Chapter 8:Nested Nonlinear Multiscale Frameworks for the Analysis of Thick-Section Composite Materials and Structures Rami Haj-Ali Georgia Institute of Technology,Atlanta,GA 30332-0355,USA rami.haj-ali@ce.gatech.edu 8.1 Introduction This chapter presents nonlinear and time-dependent multiscale frameworks for the analysis of thick-section and multilayered composite materials and structures.Nested and hierarchical three-dimensional (3D)micromechanical models are formulated within the nonlinear analysis framework.The constitutive framework is composed of nonlinear material models for the matrix behavior,micromodels for the unidirectional lamina,and a sublaminate model for a repeating ply-stacking sequence.A unified development of a class of constant deformation cell(CDC)micromodels is presented to generate the effective nonlinear response of a unidirectional lamina from the response of its matrix and fiber constituents (subcells) Two structural modeling approaches for nonlinear analysis of laminated composites are proposed using 3D and shell nonlinear finite element(FE) analysis.The first,for the analysis of multilayered and thick-section composites,uses the 3D sublaminate model coupled with 3D FE structural models.The sublaminate represents the nonlinear effective continuum response of a through-thickness repeating stacking sequence at the FE material points(Gaussian integration points).The CDC micromodels can be employed for the different layers within the sublaminate model.The second structural approach is used for the analysis of thin-section laminated composite plates and shells in the form of a ply-by-ply.In this
Chapter 8: Nested Nonlinear Multiscale Frameworks for the Analysis of Thick-Section Composite Materials and Structures Rami Haj-Ali Georgia Institute of Technology, Atlanta, GA 30332-0355, USA rami.haj-ali@ce.gatech.edu 8.1 Introduction This chapter presents nonlinear and time-dependent multiscale frameworks for the analysis of thick-section and multilayered composite materials and structures. Nested and hierarchical three-dimensional (3D) micromechanical models are formulated within the nonlinear analysis framework. The sublaminate model for a repeating ply-stacking sequence. A unified development of a class of constant deformation cell (CDC) micromodels is presented to generate the effective nonlinear response of a unidirectional lamina from the response of its matrix and fiber constituents (subcells). Two structural modeling approaches for nonlinear analysis of laminated composites are proposed using 3D and shell nonlinear finite element (FE) analysis. The first, for the analysis of multilayered and thick-section composites, uses the 3D sublaminate model coupled with 3D FE structural models. The sublaminate represents the nonlinear effective continuum response of a through-thickness repeating stacking sequence at the FE material points (Gaussian integration points). The CDC micromodels can be employed for the different layers within the sublaminate model. The second structural approach is used for the analysis of thin-section laminated composite plates and shells in the form of a ply-by-ply. In this the matrix behavior, micromodels for the unidirectional lamina, and a constitutive framework is composed of nonlinear material models for
318 R.Haj-Ali case,the micromodels are used to represent the effective response of each layer.New stress-update solution algorithms are developed for the micromodels and the sublaminate model;they are well suited for non- linear displacement-based FE.Different applications are presented and comparisons are made with reported experimental results.The proposed micromodels are shown to be very capable of predicting the response of different composite materials and structural systems,such as multilayered laminated composites and thick-section pultruded composites.The numerical stress-update algorithms are shown to be well behaved and robust.Applications presented using the proposed frameworks indicate their suitability as practical,general material,and structural analysis tools. Unlike traditional structural materials,such as metals,composite materials add a new and exciting dimension to the engineering design process.Their effective material properties and strengths can be controlled based on the choice of the matrix and fiber materials,volume fractions, and multiaxial reinforcements,along with several other material,geometry and manufacturing parameters.Proper selection of these parameters in the design process can lead to an optimal structural design,such as a structure with minimum weight and a maximum resistance to the applied forces. Composite materials are widely used in high-performance structures where high stiffness and strength combined with low weight are required. Today,many structural components are made from composite materials, especially in the aviation industry.However,it is still rare to find a complete structure that is fully made of composite materials.This indicates that the analysis,design,and manufacturing of composite structures have not yet fully reached a satisfactory level of