Research Papers

Modeling of the Long-Term Behavior of Glassy Polymers

[+] Author and Article Information
Mathias Wallin

Division of Solid Mechanics,
Lund University,
P. O. Box 118,
SE-221 00 Lund, Sweden

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received March 23, 2012; final manuscript received August 17, 2012; published online December 21, 2012. Assoc. Editor: Yoshihiro Tomita.

J. Eng. Mater. Technol 135(1), 011001 (Dec 21, 2012) (11 pages) Paper No: MATS-12-1055; doi: 10.1115/1.4007499 History: Received March 23, 2012; Revised August 17, 2012

The constitutive model for glassy polymers proposed by Arruda and Boyce (BPA model) is reviewed and compared to experimental data for long-term loading. The BPA model has previously been shown to capture monotonic loading accurately, but for unloading and long-term behavior, the response of the BPA model is found to deviate from experimental data. In the present paper, we suggest an efficient extension that significantly improves the predictive capability of the BPA model during unloading and long-term recovery. The new, extended BPA model (EBPA model) is calibrated to experimental data of polycarbonate (PC) in various loading–unloading situations and deformation states. The numerical treatment of the BPA model associated with the finite element analysis is also discussed. As a consequence of the anisotropic hardening, the plastic spin enters the model. In order to handle the plastic spin in a finite element formulation, an algorithmic plastic spin is introduced. In conjunction with the backward Euler integration scheme use of the algorithmic plastic spin leads to a set of algebraic equations that provides the updated state. Numerical examples reveal that the proposed numerical algorithm is robust and well suited for finite element simulations.

