Constitutive Model And Finite Element Formulation For Isothermal Yield Behavior Of Polymeric Materials

A systematic introduction to the theories and formulations of the explicit finite element method As numerical technology continues to grow and evolve with industrial applications, understanding the explicit finite element method has become increasingly important, particularly in the areas of crashworthiness, metal forming, and impact engineering. Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics is the first book to address specifically what is now accepted as the most successful numerical tool for nonlinear transient dynamics. The book aids readers in mastering the explicit finite element method and programming code without requiring extensive background knowledge of the general finite element. The authors present topics relating to the variational principle, numerical procedure, mechanical formulation, and fundamental achievements of the convergence theory. In addition, key topics and techniques are provided in four clearly organized sections: • Fundamentals explores a framework of the explicit finite element method for nonlinear transient dynamics and highlights achievements related to the convergence theory • Element Technology discusses four-node, three-node, eight-node, and two-node element theories • Material Models outlines models of plasticity and other nonlinear materials as well as the mechanics model of ductile damage • Contact and Constraint Conditions covers subjects related to three-dimensional surface contact, with examples solved analytically, as well as discussions on kinematic constraint conditions Throughout the book, vivid figures illustrate the ideas and key features of the explicit finite element method. Examples clearly present results, featuring both theoretical assessments and industrial applications. Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics is an ideal book for both engineers who require more theoretical discussions and for theoreticians searching for interesting and challenging research topics. The book also serves as an excellent resource for courses on applied mathematics, applied mechanics, and numerical methods at the graduate level.

The novel finite element formulations fall into the category of geometrically exact Kirchhoff-Love beams. A prominent characteristic of this category is that the absence of shear deformation is strongly enforced by removing two degrees of freedom. Further, the corresponding beam theories exhibit not only translational but also rotational degrees of freedom and their configurations thus form a non-additive and non-commutative space. Sophisticated interpolation schemes are required that need to be tested not only for locking, spatial convergence behavior, and energy conservation, but also for observer invariance and path-independence. For the three novel beam element formulations all these properties are analytically and numerically studied and confirmed, if applicable. Two different rotation parameterization strategies are employed based on the well-known geodesic interpolation used in many Simo-Reissner beams and the lesser known split into the so-called \textit{smallest rotation} and a torsional part. Application of the former parameterization results in a mixed finite element formulation intrinsically free of locking phenomena. Additionally, the first geometrically exact Kirchhoff-Love beam element is presented, which strongly enforces inextensibility by removing another degree of freedom. Furthermore, the numerical efficiency of the new beam formulations is compared to other beam elements that allow for or suppress shear deformation. When modeling very slender beams, the new elements offer distinct numerical advantages. Standard molecular dynamics simulations, which are commonly used to study polymers, suffer from a lack of a careful mathematical basis and the use of an expensive explicit time integration scheme. To circumvent these shortcomings and to be able to simulate stretching experiments on relevant time scales, the problem is described by a stochastic partial differential equation, which can be solved using the finite element method with a backward Euler temporal discretization. In detail, the polymer is represented by a Kirchhoff-Love beam with a linear elastic constitutive model. Inertial and electrostatic forces are neglected. It is deformed by a distributed load mimicking collisions with molecules of the surrounding fluid. Naturally, this load heavily fluctuates over time and space and mean values need to be computed in a Monte Carlo manner. To vastly speed up the fitting process to experimental data in a Bayesian framework, a surrogate model based on a Gaussian process is set up, which directly computes the mean values for given material parameters. The uncertainties and correlations of the material parameters are studied and compared to the literature.