidal wave was employed to apply the indenter and was used to mimic a fatigue loading condition and the loading frequency (f=ω/2π, Eq. 2) was chosen to be 1 Hz. To keep contact between the indenter and surface of the specimens during the experiment, the minimum indentation load (P_min) was set to 20 N. The conventional tensile machine INSTRON8871 that can easily realize the fatigue loading has been designed to conduct the indentation fatigue experiment with a customized load cell, as schematized in Fig. 5. All experiments were performed at room temperature, and the indentation load-depth data were recorded automatically for each 10 loading cycles during an indentation fatigue testing.

Fig. 5

Schematic of indentation fatigue testing system

Experimental Results

For representative fatigue loadings, P_max = 200 N and P_min = 20 N were employed and the corresponding K_max = 485.26MPa ⋅ mm^1/2 and ΔK = 469.12MPa ⋅ mm^1/2 can be calculated by utilizing Eq. 22. Figure 6a shows the recorded indentation load-depth curve. The indentation depth increases with the cyclic number. Besides, there is a quick penetration of the indenter at the initial stage, and then a relative steady state of indentation penetration rate arrives. More importantly, a hysteresis loop is observed in each loading-unloading cycle and seems to remain unclosed even after 3000 cycles. Figure 6b presents the curve of the indentation depth-loading cycle, where the width of the bands is expected to associate with the elastic recovery in each loading cycle. Similar to that observation in simulations in Fig. 3d, the indentation depth propagation shows a quick increase at the beginning and then followed by a steady-state stage. Essentially, these features can be understood from the dislocation activation and retraction, and more details see section “Discussion on Deformation Mechanism.” In fact, these features are quite analogous to the first two stages of uniaxial tensile fatigue crack growth. The approximate steady-state propagation of indentation fatigue is further confirmed when the indentation fatigue depth propagation rate dh/dN is plotted as a function of number of cycles in Fig. 6c, and the steady-state propagation rate is(dh/dN)_sas denoted in Eq. 22.

Fig. 6

Indentation fatigue experiment on polycrystalline copper. (a) Indentation loadP - indentation depth h. (b) Indentation depth h and (c) indentation depth per cycle dh/dN with the applied fatigue cycle number.K_max = 485.26MPa ⋅ mm^1/2 and ΔK = 469.12MPa ⋅ mm^1/2 Xu and Yue 2006 (reprinted with the permission from Cambridge University Press)

Given (dh/dN)_s, we can examine its variation with ΔK and K_max and validate the indentation fatigue depth propagation law (i.e., Eq. 22). Figure 7 shows there is a linear variation of (dh/dN)_s with both ΔK and K_max on log-scale plots, which agrees with Eq. 22. Besides, the power exponents are m_i = 3.1 and n_i = 2.1, respectively. The higher m_i implies that the steady-state propagation of indentation fatigue is dominated by ΔK, which is also consistent with fatigue crack growth in ductile metals (Ritchie 1999). We should note the power exponents n_i and m_i may not exactly equal to the n and m obtained from uniaxial tensile fatigue crack propagation because of influences of local microstructures in indentation experiments, and this difference is expected to increase with the decrease of indentation scales.

Fig. 7

Variation of the steady-state rate of indentation fatigue depth (dd/dN)_s with the stress intensity range ΔK and maximum stress intensity K_max, measured from experiment on a polycrystalline copper Xu et al. 2009 (reprinted with the permission from Elsevier)

Extend to Overloading and Underloading

In previous sections, we have revealed the similarity between indentation fatigue depth propagation and fatigue crack growth under a normal fatigue loading spectrum. It is known that the fatigue crack growth is greatly affected by the applied fatigue loading interactions such as sudden increase of peak load followed by a normal fatigue loading conditions (referred to as overloading here) and sudden decrease of peak load followed by a normal fatigue loading conditions (referred to as underloading here). For example, the normal fatigue crack growth can be delayed and boosted by the overloading and underloading, respectively. In this section, we will continue to employ polycrystalline copper to investigate whether the overloading/underloading will lead to delay/acceleration of indentation fatigue depth propagation so as to further study its similarity to fatigue crack growth.

Figure 8a shows the indentation fatigue P-h curve under an overloading condition, and the inset illustrates the overloading spectrum, where two loading blocks with a low-high loading sequence are considered. P_max0 and P_min are the maximum load and the minimum load of the first load block, respectively. P_max1 is the maximum load of the second load block. N₀ and N₁ are the number of loading cycles of the two load blocks, respectively. The indentation depth increases with the number of cycles, and its propagation arrives at a relatively steady stage with a constant rate after an initial sharp increase in each loading block. Besides, hysteresis loops are also found in each block and seem not be closed, even after 7000 cycles in the second loading block.

