Constitutive Equation for Plastic Behavior of Hydrostatic Pressure Dependent Polymers

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Βαρ-ιησούς Χάρων Γιάγκος
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1 1/5 Constitutive Equation for Plastic Behavior of Hydrostatic Pressure Deendent Polymers by Yukio SANOMURA Hydrostatic ressure deendence in mechanical behavior of olymers is studied for the constitutive modeling with the yield surface described by the first invariant and the second invariant of stress and the nonassociated flow rule satisfying incomressible hyothesis. An internal variable theory of rate-indeendent lasticity is resented incororating isotroic hardening as a function of accumulated lastic strain. After determination of material constants under uniaxial tension and comression, the model shows that the von Mises tye effective stress lastic strain curves under multiaxial load are quite different from those under uniaxial load. The model is comared with the exerimental results in uniaxial tension and comression by Sitzig and Richmond and in torsion by Silano et. al under high ressure. Key words: Constitutive equation, Polymer, Plastic behavior, Hydrostatic ressure deendence, Nonassociated flow rule 1 Kreml 1)-3) Boyce 4)5) 6)-11) 6)-8) - 9)1) 9) Mises 12)13) 14)-16) Sitzig Richmond 1) Silano 8) FEM 1 e e = + (1) Hooke e + ν ν = 1 σ σ kkδ (2) E E E E Received ,Det. of Mech. Eng.,Tamagawa Univ.,Machida,Tokyo,

2 2/5 Fig.1 Yield locus redicted by eq.(3) for various β. 17) f = (1 β ) 3J + = 2 βi1 κ 1 (3) J 2 = ss, I1 = σ kk 2 J 2 I κ ββ (3) Mises Fig.1 (3) Y (1) g = 3J 2 (4) g & = λ (5) σ Prager & f f f = σ& + κ& = (6) σ κ & = (1 β ) 2H 2 dσ H = d 2. 2/ Y O s σ& kl + βδkl kl J2 3J2 3.5 =. =.1 =.15 = / Y s (7) σ = (1 β ) 3J 2 = βi 1/ 2 1 (8) κ = σ ( ) (9) - 2 n σ = F ( b + ) (1a) Y Y σ = σ + ( σ σ )[1 ex( c )] (1b) Y Fbnσ Y σ Y c σ σ11 = σ, σ = =, = = / 2, = (11a) (7) 1 σ& H & = (11b) 1 2β σ& H β Fig.2 (HDPE) (PP)- 18) (1a)(1b) E =.982GPa, β =.12 F = 37.5MPa, b = 1. 1, n =.244 (HDPE) σ Y = 7.MPa, σ Y = 2.MPa, c = 4. E = 1.97GPa, β =.17 F = 67.2MPa, b = 1. 1, n =.217 σ Y = 14.MPa, σ Y = 35.7MPa, c = 6. (12a) (PP) (12b) Fbnσ Y σ Y c -β (1a)(1b) (1a)(1b) -

3 3/5 Fig.2 Stress-strain curves under uniaxial tension and uniaxial comression. σ σ = σ = = σ, =, σ = 33 = 2, (7) (13a) = 1+ β σ& 2 H & = (13b) 1 3β σ& H τ γ σ = σ = = τ, = γ (7) (a) HDPE (b) PP ( 3τ& ) Uniax ial comres sion Exerimental Calculated [eq.(1a)] Calculated [eq.(1b)] Uniaxial comression Exerimental Calculated [eq.(1 a)] Calculated [eq.(1 b)] σ = (14a) 2, = γ& 1 β = (14b) 3 H (3J2) 1/2 (MPa) Biaxial comression Uniaxial comression Simle shear Biaxial tension Calcu lated [eq.(1a)] Calcu lated [eq.(1b)] Effective lastic strain Fig.3 Stress-lastic strain curves under multiaxial stress. Fig.3 (1b) (1a)(13)(14) (11) (12b) Fig.2 (1a)(1b) β=mises 8)1) Fig.4 Sitzig Richmond 1) =.11,138,276,552,828,114MPa (1b) (1a) E = MPa, β =.35 F = 4.MPa, b = 1. 1, n =.172 (HDPE) (15a) σ Y = 13.MPa, σ Y = 28.5MPa, c = 3. E = MPa, β =.5 F = 1MPa, b = 1. 1, n =.114 σ Y = 4.MPa, σ Y = 6898MPa, c = 1 (PC) (15b) 1) Fbnσ Y σ Y c -

4 4/ (a) HDPE (b) PC Comression Tension Exerimental 1) Comression Tension 552MPa 276MPa 138MPa.11MPa Exerimental 1) =114MPa 276MPa 828MPa Calculated =114MPa 828MPa 552MPa 138MPa.11MPa Calculated[eq.(1a)] Calculated[eq.(1b)] Fig.4 Tensile and comressive stress-strain curves at various hydrostatic ressures. β PC 19) (1a)(1b) Fig.5 Silano POM 8) =.11,2,4,6MPa (1a) (1b) G = MPa, β =.277 F = 182MPa, b = 1. 1, n =.366 σ Y = 3.36MPa, σ Y = 89.MPa, c = 23. (16) (1a)(1b) Shear stress(3) 1/2 (MPa) Fig =.11M Pa Exerimental 8) 6MPa 4MPa 2MPa Calculated [eq.(1a)] Calculated [eq.(1b)] Shear strain/(3) 1/2 Shear stress strain curves at various hydorostatic ressures. (1) - (2) Mises - (3)

5 5/5 1) E.Kreml, Transactions of the ASME, Journal of Engineering Materials and Technology, 11, 38, (1979). 2) (1988). 3) A (1992) 4) M.C.Boyce, D.M.Parks, A.S.Argon, Mechanics of Materials, 7, 15, (1988). 5) A-6642(2) 6) D.R.MearsK.D.Pae, and J.A.Sauer, Journal of Alied Physics4, 49 (1969). 7) S.RabinowitzI.M.Ward, Journal of Material Science, 5 29 (197). 8) A.A.Silano, K.D.Pae and J.A.Sauer, Journal of Alied Physics, 48, 476(1977). 9) C.Bauwens-Crowet, J-C.Bauwens, and G.Homés, Journal of Material Science, 7, 176 (1972). 1)W.A.Sitzig and O.Richmond, Polymer Engineering and Science, 19, 1129 (1979). 11) A.W.Christiansen, E.Bear, and S.V.Radcliffe, Phil. Mag., 24, 451 (1971). 12) P.B.Bowden and J.A.Jukes, Journal of Material Science, 7, 52 (1972). 13) R.Raghava, R.M.Caddell, and G.S.Y. Yeh, Journal of Material Science, 8, 5 (1973). 14) A.L.Gurson, Transactions of the ASME, Journal of Engineering Materials and Technology, 99, 2 (1977). 15) A.Needleman and V.Tvergaard, J. Mech.Phy. Solids, 32, 461 (1984). 16) S.Nemat-Nasser, M.M.Mehrabadi, and T.Iwakuma, (S. Nemat-Nasser ), Three-Dimensional Constitutive Relations and Ductile Fracture, 157, (1981), Noth-Holland Publishing. 17) 1112 (1999) 18), 9,3,(1999). 19), 2,Vol.,167(2).

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