Veronika Effinger

Abstract

To be approved for European public roads passenger cars must meet the legal requirements concerning pedestrian safety. One legal impact test, assessing the compliance of the car front in the event of a collision with a pedestrian, consists in shooting the upper legform impactor onto the bonnet leading edge. In order to replicate this load case in a crash simulation with predictive capability, the material behavior of Confor ® CF-45, an open-cell polyurethane foam, and one of the constituents of the upper legform impactor is of major importance. Hereby, an adequate modeling of the energy absorption in the foam and of the contact area between impactor and bonnet is important for capturing the load transfer in the simulation. The objective of this thesis is to develop a material model of open-cell polymer foams for application in pedestrian protection, which in particular considers the nonlinear visco- elastic properties of these materials and which can be used in finite element simulations with explicit time integration. In the framework of classical continuum mechanics the material modeling follows a phenomenological approach within which the isotropic stress response is additively decomposed into an equilibrium and an over stress response. While the MFCF model (Kolling u.a. (2007a), Hallquist (2016)) is adopted as a good example of a simplified model for the equilibrium response from the finite element analysis software program LS-DYNA, the focus of this thesis is the description of the over stress in the context of finite nonlinear viscoelasticity. A possible stress contribution created by the interference of the inlet and outlet of a fluid phase due to cellullar structure of the foam can be added to the modeling. However, this stress contribution is not considered in this thesis as there were no usable measurements of the air flow in Confor CF-45 available. For the understanding and classification of the material models considered in this thesis the theory of linear viscoelasticity introduces different representations of constitutive equations of viscoelastic solids which express the stress as a functional of the complete strain or strain rate history. Rheological elements are used to illustrate the differential equations, the systems of differential equations and the integral equations as well as the relations between them. The integral representation serves as a starting point for the over stress constitutive equations in this thesis and this will require the examination of numerical convolution algorithms. Time-dependent material functions included as convolution kernels contain the material properties and their shape determines how the convolution integrals can be calculated numerically. The naïve convolution algorithm with a piecewise linear material function requires to save the complete strain history, reads it from storage during each time step and reprocesses all past values to obtain the current stress response. In contrast, the recursive convolution algorithm with a Prony series as material functions exploits the characteristics of the exponential functions in order to summarize the past strain history and to reuse it in each time step with the help of a relaxation prefactor. The local convolution algorithm is based on the work of López-Fernández u.a. (2008) and requires multiple Prony series which are valid for discrete and overlapping intervals and consequently are locally defined. Thus, the characteristics of the exponential functions can be used as well and the number of terms of the individual Prony series can be reduced in comparison to the recursive convolution algorithm. In order to reduce the simulation time this work uses the strain rate instead of the strain as input for the convolution algorithm. Furthermore, the model size of the upper legform impactor and the number of timesteps in a pedestrian impact simulation require the use of the recursive or the local convolution algorithm. In order to combine convolution integrals in a constitutive equation for open-cell polymer foams in a meaningful way it is necessary to analyze the behavior of these materials and to simplify it with appropriate assumptions. Due to their cellullar structure open-cell polymer foams put special demands on the constitutive equation. As a result of the different deformation mechanisms involved, they do not only show a distinct tension-compression asymmetry with regard to the shape and magnitude of the stress-strain curve but also with respect to the transverse strain behavior. The constitutive equations of this work are based on the assumption that longitudinal, transverse and thickness strains are uncoupled which follows from the observation of a diminishing Poisson’s ratio under uniaxial compression loading. This thesis proposes different constitutive equations for the over stress which differ besides the convolution algorithm in the material nonlinearity and the choice of stress and strain rate measures. The nomenclature of the constitutive equation is additively composed by the underlying theory, the finite linear respectively nonlinear viscoelasticity (FLVE respectively FNLVE), and the individual elements such as the stress and strain rate measures, the material nonlinearity and the convolution algorithm, each in abbreviated form. For example, FLVE-TBU-recursive represents a finite linear viscoelastic constitutive equation which uses the Biot stress and the right stretch tensor as well as the recursive convolution algorithm. Two pairs of dual stress and strain tensors are used: the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain rate tensor (SE) as well as the Biot stress tensor and the rate of the right stretch tensor (TBU). Due to their definition with respect to the reference configuration all stress contributions can be summed up and