We used a physically motivated internal state variable plasticity/damage model containing a mathematical length scale to idealize the material response in finite element simulations of a large-scale boundary value problem. The problem consists of a moving striker colliding against a stationary hazmat tank car. The motivations are (1) to reproduce with high fidelity finite deformation and temperature histories, damage, and high rate phenomena that may arise during the impact accident and (2) to address the material postbifurcation regime pathological mesh size issues. We introduce the mathematical length scale in the model by adopting a nonlocal evolution equation for the damage, as suggested by Pijaudier-Cabot and Bazant in the context of concrete. We implement this evolution equation into existing finite element subroutines of the plasticity/failure model. The results of the simulations, carried out with the aid of Abaqus/Explicit finite element code, show that the material model, accounting for temperature histories and nonlocal damage effects, satisfactorily predicts the damage progression during the tank car impact accident and significantly reduces the pathological mesh size effects.
The design of accident-resistant hazmat tank cars or the improvement of existing ones requires material models that describe the physical mechanisms that occur during the accident. In the case of impact accidents such as collisions, finite deformation, and temperature histories, damage and high rate phenomena are generated in the vicinity of the impact region. Unfortunately, the majority of material models used in the finite element (FE) simulation of hazmat tank car impact scenarios do not account for such physical features (unavoidably, this will under- or overestimate, for instance, the numerical prediction of the puncture resistance of hazmat tank cars’ structural integrity). Furthermore, in the few models that do, a mathematical length scale aimed at solving the postbifurcation problem is absent. As a consequence, when one material point fails in the course of the numerical simulations, the boundary value problem for such material models changes, from a hyperbolic to an elliptical system of differential equations in dynamic problems, and the reverse in statics. In both cases, the boundary value problem becomes ill posed, Muhlhaus [
Alternatively to the shortcomings encountered in hazmat tank car impacts’ numerical simulations, we propose to use a nonlocal version of the Bammann-Chiesa-Johnson (BCJ) model, Bammann and Aifantis [
Load versus displacement curves for different temperatures and strain rates. Full lines: BCJ model predictions; point lines: experiments. Note that the model satisfactorily captures temperature and strain rate histories effects. The curves were extracted from Bammann et al. [
In this study, a dynamic nonlinear FE analysis, carried out with Abaqus FE code, is used to simulate a moving striker colliding against a stationary hazmat tank car. The structure part of this FE model is represented by Lagrangian elements obeying the nonlocal version of the BCJ model, while the fluid part is represented by Eulerian elements. The objective of this study is twofold: (1) use a high fidelity material model to idealize the physics occurring during the impact accident and (2) rectify the computational drawbacks (postbifurcation mesh dependence issues) for this model on a large-scale boundary value problem. The resulting numerical simulations of hazmat tank car impact scenarios which account for nonlocal damage and temperature history effects predict satisfactorily the tank car failure process independently of the element size. Although several other studies have addressed successfully the mesh dependence issues in finite element computations of problems involving dynamic failure of materials, as in for instance, Voyiadjis et al. [ The first section provides a summary of the equations of the BCJ model and its nonlocal extension. The second section discusses several methods to numerically implement the integral-type nonlocal damage into existing Abaqus FE BCJ model subroutines. The main difficulty encountered in this implementation relates to the double loop over the integration points required by the calculation of several convolution integrals, which might otherwise dismantle the architecture of the entire code. The last section is devoted to numerical applications of the local and nonlocal versions of the BCJ model on hazmat tank car shell impact accident simulations.
The BCJ model is a physicallybased plasticity model coupled with the Cocks and Ashby [
The deformation gradient is multiplicatively decomposed into terms that account for the elastic, deviatoric inelastic, dilatational inelastic, and thermal inelastic parts of the motion. For linearized elasticity, the multiplicative decomposition of the deformation gradient results in an additive decomposition of the Eulerian strain rate into elastic, deviatoric inelastic, dilatational inelastic, and thermal inelastic parts. The constitutive equations of the model are written with respect to the intermediate (stress-free) configuration defined by the inelastic deformation such that the current configuration variables are corotated with the elastic spin. The pertinent equations of the BCJ model are expressed as the rate of change of the observable and internal state variables and consist of the following elements.
