170 Y.W.Kwon strain vector consisting of particle and matrix strains to the unit cell strain vector(effective composite strain). Equation(4.11)is used for the Stiffness Loop while (4.12)is used for the Stress Loop.Once the microlevel strains are computed from (4.12),the constitutive equation of the particle and matrix material,respectively,is used at the microlevel to compute the microlevel stresses.Then,damage and failure criteria are applied to the microlevel stresses and strains. 4.2.3 Damage Mechanics and Crack Initiation Criterion One of the advantages of applying damage and failure criteria at the microlevel is that even if the composite material has a different particle volume fraction,it is not necessary to obtain new composite material strength data from an experiment.Furthermore,all damage or failure modes can be simplified at the microlevel so that the damage or failure mechanism can be understood more clearly.For example,damage in a particulate composite can be classified into three categories as illustrated in Fig.4.3:particle breakage,matrix cracking,and particle/matrix interface debonding.The interface debonding may be considered as matrix cracking at the boundary of particles.Different damage or failure criteria may be used for different damage or failure modes.For example,an isotropic damage theory can be applied to the matrix material if the material is isotropic and the damage progression is also assumed to be isotropic. An experimental study of crack initiation and growth from a round notch tip in a composite showed that a crack initiated at the notch tip and Matrix Crack Interface Particle Debonding Crack Fig.4.3.Different damage at the microlevel of a particulate composite
strain vector consisting of particle and matrix strains {ε} to the unit cell strain vector (effective composite strain) { } ε . Equation (4.11) is used for the Stiffness Loop while (4.12) is used for the Stress Loop. Once the microlevel strains are computed from (4.12), the constitutive equation of the particle and matrix material, respectively, is used at the microlevel to compute the microlevel stresses. Then, damage and failure criteria are applied to the microlevel stresses and strains. 4.2.3 Damage Mechanics and Crack Initiation Criterion One of the advantages of applying damage and failure criteria at the microlevel is that even if the composite material has a different particle volume fraction, it is not necessary to obtain new composite material strength data from an experiment. Furthermore, all damage or failure modes can be simplified at the microlevel so that the damage or failure mechanism can be understood more clearly. For example, damage in a particulate composite can be classified into three categories as illustrated in Fig. 4.3: particle breakage, matrix cracking, and particle/matrix interface debonding. The interface debonding may be considered as matrix cracking at the boundary of particles. Different damage or failure criteria may be used for different damage or failure modes. For example, an isotropic damage theory can be applied to the matrix material if the material is isotropic and the damage progression is also assumed to be isotropic. An experimental study of crack initiation and growth from a round notch tip in a composite showed that a crack initiated at the notch tip and Fig. 4.3. Different damage at the microlevel of a particulate composite Particle Crack Matrix Crack Interface Debonding 170 Y.W. Kwon
Chapter 4:Multiscale and Multilevel Modeling of Composites 171 grew until it reached a certain size.Then,the crack tip became blunted for a while until it propagated further.Using the computational model,it was hoped to predict the initial crack length before blunting and subsequent crack propagation.The crack size before the initial blunting is called the initial crack length.To predict such an initial crack size,the damage mechanics were used along with the criterion described below. Let us consider a perforated plate under tension.Because of stress concentration,the stress very near the hole is much greater than the nominal value.Such high stress occurring very near the hole also results in damage at that location earlier than other locations.As the damage progresses very near the hole,the material at the same location becomes softer with greater damage.This means that even though the strain at very near the hole continues to grow with damage growth,the stress at the same location becomes lower with softer materials.Eventually,the stress very near the hole becomes lower than that in other locations until the stress at the tip of the hole goes down to nil.Such a process for stress reduction along with an increase of damage is illustrated in Fig.4.4. The case in Fig.4.4d indicates damage saturation at the edge of the hole so that the stress there becomes nil.This means a crack can initiate from the hole edge at the onset of damage saturation.Then,the main question for the initial crack size is how far the crack will propagate from the initiation to form an initial crack before blunting.To answer this question, the material behavior near the hole edge is examined.This investigation shows that the material very near the hole edge has material softening.In other