**2. Surface contact damage model**

The first part of the modeling work is on the surface contact damage initiation using a microscale approach. Since the deformation state at the contact point is highly anisotropic, the deformation mechanism of single crystal plasticity is enforced in this stage. The

under different loading modes including mode I, mode II and mixed mode (I & II) was studied by Dessureautt and Spelt [39]. It was observed that the debonding rate was the

In this chapter, the emphasis on mechanics analysis will be put on the damage initiation and propagation from the debonding of particle/matrix interface. Both macro- and micro-scale analysis will be performed. The macroscale approach based on continuum mechanics will be used to obtain the stress field in the elastic-plastic region within the matrix in front of the debonded particle. Treating the debonded region as a crack, stress intensity solutions can be obtained. In the plastic zone just ahead of the debonded particle, the microscale approach will be used to find the stress solutions. In the classical plasticity theory, the material property at the crack tip is considered to be isotropic and the maximum stress in the plastic zone is assumed to be the yield strength of the material. In this work, the particle-matrix interaction is modeled as surface contact and multiscale approaches are used in the

Why the particle-matrix interaction and the debonding in the interface region can be modeled as contact damage under indentation load? The rationale is evident by examining the damage zone. As shown in Figure 1, the particle inclusion is debonded from pearlitic steel matrix. Along the main crack propagation direction (marked as *x*-direction), two distinct slip regions are found. These regions are denoted as *Region I* and *Region II*. In each of these regions, persistent slip lines are found. Although there are also some other slip zones around the particle, the predominant slip activities that determine the main crack speed are from *Region I* and *Region II*. Therefore, with a simplified model, the slip in these two regions can be seen as generated by indentation. The hard particle is equivalent to an indenter.

Fig. 1. Scanning electron microscopic image showing contact damage induced slip zones

The first part of the modeling work is on the surface contact damage initiation using a microscale approach. Since the deformation state at the contact point is highly anisotropic, the deformation mechanism of single crystal plasticity is enforced in this stage. The

around a debonded particle. The main crack propagation is along *x*-axis.

**2. Surface contact damage model** 

greatest under mixed-mode conditions.

modeling and experiments.

deformation of the material in the indenter tip region due to the motion of dislocation on different slip systems will be described. Based on such a consideration, we assume that the stresses at the boundary between the elastoplastic region and the plastic zone propagate into the plastic zone. The magnitudes of the stress components are determined. The primary slip lines are assumed to be collinear with the dislocation motion directions. The second part of this section is specifically on the contact damage propagation. Once a short crack from the interface debonding starts growing, how to characterize the fatigue crack growth resistance becomes an important issue. A simulated crack (indenter penetration depth) is used to study the contact damage propagation kinetics. The specific energy of damage, a parameter which is used to characterize the resistance of the material to contact damage, is defined. The relationship between energy release rate and the specific energy of damage is established.

#### **2.1 Contact damage initiation stage: microscale approach**

It is assumed that the matrix is elastic-plastic so that in-plane slip is the prevailing plastic deformation mechanism. *S* is the unit vector parallel to the slip direction. *N* is the unit vector along the slip plan normal. To use indentation to simulate the debonding, the partial debonding and the fully debonded states, as shown in Figures 2(a) and 2(b), respectively, can be treated by the cases with indenter partially penetration and retreating. In order to find the stress solution, the debonded region is considered as a crack. A stress intensity approach is applied to find an approximate solution. Figures 2(c) shows both the global and the local coordinates for deriving the stress solutions in the slip regions.

Fig. 2. Sketches for particle/matrix interface debonding analysis: (a) partial debonding, (b) complete debonding, (c) configuration of global, local coordinates related to the slip direction and slip plane normal vectors.