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Haward, R. N., and Thackray, G., 1968, “The Use of a Mathematical Model to Describe Isothermal Stress-Strain Curves in Glassy Thermoplastics,” Proc. R. Soc. London, Ser. A, 302, pp. 453–472. [CrossRef]
Gibson, A. G., Hope, P. S., and Ward, I. M., 1980, “The Hydrostatic Extrusion of Polymethylmethacrylate,” J. Mater. Sci., 15, pp. 2207–2220. [CrossRef]
Stokes, V. K., and Nied, H. F., 1986, “Solid Phase Sheet Forming of Thermoplastics, Parts I and II,” J. Eng. Mater. Technol., 108, pp. 107–118. [CrossRef]
Bowden, P. B., and Raha, S., 1970, “The Formation of Micro Shear Bands in Polystyrene and Polymethylmethacrylate,” Philos. Mag., 22, pp. 463–482. [CrossRef]
G'Sell, C., and Jonas, J. J., 1981, “Yield and Transient Effects During the Plastic Deformation of Solid Polymers,” J. Mater. Sci., 16, pp. 1956–1974. [CrossRef]
Arruda, E. M., Boyce, M. C., and Quintus-Bosz, H., 1993, “Effects of Initial Anisotropy on the Finite Strain Deformation Behavior of Glassy Polymers,” Int. J. Plast., 9, pp. 783–811. [CrossRef]
Arruda, E. M., and Boyce, M. C., 1993, “Evolution of Plastic Anisotropy in Amorphous Polymers During Finite Straining,” Int. J. Plast., 9, pp. 697–720. [CrossRef]
Krempl, E., and Khan, F., 2003, “Rate(Time)-Dependent Deformation Behavior: An Overview of Some Properties of Metals and Solid Polymers,” Int. J. Plast., 19, pp. 1069–1095. [CrossRef]
Zari, F., Naït-Abdelaziz, M., Woznica, K., and Gloaguen, J.-M., 2007, “Elasto-Viscoplastic Constitutive Equations for the Description of Glassy Polymers Behavior at Constant Strain Rate,” J. Eng. Mater. Technol., 129, pp. 564–571. [CrossRef]
Dreistadt, C., Bonnet, A. S., Chevrier, P., and Lipinski, P., 2009, “Experimental Study of the Polycarbonate Behaviour During Complex Loadings and Comparison with the Boyce, Parks and Argon Model Predictions,” Mater. Des., 30, pp. 3126–3140. [CrossRef]
James, H. M., and Guth, E., 1943, “Theory of the Elastic Properties of Rubber,” J. Chem. Phys., 11, pp. 455–481. [CrossRef]
Treloar, L. R. G., 1946, “The Elasticity of a Network of Long-Chain Molecules: Part III,” Trans. Faraday Soc., 42, pp. 83–94. [CrossRef]
Boyce, M. C., Parks, D. M., and Argon, A. S., 1988, “Large Inelastic Deformation of Glassy Polymers, Part I: Rate-Dependent Constitutive Model,” Mech. Mater., 7, pp. 15–33. [CrossRef]
Arruda, E. M., and Boyce, M. C., 1991, “Anisotropy and Localization of Plastic Deformation,” Proceedings of Plasticity, J. P.Boehler and A. S.Khan, eds., Elsevier Applied Science, London, 483 pp.
Anand, L., and Ames, N. M., 2006, “On Modeling the Micro-Indentation Response of an Amorphous Polymer,” Int. J. Plast., 22, pp. 1123–1170. [CrossRef]
Silberstein, M. N., and Boyce, M. C., 2010, “Constitutive Modeling of the Rate, Temperature, and Hydration Dependent Deformation Response of Nafion to Monotonic and Cycling Loading,” J. Power Sources, 195, pp. 5692–5706. [CrossRef]
Treloar, L. R. G. and Riding, G., 1979, “A Non-Gaussian Theory for Rubber in Biaxial Strain. I Mechanical Properties,” Proc. R. Soc. London, Ser. A., 369, pp. 261–280. [CrossRef]
Wu, P. D., and van derGiessen, E., 1993, “On Improved Network Models for Rubber Elasticity and Their Applications to Orientation Hardening in Glassy Polymers,” J. Mech. Phys. Solids, 41(3), pp. 427–456. [CrossRef]
Beatty, M. F., 2003, “An Average-Stretch Full-Network Model for Rubber Elasticity,” J. Elast., 70, pp. 65–86. [CrossRef]
Arruda, E. M. and Boyce, M. C., 1993, “A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,” J. Mech. Phys. Solids, 41, pp. 389–412. [CrossRef]
Govaert, L. E., Timmermans, P. H. M., and Brekelmans, W. A. M., 2000, “The Influence of Intrinsic Strain Softening and Strain Localization in Polycarbonate: Modeling and Experimental Validation,” J. Eng. Mater. Technol., 122, pp. 177–185. [CrossRef]
Dupaix, R. B., Palm, G., and Castro, J., 2006, “Large Strain Mechanical Behavior of Polymethylmethacrylate (PMMA) Near the Glass Transition Temperature,” J. Eng. Mater. Technol., 128, pp. 559–563. [CrossRef]
Negahban, M., Goel, A., Delabarre, P., Feng, R., and Dimick, A., 2006, “Experimentally Evaluating the Equilibrium Stress in Shear of Glassy Polycarbonate,” J. Eng. Mater. Technol., 128, pp. 537–542. [CrossRef]
Wang, M. C., and Guth, E. J., 1952, “Statistical Theory of Networks of Non-Gaussian Flexible Chains,” J. Chem. Phys., 20(7), pp. 1144–1157. [CrossRef]
Argon, A. S., 1973, “A Theory for the Low-Temperature Plastic Deformation of Glassy Polymers,” Philos. Mag., 28(4), pp. 839–865. [CrossRef]
Cohen, A., 1991, “A Pade Approximant to the Inverse Langevin Function,” Rheol. Acta, 30(3), pp. 270–273. [CrossRef]
Agah-Tehrani, A., Lee, E. H., Mallett, R. L., and Onat, E. T., 1987, “The Theory of Elastic-Plastic Deformation at Finite Strain With Induced Anisotropy Modeled as Combined Isotropic-Kinematic Hardening,” J. Mech. Phys. Solids, 35(5), pp. 519–539. [CrossRef]
G'Sell, C., and Gopez, A. J., 1985, “Plastic Banding in Glassy Polycarbonate Under Plane Simple Shear,” J. Mater. Sci., 20, pp. 3462–3478. [CrossRef]
Hasan, O. A., and Boyce, M. C., 1995, “A Constitutive Model for the Nonlinear Viscoelastic Viscoplastic Behavior of Glassy Polymers,” Polym. Eng. Sci., 35, pp. 331–344. [CrossRef]
Cao, K., Wang, Y., and Wang, Y., 2012, “Effects of Strain Rate and Temperature on the Tension Behaviour of Polycarbonate,” Mater. Des., 38, pp. 53–58. [CrossRef]
Weber, G., and Anand, L., 1990, “Finite Deformation Constitutive Equations and a Time Integration Procedure for Isotropic, Hyperelastic-Viscoplastic Solids,” Comput. Methods Appl. Mech. Eng., 79, pp. 173–202. [CrossRef]
Steinmann, P., and Stein, E., 1996, “On the Numerical Treatment and Analysis of Finite Deformation Ductile Single Crystal Plasticity,” Comput. Methods Appl. Mech. Eng., 129, pp. 235–254. [CrossRef]
Miehe, C., 1998, “Comparison of Two Algorithms for the Computation of Fourth-Order Isotropic Tensor Functions,” Comput. Struct., 66(1), pp. 37–43. [CrossRef]
Ekh, M., and Runesson, K., 2001, “Modeling and Numerical Issues in Hyperelasto-Plasticity With Anisotropy,” Int. J. Solids Struct., 38, pp. 9461–9478. [CrossRef]
Wallin, M., and Ristinmaa, M., 2005, “Deformation Gradient Based Kinematic Hardening Model,” Int. J. Plast., 21, pp. 2025–2050. [CrossRef]
Shampine, L. F., Reichelt, M. W., and Kierzenka, J. A., 1999, “Solving Index-1 DAEs in MATLAB and Simulink,” SIAM Rev., 41(3), pp. 538–552. [CrossRef]