words,the slope of the stress-strain curve at the material softening zone becomes negative.This implies that the material softening zone is unstable.As a result,the crack initiated at the hole edge at the onset of damage saturation is expected to grow through the unstable material zone. This indicates that the initial crack size is equal to the length of the unstable material zone in front of the hole edge,as indicated by le in Fig.4.4d.In summary,the criterion to predict the initial crack length is stated below: At the onset of damage saturation at the edge of a hole,i.e.,the stress becomes nil at that location,the length of the unstable material zone, i.e.,material softening zone,is the initial crack size. This criterion was tested against experimental data.The predicted results agreed very well with experimental results.For example,a parti- culate composite made of hard particles embedded in a very soft matrix
grew until it reached a certain size. Then, the crack tip became blunted for a while until it propagated further. Using the computational model, it was hoped to predict the initial crack length before blunting and subsequent crack propagation. The crack size before the initial blunting is called the initial crack length. To predict such an initial crack size, the damage mechanics were used along with the criterion described below. Let us consider a perforated plate under tension. Because of stress concentration, the stress very near the hole is much greater than the nominal value. Such high stress occurring very near the hole also results in damage at that location earlier than other locations. As the damage progresses very near the hole, the material at the same location becomes softer with greater damage. This means that even though the strain at very near the hole continues to grow with damage growth, the stress at the same location becomes lower with softer materials. Eventually, the stress very near the hole becomes lower than that in other locations until the stress at the tip of the hole goes down to nil. Such a process for stress reduction along with an increase of damage is illustrated in Fig. 4.4. The case in Fig. 4.4d indicates damage saturation at the edge of the hole so that the stress there becomes nil. This means a crack can initiate from the hole edge at the onset of damage saturation. Then, the main question for the initial crack size is how far the crack will propagate from the initiation to form an initial crack before blunting. To answer this question, the material behavior near the hole edge is examined. This investigation shows that the material very near the hole edge has material softening. In other words, the slope of the stress–strain curve at the material softening zone becomes negative. This implies that the material softening zone is unstable. As a result, the crack initiated at the hole edge at the onset of damage saturation is expected to grow through the unstable material zone. This indicates that the initial crack size is equal to the length of the unstable material zone in front of the hole edge, as indicated by lc in Fig. 4.4d. In summary, the criterion to predict the initial crack length is stated below: At the onset of damage saturation at the edge of a hole, i.e., the stress becomes nil at that location, the length of the unstable material zone, i.e., material softening zone, is the initial crack size. This criterion was tested against experimental data. The predicted results agreed very well with experimental results. For example, a particulate composite made of hard particles embedded in a very soft matrix Chapter 4: Multiscale and Multilevel Modeling of Composites 171
172 Y.W.Kwon material was studied.In this case,because particles are much stronger than the matrix material,a crack formed in the matrix material.Hence,the multiscale technique described in Sect.4.2.1 was applied to the particulate composite,and the damage mechanics along with the proposed initial crack length criterion was applied to the matrix material level stresses and strains.The difference between the experimental and predicted results of the initial crack lengths formed at the edge of holes was almost uniformly between 5 and 10%.Figure 4.5 shows an initial crack formed in a parti- culate composite [30,311. Distance from the hole edge Distance from the hole edge (a) (b) le Distance from the hole edge Distance from the hole edge (c) (d) Fig.4.4.Stress plots from the edge of a hole.Each stress plot from (a)to (d)is associated with increase of damage very near the hole along with load increase: (a)no damage state,(b)and (c)progressive damage states,and (d)saturated damage state
material was studied. In this case, because particles are much stronger than the matrix material, a crack formed in the matrix material. Hence, the multiscale technique described in Sect. 4.2.1 was applied to the particulate composite, and the damage mechanics along with the proposed initial crack length criterion was applied to the matrix material level stresses and strains. The difference between the experimental and predicted results of the initial crack lengths formed at the edge of holes was almost uniformly between 5 and 10%. Figure 4.5 shows an initial crack formed in a particulate composite [30, 31]. (a) (b) (c) (d) Fig. 4.4. Stress plots from the edge of a hole. Each stress plot from (a) to (d) is associated with increase of damage very near the hole along with load increase: (a) no damage state, (b) and (c) progressive damage states, and (d) saturated damage state Stress Distance from the hole edge Stress Distance from the hole edge Stress Distance from the hole edge Stress Distance from the hole edge lc 172 Y.W. Kwon