reliability.Therefore,there is still a need to improve and introduce new analysis and design approaches that can predict the nonlinear and damage behavior of composites. Recently,the use of composite technology in civil and infrastructure applications,such as bridges and construction joints,has been advocated. However,there are two major obstacles standing in the way:the relatively high manufacturing cost and the lack of sufficient predictive models to provide information on the behavior of such structures over their lifespan. Nevertheless,in some cases,the relatively high cost of using composite materials can be justified.For example,the use of composite materials in bridges can eliminate the need to reinforce the concrete with steel bars that are subject to corrosion,thereby prolonging the lifespan of the bridge
comparisons are made with reported experimental results. The proposed micromodels are shown to be very capable of predicting the response of different composite materials and structural systems, such as multilayered laminated composites and thick-section pultruded composites. The numerical stress-update algorithms are shown to be well behaved and robust. Applications presented using the proposed frameworks indicate their suitability as practical, general material, and structural analysis tools. Unlike traditional structural materials, such as metals, composite materials add a new and exciting dimension to the engineering design process. Their effective material properties and strengths can be controlled based on the choice of the matrix and fiber materials, volume fractions, and multiaxial reinforcements, along with several other material, geometry and manufacturing parameters. Proper selection of these parameters in the design process can lead to an optimal structural design, such as a structure with minimum weight and a maximum resistance to the applied forces. Composite materials are widely used in high-performance structures where high stiffness and strength combined with low weight are required. Today, many structural components are made from composite materials, especially in the aviation industry. However, it is still rare to find a complete structure that is fully made of composite materials. This indicates that the analysis, design, and manufacturing of composite structures have not yet fully reached a satisfactory level of reliability. Therefore, there is still a need to improve and introduce new analysis and design approaches that can predict the nonlinear and damage behavior of composites. Recently, the use of composite technology in civil and infrastructure applications, such as bridges and construction joints, has been advocated. However, there are two major obstacles standing in the way: the relatively high manufacturing cost and the lack of sufficient predictive models to provide information on the behavior of such structures over their lifespan. Nevertheless, in some cases, the relatively high cost of using composite materials can be justified. For example, the use of composite materials in bridges can eliminate the need to reinforce the concrete with steel bars that are subject to corrosion, thereby prolonging the lifespan of the bridge R. Haj-Ali micromodels and the sublaminate model; they are well suited for nonlinear displacement-based FE. Different applications are presented and case, the micromodels are used to represent the effective response of each layer. New stress-update solution algorithms are developed for the 318
Chapter 8:Nested Nonlinear Multiscale Frameworks 319 which can compensate for the higher cost of the composite structure.In addition,it is evident that advances in mass manufacturing of composite materials will drive costs down.This provides additional incentive to continue the research on the behavior of composite structures in civil and infrastructure applications. The tremendous advances in computer technology that have taken place over the last two decades have made possible the development of computational tools that routinely employ nonlinear analysis for practical engineering applications.The use of nonlinear stress-strain relations,such as those provided by plasticity and other inelastic models,is now considered a standard engineering practice.However,nonlinear structural modeling approaches that use 3D analysis are not widespread for laminated composites.This is due to many factors.Laminated composites are often considered as brittle materials without accounting for their nonlinear behavior.Therefore,elastic structural analysis and design are often considered sufficient.Furthermore,many laminated structures are thin shell structures that can be idealized using plane-stress constitutive models.However,nonlinear 3D structural analyses may be needed to produce reliable structural designs.Even in the case of thin shell structures,a realistic nonlinear 3D constitutive model is needed to depict accurately the structural response in the presence of edge effects and structural discontinuities.These discontinuities,such as crack tips,holes, and cutouts,usually have a significant impact on the response of the