Fig. 8

(a) Indentation depth and (b) indentation depth per cycle - number of cycles curves under a low-high overloading (inset in a). P_max0 = 500 N, P_max1 = 700 N, N₀ = 2000 cycles, N₁ = 7000 cycles. (c) Indentation depth and (b) indentation depth per cycle - number of cycles curves under a high-low underloading (inset in c). P_max0 = 600 N, P_max1 = 550 N, N₀ = 2000 cycles, N₁ = 7000 cycles Xu et al. 2007a (reprinted with the permission from Elsevier)

The indentation depth-loading cycle is plotted in Fig. 8b, and it further indicates the approximately constant evolution of indentation depth propagation after initial rapid adjustment in each loading block, similar to that under a single normal fatigue loading block. Figure 8b further shows the rate of indentation depth propagation dh/dN with the number of fatigue loading. In each block, a clear steady stage with a constant dh/dN is observed after a quick decrease. More importantly, when the maximum load increases from 500 N in the first loading block to 700 N in the second loading block, a dh/dN shows an obvious increase first and then gradually decreases until to a new higher steady state. Experiments further indicate that the new steady state is nearly independent of the maximum loads in the second load block, except that it needs more cycles to reach the new steady state for the larger maximum load, as shown in Fig. 8b, where the maximum load changes from 700 N to 800 N in the second load block. Generally, a larger increase of the maximum load will require more number of cycles to eliminate the effect of load interaction. The enhancement of the indentation depth propagation by the overloading further shows similarity with that in conventional fatigue crack propagation (Tvergaard 2005; Sadananda and Vasudevan 2003; Huang and Ho 2000, 2003; Borrego et al. 2003; Kumar et al. 1996).

Similar to overloading conditions, the underloading condition also consists of two fatigue loading blocks with a decrease of the maximum load, i.e., a high-low loading sequence in the two load blocks, as illustrated in the inset in Fig. 8c. Consider a decrease of the maximum load from P_max0 = 600 N to P_max1 = 550 N with N₀ = 2000 cycles, and N₁ = 7000 cycles, Fig. 8c shows the indentation depth - number of cycles. The indentation depth keeps increasing instead of ceasing when the maximum load decreases. Besides, similar to that under overloading conditions, dh/dN arrive at a new steady state after lowering the maximum load, yet within a very short cyclic numbers, as shown in Fig. 8d. The steady state dh/dN for the new maximum load is obtained in the second cyclic loading block and smaller than that for the first cyclic loading block, indicating a delayed propagation of indentation depth. When the decrease of the maximum load is much larger (from P_max0 = 600 N to P_max1 = 500 N), Fig. 8d shows that the indentation depth propagation becomes slower. The delay of indentation fatigue depth propagation due to underloading is also similar to those findings in conventional fatigue crack propagation.

Microstructural Observation

Figure 9 presents the optical image of indentation on polycrystalline copper after cyclic loadings (Fig. 8c for indentation fatigue load-depth curve), where P_max0 = 600 N, P_max1 = 550 N, N₀ = 2000 cycles, N₁ = 7000 cycles. Pile-up, gap, and wrinkles (i.e., overlapping layers) are observed, similar to those under monotonous loading conditions. Generally, the pile-up or sink-in is caused by material hardening and when the hardening exponent of materials, n, is less than 0.33 (Storakers and Larsson 1994; Taljat and Pharr 2004), the pile-up will appear near the flat punch indentation (Storakers and Larsson 1994). For the polycrystalline copper used in our experiment, n ≈ 0.3 < 0.33was measured from our uniaxial tensile testing (Xu et al. 2006), and its cyclic hardening exponent n ≈ 0.1is also less than 0.33, which agrees well with the appearance of pile-up. Given the constant contact area between the flat punch indenter and surface of materials, deformation of materials will deviate from the indenter in accommodation with the pile-up, leaving a gap between pile-up and indentation. This gap will increase with the continuous sinking of indenter into the materials, which also agrees with the finite element analysis (Xu and Yue 2007). Wrinkles reflect the slip-steps of shear bands surrounding the indentation, and the formation and propagation of these shear bands can be deemed the major plastic deformation mechanism in the present indentation fatigue on polycrystalline copper. The closer to the indentation, the more wrinkles due to the more severe plastic deformation. From crystal plasticity theory point of view, the wrinkles result from activation of multiple slip systems, and these wrinkles are also referred to multislip-steps in quasi-static indentation (Lloyd et al. 2005; Zaafarani and Raabe 2008; Nibur and Bahr 2003).