Chapter 4:Multiscale and Multilevel Modeling of Composites 173 Fig.4.5.Specimen under tensile loading.The figure on the right shows initial cracks formed at the edges of the hole 4.2.4 Study of Microstructural Inhomogeneity Because the multiscale technique presented previously uses the material properties at the constituent material level,i.e.,microscale level,it is easy to model inhomogeneous microstructure,such as a nonuniform particle distribution inside a composite.Even if the overall stiffness of a composite is much less dependent on the actual particle distribution,the effective strength of the composite depends on the particle distribution.For different particle distributions of the same amount of volume fraction,result in different local stresses which control failure at different load levels. An experimental study was conducted for a particulate composite to determine the particle distribution.In this study,a large square specimen was cut into smaller sizes of square specimens subsequently.Then,at each level of cutting,the same sizes of specimens were examined using an X- ray technique to measure the particle volume fraction of every respective specimen.If the particle volume fraction were uniform in the original specimen,all smaller specimens would have the same particle volume fraction.However,as expected in a real specimen,there was a deviation of the particle volume fraction which is directly related to the mean intensity of the X-ray passing through each composite specimen.Therefore,the standard deviation was computed for each size of small samples.The study indicated that as the specimen size became smaller than a critical size,the standard deviation of the mean intensity of X-ray began to increase significantly.This result implies that an average particle distribution is quite uniform over a domain size greater than a critical size. To model such an inhomogeneity of particle distribution,a composite specimen was divided into a number of domains with each domain size equal to the critical size.Then,the particle volume fraction was assumed to be the same for each domain.Every domain was further divided into much smaller subdomains.Particle volume fractions were varied randomly
Fig. 4.5. Specimen under tensile loading. The figure on the right shows initial cracks formed at the edges of the hole 4.2.4 Study of Microstructural Inhomogeneity Because the multiscale technique presented previously uses the material properties at the constituent material level, i.e., microscale level, it is easy to model inhomogeneous microstructure, such as a nonuniform particle distribution inside a composite. Even if the overall stiffness of a composite is much less dependent on the actual particle distribution, the effective strength of the composite depends on the particle distribution. For different particle distributions of the same amount of volume fraction, result in different local stresses which control failure at different load levels. An experimental study was conducted for a particulate composite to determine the particle distribution. In this study, a large square specimen was cut into smaller sizes of square specimens subsequently. Then, at each level of cutting, the same sizes of specimens were examined using an Xray technique to measure the particle volume fraction of every respective specimen. If the particle volume fraction were uniform in the original specimen, all smaller specimens would have the same particle volume fraction. However, as expected in a real specimen, there was a deviation of the particle volume fraction which is directly related to the mean intensity of the X-ray passing through each composite specimen. Therefore, the standard deviation was computed for each size of small samples. The study indicated that as the specimen size became smaller than a critical size, the standard deviation of the mean intensity of X-ray began to increase significantly. This result implies that an average particle distribution is quite uniform over a domain size greater than a critical size. To model such an inhomogeneity of particle distribution, a composite specimen was divided into a number of domains with each domain size equal to the critical size. Then, the particle volume fraction was assumed to be the same for each domain. Every domain was further divided into much smaller subdomains. Particle volume fractions were varied randomly Chapter 4: Multiscale and Multilevel Modeling of Composites 173
174 Y.W.Kwon Uniform Nonuniform Strain Fig.4.6.Comparison of stress-strain curves for uniform and nonuniform particle distribution cases 0.9 0.6 0.2 (a) Fig.4.7.(continue)
Strain Stress Uniform Nonuniform Fig. 4.6. Comparison of stress–strain curves for uniform and nonuniform particle distribution cases Fig. 4.7. (continue) 174 Y.W. Kwon