structure,because damage will typically initiate at and propagate from these locations.Therefore,it is important to develop nonlinear and three- dimensional material models to properly simulate the structural behavior with local nonlinear and damage responses. Macroscale nonlinear constitutive models can be formulated directly at the lamina level.On the other hand,micromechanical models of nonlinear lamina behavior,which explicitly recognize the fiber and matrix con- stituents,are appealing because they can provide more detailed response information than macromechanical models.They are also potentially simpler to formulate because they operate at a more fundamental level than macromechanical models.However,the direct use of micromechanical models in practical nonlinear analysis of laminated structures requires compromise between accuracy and computational effort. This chapter reviews multiscale material and structural frameworks that allow the application of several micromechanical models while
computational tools that routinely employ nonlinear analysis for practical engineering applications. The use of nonlinear stress–strain relations, such as those provided by plasticity and other inelastic models, is now considered a standard engineering practice. However, nonlinear structural modeling approaches that use 3D analysis are not widespread for laminated composites. This is due to many factors. Laminated composites are often considered as brittle materials without accounting for their nonlinear behavior. Therefore, elastic structural analysis and design are often considered sufficient. Furthermore, many laminated structures are thin shell structures that can be idealized using plane-stress constitutive models. However, nonlinear 3D structural analyses may be needed to produce reliable structural designs. Even in the case of thin shell structures, a realistic nonlinear 3D constitutive model is needed to depict accurately the structural response in the presence of edge effects and structural discontinuities. These discontinuities, such as crack tips, holes, and cutouts, usually have a significant impact on the response of the structure, because damage will typically initiate at and propagate from these locations. Therefore, it is important to develop nonlinear and threedimensional material models to properly simulate the structural behavior with local nonlinear and damage responses. Macroscale nonlinear constitutive models can be formulated directly at the lamina level. On the other hand, micromechanical models of nonlinear lamina behavior, which explicitly recognize the fiber and matrix constituents, are appealing because they can provide more detailed response information than macromechanical models. They are also potentially simpler to formulate because they operate at a more fundamental level than macromechanical models. However, the direct use of micromechanical models in practical nonlinear analysis of laminated structures requires compromise between accuracy and computational effort. This chapter reviews multiscale material and structural frameworks that allow the application of several micromechanical models while The tremendous advances in computer technology that have taken place over the last two decades have made possible the development of materials will drive costs down. This provides additional incentive to continue the research on the behavior of composite structures in civil and infrastructure applications. which can compensate for the higher cost of the composite structure. In addition, it is evident that advances in mass manufacturing of composite Chapter 8: Nested Nonlinear Multiscale Frameworks 319
320 R.Haj-Ali performing nonlinear structural analysis.A class of simple 3D micro- models that strike a reasonable balance between accuracy and simpli- city is reviewed.These nonlinear micromechanical models,e.g.,for a unidirectional lamina,are incorporated into a hierarchical framework that is suitable for FE analysis.The structural analysis includes both nonlinear material and geometric effects.The hierarchical nature of this framework allows the use of several alternative combinations of material and structural modeling approaches.The nonlinear material behavior can arise from different sources:matrix nonlinear constitutive behavior,micro- failure effects,e.g.,matrix microcracking,fiber-matrix debonding,and fiber failure,e.g.,fiber buckling.Several examples of structural analyses are presented and compared with experimental results where possible. 