**Energy Dissipation Criteria for Surface Contact Damage Evaluation**

Yong X. Gan

*Department of Mechanical, Industrial and Manufacturing Engineering, College of Engineering, University of Toledo, USA* 

### **1. Introduction**

142 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

Wada, A.; Motogi, S.; Fukuda, T. (1999). Damage mechanics approach to nonlinear behavior

234

of FRP laminates with cracking layers. *Adv. Composite Mater*., Vol 8, No. 3, pp. 217-

This chapter presents the energy dissipation approach for analyzing surface contact damages in various materials, including composite materials. As known, surface contact is a very common phenomenon, which can be found in daily life and many scientific and engineering problems. The contact of different bodies can be modeled as indentation. Analysis of indentation and modeling of the deformation states of indented materials are often difficult because of the complexity of stress distributions within indentation zones. It is also very difficult to evaluate stress states in regions underneath an indented zone. Instrumental indentation has been performed on various materials including composite materials. Experimental studies on indention of coatings and brittle materials have been reported extensively, but the criterion for evaluating the extent of damage is not unified. Ductile materials deform relatively stable in indentation processes. While brittle materials are sensitive to compressive contact loadings in view of the formation of surface cracks. Therefore, it is difficult to find a unified stress or strain based damage criterion to characterize the damage evolution. Energy dissipation analysis may be more accurate to describe the deformation behavior of such materials. Specifically, under wedge indentation, the analysis should be investigated because the stress field has the singularity which limits the applicability of the strength criterion. In this chapter, the load-displacement relations with elastic-plastic responses of the materials associated with the indentation processes will be obtained to calculate the hysteresis energy. Lattice rotation measurement using electron backscatter diffraction (EBSD) technique will be performed in the region ahead of the indenter tip to measure the dimension of the contact damage zone (CDZ) and the results will be used to define the length scales in contact deformation. A unified criterion using the hysteresis energy normalized by the length scales will be established.

Damage evolution in composite materials is very sensitive to the interaction of reinforcements and matrices in interface regions. For example, the development of damage in glass particle and fiber reinforced epoxy composite materials is strongly influenced by the interface debonding conditions [1]. However, the exact effect of bonding conditions on the performance of particle filled composite materials is still not fully understood. Kawaguchi and Pearson [2] reported that strong matrix-particle adhesion may lower the fatigue crack propagation resistance. While the studies on Si3N4 nanoparticle filled epoxy composites

Energy Dissipation Criteria for Surface Contact Damage Evaluation 145

typically shows an initial hardening process, while at high temperatures, only cyclic

Fatigue tests on reinforced titanium composites revealed various interface damage mechanisms [20-27]. Shear frictional sliding [20], interfacial debonding [21], fiber bridging [22], surface embrittlement [23], matrix ligament premature ductile shear [24], and crack deflection [25] are typical damage mechanisms observed. These damage mechanisms could occur simultaneously depending on loading modes, but debonding always exists and is considered as the major mechanism. A stress-based criterion for predicting the debonding behavior was proposed [22]. Rios, Rodopoulos and Yates [26] assessed the initial and final damage states caused by interface debonding and fiber bridging to determine the damage accumulation rates in SiC fiber reinforced titanium composite. Their method was used for damage tolerant fatigue design. Residual stiffness and the post-fatigued tensile strength as a function of microstructural damage were obtained through computer simulation, and the interfacial frictional stress and the critical crack length were also calculated [27]. Under combined thermal and mechanical fatigue loading, carbon fiber/Al and SiC fiber/Al composites were found to fail by a ratchetting mechanism, which is characterized by the progressive plastic deformation increasing with the number of cycles, even at stress levels far below the yield stress [28]. It is further found that the main phenomenon leading to composite

failure is ratchetting at high load levels and interface degradation at low load levels.

compared to SiC particles, was observed in particle reinforced A356 casting alloy [33].

