Chapter 8:Nested Nonlinear Multiscale Frameworks 327 (8.16) B(N) The stress analysis of a micromechanical unit cell becomes a straight- forward procedure as a result of this formulation.Given an average strain increment and the history of deformations in the subcells,the strain- interaction matrices are formed using (8.16).The strain increments are subsequently formed in each of the subcells followed by the corresponding stress increments.This procedure is a linearized incremental stress analysis and will be referred to as the trial state.If only this linearized trial analysis is used,two types of error will result at each trial increment and will accumulate during the analysis.It is important to mention,however,that the strain compatibility and traction continuity constraints are exactly satisfied by the trial state which is composed of tangential approximations. The first error occurs in the strain increments because the strain-interaction matrices are derived using the tangent stiffness matrices of the subcells at the beginning of the increment.The second error occurs as a result of using the tangent stiffness to compute the stress increment.Therefore,a correction scheme must be used to accurately account for the nonlinear constitutive (with or without damage)material behavior (prediction)and its associated error in the incremental micromechanical equations.New general correction algorithms have been derived for different CDC type micromechanical models with nonlinear and time-dependent behavior, e.g,[16,18,20,22-24. A four-cell micromodel is formulated next using the previous tan- gential and stress-update formulations.This model was originally for- mulated using the method of cells(MOCs),e.g.,[2-6].Aboudi's MOC or GMOC has been shown to be well suited for highly nonlinear matrix response,such as that exhibited by metal matrix composites.However, integration of the MOC formulation in general 3D analysis of composite structures can be tremendously enhanced using the proposed numerical formulation because of the large computational effort that is needed to be performed at each material point(Gaussian point)of the FEA.Therefore,it is important to employ the above efficient stress-update and stress- correction formulations for this model that are suitable for nonlinear
(1) (2) 1 ( ) . N B B A D B − ⎡ ⎤ ⎢ ⎥ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ # (8.16) strain increment and the history of deformations in the subcells, the straininteraction matrices are formed using (8.16). The strain increments are subsequently formed in each of the subcells followed by the corresponding stress increments. This procedure is a linearized incremental stress analysis and will be referred to as the trial state. If only this linearized trial analysis is used, two types of error will result at each trial increment and will accumulate during the analysis. It is important to mention, however, that the strain compatibility and traction continuity constraints are exactly satisfied by the trial state which is composed of tangential approximations. The first error occurs in the strain increments because the strain-interaction matrices are derived using the tangent stiffness matrices of the subcells at the beginning of the increment. The second error occurs as a result of using the tangent stiffness to compute the stress increment. Therefore, a correction scheme must be used to accurately account for the nonlinear constitutive (with or without damage) material behavior (prediction) and its associated error in the incremental micromechanical equations. New general correction algorithms have been derived for different CDC type micromechanical models with nonlinear and time-dependent behavior, e.g., [16, 18, 20, 22–24]. The stress analysis of a micromechanical unit cell becomes a straightforward procedure as a result of this formulation. Given an average or GMOC has been shown to be well suited for highly nonlinear matrix response, such as that exhibited by metal matrix composites. However, integration of the MOC formulation in general 3D analysis of composite structures can be tremendously enhanced using the proposed numerical formulation because of the large computational effort that is needed to be performed at each material point (Gaussian point) of the FEA. Therefore, it is important to employ the above efficient stress-update and stresscorrection formulations for this model that are suitable for nonlinear A four-cell micromodel is formulated next using the previous tangential and stress-update formulations. This model was originally formulated using the method of cells (MOCs), e.g., [2–6]. Aboudi’s MOC Chapter 8: Nested Nonlinear Multiscale Frameworks 327