(i) A hypoelasticity law connecting the elastic strain rate to an objective time derivative Cauchy stress tensor is given by
(ii) The decomposition of both the skew symmetric and symmetric parts of the velocity gradient into elastic and inelastic parts for the elastic stretching rate
(iii) Next, the equation for the plastic spin
For the plastic flow rule, a deviatoric flow rule [
There are several choices for the form of
(iv) The evolution equations for the kinematic and isotropic hardening internal state variables are given in a hardening minus recovery format by
(v) To describe the inelastic response, the BCJ model introduces nine functions which can be separated into three groups. The first three are the initial yield, the hardening, and the recovery functions, defined as
(vi) The evolution equation for damage, credited to Cocks and Ashby [
(vii) The last equation to complete the description of the model is one that computes the temperature change during high strain rate deformations, such as those encountered in high rate impact loadings. For these problems, a nonconducting (adiabatic) temperature change following the assumption that 90% of the plastic work is dissipated as heat is assumed. Taylor and Quinney [
The empirical assumption in (
Following suggestion Pijaudier-Cabot and Bazant [
The evolution equation of this variable is given by a convolution integral including a bell-weighting function the width of which introduces a mathematical length scale:
The new evolution equation for the damage, (
Nonetheless, the numerical implementation of the new evolution equation for the damage rate into an existing FE code is not an easy task because of the double loop over integration points required by the calculation of several convolution integrals, which may potentially compromise the entire architecture of the existing code. The following section is devoted to explaining this implementation.
The numerical implementation of the original BCJ model into an FE code such as Abaqus has been extensively addressed in Bammann et al. [
To implement the nonlocal extension of the BCJ model in the implicit version of the code, we have computed the nonlocal damage increment at elastic-plastic convergence by means of the built-in subroutine URDFIL, which stores all the variables necessary for the convolution operation performs this operation. This method involves all the integration points of the finite element model considered. However, since the Gaussian influence weighting function vanishes for integrations points that are outside the region limited by the characteristic length scale, only the integration points inside that zone matter in the calculation of the convolution integral. The nonlocal damage increment are stored for all the integration points involved in the FE model and are used to calculate the damage at time
Comparison of the damage distribution of a double-notched edge specimen numerical tensile tests using the local BCJ model for two different mesh sizes, coarse (a) and fine (b) meshes. Note the localization of the damage in a layer of width one element size for the two meshes.
Comparison of the damage distribution of double-notched edge specimen numerical tensile tests using the nonlocal BCJ model, coarse (a) and fine (b) meshes; the value of the length scale used in the simulations is 9 mm. The nonlocal damage method has spread the damage to neighboring regions; also, note the similarity between the damage pattern in the two cases.
To assess the validity of the method to regularize hazmat tank car impact boundary value problems requires the implementation of the nonlocal damage rate in the explicit version of the BCJ model subroutine. One option of doing so consists of branching outside the “nblock loop” in the subroutine VUMAT, which allows the computation of the nonlocal damage rate at each integration point using the coordinates, the local damage velocities, and the weights of “nblock” integration points. This method also successfully avoids the localization problems arising in the numerical simulations of double-notched edge specimen tensile tests and therefore is excepted to reduce, if not completely remove, the mesh sensitivity issues arising in the hazmat tank car impacts numerical simulations.
The goal of the hazmat tank car shell impact FE simulations section is to predict numerically the hazmat tank car structural failure process independently of the mesh size effects, as encountered in numerical computations based on material models involving softening, just like the BCJ material model. No comparison with experimental results is needed for this purpose. We describe later the impact accident physical problem and the associated FE model in Abaqus, and we present different numerical predictions of the damage created by the impact.
The problem considered here is schematized in Figure
Geometry of the
Schematization of the hazmat tank car impact problem.
The FE model of the problem consists of an Eulerian mesh representing the fluid in the tank car and a Lagrangian mesh idealizing the tank, the jacket, and the impactor. The impactor consists of R3D4 rigid elements, while the tank and the jacket are meshed using solid C3D8R elements. For the sake of simplicity, the space between the tank and the jacket is assumed to be empty. The two legs are rigid and are idealized with squared analytical surfaces on which reference points located below the jacket are assigned. Each reference point is kinematically coupled with a definite set of nodes on the jacket and the tank car. The contact algorithm available in Abaqus/Explicit FE code is used to account for all possible contacts between the different parts in the model.
