Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber- Reinforced Composites Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber-Reinforced Composites M.G. Lee, D. Kim, K. Chung *, J. R. Youn and. J. Kang School of Materials Science and Engineering, Seoul National University, 56- Shinlim-dong, Kwanak-gu, Seoul 5-74, Korea Research Institute of Advanced Materials, Seoul National University 56-, Shinlim-dong, Kwanak-gu, Seoul, 5-74, Korea Received: 8 May 003 Accepted: 3 November 003 SUMMARY In order to describe the Bauschinger and transient behavior of orthotropic fiberreinforced composite solids, a combined tropic-kinematic hardening law based on the non-linear kinematic hardening rule was considered here, in particular, based on the Chaboche type law. In this modified constitutive law, the antropic evolution of the back-stress was properly accounted for. Also, to represent the orthotropy of composite materials, Hill s 948 quadratic yield function and the orthotropic elasticity constitutive equations were utilized. Furthermore, the numerical formulation to update the stresses was also developed based on the incremental deformation theory for the boundary value problems. Numerical examples confirmed that the new law based on the antropic evolution of the back-stress complies well with the constitutive behavior of highly antropic materials such as fiber-reinforced composites. INRODUCION Recently, fiber-reinforced composites have been commonly used as high performance materials, which are subjected to various ranges of stresses, temperatures and loading conditions. As for their mechanical properties, fiber-reinforced composites are usually treated as linear elastic materials since fibers provide the majority of the strength and stiffness. However, more sophisticated description requires consideration of some form of plasticity, viscoelasticity, or both (viscoplasticity). Experimental studies under static loading confirm that fiberreinforced composites show elasto-plastic behavior including the Bauschinger effect and transient behavior, hysteresis loops under cyclic loading and so on. Although significant progresses have been made over the years, little work has been done to implement those sophisticated material behaviors for structural applications. his article deals with * Corresponding author: kchung@snu.ac.kr the theory of plasticity for antropic fiber-reinforced composites, especially for orthotropic ones. he antropic yield function and the combined type of tropic-kinematic hardening have been considered based on elasto-plasticity. Previous efforts to implement plasticity considered three types of hardening assumptions, which are schematically illustrated in Figure 3-5 : tropic, kinematic and combined type hardening. For example, Leewood et. al. 6 used the tropic hardening to describe plastic properties of fibrous metal composites. Isupov et al. 7 dealt with the micromechanical analysis of plastic deformation processes of metal matrix composites composed of elastic reinforcing elements and the plastic matrix, based on the generalized antropic Prager kinematic hardening law. Recently, Sarbayev 8 described the plasticity theory for antropic composites utilizing the kinematic hardening of the antropic quadratic yield function. However, few of the previous works properly described the Bauschinger and transient behaviors during unloading. In this work, in order to describe Polymers & Polymer Composites, Vol., No. 3, 004 5
M.G. Lee, D. Kim, K. Chung, J. R. Youn and. J. Kang Figure. Schamatic view of three types of hardening laws:(a) Isotropic hardening (b) Kinematic hardening (c) Isotropic-kinematic hardening (a) (b) (c) the Bauschinger effect and transient behavior under the cyclic loading, two types of combined tropickinematic hardening laws were developed and compared by modifying the non-linear kinematic hardening law previously proposed by Chaboche 9. In the first law, the antropy of the yield function was considered but no particular antropic evolution of the back-stress is considered, as commonly done for antropic metals 0 : the tropic back-stress evolution. In the second law, besides the antropic yield function, the antropic evolution of the backstress was also implemented considering the directionality of the back-stress evolution: the antropic back-stress evolution. In order to account for the orthotropy of composites, Hill s 948 quadratic yield function was utilized. he procedure to separate the tropic and kinematic hardening curves from the simple tension test data is also discussed along with the stress update scheme based on the incremental deformation theory for the boundary value problem. CONSIUIVE EQUAIONS Elastic Behavior Consider uni-directionally-laminated composites in the - plane under the plane stress condition as 6 Polymers & Polymer Composites, Vol., No. 3, 004
Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber- Reinforced Composites shown in Figure. he stress and strain states are described by vectors σ σ, σ, σ and = ( ) ε ε e, ε e, ε e for the plane stress state. he stressstrain relationship for orthotropic materials is = ( ) σ σ = σ σ e Q Q ε ε ε = 0 e ε = e e Q Q 0 D e 0 0 Q66 where the reduced stiffness Q ij are Q Q () E ν E νe =, Q( = Q) = =, ν ν ν ν ν ν E =, Q = G ν ν 66 () where E, E, G, ν, ν are Young s moduli in the and directions, shear moduli in the - plane and Poisson s ratios, respectively. Antropic Yield Function In order to describe the initial antropic yield stress surface, Hill s 948 antropic yield criterion 0 for the plane stress state is considered; i.e., Φ = f( σ) σ = σ Bσ σ = 0 (3) b b where B = b b 0 0 0 0 b 66 and σ is the equivalent stress measuring the size of the yield surface as the first order homogeneous function, while f is the nd order homogeneous function. Flow and Evolution Rules Flow Rule with Isotropic Back-Stress Evolution he yield surface of the combined type is described by f( σ a) σ = 0 (4) Figure. Coordinate definition for uni-directionally laminated composites Polymers & Polymer Composites, Vol., No. 3, 004 7
M.G. Lee, D. Kim, K. Chung, J. R. Youn and. J. Kang where α is the back-stress by which the current yield surface is translated from an initial position. he plastic work increment becomes d α dα = dε d ε α hd εα = (0) dw = σdε p = (σ-α)dε p + αdε p = dw + dw α (5) where dε p is the plastic strain increment. he effective quantities are defined considering the following modified plastic work equivalence relationship; i.e., p dw = σ α dε σ dε (6) ( ) = where dε is the effective plastic strain increment which is conjugated with σ. In Equation (6), σ is obtained by replacing σ with σ α. As for the effective back-stress increment, dα, the value is obtained from the initial effective stress by replacing σ with dα. he definitions of the effective quantities for the stress, the conjugate plastic strain increment, and the back stress increment are for any antropic yield surfaces, which are expressed as first-order homogeneous functions. When plastic deformation and loading are monotonically proportional (from an initial state), p dw = αdε = αdε α (7) where α = d α, therefore, dw = ( σ + α) dε. In the Chaboche model, the back stress increment is composed of two terms to differentiate the hardening behaviors during loading and unloading. dα = dα - dα (8) he magnitudes of the back stress increments, dα, is obtained by substituting the back-stress increment into the yield surface f. hen, Note that α = α ( ε), while α = α ( ε) in general. Under the simple tension test condition, equations (8)-(0) lead to dα = hd ε hd ε α () If both h and h are assumed constant, equation (0) is the st order linear differential equation for α, whose solution becomes h ε α = ( ) e h () h where, ε = d ε. After α ( ε ) is measured from the simple tension test involving loading and unloading, h and h are obtained by comparing equation () with the measured α( ε) 3. Flow Rule with Antropic Back-Stress Evolution he tropic back-stress evolution law described in equation (9) and (0) is effective when the kinematic hardening parameters are available for only one direction. However, to account for the directional difference of the back-stress evolution for the highly antropic materials such as fiber-reinforced composites, the following antropic back-stress evolution law is proposed here, by modifying the tropic evolution rule: ( ) σ α dα = C dε Hαdε = ( Cn Hα) dε σ (3) or dα = C n H α dε ( ) ij ijkl kl ijkl kl but ( ) = ( ) dα dα d d σ α ε ε σ hd ε σ α = σ (9) where C and H are fourth order tensors which contain material parameters to be experimentally determined for back-stress evolutions. When the plane stress condition for orthotropic materials is considered, equation (3) becomes 8 Polymers & Polymer Composites, Vol., No. 3, 004
Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber- Reinforced Composites dα dα dα = where stands for a trial state and the subscript denotes the process time step. Also, preserving the plastic quantities as the previous values, C C C C 0 0 0 n 0 n C n 33 H H H H 0 0 0 0 H 33 α α dε α (4) ε = ε, α = α (7) n+ n n+ n If the following yield condition is satisfied with the trial values for a prescribed elastic tolerance; i.e., which can be further simplified assuming C = C = H = H = 0. Under such circumstances, the following relationships are obtained for monotonously proportional experiments: ( α, α, α) = C C C 33 ( e ), e, H H H H ε H ε H ε 33 ( ) ( e ) 33 (5) he six material constants in equation (5) are obtained from the curve fitting of actual stress-strain curves in three deformation modes: simple tension tests in the and directions and the pure shear test. SRESS UPDAE SCHEME In this paper, the additive decoupling of the total strain increments into the elastic and plastic strain increments, dε = dε p + dε e, and the associated flow rule based on the normality rule are assumed. For the numerical formulation, the incremental deformation theory was applied to the elasto-plastic formulation based the materially embedded coordinate