Grahic Jump Location
Fig. 1

Rheological illustration of the BPA model. The model is governed by the elements: (a) elastic spring, (b) viscoplastic dashpot, and (c) nonlinear Langevin spring.

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Fig. 2

The idealized chain structure, according to the 8-chain model. The dimension of the 8-chain cube is a0 and φ0 denotes the initial orientation angle between a chain and the principal axes. The unit vectors eα, α = 1,2,3, align with the principal directions of V¯p.

Grahic Jump Location
Fig. 3

The stress–strain response of the BPA model during uniaxial compression. The experimental results are taken from Ref. [10].

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Fig. 4

The stress–strain response of the BPA model during uniaxial compression for repeated unloadings to π = 1.2 MPa. The experimental results are taken from Ref. [10].

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Fig. 5

Rheological illustration of the EBPA model. The model is governed by the elements: (a) elastic spring, (b) two viscoplastic dashpots, and (c) nonlinear Langevin spring.

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Fig. 6

Stress–strain curves for uniaxial compression of PC. Experimental data is taken from Ref. [7].

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Fig. 7

Stress–strain curves for plane strain compression of PC. Experimental data is taken from Ref. [7].

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Fig. 8

Stress–strain curves for simple shear of PC. Experimental data is taken from Ref. [5].

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Fig. 9

True stress versus true strain for uniaxial compression of bisphenol A polycarbonate. Experimental data is taken from Ref. [10].

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Fig. 10

Athermal shear strength, s, and its components s1 and s2 versus true strain, according to the EBPA model. The backstresses, βBPAdev and βEBPAdev, in the direction of the applied load.

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Fig. 11

Uniaxial compression responses for bisphenol A polycarbonate, according to (a) the BPA and (b) the EBPA model. The repeated unloadings are performed to π = 59 MPa. Experimental data is taken from Ref. [10].

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Fig. 12

Uniaxial compression responses for bisphenol A polycarbonate, according to (a) the BPA and (b) the EBPA model. The repeated unloadings are performed to π = 1.2 MPa. Experimental data is taken from Ref. [10].

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Fig. 13

The force–displacement response of the cylindrical bar. The deformation rate, u·/L = 0.001 1/s, was used in the simulation. s0 is a constitutive parameter given in Table 3.

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Fig. 14

(a) The geometry and finite element mesh of the cylindrical bar. The deformed mesh and the plastic stretch, λ¯p, at (b) u/L = 0.08, (c) u/L = 0.10, and (d) u/L = 0.18.

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Fig. 15

(a) The geometry of the specimen subjected to combined tension and bending. Visualization of the plastic stretch, λ¯p, in the deformed meshes at the end of (b) loading (by EBPA), (c) unloading (by EBPA), and (d) dwell period of two months (by both models).

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Fig. 16

The displacement component, ux versus time during unloading and the dwell period of two months. Solid and dashed line present the EBPA and BPA response, respectively. The markers, ▪ and ♦, indicate the final position.




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