The Eulerian mesh is based on the volume-of-fluid (VOF) method. The VOF method, widely used in computational fluid dynamics problems, tracks and locates the fluid-free surface; it belongs to the class of Eulerian methods that are characterized by either a stationary or moving mesh. The VOF method suitably captures the change of the fluid interface topology. In this method, the material in each element is tracked as it flows through the mesh using the element volume fraction (EVF), a unique parameter for each element and each material. The material parameters used in the simulations are determined from tension, compression, and torsion tests under different constant strain rate and temperatures. We used the material parameters provided in Horstemeyer et al. [
The stationary part of the problem consists of a tank car surrounded by a jacket. The dimensions of the tank car and the jacket are the same as in the works of Tang et al. [
This section presents evidence of the integral nonlocal damage method to reduce mesh dependence issues which arise in the FE solution of hazmat tank car impact accident problems. To that end, the calculations were performed on three different meshes, coarse, medium, and fine meshes (see Figure
Different mesh discretizations of the tank car: (a) coarse mesh, (b) medium mesh, and (c) fine mesh. The meshes are made of solid C3D8R elements. There are four C3D8R elements in the thickness of the tank to develop a bending resistance. The size of the elements in the impact zone is 75 mm, 50 mm, and 35 mm for the coarse, medium and fine meshes, respectively.
Due to the calculation of the integral nonlocal evolution equation of the damage, the CPU time required by the nonlocal calculations is higher than the case with the local model. It usually takes up to 12 hours to complete the simulation of the impact problem with nonlocal damage using the coarse mesh (59049 integration points for the structure part of the model). The medium (80373 integration points for the structure part of the model) and fine meshes (158037 integration points for the structure part) simulation times are longer, 15 hours for the medium mesh and almost 20 hours for the refine mesh. This long time period is also due to the presence of the fluid, which is modeled by the element volume fraction (EVF) technique available in ABAQUS.
Figure
An illustration of the mesh size effects in the FE simulations of hazmat tank car using the local BCJ model. The figures show the damage pattern on the inner surface of the tank located behind the impact region for a coarse (a), medium (b) and fine (c) mesh at the same time. Note the reduction of the damage pattern size with decreasing mesh element size.
Illustration of the mesh size effects in the FE simulations of hazmat tank car using the local BCJ model. The figures show the damage pattern on the outer surface of the tank located in the impact region for a coarse (a) and fine (b) mesh at the same time. Note the difference between the damage pattern for the meshes.
Figure
An illustration of the introduction of nonlocal damage effects in the BCJ model to reduce the mesh size effects in the FE simulations of hazmat tank car. The contour plots of the damage on the inner face of the tank are shown for three different meshes at the same time as in Figure
Also remarkable from Figure
Another illustration of the mitigation of mesh size effects in the FE simulations of hazmat tank car using nonlocal BCJ model. The figures show the damage pattern on the outer surface of the tank body located in the impact region for coarse (a) and fine (b) meshes at the same time. Note the similarity between the damage patterns.
With the aid of Abaqus/Explicit FE code, we have presented a dynamic nonlinear FE simulation of a hazmat tank car shell impact. In these simulations, the tank car body material was idealized with a physically motivated internal state variable model containing a mathematical length scale, the nonlocal BCJ model. This model consists of all the constitutive equations of the original BCJ model, except for the evolution equation for the damage, which is modified into a nonlocal one. Numerical methods to implement this equation in existing BCJ model subroutines were also presented.
The results of the simulations show the ability of the damage delocalization technique to significantly reduce the pathological mesh dependency issues while predicting satisfactory hazmat tank car impact failure, provided that nonlocal damage effects are coupled with temperature history effects (adiabatic effects). However, the performance of our approach will fully demonstrate its power when the damage progression it predicts reproduces the real-world hazmat tank car impact damage progression. This comparison with experimental results will be investigated in the near future.
The study was supported by the US Department of Transportation, Office of the Secretary, Grant no. DTOS59-08-G-00103. Dr. Esteban Marin and Mrs. Clémence Bouvard are gratefully thanked for their insightful discussions on the topic.