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Material parameters identification: an inverse modeling methodology applicable for thermoplastic materials.

INTRODUCTION

Polymers are widely used in the transport industry, specially, when structural components and passengers/pedestrian safety are in focus. The experience in using polymers in impact protection and structural systems, however, is limited, and there are several challenges which call for research. One of the most obvious is the lack of robust material models in commercial finite element codes and in order to address this issue a new hyperelastic-vis-coplastic constitutive model has been developed by Polanco-Loria et al. (1-3). This model, limited to isothermal conditions, is able to handle finite deformations, pressure and rate sensitivity and nonisochoric plastic flow

Development of new material models also involves experimental efforts where well defined material tests are required for identification of the parameters in the model. Thereafter, it is common to validate the proposed material model by doing numerical predictions of a component test which serves as an independent check of the capabilities of the model. At present, the static uniaxial tension test is the simplest mechanical test used for material characterization. However, for polymeric materials measurements of the true tensile stress--strain are difficult, in particular, when neck propagation and volumetric plastic strains are present during the deformation process. It seems that only optical-based systems should be used for a reliable material characterization when dealing with polymeric materials, as one can confirm from the literature (4-10).

Conversely, extraction of experimental data and its utilization for material parameter identification purposes can be cumbersome because constitutive relations of polymers are strain, strain rate, and temperature dependent. In particular after necking (e.g., strain localization), two basic problems are commonly observed: a nonhomogenous increase of the local strain rate when compared to the global applied one; and a nonhomogenous increase of temperature due to the low heat conduction capacity of polymers (e.g., adiabatic process). In addition, the variables involved in the thermo-mechanical characterization are highly coupled; consequently, material identification by trial and error procedures can be inefficient for such complex materials. Alternatively, nonlinear inverse computational methods have been applied to identify material parameters (see for instance references (11-16)) in cases where the parameters are not directly measurable or the deformations fields are of heterogeneous nature (e.g., complex boundary conditions, loss of homogeneity during the deformation process, among others). In addition, these nonlinear mathematical techniques are largely used during the design and performance of complex structures (e.g., crashworthiness) and the numerical simulation of complex manufacturing processes (e.g., forming. extrusion, molding). The basic principle of the inverse method, for parameter identification purposes, is illustrated in Fig. 1.

The objective of this work is to describe a simple inverse modeling methodology to identify the material parameters of the recently developed constitutive model for thermoplastic materials [3]. For this purpose, an experimental program including uniaxial tensile and compression tests is performed on a mineral - and rubber-modified PP compound. In addition, the optimization software LS-OPT (17) is used to demonstrate the identification procedure.

The modified Boyce-Raghava Model For Thermoplastic Materials

The model presented here, limited to isothermal conditions, is a physically-based constitutive model, involving the typical mechanisms of the elastic behavior of polymers, i.e. relative rotation around backbone carbon-car-bon bonds and entropy change by uncoiling molecule chains. In addition, viscoplastic flow associated with relative movement between molecules is included. Historically, the development of this model goes back to the work by Haward and Thackray (18) and further developed by Boyce (19) and Boyce et al. (20), who assumed that the total stress was the sum of an intermolecular and intramolecular contribution denoted Part A and Part B, respectively.

Part A. An inter-molecular barrier to deformation related to relative movement between molecules.

Part B. A network (entropic) resistance related to straightening of the molecule chains. According to the rheological model shown in Fig. 2, the two resistances A and B are assumed to have the same deformation gradient, i.e., F = [F.sub.A] = [F.sub.B], while the Cauchy stress tensor is obtained by summing the contributions from Parts A and B. The deformation gradient FA (Part A) is decomposed into the elastic and the plastic parts, i.e. [F.sbu.A] = [F.sub.A.sup.e] * [F.sup.p.sub.A]. Similarly, the Jacobian is decomposed in a multiplicative way, i.e. [J.sub.A] = det [F.sub.A] =[J.sub.A.sup.e] [J.sub.A.sup.p] =J. Next, a compressible Neo-Hookean material is chosen for the elastic part of the deformation, and the Cauchy stress tensor [[sigma].sub.A] reads

