Experiments and Identifications for Finite Polymer Inelasticity
2
Experiments and Identifications for Finite Polymer Inelasticity Joel M´ endez ∗ , Serdar G ¨ oktepe, and Christian Miehe Institut f¨ ur Mechanik (Bauwesen) Lehrstuhl I, Universit¨ at Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany The constitutive theories intended to quantitatively account for the complicated material response exhibited by polymers in- clude, in general, adjustable material parameters. These must be identified from experimental data obtained from the material under consideration. This contribution presents the complete procedure studying the behavior of polymers at large strains in three basic steps: i) Accomplishment of homogeneous and 3-D inhomogeneous experiments under different deformation con- ditions. ii) Identification of the material parameters of a constitutive model by means of gradient–based optimization methods with respect to the homogeneous experimental data. iii) Validation of the identified material parameters by comparing 3-D FE simulations to the inhomogeneous experimental data. 1 Introduction and Parameter Identification The two types of polymers considered in this work are: i) a hydrogenated nitrile butadiene rubber (HNBR50) having a viscoelastic mechanical behavior, and ii) a glassy polymer (a type of polycarbonate denominated as Makrolon 2607) showing a mechanical response related to viscoplasticity. Constitutive models in general, and those shown here in particular, contain some adjustable material parameters which must be identified. Parameter identification as an optimization problem is generally not well–posed. It is tackled by means of methods that can be roughly classified as gradient–free or gradient–based, with the latter being applied in our study. We start from the so-called inverse problem of parameter identification which reads: find a set of material parameters κ , such that a mathematical model z (κ, f , u)=0 with a given input quantity f and an output quantity u(κ)= u exp . Comparison of u exp (displacements obtained via experiments) and u (obtained from simulations) defines an overdetermined problem, solved by the least squares minimization method: find κ , such that z (κ, f , u)= 0 for a given f and f (κ) := 1 2 u(κ) − u exp 2 → MIN with κ ∈S := {κ | h(κ)= 0 and g(κ) ≤ 0}. In order to be able to treat the inequality constraints g(κ) ≤ 0 as equality constraints and then include them in a non-linear system of equations, an active set strategy is used: g W (κ)= 0 ⊂ g(κ). Associated to this non-linear mathematical (primary) optimization problem there exists a dual Lagrange function L(κ, µ, λ W )= f (κ)+ µ T h(κ)+ λ T W g W (κ) → STAT. (1) with the Lagrange multipliers µ := {µ 1 , ..., µ n h } T and λ := {λ 1 , ..., λ n h } T . The inclusion of the necessary Karush–Kuhn– Tucker (KKT) optimality constraints allows us to recast the dual problem into a non-linear system of equations G(κ, µ, λ W )= ⎡ ⎣ ∇ κ L ∇ µ L ∇ λ W L ⎤ ⎦ = ⎡ ⎣ ∇ κ f (κ)+ ∇ T κ h(κ)µ + ∇ T κ g W (κ)λ W h(κ) g W (κ) ⎤ ⎦ = 0 (2) which will then be linearized via the Newton’s method. Its iterative solution is known as sequential quadratic programming or SQP method. For further application of this method to material parameter identification, as well as its extension to the use of quasi Newton methods to assure the positive definiteness of the Hessian matrix needed for linearization see Scheday and Miehe [6], Scheday [5], and the citations therein. This methodology will be now applied together with constitutive theories and homogeneous experiments for the characterization of polymers. 2 Experiments and Simulations on Finite Polymer Inelasticity Finite Rubber Viscoelasticity. As a micro-mechanically motivated constitutive theory for finite rubber elasticity and vis- coelasticity, the recent models proposed by Miehe, G¨ oktepe and Lulei [4] and Miehe and G¨ oktepe [3] are applied. They are based on a new micro-structure symbolized by a micro-sphere representing a continuous distribution of chain orientations in space. The micro-mechanical response of each chain in a two-dimensional constrained environment defines the micro- kinematic variables in the form of the stretch of the chain and the contraction of the cross section of a micro-tube that contains the chain. Once the viscoelastic constitutive theory has been selected, homogeneous and inhomogeneous experiments are car- ried out on the elastomer HNBR50. The results obtained are shown in Fig.1. First of all, parameters for the elastic part must be identified, for which long relaxation experiments similar to the one shown in Fig.1b are carried out. From these experiments, the equilibrium response for uniaxial and equibiaxial tension tests shown in Fig.1a are obtained (discrete points). Using these ∗ Corresponding author: e-mail: [email protected], Phone: +00 49 711 685 66325, Fax: +00 49 711 685 66347 PAMM · Proc. Appl. Math. Mech. 6, 401–402 (2006) / DOI 10.1002/pamm.200610181 © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Transcript of Experiments and Identifications for Finite Polymer Inelasticity
Experiments and Identifications for Finite Polymer Inelasticity
Joel Mendez∗, Serdar Goktepe, and Christian Miehe
Institut fur Mechanik (Bauwesen) Lehrstuhl I, Universitat Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany
The constitutive theories intended to quantitatively account for the complicated material response exhibited by polymers in-clude, in general, adjustable material parameters. These must be identified from experimental data obtained from the materialunder consideration. This contribution presents the complete procedure studying the behavior of polymers at large strains inthree basic steps: i) Accomplishment of homogeneous and 3-D inhomogeneous experiments under different deformation con-ditions. ii) Identification of the material parameters of a constitutive model by means of gradient–based optimization methodswith respect to the homogeneous experimental data. iii) Validation of the identified material parameters by comparing 3-DFE simulations to the inhomogeneous experimental data.