converted in the same way to the Cauchy stress tensor with reference to the current configuration. The material nonlinearity of the examined models is based on two approaches. Either the results of multiple convolution integrals of different powers are summed up or the material function within one convolution integral is complemented by an additional dependency upon one or multiple functions. The sum of multiple convolution integrals results from the simplification of the Green-Rivlin model (GR) for foams which transforms the multiple integrals of fifth order into single integrals of fifth power using the product form of the material function. Materially nonlinear constitutive equations on the basis of one convolution are created with three types of rate dependent coefficients for the Prony series. The first type uses an equivalent strain rate (EQV) which corresponds to the absolute value of the excitation strain rate in an uniaxial test. The second and third type transform the constitutive equation into the principal coordinate system (HS) of the strain rate tensor and the material function is evaluated for each principal direction with the corresponding principal strain rate. The third type differs from the second type as the assessment in each principal direction is complemented by a switch which distinguishes between loading and unloading (HS+). As a result a modular material routine is established which provides a multitude of constitutive equations. In order to determine which constitutive equations are able to reproduce the test data of open-cell polymer foams the stress responses to standard excitations are examined. Hereby, the preferred strategy is to identify the parameters for a uniaxial test at one nominal strain rate with the help of a linear material model and to use the extended nonlinear material model in order to unify the parameters of the individual tests into one single parameter set. This procedure can only be applied to the constitutive equation FLVE-TBU which is able to replicate every individual tension and compression test with different parameters and which can afterwards be combined with the three different types of extended material functions into FNLVE-TBU-EQV, FNLVE-TBU-HS or FNLVE-TBU-HS+. The stress response of these models is then evaluated for tension and compression tests at different strain rates, for multiaxial tests such as the biaxial tension test, the biaxial tension compression test and shear tests as well as for cyclic uniaxial tension, tension-compression and uniaxial compression tests. Out of the three materially nonlinear constitutive equations only the model FNLVE-TBU-HS+ is able to differentiate the uniaxial tests at different strain rates, to distinguish between tensile and compressive loading and to prevent that a load reversal automatically generates a switch from tension to compression parameters. To identify the parameters of the constitutive equation a comprehensive test program was conducted for Confor CF-45. The experimental characterization reveals different influence factors on the mechanical behavior of the open-cell PU foam. Besides the influence of the air flow which induces a dependency upon the specimen size the material shows a high sensitivity in the neighborhood of the glass transition temperature upon any changes of temperature or moisture. Hence, a comparison of measurements conducted under atmospheric conditions and in vacuum cannot be used to determine the air flow. The tension and compression tests under normal conditions and at specimen size of 50x50x30mm3 show a distinct strain rate dependency in the range of strain rates between 1e-61/ms and 2e-11/ms and anomalies in comparison to other PU foams. Compression tests reveal a stress peak between the initial slope and the plateau stress and the increase of the absolute stress which is usually associated with the densification of the foam material already occurs for low compression values. In order to consider these peculiarities the parameter identification deviates form the defined stress decomposition into equilibrium stress on the basis of quasistatic tests and in the over stress as the difference between dynamic and quasistatic testing for the compression tests. With this modification it is possible to reproduce the compression tests with the model GG+FNLVE-TBU-HS(+)-recursive with high conformity and the tension tests with very high conformity. Thereby, the stress answer of the model GG+FNLVE-TBU-HS (+) -recursive is additively composed by the equilibrium response GG and the viscoelastic overstress FNLVE-TBU-HS(+)-recursive. The modeling with GG+FNLVE-TBU-HS+-recursive and the selected parameter set prove to be robust in the simulation of the material tests, of the upper legform impact test in pedestrian protection and of two component tests. Due to the decomposition of the stress response into equilibrium and over stress an instantaneous and time-delayed recovery of the foam material takes place and the elastic part of the internal energy can be recaptured. By capturing theses viscoelastic effects the new modeling exceeds by far the current state of the art. A final validation of the model with a component test is not available as Confor CF-45 is not produced anymore and hence the component test as well as the test to model the influence of the air flow could not be conducted. However, the modeling can be transferred to other open-cell polymer foams and in particular to the succession products of Confor CF-45 as part of the upper legform impactor which possess similar mechanical properties according to their specifications. Furthermore, the modular material routine represents a solid starting point to model nonlinear viscoelastic over stress for other materials at finite deformations. In particular materials with a non-zero Poisson’s ratio could be considered.