8.2 Multiscale Analysis of Laminated Composite Structures A general 3D multiscale framework is proposed for the nonlinear analysis of laminated composite structures.Figure 8.1 illustrates the proposed analysis framework for multilayered structures using 3D or shell-based structural FE models.In the case of a 3D FE structural model,a sublaminate model is formulated to represent the nonlinear effective continuum response at each material point(Gaussian point)[24,25,33, 34].The sublaminate model is used to generate a 3D through-thickness effective response of a representative stacking sequence. In the case of shell elements,Fig.8.1 illustrates that each layer is explicitly modeled with one or more integration points under plane-stress condition;and the sublaminate model is reduced to the classical lamination theory in this case.Constant transverse shear,cross-sectional stiffness is assumed for the shell elements.This assumption is valid where the trans- verse stresses in the different layers are very small compared to the in-plane stresses.The 3D micromechanical models provide for the effective nonlinear constitutive behavior for each Gaussian point.The shell element's effective through-thickness response is generated at select integration points on its reference surface by integrating the effective micromechanical response over all Gaussian points,as shown in Fig.8.1
a unidirectional lamina, are incorporated into a hierarchical framework that is suitable for FE analysis. The structural analysis includes both nonlinear material and geometric effects. The hierarchical nature of this framework allows the use of several alternative combinations of material and structural modeling approaches. The nonlinear material behavior can arise 8.2 Multiscale Analysis of Laminated Composite Structures A general 3D multiscale framework is proposed for the nonlinear analysis of laminated composite structures. Figure 8.1 illustrates the proposed analysis framework for multilayered structures using 3D or shell-based structural FE models. In the case of a 3D FE structural model, a sublaminate model is formulated to represent the nonlinear effective continuum response at each material point (Gaussian point) [24, 25, 33, 34]. The sublaminate model is used to generate a 3D through-thickness effective response of a representative stacking sequence. In the case of shell elements, Fig. 8.1 illustrates that each layer is explicitly modeled with one or more integration points under plane-stress condition; and the sublaminate model is reduced to the classical lamination theory in this case. Constant transverse shear, cross-sectional stiffness is in-plane stresses. The 3D micromechanical models provide for the effective nonlinear constitutive behavior for each Gaussian point. The shell element’s effective through-thickness response is generated at select integration points on its reference surface by integrating the effective micromechanical response over all Gaussian points, as shown in Fig. 8.1. R. Haj-Ali from different sources: matrix nonlinear constitutive behavior, microfailure effects, e.g., matrix microcracking, fiber–matrix debonding, and fiber failure, e.g., fiber buckling. Several examples of structural analyses are presented and compared with experimental results where possible. models that strike a reasonable balance between accuracy and simplicity is reviewed. These nonlinear micromechanical models, e.g., for performing nonlinear structural analysis. A class of simple 3D microassumed for the shell elements. This assumption is valid where the transverse stresses in the different layers are very small compared to the 320
Chapter 8:Nested Nonlinear Multiscale Frameworks 321 Multi-Scale Nonlinear Analysis Framework of Laminated Composites Shell model 3D model Shell element 3D brick element Material point 3 Material point Homogenized (Piane Stress) Through-thickness homogenization /Homogeniced /(3D) Sublaminate Model Lamina Homogenized (3D) 3 Homogenizcd Micromechanical model (Plane Stress) Fiber Matrix Homogenized (3D) Fig.8.1.A multiscale micromechanical-structural framework for nonlinear and viscoelastic analysis of laminated composite structures(adapted from [23])
Fig. 8.1. A multiscale micromechanical–structural framework for nonlinear and viscoelastic analysis of laminated composite structures (adapted from [23]) Chapter 8: Nested Nonlinear Multiscale Frameworks 321
322 R.Haj-Ali 8.3 A Simplified Class of Micromechanical Constitutive Models A unified approach for defining and characterizing a simple and phenomenological class of nonlinear micromechanical models for fiber composites is presented in this section.A unified development of a class of CDC micromodels,or unit cell (UC),is presented to generate the effective nonlinear continuum response from the average response of its matrix and fiber constituents(subcells).The main advantage of these simple multicell models lies in their ability to generate the full 3D effective stress-strain response of fiber composites in a form that is suitable for integration into finite element structural analysis.The first part of this section sets out some general definitions and relations that are valid for all the micro- models in this class.Specific micromodels are presented in the later part of this section and through this chapter. The objective of the CDC models is to generate the nonlinear effective stress-strain relations by employing a simple geometrical representation of the unit cell geometry and satisfy traction and displacement continuity between the cells in an average sense.Few assumptions are made at this stage regarding the fiber and matrix constitutive relations;specific material nonlinear constitutive behavior is