[[sigma].sub.A] = 1/[J.sub.A.sup.e][[lambda]ln][J.sub.A.sup.e]I + ([B.sup.e.sub.A - I]) (1)

where [lambda] and [mu] are the classical Lame constants of the linearized theory, [B.sup.e.sub.A] = [F.sub.A.sup.e] * [([Fsup.e.sub.A]).sup.T] is the elastic left Cauchy-Green deformation tensor, and I is the second order unit tensor. The coefficients [lambda] and [mu] may alternatively be expressed as functions of Young's modulus E and Poisson's ratio v. The yield criterion is assumed in the form [f.sub.A] = [[bar.[sigma]].sub.A] - [[sigma].sub.T] = 0, where [[sigma].sub.T] is the uniaxial yield stress in tension. The equivalent stress [[bar.[sigma]].sub.A] accounts for the pressure-sensitive behavior and is defined according to Raghava et al. (21) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The material parameter a = [[sigma].sub.C]/[[sigma].sub.T] [greater than or equal to] 1 describes the pressure sensitivity, where [[sigma].sub.C] is the uniaxial compressive yield strength of the material, [I.sub.1A] and [J.sub.2A] are the stress invariants related to respectively the volumetric and the deviatoric Cauchy stress tensor. To control the plastic dilatation, a nonassociative How rule is introduced where a Raghava-like plastic potential [g.sub.A] is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [beta] [greater than or equal to] 1 is a material parameter introduced to control the volumetric plastic strain. The flow rule gives the plastic rate-of-deformation tensor [D.sup.p.sub.A], as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where the equivalent plastic strain rate y, is chosen as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

here the two coefficients C and &q are material parameters easy to identify from uniaxial strain-rate tests.

The Part B includes the deformation gradient [F.sub.B] = [F.sub.A] = F, representing the network orientation. The network resistance is assumed to be hyperelastic. The Cauchy stress-stretch relation is used as the original definition of Boyce et al. [20]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where.J = [J.sub.B] = det FB is the Jacobian, and [L.sub.-1] is the inverse of the Langevin function defined as L (x) = coth x-l/x. The equivalent distorlional stretch is y and [B*.sub.B] =[F*.sub.B][[F*.sub.B].sup.T] is the distortional left Cauchy-Green deformation tensor. In this Equation [F*.sub.B] = [J.sub.B.sup.-1/3] represents the distortional part of [F*.sub.B]. There are two material parameters describing the network resistance: [C.sub.R] is the initial elastic modulus of Part B and N can be interpreted as the number of "rigid links" between the entanglements of the molecule chains (which can be related to a maximum attainable stretching).

Finally, the Cauchy stress tensor for the material is obtained by summing the contributions of Parts A and B. i.e.

[sigma] = [[sigma].sub.A] + [[sigma].sub.B] (7)

The stress update algorithm applied herein is valid for general rate-dependent hyperelaslic-plaslic formulations. The update scheme was proposed by Moran et al. (22) and is semiimplicit. Since the constitutive model is developed for explicit finite element analysis, it is assumed that the time steps are small and that a semiimplicit algorithm is sufficiently accurate. The constitutive model was implemented as a user-defined model in the finite element code LS-DYNA (23). Finally, because the source of inspiration of the present model is based on the work of Boyce and Raghava with some new features the model will be referred here as the modified Boyce-Raghava (MBR) model. Some examples of structural applications (e.g., three point bending beams and centrally loaded plates) with an in-depth discussion of results and parameters identification are provided in the original reference [3].

EXPERIMENTAL DATA USING A PP COPOLYMER MATERIAL

Introduction

To illustrate the material parameter identification process the experimental results presented here are complemented with the data obtained by Grytten el al. (24). The material investigated in this study is a commercial impact-modified PP used for injection molded automotive exterior parts. This is a 20% mineral filled and rubber-modified PP compound. Tensile injection molded test specimens (Type 1A of the ISO 527-2:1993) of 4 mm thickness were used. For DIC measurements a random black and white speckle pattern was applied to both the front and lateral side of the specimen prior to testing using mat spray paint. The commercial DIC system Vic3D was used to measure the displacement field during loading.