328 R.Haj-Ali X:h h -h 出E () (2) fiber matrix 比比 (3) (4) -b matrix i matrix X2 X) Idealized Lamina with periodic arrays of long fibers(recangular cross-sections) X1 Fig.8.2.Unit cell micromodel for unidirectional reinforced composites structural analysis.Next,an incremental formulation of the four-cell model is presented in terms of the average stresses and strains in the subcells. New stress-update and stress-correction algorithms are developed which significantly reduce the computational effort that is needed.The new algorithms are formulated given a constant average strain rate for each time step,which makes them suitable for integration with FE constitutive framework. The micromechanical model is shown in Fig.8.2.The unidirectional composite,which consists of long fibers arranged unidirectionally in the matrix system,is idealized as doubly periodic array of fibers with rectangular cross-section.A quarter UC that consists of four subcells is modeled due to symmetry.The first subcell is a fiber constituent,while subcells 2-4 represent the matrix constituents.The long fibers are aligned in the xi-direction.The other cross-section directions are referred to as the transverse directions.The x3-direction is called the out-of-plane axis or lamina thickness direction.The total volume of the UC is taken to be equal to one.The volumes of the four subcells are: V=bh,V=h(1-b),V3=b1-h),'4=(1-h)1-b). (8.17) The notations used for the stress and strain vectors are: do)={do11,do22,do3,dt2,dt13,dt23},a=1,,4, (8.18) de={ds,ds22,de3,dy12,dy13,dy23},k=1,,6. The 3D nonlinear constitutive integration for the fiber and matrix constituents is performed separately for each subcell.The fiber is linear elastic and transversely isotropic,while the matrix medium is viscoelastic
structural analysis. Next, an incremental formulation of the four-cell model is presented in terms of the average stresses and strains in the subcells. New stress-update and stress-correction algorithms are developed which significantly reduce the computational effort that is needed. The new algorithms are formulated given a constant average strain rate for each time step, which makes them suitable for integration with FE constitutive framework. The micromechanical model is shown in Fig. 8.2. The unidirectional composite, which consists of long fibers arranged unidirectionally in the matrix system, is idealized as doubly periodic array of fibers with rectangular cross-section. A quarter UC that consists of four subcells is modeled due to symmetry. The first subcell is a fiber constituent, while subcells 2–4 represent the matrix constituents. The long fibers are aligned in the x1-direction. The other cross-section directions are referred to as the transverse directions. The x3-direction is called the out-of-plane axis or lamina thickness direction. The total volume of the UC is taken to be equal to one. The volumes of the four subcells are: 12 3 4 V bh V h b V b h V h b = = − = − =− − , (1 ), (1 ), (1 )(1 ). (8.17) The notations used for the stress and strain vectors are: ( ) 11 22 33 12 13 23 ( ) 11 22 33 12 13 23 d {d ,d ,d ,d ,d ,d }, 1, , 4, d {d ,d ,d ,d ,d ,d }, 1, ,6. k k k α α σ σσστττ α ε εεεγγγ = = … = = … (8.18) The 3D nonlinear constitutive integration for the fiber and matrix constituents is performed separately for each subcell. The fiber is linear elastic and transversely isotropic, while the matrix medium is viscoelastic. R. Haj-Ali Fig. 8.2. Unit cell micromodel for unidirectional reinforced composites 328
Chapter 8:Nested Nonlinear Multiscale Frameworks 329 The homogenization of the micromodel should satisfy displacement and traction continuity.Perfect bond is assumed along the interfaces of the subcells.In the fiber direction,the four subcells satisfy the same strain continuity relation.The axial average stress definition is used as a second independent relation to relate the effective axial stress to the stresses in the subcells.The following equations summarize the relations in the axial mode ds)ds2)ds3)=ds(4)=d5, Vdo"+'do2+'do+'do4=dō， (8.19) where overbar denotes an overall average quantity over the unit cell Along the interfaces between the subcells with normal in the x2- direction,the in-plane stress components o22 and ni2 must satisfy traction continuity conditions.The total strain components 82 and 72 from subcells 1 and 2 and subcells 3 and 4,respectively,should also satisfy strain compatibility conditions.These relations are written in an incremental form as: do=do, do》=do, (8.20) Vi+V2 de+dg=d返， V+' +'a do-do, dos=do, (8.21) W+2 +3 V,de+ V+' V de=d Considering interfaces between subcells with normal in the x3- direction,the out-of-plane stress components o33 and 7i3 must satisfy
