THE INCREMENTAL DAMAGE THEORY OF PARTICULATE- REINFORCED COMPOSITES - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS THE INCREMENTAL DAMAGE THEORY OF PARTICULATE- REINFORCED COMPOSITES WITH A DUCTILE INTERPHASE Y.P. Jiang 1 *, K. Tohgo 2 1 Department of Engineering Mechanics, Hohai University, Nanjijng, China 2 Department of Mechanical Engineering, Shizuoka University, Hamamatsu, Japan * Corresponding author( ypjiang@nuaa.edu.cn ) Keywords : FEM; Debonding; Particle-reinforced composite; Ductile interphase particle volume fraction on the overall stress − strain 1 Introduction response of PRCs were studied systematically. Also, Particulate-reinforced composites (PRCs) are a unit-cell (UC) based FEM was performed. becoming more and more attractive in the modern 2 Incremental damage theory of PRC with a industry. Many properties of PRCs are influenced by ductile interphase particle size, which attributes to the significant modification of microstructures by the introduction The adopted composite system consists of three of inorganic particles. In explaining particle size phases of particle, matrix and interphase between effect, the interphase is undoubtedly one of the most them. The interphase concentration f I is related to important factors [1]. Quite a number of researches that of particles f P by ( ) have been published to date for studying the impact = + 3 − (1) f f 1 2 t d 1 I P P of the interphase, and the readers can refer our here, d P and t are particle diameter and interphase previous work [2]. Jiang et al. have systematically thickness, respectively. f P0 is the initial content of investigated the effects of the interphase on the particles, and the initial loading of the interphase is stiffness, elastic − plastic and damage behaviors of determined by Eqn. (1). PRCs by using FEM, theoretical and experimental methods [2-3]. The elastic − plastic deformation of an 2.1 Constitutive relations of the constituents interphase was found to have a great influence on The elastic incremental stress − strain relations of the mechanical behavior of PRCs. Ruiz-Navas et al. the constituents follow as: [4] found that Al+Ti 5 Si 3 -Cu composites exhibit the σ = C ε , i = ν d i i d ( E , ): M, I or P (2) superior mechanical properties as compared to i i i where, the symbol “:” is contraction product, d σ i and Al+Ti 5 Si 3 composites. Lauke [5] numerically d ε i are the incremental stress and strain, respectively, analyzed the interfacial adhesion strength between a and C i ( E i , ν i ) is the stiffness tensor. E i and ν i are coated particle and a polymer matrix material, and indicated the influence of a ductile interphase on the Young’s modulus and Poisson’s ratios of local stress field. Wang and Yang [6] have employed constituents, respectively. The plastic deformations of the constituents are described by the Prandtl − FEM analysis to simulate the behavior of energy dissipation for the PRCs with a ductile interphase. Ruess equation (the J 2 -flow theory) as, ′ ′ For the PRCs, Tohgo and Chou [7] and Tohgo and σ = ν ε , i = i i d C (E , ): d M, I or P (3) i i i Weng [8] proposed an ID theory of PRCs taking into where account the plasticity of a matrix and progressive ν + ′ debonding damage of particles based on Eshelby's E E (2H ) ′ = ′ ν = i i i i E , equivalent inclusion method and Mori − Tanaka's + ′ + ′ i i 1 E H 1 E H i i i i mean field concept. In order to fully study the effect E' i and ν' i represent tangent Young’s moduli and of an interphase, it is necessary to extend Tohgo’s tangent Poisson's ratios of the constituents under ID theory to the three − phase case with a ductile elastic-plastic deformation. H ' i shows the work- interphase. hardening ratio of each phase, Based on the previous studies [2-3], a ductile ′ = σ ε i pl i H d ( d ) interphase was introduced and studied in the frame i e e ′ ′ σ = σ σ i i i of ID theory [7]. Numerical computations of the 3 2( ) ( ) e kl kl stress − strain relations under uniaxial tension were ε = ε ε pl i pl i pl i ( d ) 2 3( d ) ( d ) carried out for different microstructures. Influences e kl kl of debonding, interphase properties, particle size and