In a contact cycle, supposing that plane-strain conditions hold, the non-zero components of the stress field ahead of the particle are calculated by fracture mechanics as

$$
\sigma\_{xx} = \frac{K\_l}{\sqrt{2\pi r}} \cos\left(\frac{\theta}{2}\right) \left[1 - \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right)\right] \tag{1a}
$$

$$
\sigma\_{xx} = \sigma\_{xx} = \frac{K\_l}{\sqrt{2\pi r}} \cos\left(\frac{\theta}{2}\right) \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right) \tag{1b}
$$

$$
\sigma\_{zz} = \frac{K\_l}{\sqrt{2\pi r}} \cos\left(\frac{\theta}{2}\right) \left[1 + \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right)\right] \tag{1c}
$$

$$
\sigma\_{yy} = \mu \left( \sigma\_{xx} + \sigma\_{zz} \right) \tag{1d}
$$

Energy Dissipation Criteria for Surface Contact Damage Evaluation 149

Once the stress field along the radial line *θ* = 0 is obtained, it is straightforward to find the stress state within the slip region. One of the ways is to follow the slip line analysis [40] to

In this part, evaluation of the contact damage propagation behavior based on experimentally determined irreversible work and energy dissipation is presented. The energy dissipated into damage formation is considered as the indentation penetration driving force. A materials parameter, the specific energy of damage is used as the contact damage tolerance criterion as previously introduced for some materials in [41-46]. Correlation between the contact damage tolerance and the microstructure of the material is

Considering indenter penetration region and its surrounding damage zone in the material as a thermodynamic entity, the following relationship can be obtained based on entropy and

> \* *da TS J a D dN*

comprising the indenter penetration region and the surrounding damage zone. *J\** is the energy release rate. *γ* is the specific damage of energy. *a* is the nominal indenter penetration depth or developed contact length between the indenter and the indented material. d*a*/d*N* is the indenter penetration speed. *N* is the number of indentation cycles. *D* is the rate of energy dissipation into contact damage formation associated with the damage zone

> *da D dN a J*

Under force control indentation conditions, the energy release rate *J\** can be evaluated by

*da P* 1 *dN B a*

where *P* is the potential energy (area above the unloading curve) at the indenter penetration depth *a*, and *B* is the specimen thickness. The cyclic rate of energy dissipation, *D* associated with contact damage zone evolution can be evaluated by the difference between the hysteresis energy related to indentation and the hysteresis energy dissipated into the bulk of

*Hn <sup>D</sup>*

\*

(6)

is the rate of change of the entropy of the system

(7)

(8)

*<sup>B</sup>* (9)

solve the stress components in *Region I* and *Region II*.

made.

evolution.

energy balance considerations.

where *T* is the ambient temperature and *S*

the material. It can be expressed as:

At minimum entropy, *TS* 0 . Eq. (6) can be rearranged as

where *Hn* is the hysteresis energy. Rearranging Eq. (7) yields

**2.2 Contact damage propagation stage: macroscale approach** 

where *KI* is the stress intensity factor related to the particle shape.

Other stress components are zeros, i.e. *τxy* = *τyx* = *τyz* = *τzy* = 0. Assuming the material near the tip is fully plastic, the following yielding criterion holds

$$N\Sigma S = \pm \tau\_i \tag{2}$$

where *τ<sup>i</sup>* is the shear strength of the *i*th slip system, *τ<sup>i</sup>* = *τ<sup>I</sup>* for *Region I* and *τ<sup>i</sup>* = *τII* for *Region II*. *N* is the surface normal of the slip plane, *S* is a unit vector along the slip direction. If the dislocation motion is along positive *S*, the right hand side takes positive *τi*, while in the case that the slip occurs along negative *S*, the negative sign is kept on the right hand side. **Σ** is the stress tensor. The components of *N* are *Nx*, *Ny* and *Nz*, and *S* has the components: *Sx*, *Sy* and *Sz*. Since only the in-plane slip is considered in this work, the *z*-components for both *N* and *S* are zeros. Therefore, the yield condition is

$$N\_{x}\sigma\_{xx}S\_{x} + N\_{y}\sigma\_{yy}S\_{y} = \pm \tau\_{i} \tag{3}$$

where *Sx* = cos(*φi*), *Sy* = sin(*φi*), *Nx* = -sin(*φi*), *Ny* = cos(*φi*), *φ<sup>i</sup>* = *φ<sup>I</sup>* for *Region I* and *φ<sup>i</sup>* = *φII* for *Region II*. Substituting these relations into Eq. (3) yields

$$-\frac{\sigma\_{xx} - \sigma\_{yy}}{2}\sin(2\varphi\_i) = \pm \tau\_i \tag{4}$$

