Chapter 4:Multiscale and Multilevel Modeling of Composites Young W.Kwon Naval Postgraduate School,Monterey,CA,USA 4.1 Introduction Composites have been used increasingly in various engineering applications which include,but are not limited to,the aerospace,automobile,sports,and leisure industries.To improve properties of composites so that they become stronger,stiffer,tougher,refractory,etc.,it would be very useful to design the composite materials from the atomic levels.This requires proper multiscale and multilevel modeling techniques so that those techniques can be used for the design stage of new composites as well as the analysis of existing composites.This chapter presents multiscale and multilevel modeling techniques for different kinds of composite architectures which include particle-reinforced,fiber-reinforced,and woven fabric composites The following sections describe these techniques. 4.2 Particulate Composites 4.2.1 Multiscale Analysis for Particulate Composites A particle-reinforced composite,or particulate composite,is one of the simplest forms of composites.It has particles embedded in a matrix material. As a result,the multiscale analysis hierarchy is simple for the particulate composite,as illustrated in Fig.4.1.The analysis connects the microscale, such as particles and matrix,to the mesoscale,such as the representative particulate composite,and finally to macroscale composites,such as a particulate composite structure [13,23-32].The multiscale analysis has
Chapter 4: Multiscale and Multilevel Modeling of Composites Young W. Kwon Naval Postgraduate School, Monterey, CA, USA 4.1 Introduction Composites have been used increasingly in various engineering applications which include, but are not limited to, the aerospace, automobile, sports, and leisure industries. To improve properties of composites so that they become stronger, stiffer, tougher, refractory, etc., it would be very useful to design the composite materials from the atomic levels. This requires proper multiscale and multilevel modeling techniques so that those techniques can be used for the design stage of new composites as well as the analysis of existing composites. This chapter presents multiscale and multilevel modeling techniques for different kinds of composite architectures which include particle-reinforced, fiber-reinforced, and woven fabric composites. The following sections describe these techniques. 4.2 Particulate Composites 4.2.1 Multiscale Analysis for Particulate Composites A particle-reinforced composite, or particulate composite, is one of the simplest forms of composites. It has particles embedded in a matrix material. As a result, the multiscale analysis hierarchy is simple for the particulate composite, as illustrated in Fig. 4.1. The analysis connects the microscale, such as particles and matrix, to the mesoscale, such as the representative particulate composite, and finally to macroscale composites, such as a particulate composite structure [13, 23–32]. The multiscale analysis has
166 Y.W.Kwon two routes which complement each other for a complete cycle of analysis. The first route is the Stiffness Loop,and the other is the Stress Loop.For the Stiffness Loop,effective material properties are computed for an upper scale from material and geometric properties of the neighboring lower scale.For example,the effective particulate composite material properties are calculated from the particle and matrix material and geometric properties.The Particulate Module computes the effective properties,and it is described later.Then,the effective material properties are used for structural analysis of a composite as illustrated in Fig.4.1. Stiffness Loop Particulate Finite Ele- Module ment Analysis Microlevel Mesolevel Macrolevel (particles and (particulate (composite matrix) composite) structure) Particulate Finite Ele- Module ment Analysis Stress Loop Fig.4.1.Multiscale analysis hierarchy for a particulate composite Structural analysis of the composite results in stresses,strains,and displacements at the macroscale.The stresses and strains are the composite level values,i.e.,smeared values for the particles and the matrix.It is necessary to decompose the composite structural level stresses and strains into constituent level stresses and strains to apply the damage or failure criteria to the constituent materials,such as the particles and matrix.The same module used for the Stiffness Loop,i.e.,Particulate Module is also used to compute the stresses and strains in the particle and matrix
two routes which complement each other for a complete cycle of analysis. The first route is the Stiffness Loop, and the other is the Stress Loop. For the Stiffness Loop, effective material properties are computed for an upper scale from material and geometric properties of the neighboring lower scale. For example, the effective particulate composite material properties are calculated from the particle and matrix material and geometric properties. The Particulate Module computes the effective properties, and it is described later. Then, the effective material properties are used for structural analysis of a composite as illustrated in Fig. 4.1. Fig. 4.1. Multiscale analysis hierarchy for a particulate composite Structural analysis of the composite results in stresses, strains, and displacements at the macroscale. The stresses and strains are the composite level values, i.e., smeared values for the particles and the matrix. It is necessary to decompose the composite structural level stresses and strains into constituent level stresses and strains to apply the damage or failure criteria to the constituent materials, such as the particles and matrix. The same module used for the Stiffness Loop, i.e., Particulate Module is also used to compute the stresses and strains in the particle and matrix. Microlevel (particles and matrix) Mesolevel (particulate composite) Macrolevel (composite structure) Particulate Module Finite Element Analysis Finite Element Analysis Particulate Module Stiffness Loop Stress Loop 166 Y.W. Kwon