characterized only at the more fundamental subcell level.The resulting unit cell effective stress-strain relations can be viewed,from a global/structural perspective,as a material model with microstructural constraints. It is assumed that,for a given heterogeneous periodic medium,it is possible to define a basic unit cell that represents the medium's geo- metrical and material characteristics.Each unit cell is divided into a number of subcells.Within each subcell,the spatial variation of the displacement field is assumed such that the stresses and deformations are spatially uniform in each subcell.Traction continuity at an interface between subcells can,therefore,be satisfied only in an average sense. Some general definitions and linearized formulations are established that are applicable to any CDC micromodel. The volume average stress over the unit cell is defined as 元，=，e业-2ea=2 (8.1)
8.3 A Simplified Class of Micromechanical Constitutive Models A unified approach for defining and characterizing a simple and phenomenological class of nonlinear micromechanical models for fiber composites is presented in this section. A unified development of a class of CDC micromodels, or unit cell (UC), is presented to generate the effective nonlinear continuum response from the average response of its matrix and fiber constituents (subcells). The main advantage of these simple multicell models lies in their ability to generate the full 3D effective stress–strain response of fiber composites in a form that is suitable for integration into finite element structural analysis. The first part of this section sets out The objective of the CDC models is to generate the nonlinear effective stress–strain relations by employing a simple geometrical representation of the unit cell geometry and satisfy traction and displacement continuity between the cells in an average sense. Few assumptions are made at this stage regarding the fiber and matrix constitutive relations; specific material nonlinear constitutive behavior is characterized only at the more fundamental subcell level. The resulting unit cell effective stress–strain relations can be viewed, from a global/structural perspective, as a material model with microstructural constraints. It is assumed that, for a given heterogeneous periodic medium, it is number of subcells. Within each subcell, the spatial variation of the displacement field is assumed such that the stresses and deformations are spatially uniform in each subcell. Traction continuity at an interface between subcells can, therefore, be satisfied only in an average sense. Some general definitions and linearized formulations are established that are applicable to any CDC micromodel. The volume average stress over the unit cell is defined as ( ) () () ( ) ( ) ( ) 1 1 11 1 ( )d ( )d , N N ij ij ij ij V V x x V V V VV V α α α α α α α α σσ σ σ = = == = ∫ ∫ ∑ ∑ (8.1) R. Haj-Ali some general definitions and relations that are valid for all the micromodels in this class. Specific micromodels are presented in the later part of this section and through this chapter. possible to define a basic unit cell that represents the medium’s geometrical and material characteristics. Each unit cell is divided into a 322
Chapter 8:Nested Nonlinear Multiscale Frameworks 323 where N is the number of subcells and I is the unit cell volume.A similar definition applies for volume average strain The superscript a denotes the subcell number.An overbar denotes a unit cell average quantity.The variables x andx are the unit cell global and the subcell local coordinates,respectively.Stress and strain are uniform within each subcell by definition.Therefore,using matrix notation N 6= P F= (8.2) a=l where the stresses and strains are now written as vectors. Next,a strain-concentration or strain-interaction fourth rank tensor B is defined for each subcell,which relates the subcell strain increment to the unit cell average strain increment de=BdEu. (8.3) It is important to emphasize that the interaction matrices are unknown at this stage;they will be determined later in this section by solution of the unit cell governing equations.It can be easily shown that a subcell strain- interaction matrix is usually a function of the tangent stiffness and the relative volumes of all subcells. Using the incremental form of(8.2)with(8.3),expressed in matrix notation,the average strain increment of the unit cell is: (8.4) Since (8.4)must hold for an arbitrary average strain increment ds,the following relations must be satisfied 2B=1md2.(8-=0 (8.5)