Four tensile tests were carried out in a Zwick Z250 universal test machine at 23[degrees]C using constant cross head speeds of 1, 10, 100, and 500 mm/min. The length of the narrow portion of the specimen was 80 mm, resulting in nominal strain rates of 2 X [l0.sub.-4], 2 X [10.sub.-3] 2 X [10.sub.2]. and 1 X [10.sub.-1] [s.sub.-1], respectively. The influence of the strain rate response on the four nominal stress-strain curves is illustrated in Fig. 3a while Fig. 3b indicates the increase of the nominal stress with strain rate (taken at a strain level of 3%). The load was measured with a 2.5 kN load cell and the signal was logged using a 12 bit National Instruments DAQCard 6062E. DIC technique was used in two of the four tensile tests. The experimental program included in addition, compression tests on a cubic specimen of 4 X 4 X 4 [mm.sub.3] at a loading speed of 2 mm/min. No DIC measurements were used for the compression tests.

Global Strain Rate Sensitivity Based on Tensile Tests

For the DIC measurements an "area of interest" (AOI) was defined for each of the two faces visible to the two cameras used. Inside the AOI a "critical" seclion was always monitored where all post-processing data was taken from. This "critical" section was adopted based on observations of repeated events of necking occurrence for the geometry and material used. A more in-depth description of the test procedure can be seen in the work of Grytten et al. (24). Other alternatives (e.g., introduction of a smooth notch (5), (7) or a geometrical imperfection [4, 6, 8]) have been used to define the "critical" section.

For the purpose of this work only one tensile test (actually two parallels) with DIC technique is used here to deduce the true stress-strain curve, as reported in (24). Indeed, the uniaxial tensile test loaded at a nominal strain rate of 2 X [10.sub.-2] [s.sub.-1] is considered in this work. A full-field measurement of deformation is important because many polymers present volumetric plastic contribution during plastic (low [(4), (6-9), (24-27) requiring the monitoring of the actual cross section for true stress calculations purposes.

Different video system solutions have been reported in the literature attempting to measure the true stress-strain relation of polymers. With only one camera the images can only give the strain components at one surface of the specimen, i.e., at one coordinate plane. In the literature, true stresses have been calculated using the transversal strain and assuming transversal isotropy (4), (5), (7), (8), (24), (26), (27) or using of one camera with a right-angle prism (measuring strain fields on both sides of a rectangular specimen) (9), (10). However, the results presented here (24) are based on a set-up including 3D DIC with two cameras and stereo vision. Hence, the common assumption of transversal isotropy is not required in this set-up. Two cameras based system were also used in (6).

The true stress-strain curve deduced from DIC analysis is provided in Fig. 4a. This figure represents the average response of two parallels. The stress-strain curve is also characterized by some discrete black points which will be used later as the target solution for the identification procedure. Figure 4b illustrates the AOI adopted in this test. As previously mentioned, this test was run under displacement control with a nominal strain rate of 2 X [10.sub.-2] [s.sub.-1]. Generally, because of necking and cold drawing process, local strain rate is function of the position (section) along the coupon as well as time, thus, each cross section is submitted to local strain rale variations during the deformation process. In other words, each of the experimental points that conforms the deduced true stress-strain curve (see Fig. 4) are associated to a strain rate history the cross section has been exposed to. The longitudinal local strain rale history obtained during the deformation process for the material investigated here is illustrated in Fig. 5 and calculated according to a backward finite difference scheme:

[epsilon] = ([[epsilon].sub.t] +[DELTA]t - [[epsilon].sub.t])/[DELTA]t (8)

This information is crucial for the correct parameter identification of material models, as it will be explained later. An interesting aspect of the strain rate history of the material investigated here is that both global and local strain rate values are about of the same order, thus indicating than necking phenomenon occurs in a gradual manner. This observation can also be deduced from the smooth softening behavior of the true stress-strain curve of Fig. 4. Contrary, other observations have been reported in the literature where large deviations between local and global strain rates have been measured (6), (8), (10).

Because of the low frame rate of the cameras used it was not possible to capture precisely the strain level at failure, thus, the last point reported in Fig. 4 ([epsilon] = 1.13) represents a strain level close to failure (see also the last point of Fig. 5, where a tendency of strain rate increase can be observed).