Chapter 4:Multiscale and Multilevel Modeling of Composites 167 Sections 4.2.2 and 4.2.3 describe,respectively,the Particulate Module and the damage mechanics and crack initiation criterion used for the study. 4.2.2 Particulate Module A representative unit cell is used for the present module.The purpose of this module is twofold.The first is to predict the effective stiffness of a particulate composite from the particle and matrix material properties as well as their geometric data.The second is to determine the microlevel stresses and strains occurring in the particle and matrix from the stresses and strains of the composite level.As a result,the module is used for both the Stiffness Loop and the Stress Loop. To develop a representative unit cell for a particulate composite,a single representing particle surrounded by a matrix material is assumed.In general,every particle may have a different shape;however,the shape of the representative particle is simplified.A spherical shape would be a reasonable assumption.However,for mathematical simplicity,a cubic shape is assumed. A microscale analysis of different shapes of particles using the boundary element method [20]showed that the effective material properties of the composite were insensitive to the actual shape of the particle.However, the same study indicated that the microscale stresses were rather sensitive to the particle shape.The sensitivity was mostly due to the stress concentration at sharp comners.For actual composites,each particle has a different shape and stress concentration.Practically,there is no way to consider all those different particle shapes and their stress concentration effects.A possible solution to this complex problem is using statistical mechanics.However, that approach is also very time-consuming.If it is assumed that the macro- level failure strength is more or less uniform for test coupons made out of the same particulate composite,the local stress concentration effects due to different shapes of particles may be smeared out in the composite test coupons.In this regard,a more regular shape of particle in the representative unit cell may be considered.Furthermore,the average values of stresses for the representative particle will be computed.This also makes the actual shape of particle less relevant for the unit cell. Figure 4.2 shows the representative unit cell.With the assumption of symmetry,only one-eighth of the unit cell is shown in the figure,where the representative embedded particle is denoted by subcell a.The surrounding matrix material is represented by subcells b to h as illustrated in the figure.To clearly represent the relative positions of all subcells within the unit cell,the subcells are shown independently in the figure with
Sections 4.2.2 and 4.2.3 describe, respectively, the Particulate Module and the damage mechanics and crack initiation criterion used for the study. 4.2.2 Particulate Module A representative unit cell is used for the present module. The purpose of this module is twofold. The first is to predict the effective stiffness of a particulate composite from the particle and matrix material properties as well as their geometric data. The second is to determine the microlevel stresses and strains occurring in the particle and matrix from the stresses and strains of the composite level. As a result, the module is used for both the Stiffness Loop and the Stress Loop. To develop a representative unit cell for a particulate composite, a single representing particle surrounded by a matrix material is assumed. In general, every particle may have a different shape; however, the shape of the representative particle is simplified. A spherical shape would be a reasonable assumption. However, for mathematical simplicity, a cubic shape is assumed. A microscale analysis of different shapes of particles using the boundary element method [20] showed that the effective material properties of the composite were insensitive to the actual shape of the particle. However, the same study indicated that the microscale stresses were rather sensitive to the particle shape. The sensitivity was mostly due to the stress concentration at sharp corners. For actual composites, each particle has a different shape and stress concentration. Practically, there is no way to consider all those different particle shapes and their stress concentration effects. A possible solution to this complex problem is using statistical mechanics. However, that approach is also very time-consuming. If it is assumed that the macrolevel failure strength is more or less uniform for test coupons made out of the same particulate composite, the local stress concentration effects due to different shapes of particles may be smeared out in the composite test coupons. In this regard, a more regular shape of particle in the representative unit cell may be considered. Furthermore, the average values of stresses for the representative particle will be computed. This also makes the actual shape of particle less relevant for the unit cell. Figure 4.2 shows the representative unit cell. With the assumption of symmetry, only one-eighth of the unit cell is shown in the figure, where the representative embedded particle is denoted by subcell a. The surrounding matrix material is represented by subcells b to h as illustrated in the figure. To clearly represent the relative positions of all subcells within the unit cell, the subcells are shown independently in the figure with Chapter 4: Multiscale and Multilevel Modeling of Composites 167