Modeling fatigue debonding have been performed by many researchers [34-39]. In Gradin and BÄ*a*cklund's work [34], a unit cell model containing a steel bar and a co-centric epoxy cylinder was used to study the progressive de-bonding between the fiber and the matrix. Energy release rate was correlated to the interfacial debonding length. While in the work shown in [35-37], void formation and growth due to fatigue loading was characterized by the tensile stress at the interface. Three distinguishable debonding stages, two transient ones separated by a steady stage, were defined by Botsis and Zhao [38]. Stress intensity factor may be used to distinguish the steady and the transient stages because the total stress intensity factor was found to be approximately constant at the steady state. Debonding

Short crack growth behavior in steels containing different particle inclusions including Al2O3, MnS and Ti3N4 was studied by finite element method [29]. Crack-tip displacements and energy release rates were taken as the driving forces. It was found that the energy release rate is the highest for the Al2O3 inclusion case with a short through thickness crack. Li and Ellyin [30] studied the fatigue damage and the localization in Al2O3 particulate reinforced aluminum composites. The primarily damage forms are particle debonding, fractured particles and matrix cracks. Mesoscale reinforcement defects, such as a clump of large particles were also found causing damage localization. These defects were assumed to be the reason for short crack initiation and extension. In Murtaza and Akid's work on steel [31], it is reported that debonding at the matrix/inclusion interface is the major mechanism for the formation of short cracks. Stress redistribution at interfaces in alumina/aluminum multilayered composites was investigated [32]. The effects of interfacial debonding or of plastic slip in the metal phase adjacent to strongly bonded interfaces were considered. The results of stress measured around the crack reveal that debonding is much more effective than slip in reducing the stress ahead of the crack. Interaction of short fatigue crack with different types of particles was studied. Stronger interaction of fatigue crack with Si particles, as

softening is found [19].

under sliding wear conditions showed that the strong interfacial adhesion between Si3N4 nanoparticles and the matrix reduced the wear rate of the composites [3]. Damage in the form of debonding in coated fiber reinforced composites under tension-tension cyclic load was investigated [4]. The bi-interfacial debonding (fiber/coating and coating/matrix) behavior was analyzed using a double shear-lag model. Based on this model, the debond growth rate and strain energy were calculated by finite element method. Non-uniform damage of coating materials was accounted in the analysis. There exists two-interface coupling in debonding. It was found that the strength and thickness of coating materials are the major factors controlling the bi-interfacial crack growth. Numerical simulation of progressive damage evolution in fiber reinforced composites was performed to understand interface stress statistics and the fiber debonding paths development [5]. A meso cell including several hundred inclusions was used to account for the micro structure statistics of the composites. Both the local stress and effective elastic moduli of disordered fibrous composites were computed.

Micromechanics based approaches have been used for debonding damage analysis [6-10]. Cavallini, Bartolomeo, and Iacoviello [6] investigated the damage in three different ferriticpearlitic ductile cast irons with the main focus on graphite nodules debonding. Chan, Lee and Nicolella et al. [7] studied the near-tip fracture processes of nanocomposites under cyclic loads. It is found that particle bridging, debonding at the poles of particle/matrix interface, and crack deflection around the particles are the major micromechanics responses to cyclic loadings. Environmental conditions on the subcritical debond-growth rates were also examined [8]. Temperature and relative humidity are sensitive factors. Long term exposure to a moist environment resulted in the time-dependent decrease in adhesion between matrices and reinforcements. Three different interfacial damage models including the shear lag model, the linear degradation model and the modified power degradation model were used to describe the bond decay at steel/concrete interface [9]. The role of internal friction in resisting interfacial debonding was addressed. Micro-level damage in discontinuous fiber reinforced composites were found in the forms of fiber/matrix interfacial debonding and fiber failure [10]. The Weibull damage law was used to predict the microscopic damage behavior of composites with different fiber contents and orientations.