Eq. (4) provides the yield function related to the slip angle and the stress field when the material is in a fully-plastic state. For the partial debonding case, along the radial line *θ* = 0, the in-plane stresses are

$$
\sigma\_{\text{xx}} = \frac{K\_I}{\sqrt{2\pi r^\*}} \tag{5a}
$$

$$
\boldsymbol{\tau}\_{xy} = \mathbf{0} \tag{5b}
$$

$$
\sigma\_{yy} = \pm \frac{2\tau\_i}{\sin(2\phi\_i)} + \sigma\_{xx} \tag{5c}
$$

where *r\** is the distance from the origin to an arbitrary point on the *θ* = 0 radial line, and 1.12 *K ac <sup>I</sup>* .

<sup>3</sup> cos 1 sin sin 2 2 22

 

Other stress components are zeros, i.e. *τxy* = *τyx* = *τyz* = *τzy* = 0. Assuming the material near the

*N S <sup>i</sup>* 

where *τ<sup>i</sup>* is the shear strength of the *i*th slip system, *τ<sup>i</sup>* = *τ<sup>I</sup>* for *Region I* and *τ<sup>i</sup>* = *τII* for *Region II*. *N* is the surface normal of the slip plane, *S* is a unit vector along the slip direction. If the dislocation motion is along positive *S*, the right hand side takes positive *τi*, while in the case that the slip occurs along negative *S*, the negative sign is kept on the right hand side. **Σ** is the stress tensor. The components of *N* are *Nx*, *Ny* and *Nz*, and *S* has the components: *Sx*, *Sy* and *Sz*. Since only the in-plane slip is considered in this work, the *z*-components for both *N* and *S*

*N SN S x xx x*

*xx yy*

 *y yy y* 

where *Sx* = cos(*φi*), *Sy* = sin(*φi*), *Nx* = -sin(*φi*), *Ny* = cos(*φi*), *φ<sup>i</sup>* = *φ<sup>I</sup>* for *Region I* and *φ<sup>i</sup>* = *φII* for

sin(2 ) <sup>2</sup>

Eq. (4) provides the yield function related to the slip angle and the stress field when the material is in a fully-plastic state. For the partial debonding case, along the radial line *θ* = 0,

 

*i i*

(4)

(5a)

(5b)

(5c)

\* 2 *<sup>I</sup> xx K r*

 2 sin 2 *i yy xx i*

0 *xy* 

where *r\** is the distance from the origin to an arbitrary point on the *θ* = 0 radial line, and

 

*<sup>I</sup> xz zx*

*<sup>I</sup> zz K r*

where *KI* is the stress intensity factor related to the particle shape.

 

tip is fully plastic, the following yielding criterion holds

*Region II*. Substituting these relations into Eq. (3) yields

are zeros. Therefore, the yield condition is

the in-plane stresses are

 1.12 *K ac <sup>I</sup>* 

.

*K r*

> 

<sup>3</sup> cos sin sin 2 22 2

 

 

*yy xx zz* (1d)

(1b)

(2)

*<sup>i</sup>* (3)

(1c)

Once the stress field along the radial line *θ* = 0 is obtained, it is straightforward to find the stress state within the slip region. One of the ways is to follow the slip line analysis [40] to solve the stress components in *Region I* and *Region II*.

#### **2.2 Contact damage propagation stage: macroscale approach**

In this part, evaluation of the contact damage propagation behavior based on experimentally determined irreversible work and energy dissipation is presented. The energy dissipated into damage formation is considered as the indentation penetration driving force. A materials parameter, the specific energy of damage is used as the contact damage tolerance criterion as previously introduced for some materials in [41-46]. Correlation between the contact damage tolerance and the microstructure of the material is made.