Volumetric Strain Response

Considering the same "critical" section and the 3D DIC set-up, transversal strains along the width and the thickness of the tensile test specimen could be measured. True strain measurements (transversal vs. longitudinal) are presented in Fig. 6, where it is possible to observe the close similarity between strain values along the width and thickness direction. Hence, confirming that for this material, the transversal isotropy condition during the deformation process. On the other hand, utilizing the definition of the elastic Poisson's ratio during the whole deformation process, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] its evolution as a function of the longitudinal strain ([epsilon].sub.longitudinal) can be constructed, see Fig. 7. In the elastic range a Poisson's ratio close to 0.4 can be deduced while a smoothly reduction until an asymptotic value close to 0.20 is observed in the inelastic regime. Thus, indicating that the PP copolymer material studied here presents large volumetric plastic dilatation, due, probably, to damage and crazes creation at the micro scale level.

These results are summarized, for identification purposes, in Fig. 8, where the total volumetric strain ([[epsilon].sub.vol] = [[epsilon].sub.width] + [[epsilon].sub.thickness] + [[epsilon].sub.longitudinal]) is plotted against the longitudinal strain [[epsilon].sub.longitudinal]. This figure represents the average response of two parallels. The averaged response is also characterized by some discrete points (black) which will be used later as the target solution for identification of the plastic potential parameier [beta], see Eq. 3.

Compression Tests

Two compressive tests on the PP copolymer were performed on cubic specimens. These tests were carried out under displacement control, applying a constant velocity of 2 mm/min to the machines upper cross-head. This velocity was kept constant during the entire test. The rate of deformation corresponds to a nominal strain rate of 0.00833 [s.sub.-1]. The cubic samples had a main dimension of 4 mm per side.

The averaged force-displacement curve of the two tests is shown in Fig. 9; as registered by the servohydraulic machine. It is well known, that because of friction and planarity is very difficult to perform uniaxial compression tests with homogeneous stress and strain distributions. Consequently, DIC techniques are preferred to capture the potential strain in homogeneities that the boundary conditions could have introduced. Unfortunately, DIC was not used in the compression tests and it was necessary to assume that the deformation is uniform during the first stage of the test. Hence, it is believed that until strain levels of 10-20% the registered measurements, from the servohydraulic machine, characterize the compressive behavior of the material relatively well. Therefore, only one portion of the stress-strain curve in compression will be used for material identification purposes (indicated with the black points in Fig. 9).

INVERSE MODELING STRATEGY

Introduction

Material parameters identification by inverse modeling relies on, basically, a nonlinear optimization technique where a certain objective function is minimized[12]. Commonly, the objective function is assumed as the residual norm between numerical and experimental results. The establishment of the target (e.g., experiments) solution depends largely on the material and measurement technique used, among other things. For material identification purposes three types of solution strategies are usually encountered in the literature [11-13, 15, 16]:

* Global approach based on force-displacement response

* Local approach based on stress-strain response

* Mixed approach based on both global and local responses

It is worth to mention that, because of the necessity of modeling the whole specimen (or part of it) the first alternative demands relatively high computational resources while the third requires, both, advanced experimental and computational resources. On the other hand, the second option imposes access to the stress-strain history in one or various material points of the specimen. This information is obtainable with the DIC technique as employed in this work. The computational resources are minimal because the identification procedure uses only one finite element to characterize the material point behavior. Alternatively, material model driver, where strain rate history is prescribed, could have been used.

One important aspect of the identification procedure is the correct data transfer between experiments and FE loading conditions, in particular for the second option strategy. Indeed, because of necking and strain rate effects global and local strains can be quite different. Therefore, any true stress-strain curve to be used for material characterization should have a strain rate history associated to it (28). In essence, the applied displacement control of the finite element used must follow the experimental local strain and the time history according to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where L represents any arbitrary initial element length of the finite element used, [[epsilon](t).sub.local] the measured local strain and u(t) the displacement history to be applied, see Fig. 10.

Finally, we should mention that this local strategy can be enhanced by including more "material points," to represent the stress-strain behavior of different cross sections of the specimen, each of them with their corresponding strain rate history.