Fig. 9

Optical microscope (OM) indentation image of polycrystalline copper (top view) and its sketch map (slide view) Xu et al. 2007a (reprinted with the permission from Elsevier)

Figure 10 shows the SEM image of cross section of indentation. Pile-up and gap are observed more clearly. Besides, some light and dark cyclic lines with radii that decrease with increasing distance from the indenter are also observed. Those lines are expected to be dependent of the slip lines beneath the flat punch indenter, which is considered major deformation mechanism (Hill 1998). Figure 10 (i–iii) shows the local deformation near the indentation, highlighted in the boxes. Figure 10i shows the SEM images of pile-up around the samples. The pile-up is flat at first and then vanishes with the distance far from the indentation. The flat transition is expected to be related with the amount of plastic deformation of materials around the indenter, and a cavity layer or cavity strip is similar to the form of a pile-up in geometry. From Fig. 10i, many cavities appear along the cyclic lines, and even some cracks in higher density cyclic lines near the rim of indentation. These cavities are expected to be nucleated at an early stage of indentation fatigue and accumulate to cracks, and then further spread with the increase of the number of loading cycles. Once the cracks interact, peel-offs of materials will happen, especially near the rim of indentation with higher cyclic lines associated with a high stress concentration. Figure 10iii gives the SEM images in the regions below the indentation. More cavities and cracks are found at a distance of about ∼15um far from the free end of indentation. The orientation of these cracks is in line with the dark and light cyclic lines, which are approximately level near the end of the indentation and steeper further from the end of indentation.

Fig. 10

SEM image of the cross section of indented polycrystalline copper sample under a high-low underloading (inset in Fig. 8c) and higher magnification (i) around the pile-up on the surface, (ii) near the corner of indentation, (iii) below the indentation; P_max0 = 600 N, P_max1 = 550 N, N₀ = 2000 cycles, N₁ = 7000 cycles Xu et al. 2007a (reprinted with the permission from Elsevier)

Discussion on Deformation Mechanism

In the viewpoint of indentation load–indentation depth curves, the increase of indentation depth results from deformation ratcheting behavior of materials during cyclic indentation due to a nonzero mean fatigue stress. The ratcheting behavior reflects the accumulation of plastic deformation preceding in one direction. Besides, the nucleation and development of the cracks further facilitate the development of plastic zone, leading to a continuous increase of indentation depth.

When a new loading block is applied following a normal fatigue loading history, the retardation or acceleration of indentation fatigue depth propagation is caused by interactions of loading blocks and can be understood by considering activation of dislocations. For example, the effective applied stress σ_eff acting on mobile dislocation is

$$ {\sigma}_{eff}={\sigma}_{app}-{\sigma}_r $$

(23)

Where σ_app is the maximum fatigue stress and σ_r is the residual stress. For example, when the maximum load increases (i.e., overloading), the effective applied stress σ_eff is very large, meanwhile the residual stress σ_r is small, leading to a high indentation depth per cycle. At the same time, this difference will require numbers of loading cycles to increase the residual stress σ_runtil reaching a new balance with a new steady state of indentation depth propagation. Hence, the greater the overloading amplitude, the more number of cycles are needed to remove the effect of overloading, consistent with experimental results in Fig. 8b. Besides, with the increase of the effective applied stress σ_eff, the dislocation density becomes intensive, the initial indentation depth per cycle will be very large and then gradually reduces and approaches to a new steady-state value with the arrival of new balance. The variation of the residual stress σ_r will lead to rearrangement of the dislocations beneath the indenter. The larger increase of the maximum load, the more cyclic numbers of loading to rearrange dislocation, and the larger steady indentation depth propagation ratio. These interpretations further show the similarity with the generation and annihilation of dislocations around the crack tip under the fatigue loadings.

Similarly, the decrease in the maximum cyclic load after reaching steady indentation depth ratio will reduce the residual stress σ_r. Consequently, underloading with a smaller maximum load produces a smaller effective stress σ_eff for plastic zone propagation, leading to a decreased indentation depth propagation after initial adjustments, as shown in Fig. 8d, which is also similar to that of fatigue crack growth under fatigue loadings.