The homogenization of the micromodel should satisfy displacement and traction continuity. Perfect bond is assumed along the interfaces of the subcells. In the fiber direction, the four subcells satisfy the same strain continuity relation. The axial average stress definition is used as a second independent relation to relate the effective axial stress to the stresses in the subcells. The following equations summarize the relations in the axial mode (1) (2) (3) (4) 11 1 1 1 (1) (2) (3) (4) 11 21 31 41 1 d d d d d, VV V V d d d d d, εεεεε σ σ σ σσ ==== +++ = (8.19) where overbar denotes an overall average quantity over the unit cell. Along the interfaces between the subcells with normal in the x2- direction, the in-plane stress components σ22 and τ12 must satisfy traction continuity conditions. The total strain components ε22 and γ12 from subcells 1 and 2 and subcells 3 and 4, respectively, should also satisfy strain compatibility conditions. These relations are written in an incremental form as: (1) (2) 2 2 (3) (4) 2 2 1 2 (1) (2) 2 22 12 12 3 (3) 4 (4) 2 22 34 34 d d, d d, d d d, d d d, V V VV VV V V VV VV σ σ σ σ ε ε ε ε ε ε = = + = + + + = + + (8.20) (1) (2) 4 4 (3) (4) 4 4 1 2 (1) (2) 4 44 12 12 3 (3) 4 (4) 4 44 34 34 d d, d d, d d d, d d d. V V VV VV V V VV VV σ σ σ σ ε ε ε ε ε ε = = + = + + + = + + (8.21) Considering interfaces between subcells with normal in the x3- direction, the out-of-plane stress components σ33 and τ13 must satisfy Chapter 8: Nested Nonlinear Multiscale Frameworks 329
330 R.Haj-Ali traction continuity conditions.The total strain components 83 and y3 from subcells 1 and 3 and subcells 2 and 4,respectively,should also satisfy strain compatibility conditions.These relations are expressed in incre- mental form as: do)=do, dof)=do, (8.22) V+' -dc0》=da, d6g+ ,+' V de=ds do=do, do=do, 5d"+d=d返， (8.23) + + de0+de"=d运， '+V +V4 Finally,both types of interfaces should satisfy transverse shear stress continuity.Therefore,the transverse shear stresses in the four subcells are equal to the effective transverse shear stress.The transverse shear strains from the four subcells in the average strain definition are used to express the relations with the effective transverse shear strain of the UC.The transverse shear relations are summarized as: dog)=dod)=dod=dod=dos. (8.24) Vids+vds)+Vads)+Vadse)=ds Equations(8.19)(8.24)along with the stress-strain relations within each fiber and matrix subcells complete the micromechanical formulation of the unidirectional lamina.These relations are used in incremental (rate)form due to the nonlinear constitutive relations in the matrix subcells.Next,the
traction continuity conditions. The total strain components ε33 and γ13 from subcells 1 and 3 and subcells 2 and 4, respectively, should also satisfy (1) (3) 3 3 (2) (4) 3 3 1 (1) 3 (3) 3 33 13 13 2 4 (2) (4) 3 33 24 24 d d, d d, d d d, d d d, V V VV VV V V VV VV σ σ σ σ ε ε ε ε ε ε = = + = + + + = + + (8.22) (1) (3) 5 5 (2) (4) 5 5 1 (1) 3 (3) 5 55 13 13 2 4 (3) (4) 5 55 24 24 d d, d d, d d d, d d d. V V VV VV V V VV VV σ σ σ σ ε ε ε ε ε ε = = + = + + + = + + (8.23) Finally, both types of interfaces should satisfy transverse shear stress continuity. Therefore, the transverse shear stresses in the four subcells are equal to the effective transverse shear stress. The transverse shear strains from the four subcells in the average strain definition are used to express the relations with the effective transverse shear strain of the UC. The transverse shear relations are summarized as: (1) (2) (3) (4) 66 66 6 (1) (2) (3) (4) 16 26 36 46 6 d d d d d, VV V V d d d d d. σσσσσ ε ε ε εε ==== +++ = (8.24) Equations (8.19)–(8.24) along with the stress–strain relations within each fiber and matrix subcells complete the micromechanical formulation of the unidirectional lamina. These relations are used in incremental (rate) form due to the nonlinear constitutive relations in the matrix subcells. Next, the R. Haj-Ali mental form as: strain compatibility conditions. These relations are expressed in incre- 330