Considering indenter penetration region and its surrounding damage zone in the material as a thermodynamic entity, the following relationship can be obtained based on entropy and energy balance considerations.

$$T\dot{S} = \left(f^\* - a\gamma\right)\frac{da}{dN} + D\tag{6}$$

where *T* is the ambient temperature and *S* is the rate of change of the entropy of the system comprising the indenter penetration region and the surrounding damage zone. *J\** is the energy release rate. *γ* is the specific damage of energy. *a* is the nominal indenter penetration depth or developed contact length between the indenter and the indented material. d*a*/d*N* is the indenter penetration speed. *N* is the number of indentation cycles. *D* is the rate of energy dissipation into contact damage formation associated with the damage zone evolution.

At minimum entropy, *TS* 0 . Eq. (6) can be rearranged as

$$\frac{da}{d\mathcal{N}} = \frac{D}{a\mathcal{N} - \mathcal{J}}\tag{7}$$

Under force control indentation conditions, the energy release rate *J\** can be evaluated by

$$\frac{da}{dN} = \frac{1}{B} \frac{\partial P}{\partial a} \tag{8}$$

where *P* is the potential energy (area above the unloading curve) at the indenter penetration depth *a*, and *B* is the specimen thickness. The cyclic rate of energy dissipation, *D* associated with contact damage zone evolution can be evaluated by the difference between the hysteresis energy related to indentation and the hysteresis energy dissipated into the bulk of the material. It can be expressed as:

$$D = \frac{H\_n}{B} \tag{9}$$

where *Hn* is the hysteresis energy. Rearranging Eq. (7) yields

$$\frac{\int^\*}{a} = \gamma - \frac{D}{a\left(\frac{da}{dN}\right)}\tag{10}$$

Energy Dissipation Criteria for Surface Contact Damage Evaluation 151

There exists difficulty in measuring the actual damage zone size by direct visual observation. We examined indented copper crystal using scanning electron microscopy (SEM) and measured the damage zone size. The copper polycrystal was etched in warm HCl/SnCl4 solution. Further investigation of the damage zone using electron backscattering diffraction (EBSD) technique to reveal the contact damage zone in single crystal copper was

The indentation cyclic load vs time is shown in Figure 4(a). Time-dependent indentation penetration depth was recorded and shown in Figure 4(b). The relation of the indentation load v.s. the indenter penetration depth at a typical cycle is shown in Figure 4(c). From the load-displacement curves, we can calculate the potential energy and the hysteresis energy associated with the contact damage processes as schematically shown in Figure 4(d). The indentation penetration depth, *a*, versus the number of indentation cycles, *N*, for three steels was plotted. The slope of the *a* versus *N* curves was used to calculate d*a*/d*N*, and establish

Fig. 4. Calculating energy dissipation terms from indentation test data: (a) cyclic loading profile, (b) time-dependent displacement, (c) load-displacement relationship, (d) illustration

showing how to determine the potential energy and hysteresis energy.

the relationship of indentation speed, d*a*/d*N*, and indentation depth, *a*.

also performed.

**4. Results and discussion** 

The quantities *J\**, d*a*/d*N*, and *a*, can be obtained from indentation experiments. The relationship expressed in Eq. (10) can be plotted in a two dimensional domain, directly giving the value of the specific energy of damage, *γ*, which is the intercept of the straight line. *γ* can be used as a material property related parameter. By examining Eq. (10), as the contact damage propagates, the energy release rate increases, thus the change of the left term *J\*/a* can be leveled by both the increasing of *J\** and the indentation penetration depth, *a*. The variation of the term in the right side of Eq. (10), *D/*[*a*(*da/dN*)], depends on several factors. These are the indentation depth, *a*, the indentation speed, d*a*/d*N* and *D*, the cyclic rate of energy associated with the damage formation. The indentation speed changes with the indentation depth. From energy balance analysis, it is clear that the value of *D* changes with the indentation depth, *a*. Thus, the variation of *D* is well balanced by the change in both *a* and d*a*/d*N*. Thus, on the *J\*/a* vs *D/*[*a*(*da/dN*)] plot, a straight line which is almost parallel to the *D/*[*a*(*da/dN*)] axis can be obtained.