Crack initiation or small crack growth plays a critical role in interface debonding [11]. In small crack growth, plasticity-induced crack closure was observed, but the effect of crack closure in fatigue crack growth predictions was less than the estimation by the classical approaches [12]. In addition to crack closure, the shear deformation of matrix ahead of a small crack slows down the interfacial debonding rate [13]. Interface debonding controlled small crack growth behavior depends on the stress levels [14], and loading rate [15-16]. Microdebonding or subcritical debonding behavior is also dependent on surface chemistry [17] and temperature [18, 19]. To evaluate the surface chemistry effect, subcritical debonding of thin polymer layers from inorganic dielectrics was studied using selected amino- and vinyl- functional silane adhesion promoters [17]. Due to the surface modification, the failure occurs not at the interface but in a region very close to the interface. The effect of temperature on debonding is especially significant in metal matrix composite materials [20- 34]. At elevated temperatures, thermomechanical fatigue accounts for the failure of these materials. Alternating plastic shearing of the interface takes place under combined mechanical and thermal stresses [18]. At low temperatures, metal matrices such as Al

under sliding wear conditions showed that the strong interfacial adhesion between Si3N4 nanoparticles and the matrix reduced the wear rate of the composites [3]. Damage in the form of debonding in coated fiber reinforced composites under tension-tension cyclic load was investigated [4]. The bi-interfacial debonding (fiber/coating and coating/matrix) behavior was analyzed using a double shear-lag model. Based on this model, the debond growth rate and strain energy were calculated by finite element method. Non-uniform damage of coating materials was accounted in the analysis. There exists two-interface coupling in debonding. It was found that the strength and thickness of coating materials are the major factors controlling the bi-interfacial crack growth. Numerical simulation of progressive damage evolution in fiber reinforced composites was performed to understand interface stress statistics and the fiber debonding paths development [5]. A meso cell including several hundred inclusions was used to account for the micro structure statistics of the composites. Both the local stress and effective elastic moduli of disordered fibrous

Micromechanics based approaches have been used for debonding damage analysis [6-10]. Cavallini, Bartolomeo, and Iacoviello [6] investigated the damage in three different ferriticpearlitic ductile cast irons with the main focus on graphite nodules debonding. Chan, Lee and Nicolella et al. [7] studied the near-tip fracture processes of nanocomposites under cyclic loads. It is found that particle bridging, debonding at the poles of particle/matrix interface, and crack deflection around the particles are the major micromechanics responses to cyclic loadings. Environmental conditions on the subcritical debond-growth rates were also examined [8]. Temperature and relative humidity are sensitive factors. Long term exposure to a moist environment resulted in the time-dependent decrease in adhesion between matrices and reinforcements. Three different interfacial damage models including the shear lag model, the linear degradation model and the modified power degradation model were used to describe the bond decay at steel/concrete interface [9]. The role of internal friction in resisting interfacial debonding was addressed. Micro-level damage in discontinuous fiber reinforced composites were found in the forms of fiber/matrix interfacial debonding and fiber failure [10]. The Weibull damage law was used to predict the microscopic damage behavior of composites with different fiber contents and orientations. Crack initiation or small crack growth plays a critical role in interface debonding [11]. In small crack growth, plasticity-induced crack closure was observed, but the effect of crack closure in fatigue crack growth predictions was less than the estimation by the classical approaches [12]. In addition to crack closure, the shear deformation of matrix ahead of a small crack slows down the interfacial debonding rate [13]. Interface debonding controlled small crack growth behavior depends on the stress levels [14], and loading rate [15-16]. Microdebonding or subcritical debonding behavior is also dependent on surface chemistry [17] and temperature [18, 19]. To evaluate the surface chemistry effect, subcritical debonding of thin polymer layers from inorganic dielectrics was studied using selected amino- and vinyl- functional silane adhesion promoters [17]. Due to the surface modification, the failure occurs not at the interface but in a region very close to the interface. The effect of temperature on debonding is especially significant in metal matrix composite materials [20- 34]. At elevated temperatures, thermomechanical fatigue accounts for the failure of these materials. Alternating plastic shearing of the interface takes place under combined mechanical and thermal stresses [18]. At low temperatures, metal matrices such as Al

composites were computed.

typically shows an initial hardening process, while at high temperatures, only cyclic softening is found [19].