where N is the number of subcells and V is the unit cell volume. A similar definition applies for volume average strain ij ε . The superscript α denotes the subcell number. An overbar denotes a unit cell average quantity. The variables x and ( ) x α are the unit cell global and the subcell local coordinates, respectively. Stress and strain are uniform within each subcell by definition. Therefore, using matrix notation ( ) ( ) ( ) ( ) ( ) 1 11 1 1 , ,, N NN V VV V V V α α α αα α αα σ σε ε = == = == ∑ ∑∑ (8.2) where the stresses and strains are now written as vectors. Next, a strain-concentration or strain-interaction fourth rank tensor B is defined for each subcell, which relates the subcell strain increment to the unit cell average strain increment () () d d. ij ijkl kl B α α ε = ε (8.3) It is important to emphasize that the interaction matrices are unknown at this stage; they will be determined later in this section by solution of the unit cell governing equations. It can be easily shown that a subcell straininteraction matrix is usually a function of the tangent stiffness and the relative volumes of all subcells. Using the incremental form of (8.2) with (8.3), expressed in matrix notation, the average strain increment of the unit cell is: ( ) ( ) ( ) ( ) 1 1 1 1 d d d. N N V V B V V α α α α α α ε ε ε = = = = ∑ ∑ (8.4) Since (8.4) must hold for an arbitrary average strain increment dε , the following relations must be satisfied ( ) ( ) ( ) ( ) 1 1 1 and ( ) 0, N N VB I V B I V α α α α α α = = ∑ ∑ = − = (8.5) Chapter 8: Nested Nonlinear Multiscale Frameworks 323
324 R.Haj-Ali where is a unit matrix.The second relation in (8.5)follows from the first relation due to the volume sum relation expressed in (8.2).The matrix representation of the strain-concentration tensor is not symmetric.Next, the incremental stress-strain relations are used to express the stress increment in each of the subcells do(a)=c(@dg(a)=c(aB(adE, (8.6) where C)is the current tangent stiffness matrix of the subcell.The incremental form of the average stress can be expressed,using(8.6),as: (8.7) Equation(8.7)can be expressed as do=C'ds, (8.8) where C*is the unit cell effective tangent stiffness matrix defined by: C(aB(a) (8.9) An alternative for deriving the stiffness matrix is to use the second variation of the strain energy density.This is demonstrated by the following relations: de'da de dedg.(.0) Substituting (8.8)into the left-hand side of(8.10),the unit cell stiffness matrix is expressed as:
where I is a unit matrix. The second relation in (8.5) follows from the first relation due to the volume sum relation expressed in (8.2). The matrix representation of the strain-concentration tensor is not symmetric. Next, the incremental stress–strain relations are used to express the stress increment in each of the subcells () () () () () d d d, C CB α α α αα σ = = ε ε (8.6) where ( ) C α is the current tangent stiffness matrix of the subcell. The incremental form of the average stress can be expressed, using (8.6), as: ( ) () () ( ) ( ) 1 1 1 1 d d . N N V VC B V V α α α α α α α σ σ ε = = = = ∑ ∑ (8.7) Equation (8.7) can be expressed as * d d, σ = C ε (8.8) where C* is the unit cell effective tangent stiffness matrix defined by: * ( ) ( ) ( ) 1 1 . N C VC B V α α α α = = ∑ (8.9) An alternative for deriving the stiffness matrix is to use the second variation of the strain energy density. This is demonstrated by the following relations: T T T () () T () () () ( ) ( ) 1 1 1 1 dd d d d d . N N V V B C B V V α α α αα α α α α ε σ εσ ε ε = = ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ ∑ ∑ (8.10) Substituting (8.8) into the left-hand side of (8.10), the unit cell stiffness matrix is expressed as: 324 R. Haj-Ali
Chapter 8:Nested Nonlinear Multiscale Frameworks 325 B()C()B(a) (8.11) The two expressions for the effective stiffness matrix in (8.9)and (8.11) must be identical.It can be easily verified that,since the strain- concentration matrices B)satisfy the relations in(8.5),the two stiffness expressions are,in fact,identical.Equation (8.11)shows that the unit cell stiffness matrix C*is symmetric provided that the stiffness matrix of each of the subcells C()is also symmetric.However,it is interesting to note that this property is not explicitly apparent by a first examination of the expression in (8.9). Up to this stage,the properties of the strain-interaction matrices and the expression for the unit cell effective stiffness matrix have been dealt with.The only assumption that was made is that the subcells have uniform stress and strain.Therefore,these linearized relations are general for any CDC micromodel.To derive the strain-interaction matrices for a unit cell, the traction and displacement continuity conditions must be imposed,and stress-strain relations must be invoked.The fact that the strains and stresses are uniform in every subcell makes it possible to express the traction and displacement continuity conditions directly in terms of the average stress and strain vectors.The term strain compatibility will be used here to describe the relations between the strains in the subcells which satisfy displacement continuity in an average fashion.The combined set of equations that describe the strain compatibility and the traction continuity equations(micromechanical constraints)can ultimately be written in a general incremental form as: dR.=C,(doa,dea,dE,'a,a=1,2,,N)=0,i=l,,n.(8.12) Equation(8.12)is used to generate the strain-interaction matrices for the subcells.The incremental form of the stress-strain relations in the subcells (8.6)is used to express the constraints in terms of the incremental strains dR=E,(Ca,dea,dE,'a,a=1,2,,N)=0,j=l,m.(8.13)