Here, σ e i and ( d ε e pl ) i are the von Mises equivalent (4) The progressive damage is described by a decrease in the intact particle content and an stress and incremental equivalent plastic strain, respectively. ( σ ' kl ) i the deviatoric stress component, increase in the debonded particle concentration. and ( d ε kl pl ) i the incremental plastic strain. Eqn. (3) is The constitutive equations of an isotropic PRC are expressed in the form of the hydrostatic strictly valid in the case of monotonic proportional ( d ε kk − d σ kk ) and deviatoric parts ( d ε ' ij − d σ ' ij ). So, the loading. In the composite system, the stress and incremental strain d ε ( d ε kk , d ε ' ij ) − stress d σ ( d σ kk , d σ ' ij ) strain of particles, interphase and matrix are denoted relation of the composite is given by, with superscripts "P", "I" and "M", respectively, and without any superscripts for the composite. 1 1 = + εσσ 3 P d d df , κ κ kk kk kk P 3 2.2 Incremental damage theory with a ductile t d interphase 1 1 ′ = + ′ εσσ 2 P d d df (4) µ µ ij ij ij P σ σ + σ 2 d t d where [ ] κ = κ − γ + αγ M κκ 1 (1 ) Intact Particle t σ ε P P , [ ] κ = κ − α − − α γ M κ (1 ) 1 (1 ) d Damaged   µ = µ − γ + βγ M μμ 1 (1 )  Particle  t Interphase µ = µ − β  − − β γ  σ ε (1 ) 1 (1 ) I I   , M μ d Matrix and σ ε M M , κ − κ κ − κ d f ( ) f f ( ) γ = − − + p p m p Intact Particle Volume Intact Particle Volume I0 I m − κ κ + κ − κ α κ + κ − κ α − α Fraction: f Fraction: f df ( ) ( ) (1 ) P P P m p m M I m Damaged Particle Damaged Particle + µ − µ d d µ − µ d Volume Fraction: Volume Fraction: f df f f ( ) f f ( ) P P P γ = − − + p p m p I0 I m (a) State before the incremental (b) State after the incremental μ µ + µ − µ β µ + µ − µ β − β ( ) ( ) (1 ) deformation deformation m p m m I m + ν − ν Fig. 1 The states of composite undergoing damage 1 4 5 1 2 α = β = M M , process before and after incremental deformation, df P is a − ν − ν 31 15 1 volume fraction of the reinforcements damaged in the M M Here, κ i and μ i are bulk and shear modulus of three incremental deformation. constituents, respectively, which are related to Fig. 1 shows the states before and after an Young’s modulus E i and Poisson’s ratio ν i by incremental deformation of a representative volume E E κ = µ = i , i (5) element during the damage process, where a − ν + ν i i 3(1 2 ) 2(1 ) constant macroscopic stress σ and its increment d σ i i The incremental stresses of the three phases d σ P , are applied on the boundary of RVE. All the d σ M and d σ I are expressed by debonding damage is supposed to occur between κ + ( σσ ) p particle and interphase. The states before the d df = σ p p kk kk p d [ ] incremental deformation are described in terms of kk − − α γ  κ + κ − κ α  1 (1 )  ( )  κ m p m the intact particle content f P and damaged particle ′ ′ µ + ( σσ ) p d df d . The initial damaged particle content f P d content f P ′ = σ p p ij ij p d  − − β γ   µ + µ − µ β  ij =0, and f P = f P0 . df P denotes the content of particles  1 (1 )   ( )  μ m p m debonded in an incremental deformation. The states 1 = + σ ( σσ )1 (1 M P after the deformation can expressed by the intact d d df − − α γ kk kk kk P ) particles loading f P − df P and damaged particles κ d + df P . Some necessary assumptions are 1 loading f P ′ ′ ′ = + σ ( σσ ) 1 (1 M P d d df − − β γ ij ij ij P firstly given: ) μ (1) All the constituents and composites are isotropic. κ + ( σσ ) p d df = σ I I kk kk P d (2) The debonding damage is controlled by the [ ][ ] kk − − α γ κ + κ − κ α 1 (1 ) ( ) κ m I m critical stress of particles. ′ ′ µ + ( σσ ) p d df (3) After the damage, particle stress reduces to zero. ′ = σ I I ij ij p (6) d [ ] ij  − − β γ  µ + µ − µ β  1 (1 )  ( ) μ m I m