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