T * ( ) ( ) ( ) ( ) 1 1 . N C VB CB V α α α α α = = ∑ (8.11) The two expressions for the effective stiffness matrix in (8.9) and (8.11) must be identical. It can be easily verified that, since the strainconcentration matrices ( ) B α satisfy the relations in (8.5), the two stiffness expressions are, in fact, identical. Equation (8.11) shows that the unit cell stiffness matrix C* is symmetric provided that the stiffness matrix of each of the subcells ( ) C α is also symmetric. However, it is interesting to note that this property is not explicitly apparent by a first examination of the expression in (8.9). Up to this stage, the properties of the strain-interaction matrices and the expression for the unit cell effective stiffness matrix have been dealt with. The only assumption that was made is that the subcells have uniform stress and strain. Therefore, these linearized relations are general for any CDC micromodel. To derive the strain-interaction matrices for a unit cell, the traction and displacement continuity conditions must be imposed, and stress–strain relations must be invoked. The fact that the strains and stresses are uniform in every subcell makes it possible to express the traction and displacement continuity conditions directly in terms of the average stress and strain vectors. The term strain compatibility will be used here to describe the relations between the strains in the subcells which satisfy displacement continuity in an average fashion. The combined set of equations that describe the strain compatibility and the traction continuity equations (micromechanical constraints) can ultimately be written in a general incremental form as: () () ( ) d (d ,d ,d , , 1,2, , ) 0, 1, , . R C V Nin i α α σ α = = σεε α … = = … (8.12) Equation (8.12) is used to generate the strain-interaction matrices for the subcells. The incremental form of the stress–strain relations in the subcells (8.6) is used to express the constraints in terms of the incremental strains () () ( ) d ( ,d ,d , , 1,2, , ) 0, 1, , . R EC V j N j m α α ε α = = εε α … = = … (8.13) Chapter 8: Nested Nonlinear Multiscale Frameworks 325
326 R.Haj-Ali The subset of (8.13)that represents the strain compatibility constraints satisfies(8.4).Equation(8.13)forms a set of linear equations in terms of the unknown incremental strain vectors for each of the subcells.The current state of the linearized micromechanical equations can be arranged in terms of these unknowns and the known values,the current tangent stiffness matrices,and the unit cell strain vector,and represented,in a general matrix form,as: de(2) A D {d}. (8.14) 6x1 ds(w) 6Nxl 6Nx6N 6Nx6 Equation(8.14)can be rearranged by dividing the subcells'strain components into two dependent groups with (m)and (n)number of components,respectively,to yield a new compact form that can be solved numerically in an efficient manner.The general structure of the linearized micromechanical equations for the CDC class of micromodels is: dr ab D (mx1) (8.15) (nx6) The bar notation over the components of the (4)matrix denotes the new arrangement of the terms of the original matrix.Once (8.14)or (8.15)is solved,the incremental stress in each of the subcells and the average stress of the unit cell can be back-calculated using the incremental stress-strain relations.The incremental strain-concentration matrices are expressed, using(8.3)and (8.14),by
The subset of (8.13) that represents the strain compatibility constraints satisfies (8.4). Equation (8.13) forms a set of linear equations in terms of the unknown incremental strain vectors for each of the subcells. The current state of the linearized micromechanical equations can be arranged in terms of these unknowns and the known values, the current tangent stiffness matrices, and the unit cell strain vector, and represented, in a general matrix form, as: (1) (2) 6 1 ( ) 6 1 6 6 6 6 d d {d }. d N N N N N A D ε ε ε ε × × × × ⎡ ⎤⎡ ⎤ ⎧ ⎫ ⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎥ ⎪ ⎪ ⎨ ⎬ = ⎥ ⎪ ⎪ ⎥ ⎪ ⎪ ⎥ ⎩ ⎭ ⎥ ⎣ ⎦⎣ ⎦ # (8.14) Equation (8.14) can be rearranged by dividing the subcells’ strain components into two dependent groups with (m) and (n) number of components, respectively, to yield a new compact form that can be solved numerically in an efficient manner. The general structure of the linearized micromechanical equations for the CDC class of micromodels is: { } ( ) ( 1) ( ) ( 1) ( 6) (6 1) ( 1) ( 1) ( 6) ( ) () d d d . d d 0 ab a m m a m m m n m ba bb b n n nm nn n R I A D R A A ε σ ε ε ε × × × × × × × × × × × ⎧ ⎫ ⎧⎫ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ ⎪⎪ ⎢ ⎥ ⎢ ⎥ ⎨ ⎬ ⎨⎬ = = ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎩⎭ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ (8.15) The bar notation over the components of the (A) matrix denotes the new arrangement of the terms of the original matrix. Once (8.14) or (8.15) is solved, the incremental stress in each of the subcells and the average stress of the unit cell can be back-calculated using the incremental stress–strain relations. The incremental strain-concentration matrices are expressed, using (8.3) and (8.14), by 326 R. Haj-Ali