168 Y.W.Kwon lines and springs denoting their connections to neighboring subcells.The lines indicate continuous material between any two neighboring materials while the springs denote any potential interface effect between the particle and the matrix.The spring constant can be adjusted with either a strong or weak interface.The present model can have three different interface material properties along the three directions.However,for an isotropic material and damage behavior,all interface material properties,i.e.,the spring constants,will be assumed to be the same.The size of the particle subcell a is ()3 where p is the particle volume fraction of the composite. 3 Q Fig.4.2.Representative unit cell for a particulate composite For each subcell,average stresses and strains are considered for the following derivation.Stresses must satisfy the equilibrium between any neighboring subcells as shown below ou =ou,on=ouou=on>ou =ou (4.1) 02=02,2=02,02=0i,02=0, (4.2) 0=05,O3=05,O=0,O=0 (4.3) where superscript denotes the subcell identification as shown in Fig.4.2 and subscript indicates the stress component.These equations are for normal
lines and springs denoting their connections to neighboring subcells. The lines indicate continuous material between any two neighboring materials while the springs denote any potential interface effect between the particle and the matrix. The spring constant can be adjusted with either a strong or weak interface. The present model can have three different interface material properties along the three directions. However, for an isotropic material and damage behavior, all interface material properties, i.e., the spring constants, will be assumed to be the same. The size of the particle subcell a is (Vp) 1/3 where Vp is the particle volume fraction of the composite. Fig. 4.2. Representative unit cell for a particulate composite For each subcell, average stresses and strains are considered for the following derivation. Stresses must satisfy the equilibrium between any neighboring subcells as shown below 11 11 11 11 11 11 11 11 ,,,, ab cd e f gh σ ==== σ σσ σσ σσ (4.1) 22 22 22 22 22 22 22 22 ,,,, ac bd eg fh σ ==== σ σσ σσ σσ (4.2) 33 33 33 33 33 33 33 33 ,,,, ae b f cg dh σ ==== σ σσ σσ σσ (4.3) where superscript denotes the subcell identification as shown in Fig. 4.2 and subscript indicates the stress component. These equations are for normal a d b h c f g e 2 1 3 k2 k1 k3 168 Y.W. Kwon
Chapter 4:Multiscale and Multilevel Modeling of Composites 169 components of stresses.Similar equations can be written for shear compo- nents,but they are omitted here to save space. The subcell strains satisfy the following compatibility equations by assuming uniform deformation of the unit cell under periodic boundary conditions 1,+lnc哈+(6oi/k)=l,c所+lns州=l,si+nc=l,c嘴+lm1,(4.4) 1ps品+ln5+(6o2/k)=pe品+ln品=b品+1m8=1,e品+lnm品，(4.5) 1,+1n+(6o1k）=1,6$+1n=1,+1n=1,第+1第，(4.6) where ，=, (4.7) 1m=1-p (4.8) Other necessary mathematical expressions are constitutive equations for the particle and matrix materials as well as for the composite.For the present particulate composite material,both constituent materials and the effective composite material are considered as isotropic materials. Furthermore,the unit cell stresses and strains are assumed to be the volume averages of subcell stresses and strains 可，=∑yo (4.9) a=∑r, (4.10) where superimposed bar denotes the composite (unit cell level)stresses and strains,and "is the volume fraction of the nth subcell.The summation is over all subcells. Algebraic manipulation of the previous equations finally yields the two main expressions as given below [E]=[V][E][R], (4.11) {e}=[R]{E}, (4.12) in which [is the effective composite material property matrix,[V]is the matrix composed of subcell volume fractions,[E]is the matrix consisting of constituent material properties,and [R]is the matrix relating the subcell
components of stresses. Similar equations can be written for shear components, but they are omitted here to save space. The subcell strains satisfy the following compatibility equations by assuming uniform deformation of the unit cell under periodic boundary conditions ( ) 2 p 11 m 11 p 11 1 p 11 m 11 p 11 m 11 p 11 m 11 / , ab a c de fgh l l l kl l l l l l ε + + =+ =+ =+ ε σ εεεεεε (4.4) ( ) 2 p 22 m 22 p 22 2 p 22 m 22 p 22 m 22 p 22 m 22 / , ac a b de g f h l l l kl l l l l l ε ++ =+ =+ =+ ε σ εεεεεε (4.5) ( ) 2 p 33 m 33 p 33 3 p 33 m 33 p 33 m 33 p 33 m 33 / , ae a b fc gd h l l l kl l l l l l ε + + =+ =+ =+ ε σ εεεεεε (4.6) where 1/3 p p l V= , (4.7) m p l l =1 . − (4.8) Other necessary mathematical expressions are constitutive equations for the particle and matrix materials as well as for the composite. For the present particulate composite material, both constituent materials and the effective composite material are considered as isotropic materials. Furthermore, the unit cell stresses and strains are assumed to be the volume averages of subcell stresses and strains , n n ij ij n σ = ∑V σ (4.9) , n n ij ij n ε = ∑V ε (4.10) where superimposed bar denotes the composite (unit cell level) stresses and strains, and n V is the volume fraction of the nth subcell. The summation is over all subcells. Algebraic manipulation of the previous equations finally yields the two main expressions as given below eff [ ] [ ][ ][ ], E VER = (4.11) { } [ ]{ }, ε = R ε (4.12) in which [Eeff] is the effective composite material property matrix, [V] is the matrix composed of subcell volume fractions, [E] is the matrix consisting of constituent material properties, and [R] is the matrix relating the subcell Chapter 4: Multiscale and Multilevel Modeling of Composites 169
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