18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS FIBER COMPRESSIVE FAILURE CRITERION AS SHEAR BAND MODE BIFURCATION CONDITION T. Nadabe 1 *, N. Takeda 1 1 Department of Advanced Energy, The University of Tokyo, Tokyo, Japan * Corresponding author(nadabe@smart.k.u-tokyo.ac.jp) Keywords : fiber kinking, failure criterion, bifurcation, material instability, localization, progressive failure analysis, fiber reinforced composites 1 Introduction & & T n t t on S Π ⋅ = (2) Fiber kinking failure shown in Fig. 1(a) is t commonly observed in compressive failure of fiber S (3) v = v on u u reinforced composite materials. It is a band of localized shear deformation, and in many cases it is where Π is nominal stress, ρ is density, b is body considered as a result of material instability [1]. The force, n is unit normal on boundary, t is surface t localization phenomena is observed in various kinds force, v is velocity, and material derivative of any of materials such as Lüders bands in metals, necking tensor ψ is represented as ψ & . V is body and S , S t u in polymers, faults in rock mass and shear bands in is boundary surfaces. Then stress vectors t , t on A B soils, and those have been investigated as shear band the upper and lower surfaces S , S of shear band A B mode bifurcation, in which material instability due are represented using nominal stress as follows, to inelastic behavior of the material plays an & T & n t in S important role [2]. Fiber kinking in composite Π ⋅ = ( 4 ) A A A A materials is also considered to be categorized in (5) & T & n t on S Π ⋅ = those localization phenomena. For the localization B B B B phenomena, Borja [2] investigated shear band mode & are At the moment of localization onset, n and t bifurcation for general elasto-plastic solids in the continuous across the shear band, then frame of finite deformation theory. In this study, we follow the bifurcation theory given by Borja [2], and ( ) & & [ [ & ] ] T T ( 6 ) n T n 0 Π − Π ⋅ = Π ⋅ = A B obtain the localization condition in fiber kinking failure as the shear band mode bifurcation condition. where n is unit normal on initiating shear band, [ ] [ ] The obtained condition is treated as failure criterion represents the amount of discontinuity across the of fiber compressive failure, and is applied to shear band. In addition, the constitutive equation at progressive failure analysis in composite materials. finite deformation is represented as follows, & ( 7 ) T A : L Π = 2 Localization Condition in Composite Materials as Shear Band Mode Bifurcation Condition where A is tangential moduli tensor at finite deformation and L is velocity gradient. From Following the bifurcation theory shown by Borja [2], Eqs. (6-7), we obtain the localization condition in composite materials, which represents the behavior of failure in [ [ A : L ] ] ⋅ n 0 ( 8 ) = fiber kinking failure. Here, Updated Lagrangian The discontinuity of velocity gradient across the formulation is applied. First, the rate form boundary value problem at present configuration is defined as shear band is represented as [2], follows, [ [ ] L ] [ [ ] v ] n ) h ( ( 9 ) = ⊗ & & in V ( 1 ) div + b 0 Π ρ = where h is the width of shear band. From Eqs. (8-9), when tangential moduli tensor A is continuous,
it is required to be always defined with the local A : [ [ ] v ] n n 0 ( ( ) ) ( 10 ) ⊗ ⋅ = fiber direction. In finite deformation analysis, this is the problem of objective stress rate. In order to Solving Eq. (10), follow the local fiber direction which changes with a ⋅ v [ [ ] ] 0 , a = n A n ( 11 ) = the material deformation, stress rate such as Oldroyd ij k ikjl l rate should be the most appropriate, and the Tensor a is called acoustic tensor. The condition for corotational rate such as Jaumann rate has difficulty, ij the existence of non zero discontinuous velocity since rigid body rotation is not the same as the field [ ] [ v is represented as follows, ] rotation of local fiber direction because of the material deformation. Additionally, since det a 0 ( 12 ) = ij compressive failure is largely affected by local fiber direction, analysis result of compressive failure Thus the shear band mode bifurcation condition is depends on objective stress rate. In the following, obtained from the requirement of continuity of the the Oldroyd rate of Kirchhoff stress is applied as nominal traction vector on the potential shear band, objective stress rate. Then the constitutive relation of and the loss condition of the uniqueness of the composite materials is represented as follows, solution for strain field. The localization condition in fiber kinking is represented by the bifurcation ∇ ( 15 ) C : D condition in Eq. (12). τ = comp The predicted results of shear band mode bifurcation ∇ τ where is Oldroyd rate of Kirchhoff stress and D is largely depend on the constitutive description of deformation rate. In the Updated Lagrangian homogeneous deformation [2], since the localization & formulation, material derivative of nominal stress Π of deformation is closely related with the inelastic is represented as follows, behavior of the materials. In this study, as the constitutive description of fiber reinforced ∇ ( 16 ) & T L Π = τ + ⋅ σ composite materials, we apply the nonlinear deformation theory given by Tohgo et al. [3], in where σ is Cauchy stress. From Eqs. (7, 15-16), which the relation between stress rate and strain rate A : L C comp : D L is represented as follows, ( 17 ) = + ⋅ σ d C d σ = ε In this equation, second term of RHS is written as comp follows, { ( )( ) } K 1 C C 1 V C C S C − (1 3 ) = − − + comp m f f m m ( ) ( { ) } K = 1 V C C S C V C l l ( 18 ) − − + + σ = σ δ f f m m f f il lj jl ik kl where C comp C , and C are tangential moduli tensor Then Eq. (16) is represented as, f m of composites, fibers and matrix, respectively. S is ( ) A l c l ( 19 ) = + σ δ Eshelby tensor and V is fiber volume fraction. To ijkl kl ijkl jl ik kl f obtain tangential moduli tensor of matrix, the where d sym l is applied. Tensor A is ( ) = kl kl ijkl evaluation for equivalent stress of matrix is represented as, necessary. The following relation is applied to evaluate the equivalent stress of matrix from the A c ( 20 ) = + σ δ ijkl ijkl jl ik applied stress in composite materials. Therefore the acoustic tensor a in Eq. (11) is ij { ( ) } ( ) − 1 d C m S I K C C S C σ = − − + represented as follows, m f m m (1 4 ) ( ) 1 1 S I − C − d ⋅ − σ a c n n n n ( 21 ) = + σ δ m ij ikjl k l kl k l ij In addition, the fiber direction of composite In the following section, the explicit expressions for materials changes during the deformation of the compressive strength are obtained from the material. The constitutive relation is anisotropic, and bifurcation condition, and they are compared with
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