Indentation Fatigue Damage

Indentation Fatigue Strength Law

With the continuous increase in indentation load cycles, damage of indented materials will nucleate and propagate, and eventually leads to failure of materials (Lawn and Wilshaw 1975; Xu et al. 2007a; Guiberteau et al. 1993; Bhowmick et al. 2007). Consider the number of indentation cycles to failure as N_f and the amplitude of the cyclic indentation load as F_a, in this section, we will build the relationship between N_f and F_a.

For a uniaxial fatigue testing, the relationship between the amplitude of fatigue load, σ_a, and the number of cycles to failure of materials, N_f, is the well-known S-N curve and can be expressed as a power-law relationship (Suresh 1998; Basquin 1910)

$$ {\sigma}_a={\sigma}_f{\left({N}_f\right)}^n $$

(24)

where σ_f and n are the fatigue strength coefficient and exponent, respectively, and are related to the material/microstructure and testing environment/condition. In general, σ_f equals the uniaxial fracture strength of the material under quasi-static loading.

Inspired by this equation (Eq. 24), we propose that a similar power-law relationship exists for indentation fatigue damage

$$ {F}_a={F}_f{\left({N}_f\right)}^m $$

(25)

where F_f and m are the indentation fatigue strength coefficient and exponent, respectively, and both depend on the material/microstructure and testing environment/condition. In the following, we will validate this power law equation in experiments.

Experimental Validation

Material Choice and Testing Platform

Different from the employment of copper alloy, to highlight the damage, relatively brittle materials were used in this section, and they were PVC bulk material, and TiN, NiP thin films on steel (SUS304) substrate. Besides, in order to avoid the early damage by stress concentration at contact, a spherical indenter was employed (instead of a flat punch indenter or a pyramidal/conical indenter). The cyclic indentation load applied on the indenter varies in a sinusoidal manner; the load ratio, i.e., the ratio between the minimum and maximum indentation loads, was fixed at 0.1 for the thin-film specimens and fixed at 0.05 for the PVC bulk specimen. Similar to the study of indentation fatigue depth propagation on polycrystalline copper, the minimum load was above zero to keep the “compressive” contact all the time. The frequency of load is 20 Hz for thin films and 5 Hz for the bulk material. The maximum indentation force is constrained so as not to induce cracking under static/monotonic loading conditions.

The failure was monitored in situ by acoustic emission (AE) during the indentation fatigue, as schematized in Fig. 11. This technique was used to diagnose the nucleation and propagation of damage and cracks in the static and cyclic indentation tests and showed good agreement with the observations of microstructures (Yonezu et al. 2009, 2010). Since AE may also be sensitive to contact friction/noise due to cyclic indentation, the initiation of major crack, or material failure, is quantitatively defined if more than 20 new AE counts per second are detected. This ensured the detection of the initiation of major cracks (macrocracks), which is regarded as failure in this study. From the variation of the AE count, the critical number of fatigue cycles (upon failure) can be readily determined, which is regarded as the fatigue strength of the specimen under a particular testing condition.

Fig. 11

Experimental setup of spherical indentation with AE and corrosion potential fluctuation monitoring system

Results and Discussion

Indentation fatigue tests on the PVC bulk material were first performed. Figure 12a shows a typical curve of indentation fatigue depth versus the number of cycles with a maximum indentation force of 880 N. Both the maximum and minimum depth histories are shown and the difference between them is due to the elastic recovery. The indentation depth increases with the number of cycles, and after an initial stage, a steady-state indentation depth propagation rate is achieved. These features are similar to those of polycrystalline copper in section “Experimental Validation.” The accompanied AE count shows that at N = 109,245 cycles there is a significant increase of new AE counts. This implies that a macrocrack nucleates around the indentation, which is identified as the failure in the PVC, as shown in the inset in Fig. 12a. For the same specimen, more indentation fatigue tests with different load amplitudes were carried out, and the resulting number of cycles to failure obtained as a function of the fatigue load amplitude. This is shown in Fig. 12b. A linear relationship is obtained in the log-log plot, which validates the feasibility of using Eq. 25 to describe the indentation fatigue strength of the bulk material.