1 Introduction and Parameter IdentificationThe two types of polymers considered in this work are: i) a hydrogenated nitrile butadiene rubber (HNBR50) having aviscoelastic mechanical behavior, and ii) a glassy polymer (a type of polycarbonate denominated as Makrolon 2607) showinga mechanical response related to viscoplasticity. Constitutive models in general, and those shown here in particular, containsome adjustable material parameters which must be identified. Parameter identification as an optimization problem is generallynot well–posed. It is tackled by means of methods that can be roughly classified as gradient–free or gradient–based, with thelatter being applied in our study. We start from the so-called inverse problem of parameter identification which reads: find aset of material parameters κ�, such that a mathematical model z(κ, f , u) = 0 with a given input quantity f and an outputquantity u(κ) = uexp. Comparison of uexp (displacements obtained via experiments) and u (obtained from simulations)defines an overdetermined problem, solved by the least squares minimization method: find κ�, such that z(κ, f , u) =0 for a given f and f(κ) := 1
2‖u(κ) − uexp‖2 → MIN with κ ∈ S := {κ | h(κ) = 0 and g(κ) ≤ 0}. In orderto be able to treat the inequality constraints g(κ) ≤ 0 as equality constraints and then include them in a non-linear systemof equations, an active set strategy is used: gW(κ) = 0 ⊂ g(κ). Associated to this non-linear mathematical (primary)optimization problem there exists a dual Lagrange function
L(κ, µ, λW) = f(κ) + µT h(κ) + λTWgW(κ) → STAT. (1)
with the Lagrange multipliers µ := {µ1, ..., µnh}T and λ := {λ1, ..., λnh
}T . The inclusion of the necessary Karush–Kuhn–Tucker (KKT) optimality constraints allows us to recast the dual problem into a non-linear system of equations
G(κ, µ, λW) =
⎡⎣
∇κL∇µL∇λWL
⎤⎦ =
⎡⎣
∇κf(κ) + ∇Tκh(κ)µ + ∇T
κgW(κ)λWh(κ)
gW(κ)
⎤⎦ = 0 (2)
which will then be linearized via the Newton’s method. Its iterative solution is known as sequential quadratic programmingor SQP method. For further application of this method to material parameter identification, as well as its extension to the useof quasi Newton methods to assure the positive definiteness of the Hessian matrix needed for linearization see Scheday andMiehe [6], Scheday [5], and the citations therein. This methodology will be now applied together with constitutive theoriesand homogeneous experiments for the characterization of polymers.