The constitutive model requires the identification of nine material parameters. The elastic constants of part A (e.g., the Young's modulus E and the Poisson's ratio v) are identified by reproducing an averaged linear response of the actual nonlinear behavior. For this, we use the uniaxial (ensile lest submitted to the lowest loading strain rate (e.g. [[epsilon].sub.0]). Next, the parameter C is identified using the nominal stress-strain curves at different strain rates with the reference strain rate [epsilon].sub.0] taken as the lowest strain rate used in the tests. Using the true stress-strain curve and the volumetric strain response as the main targets the identification of the four variables [[sigma].sub.T], [C.sub.R], N, and [beta] is done by inverse modeling using the local response approach (second strategy). Finally, the parameter [alpha], which defines the pressure sensitivity of the yield stress, is calibrated from a uniaxial compression test using a similar local approach.

The Response Surface Methodology

Identification of the material parameters was performed with the optimization software LS-OPT (17). The optimization technique used relies on the response surface methodology (RSM), a mathematical method for constructing smooth approximations of functions in a design space.

Let us assume that the response of a system is characterized by the function f(x). This function can represent the results of finite element analyses. The RSM seeks to iteratively lit the function f(x) with the approximation, [~.f](x), in a least-square sense. This approximating function can be written as a sum of products of the interpolation functions, [[empty set].sub.i] with the undetermined coefficients (e.g., the regression constants), [a.sub.i] as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where L is the number of the interpolation functions and x is the independent variables (e.g., design variables) vector. To obtain an approximate solution we must determine values of [a.sub.i] such that f and [~.f] remains as close as possible. For this purpose, the approximation functions are evaluated at N experimental points and the unknown constants ([a.sub.i], i = 1 ... .., L) are then determined by forcing the sum of errors squared, [[epsilon].sup.2], to be minimized, according to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

By collecting f and [~.f] in a vector form and defining the error vector as [epsilon] = f - [~.f] the matrix notation of Eq. 11 can be rewritten as:

[[epsilon].sup.2] = [[epsilon].sup.T][epsilon] = [(f-[~.f]).sup.T]((f-[~.f])) = [(f-[~.Xa]).sup.T]((f-[~.Xa]))(12)

where a = [[[a.sub.1] [a.sub.2] .. .. [a.sub.L]].sup.T] is the undetermined constants vector and X is the interpolation matrix defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The undetermined coefficients, [a.sub.i] can be determined by minimizing Eq. 12 in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which yields: a = [[x.sup.T]x.sup.-1][x.sup.T]f (14)

Equation 14 defines the coefficients [a.sub.i] of the approximation function [~.f](x). which represent the best lit to the function f(x). The LS-OPT software (17) uses, in addition, an adaptive surface response generator known as the successive response surface method (SSRM). This technique focuses on the automatic creation of a subspace of the design space. By successive iterations nested subregions are generated with advancements of panning and zooming steps. A more in-depth study of this technique can be found in [17].

Finally, for this purposes of this work a linear order approximation model was used together with a D-optimality criterion for the selection of the experimental points.0

NUMERICAL IDENTIFICATION

Parameter Identification Procedure

The parameter identification procedure can become cumbersome because the seven variables, describing the nonlinear behavior, are in some degree coupled to each other and a detailed knowledge of the constitutive model is necessary to facilitate such identification. The strain rate parameters C and [[epsilon].sub.0], of our model, do not have any effect on the "hardening" shape and volumetric strain response; they only affect the expansion of the yield surface by a scaling factor. Thus, according to our strategy, these two parameters should be identified first from the nominal stress-strain data. The "plastic" variables [sigma].sub.T], [C.sub.R], and N are mainly the responsible of the true stress-strain behavior and they are weakly coupled with the volumetric response. The plastic potential is characterized by the constant [beta] which controls the volumetric plastic dilatation and the evolution of the cross section (e.g., true stress). Consequently, the uniaxial true stress-strain response is affected by the [beta] parameter.