system. Under this scheme, the strain increments in the flow formulation become discrete true strain increment while the material rotates by the incremental rotation obtained from the polar decomposition at each discrete step. As for the stress update scheme, the methods developed by Chung 4 and Simo and Hughes 5 were considered here. In this stress update scheme, the updated stress is initially assumed to be elastic for a given discrete strain increment ε. herefore, e σ = σ +D ε n+ n (6) f ( σ n+ αn+ ) σ ( εn+ )< ol. (8) the process time step n+ is considered elastic. If the above yield condition is violated, the step is considered elasto-plastic and the trial elastic stress state is taken as an initial value for the solution of the plastic corrector problem. he non-linear equation to solve for ε, which enables the resulting stresses to stay on the hardening curves at the new step n+, is ( σ α σ α α )= σ ( ε + ε) (9) f m + + n n n he equation is applied for each yield surface progressively introduced between the initial and the trial elastic stresses as shown in Figure 3. he predictor-corrector based on the Newton-Raphson method was used to solve equation (9). After obtaining the converged solution of equation (9), the stresses, back stresses and equivalent plastic strain are updated for the next step. EXAMPLE SOLUIONS For verification purposes, loading and unloading behaviors were simulated for the uni-axial tension test using the constitutive equation developed. he material parameters are chosen arbitrarily and only tropic kinematic hardening is considered for simulation. Since the same kinematic hardening parameters are used for the two back-stress evolution rules, the stress-strain curve would be identical. he numerical uni-axial loading-unloading-reverse loading curve identically obtained for both rules is shown in Figure 4. he calculated curve confirms that the constitutive laws clearly show the Bauschinger effect and transient behavior. Another numerical example utilizes experimental data obtained from the previous work 8 for the analysis Polymers & Polymer Composites, Vol., No. 3, 004 9
M.G. Lee, D. Kim, K. Chung, J. R. Youn and. J. Kang Figure 3. A schematic view of the progressive scheme to update the new stress Figure 4. An example solution for a uni-axial loading-unloading-reverse loading curve. of stress-strain curves of orthotropic fiber reinforced composites under uni-axial tension and shear tests. he longitudinal (which is usually the fiber direction) and transverse directions coincide with the coordinate system described in Figure 3. he mechanical properties which conform to the characteristics of the unidirectional glass fiber-reinforced composite8, 6 are: Elastic constants: E = 63,000 (MPa), E = 9,700 (MPa), G = 7,400 (MPa), ν = 0.3 Yield function coefficients: b = 6.9e-5, b =, b 66 = 0.39, (b : No data) Initial yield stresses: σ y = 800 (MPa), σ y = 5 (MPa), σ y = 4 (MPa) 30 Polymers & Polymer Composites, Vol., No. 3, 004
Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber- Reinforced Composites Kinematic hardening equations [8]: α = 300(-e -80ε ) (MPa), α = 87.5(-e -40ε ) (MPa), (α : No data) Fracture stress: F = 800 (MPa), F = 5 (MPa), F = 59 (MPa) In this example, the kinematic hardening without tropic hardening is assumed, which means the size of the yield surface is constant during the plastic deformation. Since the yield stress in the longitudinal direction (in the -direction) is assumed to be the same as the failure strength, the plastic deformation does not occur in the -direction. herefore, the above antropic coefficients for Hill s 948 yield function was calculated from the hardening curve in the transverse direction (in the -direction). In Figures 5-6, the measured stress-strain curves are compared with the calculated curves for the uni-axial Figure 5. Comparn of the measured stress-strain curves with those calculated using the two evolution rules of kinematic hardening for uni-axial tension and pure shear tests. A: Longitudinal (measured and calculated based on the orthotropic elasticity), B: ransverse (measured and calculated based the antropic and tropic evolution rules), C: Shear (measured and calculated based on the antropic evolution rule), D: Shear (calculated based on the tropic evolution rule) Figure 6. Comparn of the measured stress-strain curves with those calculated using the two evolution rules of kinematic hardening for uni-axial tension and pure shear tests. A: Longitudinal (measured and calculated based on the orthotropic elasticity), B: ransverse (measured and calculated based the antropic evolution rule), C: Shear (measured and calculated based on the antropic and tropic evolution rules), D: ransverse (calculated based on the tropic evolution rule) Polymers & Polymer Composites, Vol., No. 3, 004 3