Fatigue tests on reinforced titanium composites revealed various interface damage mechanisms [20-27]. Shear frictional sliding [20], interfacial debonding [21], fiber bridging [22], surface embrittlement [23], matrix ligament premature ductile shear [24], and crack deflection [25] are typical damage mechanisms observed. These damage mechanisms could occur simultaneously depending on loading modes, but debonding always exists and is considered as the major mechanism. A stress-based criterion for predicting the debonding behavior was proposed [22]. Rios, Rodopoulos and Yates [26] assessed the initial and final damage states caused by interface debonding and fiber bridging to determine the damage accumulation rates in SiC fiber reinforced titanium composite. Their method was used for damage tolerant fatigue design. Residual stiffness and the post-fatigued tensile strength as a function of microstructural damage were obtained through computer simulation, and the interfacial frictional stress and the critical crack length were also calculated [27]. Under combined thermal and mechanical fatigue loading, carbon fiber/Al and SiC fiber/Al composites were found to fail by a ratchetting mechanism, which is characterized by the progressive plastic deformation increasing with the number of cycles, even at stress levels far below the yield stress [28]. It is further found that the main phenomenon leading to composite failure is ratchetting at high load levels and interface degradation at low load levels.

Short crack growth behavior in steels containing different particle inclusions including Al2O3, MnS and Ti3N4 was studied by finite element method [29]. Crack-tip displacements and energy release rates were taken as the driving forces. It was found that the energy release rate is the highest for the Al2O3 inclusion case with a short through thickness crack. Li and Ellyin [30] studied the fatigue damage and the localization in Al2O3 particulate reinforced aluminum composites. The primarily damage forms are particle debonding, fractured particles and matrix cracks. Mesoscale reinforcement defects, such as a clump of large particles were also found causing damage localization. These defects were assumed to be the reason for short crack initiation and extension. In Murtaza and Akid's work on steel [31], it is reported that debonding at the matrix/inclusion interface is the major mechanism for the formation of short cracks. Stress redistribution at interfaces in alumina/aluminum multilayered composites was investigated [32]. The effects of interfacial debonding or of plastic slip in the metal phase adjacent to strongly bonded interfaces were considered. The results of stress measured around the crack reveal that debonding is much more effective than slip in reducing the stress ahead of the crack. Interaction of short fatigue crack with different types of particles was studied. Stronger interaction of fatigue crack with Si particles, as compared to SiC particles, was observed in particle reinforced A356 casting alloy [33].

Modeling fatigue debonding have been performed by many researchers [34-39]. In Gradin and BÄ*a*cklund's work [34], a unit cell model containing a steel bar and a co-centric epoxy cylinder was used to study the progressive de-bonding between the fiber and the matrix. Energy release rate was correlated to the interfacial debonding length. While in the work shown in [35-37], void formation and growth due to fatigue loading was characterized by the tensile stress at the interface. Three distinguishable debonding stages, two transient ones separated by a steady stage, were defined by Botsis and Zhao [38]. Stress intensity factor may be used to distinguish the steady and the transient stages because the total stress intensity factor was found to be approximately constant at the steady state. Debonding

Energy Dissipation Criteria for Surface Contact Damage Evaluation 147

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.

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

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

In a contact cycle, supposing that plane-strain conditions hold, the non-zero components of

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

 

(1a)

the stress field ahead of the particle are calculated by fracture mechanics as

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

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

the local coordinates for deriving the stress solutions in the slip regions.

direction and slip plane normal vectors.

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 greatest under mixed-mode conditions.

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 modeling and experiments.

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