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