M.G. Lee, D. Kim, K. Chung, J. R. Youn and. J. Kang tension tests along the longitudinal and transverse directions (Curves A and B, respectively) and also for the pure shear test (Curve C) in the - plane. As for the longitudinal uni-axial curves, the measured curves within the elastic range were exactly the same with the calculated ones, which were obtained from the orthotropic elasticity (Curve A). Also, the calculated transverse uni-axial and pure shear curves based the antropic evolution rule are exactly the same with the experimental curves (Curves B and C). he calculated curves based on the proposed antropic evolution rule are also compared with the curves calculated based on the tropic evolution rule in Figures 5-6. In Figure 5, the reference state of the tropic rule is the transverse uni-axial tension so that the calculated transverse uni-axial curves based on the antropic and tropic rules are the same (Curve B), while the pure shear curve based on the antropic curve (Curve C) is quite different from that calculated based on the tropic rule (Curve D). In Figure 6, the reference state of the tropic rule is the pure shear so that the calculated pure shear curves based on the antropic and tropic rules are the same (Curve C), while the transverse uni-axial curve based on the antropic curve (Curve B) is quite different from that calculated based on the tropic rule (Curve D). In Figure 7, loading and unloading behaviors are compared for those calculated based on the antropic and tropic evolution rules for the transverse uniaxial tension and the pure shear (without considering fractures). In Figure 7(a), the reference state of the tropic evolution rule is the transverse uni-axial tension so that the transverse uni-axial tension curves are the same for the tropic and antropic rules (Curve A), while the curve for the pure shear based on the antropic rule (Curve B) differs from the pure shear curve based on the tropic rule (Curve C). In Figure 7(b), the reference state of the tropic evolution rule is the pure shear so that the pure shear curves are the same for the tropic and antropic rules (Curve B), while the curve for transverse uniaxial tension based on the antropic rule (Curve A) differs from the transverse uni-axial tension based on the tropic rule (Curve C). hese example solutions shown in Figures 5~7 demonstrate that the kinematic hardening rule with antropic back-stress evolution rule is essential to properly describe the constitutive behaviors of highly antropic materials such as fiber-reinforced composites. Figure 7. Comparn of loading and unloading curves calculated based on the two evolution rules of kinematic hardening (a) A: ransverse (based on the antropic and tropic evolution rules), B: Shear (based antropic evolution rule), C: Shear (based on the tropic evolution rule) (b) A: ransverse (based on the antropic evolution rule), B: Shear (based on the antropic and tropic evolution rules), C: ransverse (based on the tropic evolution rule) 3 Polymers & Polymer Composites, Vol., No. 3, 004
Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber- Reinforced Composites CONCLUSIONS In order to develop the theory of plasticity for orthotropic fiber-reinforced composites, a new combined tropic-kinematic hardening law with the antropic back-stress evolution rule was developed under the elasto-plasticity scheme. o account for the antropic yield function, Hill s quadratic yield function was used. he stress update scheme based on the incremental deformation theory was also developed. Example solutions confirmed that the new constitutive equation well represented the Bauschinger and transient behaviors, as well as the directional dependence of the back-stress evolution for the highly antropic materials such as fiber-reinforced composites. 3. Khan A. S. and Huang S., Continuum heory of Plasticity, John Wiley & Sons, New York, (995) 4. Chung K., Ph. D. Dissertation, Stanford University, (984) 5. Simo J. C. and Hughes.J.R., Computational Inelasticity, Springer-Verlag, New York, (997) 6. Valdmanis V. M. and Mikelsons M. Ya., Mekh. Kompoz. Mater., 3, (99) 447 ACKNOWLEDGEMEN his work was supported by the Ministry of Science and echnology through the National Research Laboratory. REFERENCES. Saleeb A.F., Wilt.E., Al-Zoubi N.R. and Gendy A.S., Composites Part B, 34, (003). Jones R. M., Mechanics of Composite Materials, McGraw-Hill, New York, (975) 3. Rees D.W. A., Acta Mech., 5,(984) 5 4. Kosarchuk V. V., Kovalchuk B. I. and Lebedev A. A., Probl. Prochn., 4, (986) 50 5. Shih C. F. and Lee D., Eng. Mater. echnol., 00, (978) 94 6. Leewood A. R., Doyle J. F. and Sun C.., Comput. Struct., 5, (987) 749 7. Isupov L. P. and Rabotnov Yu. N., Izv. AN SSSR M, (985) 8. Sarbayev B. S., Computational Materials Science, 6, (99) 9. Chaboche J. L., Int. J. Plasticity,, (986) 49 0. Lee M.G., Kim D., Kim C., Wenner M.L., Wagoner R.H., Chung K., Plasticity 003, Canada, Quebec (accepted).. Hill R., he Mathematical heory of Plasticity, Oxford University Press, New York, (956). Chung. K., Richmond O., Int. J. Plasticity, 9, (993) 907 Polymers & Polymer Composites, Vol., No. 3, 004 33
M.G. Lee, D. Kim, K. Chung, J. R. Youn and. J. Kang 34 Polymers & Polymer Composites, Vol., No. 3, 004