The identification procedure of the material parameters is based on the comparison between the numerical response and the experimental data by minimizing the least square residual (LSR) according to the expressions [28]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

with respect to the design variables, x = [[[sigma].sub.T] [C.sub.R] N [beta]].sup.T] Where, [sigma].sub.T] is the yield stress in tension. [C.sub.R] and N are the "hardening" variables and [beta] is the parameter which controls the volumetric plastic strains.

Identification of the Strain Rate Parameters

The nominal stress-strain responses (see Fig. 3) of the four tensile tests are used for identification of the strain rate parameters. The tests were submitted to nominal strain rate values of 2 X [10.sup.-4], 2 X [10.sup.-3], 2 X [10.sup, -2], and 1 X [10.sup.-1] [s.sup.-1]. The lowest strain rate used is chosen as the reference strain rate, [[epsilon].sub.0] = 2 X [10.sup.-4]. In addition, for each nominal strain rate value the maximal stress (at total strain level of 3%) is taken to build up the diagram of Fig. 11. A value of C = 0.073 is found for this material.

Identification of [[sigma].sub.T], [C.sub.R], N and [beta] Using Inverse Modeling

We use the optimization software LS-OPT [17] to perform the identification procedure of the variables, [[sigma].sub.T], [C.sup.R], N and [beta]. At this stage, the elastic constants were assumed as E = 1500 MPa and v = 0.40, based on the experimental results. Next, we fixed the strain rate parameters according to the results found previously ((C = 0.073 and [[epsilon].sub.0] = 2 X [10.sup.-4])). The identification procedure is based on the minimization of two objective functions related to the true stress-strain behavior and the volumetric response. We use the default linear optimization algorithm (LOFP). The finite element used (e.g., 8 node hexahedral) is submitted to a local strain rate history according to Fig. 5. The values obtained are shown in Table 1.

TABLE 1. Identification Of material parameters.

[[sigma].sub.T](MPa)  [C.sub.R](MPa)    N     [beta]

               12.90           0.927  4.35    1.47


Furthermore, the comparisons between numerical and experimental responses are shown in Fig. 12. It should be mentioned that we assumed a pressure sensitivity value of [alpha] = 1.17 in the identification procedure. This parameter is weakly coupled to the stress-strain and volumetric response of the tensile test; hence, the values indicated in Table 1 are the outcome of the a assumed in the optimization. To illustrate this observation we perform a new identification procedure but this time assuming a value of [alpha] = 1.0. The new resulting values were [[sigma].sub.T] = 12.91, [C.sub.R] = 0.86, N = 4.27, and [beta] = 1.47 which are quite similar to these indicated in Table 1, corroborating then the slight influence of on the stress-strain and volumetric tensile responses. Figure 13 illustrates the variation of one of the design variables (e.g., the hardening modulus, [C.sub.R]) given by LS-OPT during iteration of the optimal solution.

Identification of the Pressure-Dependent Parameter

Finally, the identification of the parameter a, which defines the pressure dependence of the yield stress, is calibrated from the experimental load-displacement curve from the compression test. The optimization procedure yields a value of [alpha] = 1.168 which compares fairly well with the assumed value of [alpha] = 1.17. It should be pointed out that an iterative procedure is required if large deviations between the assumed a value (used to perform the optimization of the design variables X) and that found from the minimization of the compressive response exist. This situation was unnecessary in our case.

Numerical Verification Using the Uniaxial Tensile Test

The inverse modeling strategy used in this work is, in fact, a combination of global and local responses, in the sense that, nominal (global) stress-strain results are used to identify the strain rate parameters and true (local) stress-strain-volume curves were used to identify the four "plastic" parameters which characterize the "yielding" behavior. This section intends to show the capability of the constitutive model presented in section 2 to predict the experimental tensile response the main parameter were identified from. For this purpose we perform a numerical analysis of the tensile specimen using the finite element mesh illustrated in Fig. 14. The cross section of the specimen was 10 x 4 [mm.sup.2] while the global displacement was monitored with an extensometer of 50-mm gauge length. The set of parameters adopted (found in the previous sections) are indicated in Table 2.

TABLE 2. Material parameters for the PP copolymer.