2 Experiments and Simulations on Finite Polymer Inelasticity
Finite Rubber Viscoelasticity. As a micro-mechanically motivated constitutive theory for finite rubber elasticity and vis-coelasticity, the recent models proposed by Miehe, Goktepe and Lulei [4] and Miehe and Goktepe [3] are applied. They arebased on a new micro-structure symbolized by a micro-sphere representing a continuous distribution of chain orientationsin space. The micro-mechanical response of each chain in a two-dimensional constrained environment defines the micro-kinematic variables in the form of the stretch of the chain and the contraction of the cross section of a micro-tube that containsthe chain. Once the viscoelastic constitutive theory has been selected, homogeneous and inhomogeneous experiments are car-ried out on the elastomer HNBR50. The results obtained are shown in Fig.1. First of all, parameters for the elastic part must beidentified, for which long relaxation experiments similar to the one shown in Fig.1b are carried out. From these experiments,the equilibrium response for uniaxial and equibiaxial tension tests shown in Fig.1a are obtained (discrete points). Using these
∗ Corresponding author: e-mail: [email protected], Phone: +00 49 711 685 66325, Fax: +00 49 711 685 66347
PAMM · Proc. Appl. Math. Mech. 6, 401–402 (2006) / DOI 10.1002/pamm.200610181
© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
experimental data material parameters are identified (lines in Fig.1a). Then the viscoelastic overstress contribution of the ma-terial is identified from cyclic tension-compression tests (Fig.1c and 1d). For the numerics of the constitutive model as well asa list of the material parameters identified with the procedure presented here see [4] and [3]. Up to here the excellent capacityof the model in quantitatively describing the homogeneous response of rubber has been assessed. In order to investigate itsperformance in finite element computations, inhomogeneous experiments are required. Fig.1e shows the load-displacementdiagram of an inhomogeneous shear test using hyperboloidal rubber specimens. This inhomogeneous rubber specimen was
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Experiment
Simulation
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c) d)
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λ [−] λ [−]e) f)
Fig. 1 a) Equilibrium response obtained from long relaxation experiments such as b). c) and d) are tension-compressioncyclic tests. e) Load-displacement diagram of an inhomogeneous shear test with its correspondent pictures taken. Solid linescorrespond to experiments and dashed ones to simulations. f ) 3-D deformation fields (left) post-processed from the imagesshown in e) and their counterpart obtained by FE simulation (right).
discretized with 1152 eight-noded Q1P0 mixed brick finite elements. The dashed line in Fig.1e is the simulated force requiredto drive the model. It is seen that the model accurately resembles the experiment. At last, deformation measurement usingthe grating method is performed. For this purpose the software ARAMIS is utilized. The three-dimensional inhomogeneousdeformation fields are shown in Fig.1f together with their corresponding simulations at different steps of the deformation.
Viscoplasticity of Glassy Polymers. As a constitutive theory for polycarbonate a formulation of finite viscoplasticity inthe logarithmic strain space is used. For details of this model and the modular structure on which it is based see Miehe, Apeland Lambrecht [2] and Goktepe, Mendez and Miehe [1]. Since polycarbonate necks when subjected to tension, a suitable wayof deforming it homogeneously is under compression. Uniaxial compression and pure shear homogeneous experiments werecarried out on Makrolon 2607 specimens (solid lines in Fig.2a). For uniaxial compression, cylinders of 10mm in length and10mm in diameter were used. Cubic specimens measuring 10mm on each side were applied for pure shear deformation. The
σ [
]
[ ]
= 50
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m
λ.
Thickness reduction [%]
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0 20 40 60 80 1000.70.60.50.40.30.20.100
20
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120
140
160
11
Fig. 2 a) Homogeneous uniaxial and pure shear compression. True-stress vs. true-strain diagrams are shown. b) Load-displacement diagram obtained from tension of a flat coupon specimen. c) 3-D deformation fields (left) post-processed withARAMIS and their counterparts obtained by FE simulation (right).
material parameters for the model were identified from these two homogeneous deformation modes making the model fit theexperiments quantitatively (dashed lines in Fig.2a). For inhomogeneous experiments flat tension specimens (ISO 527) wereused. The FE simulations which showed an excellent agreement with the experimental results are depicted in Fig.2b and 1c
Acknowledgements The authors would like to thank Harish Iyer for his generous support in the course of this research. Partial supportfor this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under grant Mi 295-9/1 and SFB-404 C6.
References
[1] S. Goktepe, J. Mendez and C. Miehe, PAMM, 5, 269–270 (2005).[2] C. Miehe, N. Apel, and M. Lambrecht, Comput. Methods Appl. Mech. Engrg., 191, 5383–5425 (2002).[3] C. Miehe and S. Goktepe, J. Mech. Phys. Solids, 53, 2231-2258 (2005).[4] C. Miehe, S. Goktepe, and F. Lulei, J. Mech. Phys. Solids, 52, 2617–2660 (2004).[5] G. Scheday, Ph.D. Thesis, Institute of Applied Mechanics, University of Stuttgart (2003).[6] G. Scheday and C. Miehe, PAMM, 1, 189–190 (2002).
© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Section 6 402