Fig. 12

(a) Experimental evolution of indentation fatigue depth and AE count with number of cycles on the PVC bulk material. The inset shows the surface topography near the indentation (after failure). (b) The variation of the indentation fatigue load amplitude with number of cycles to failure for the PVC bulk material. The maximum indentation fatigue load is 880 N and the load ratio is 0.05. The frequency is 5 Hz Xu et al. 2010 (reprinted with the permission from Taylor & Francis)

Next, indentation fatigue on film/substrate systems was also carried out. Figure 13a shows the evolution of indentation depth with number of cycles for the NiP film on a steel substrate, where the maximum indentation force was 600 N. A similar characteristic of the evolution of the indentation depth with that of PVC bulk material is observed: the indentation depth rate remains steady for a number of cycles, and then undergoes a sudden change when the cycle number is about 409,860 cycles. Accordingly, a significant increase of new AE counts (well over 20 per second) was also detected, indicating macrocracking failure in the system. These findings are consistent with the post-observation (inset in Fig. 13a) where the NiP thin film develops many cracks.

Fig. 13

(a) Experimental evolution of indentation depth and AE count with number of cycle on the NiP thin film/SUS304 steel substrate system. The inset shows the surface topography near the indentation (after failure). (b) Variation of indentation fatigue load amplitude with number of cycles to failure for the Ni-P film/SUS 304 steel substrate system, as well as that of the TiN film/SUS304 substrate system. The maximum indentation fatigue load is 600 N and the load ratio is 0.1. The frequency is 20 Hz Xu et al. 2010 (reprinted with the permission from Taylor & Francis)

With the similar procedures, a series of indentation fatigue tests were also performed with different load amplitudes on the NiP and TiN film/steel substrate systems. Figure 13b shows the relationship between the indentation fatigue load amplitude and the number of cycles to their failure. In both cases, a power-law relationship (i.e., Eq. 25) is validated. Moreover, the indentation fatigue strength exponent, m, can be determined by fitting Fig. 13b, and it equals to −0.041 ± 0.005 and −0.056 ± 0.013 for TiN and NiP thin films on steel substrate, respectively. At the same time, the fatigue strength coefficient F_f can be obtained from the extrapolation and is 8.9 ± 1N and 660 ± 120N for TiN and NiP films on steel substrate, respectively, which agrees well the measurements (14.4 N and 1000 N counterparts (Yonezu et al. 2009, 2010)) at the static indentation tests on the same TiN and NiP films on steel substrate, thus suggesting an underlying bridge between the indentation fatigue strength and uniaxial fatigue strength.

Table 1 lists the further comparison of the fatigue stress exponents between the current measurements with those obtained from uniaxial tests. Although the fatigue stress exponents vary with respect to the thickness of films, they are on the same order of magnitude. Therefore, we may conclude that the indentation fatigue strength law, Eq. 25, is suitable for describing the failure behavior with measurement fatigue parameter close to those in conventional uniaxial fatigue test. Note that although the indentation fatigue failure mechanisms of PVC bulk materials, and TiN/SUS304 and NiP/SUS304 film/substrate systems are different, the general potential applicability of Eq. 25 seems to not be affected. This qualitatively echoes the fact that in conventional uniaxial fatigue tests, different failure mechanisms may be involved for different materials, e.g., for brittle and ductile materials, yet Eq. 24 holds for all common mechanisms. Thus, the direct comparison of fatigue strength exponents indicates a quantitative connection between the indentation fatigue test and uniaxial fatigue test.

Table 1 Comparisons between the fatigue strength exponents obtained from the indentation fatigue test (shaded rows) and uniaxial fatigue test in literature, for several thin film/substrate systems Xu et al. 2010 (reprinted with the permission from Taylor & Francis)

Concluding Remarks

Indentation fatigue is emerging as an alternative approach to measure the fatigue properties and to probe fatigue-induced deformation and failure mechanism of materials and structures in particular materials and structures with small volumes. This chapter presents, in theory, experiment and computation, extensive evidences of similarity between indentation fatigue depth propagation and fatigue crack growth, and builds a quantitative foundation in mechanics. The recent indentation fatigue experiments performed at the nanoscale on nanomaterials (Cavaliere 2010; Wei et al. 2008) and nanofilms (Bhat 2012) further confirmed the proposed similarity and indentation fatigue laws/equations. Further development of this indentation fatigue technique is envisioned to be integrated with multiple different measurements (e.g., AE) and/or observatory (e.g., TEM, SEM) tools to monitor in situ deformation of materials under cyclic fatigue loadings. At the same time, advanced computational models that can reproduce the experimental results of indentation fatigue will be critically important, and they may require to span several length scales to capture both indentation fatigue phenomena at the macroscale and deformation mechanism at the nanoscale. Moreover, the application of indentation fatigue to measure the cyclic/fatigue properties of broader materials such as soft matter and biological materials and structures will help to extend theories of indentation fatigue mechanics beyond the current focus of engineering metals, and may also offer new opportunities to probe mechanics of soft materials and structures.