E (MPa)  [gamma]    C    [[epsilon].sub.0]  [[sigma].sub.T](MPa)
                                   (1/s)

1500         0.40     0.073    2X[10.sup.4]             12.90

E (MPa)  [C.sub.R](MPa)    N     [beta]  [alpha]

1500           0.927       4.35   1.47     1.17


A comparison between the experimental test response and numerical simulations is shown in the force-displacement curve of Fig. 15. Deviation of about 6% (e.g., over and under predictions) in the load-carrying capacity are observed. Because there is no failure criterion introduced in the model the analysis was stopped at a displacement level of 67 mm while the specimen presented rupture at a displacement of 68 mm. Furthermore, the numerical model predicted a uniform strain distribution without any indication of strain localization; contrary, to the experiments where strain localization was observed just before failure occurrence. The strain level predicted by the numerical model was underestimated ([[epsilon].sub.xx] = 0.65). However, these results are improved if a geometrical imperfection is introduced in the FE model to trigger strain localization, as illustrated in Fig. 16. In this figure a strain level of [[epsilon].sub.xx] = 0.82 is attained and it compares relatively well to that observed at failure, [[epsilon].sub.xx] = 1.13.

Finally, it is believed that the deviations observed are more related to the robustness of the constitutive model used than, in a less degree, on the parameter identification strategy adopted. In this direction some enhancements of the constitutive model to capture hardening/softening behavior are under development. Some preliminary results indicate that introducing an isotropic hardening/softening response in Part A the numerical prediction of the tensile test is enhanced, as illustrated in Fig. 15.

CONCLUSION

Three main ingredients included in this work are: A short description of the MBR constitutive model, an experimental program, on a PP copolymer, consisting of uniaxial tension and compression tests and an inverse modeling strategy for material identification purposes. The MBR model is suitable for the analysis of thermoplastic structural components subjected to quasi-static or impact loading conditions and it requires the characterization of nine parameters. Some of the model parameters are identified directly from the experimental results. Indeed, the elastic constants (E, v) is taken from the true stress-strain curve and the lateral strain measurements, respectively. In addition, the strain rate parameters (C, [ [epsilon].sub.0]) are identified from the nominal stress-strain curves of four uniaxial tensile tests at different strain rates. An inverse modeling strategy is adopted to identify the other 5 material constants. The pressure dependent parameter (a) is calibrated by minimization of one objective function related to the force-displacement response in compression. The "plastic" variables [[sigma].sub.T], [C.sub.R], N and [beta] are found from minimizing two objective functions related to the true stress-strain behavior and the volumetric response, simultaneously.

The main experimental data, of the PP copolymer investigated here, is measured by 3D-DIC technique, where the common transversal isotropy assumption is avoided. However, such common hypothesis is verified for the material studied here. Next, volumetric plastic dilatation could be confirmed due, possibly, to damage and crazes formation during deformation process.

Finally, the inverse modeling approach is based on a local strategy where the correct data transfer between experiments and finite element loading conditions is guaranteed. Accordingly, local strain rate values, measured from DIG, are directly applied as the loading condition to the finite element used to represent the material point. The methodology presented in this work can be extended to include the local temperature history clue to adiabatic heating process in the necking zone. Indeed, measurements of local temperature jumps of 26 [degrees] C on an ABS polymer, using an infrared camera, have been reported by Louche el al. (29). With this additional information reliable identification procedures can be expected on constitutive models with temperature dependent parameters.

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M. Polanco-Loria, (1) H. Daiyan, (2) F. Grytten (2)

(1) Department of Applied Mechanics and Corrosion, SINTEF Materials and Chemistry, PB 4760 Sluppen, NO-7465, Trondheim, Norway

(2) Department of Synthesis and Properties, SINTEF Materials and Chemistry, PB 124 Blindern, NO-0314, Oslo, Norway

Correspondence to: M. Polanco-Loria; e-mail: mario.polanco@sintef.no

Contract grant sponsor: SINTEF Materials and Chemistry (internal SEP program MechPol-2008).

DOI 10.1002/pen.22102

Published online in Wiley Online Library (wileyonlinelibrary.com).

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Author:Polanco-Loria, M.; Daiyan, H.; Grytten, F.
Publication:Polymer Engineering and Science
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Geographic Code:4EXNO
Date:Feb 1, 2012
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