Abstract
The potential-of-mean-force (PMF) approach to the lattice element method (LEM) has recently been developed and utilized for simulating the fracture in heterogeneous materials and adapted to model the response of structural systems. LEM is a quasi-static approach which relies on lattice discretization of the domain via a set of particles that interact through prescribed potential functions, representing the mechanical properties of members. The approach offers unique advantages, including robustness to discontinuity and failure without the need for mesh refinement, and the ability to accurately simulate nonlinearities through the use of non-harmonic potentials. The overall goal of this research is to extend the PMF-based LEM for simulation of dynamic response. Such simulation framework provides a means for simulating nonlinear response and failure under dynamic loading that is the nature of most natural hazards and extreme conditions. Furthermore, a dynamic LEM approach opens the door for consideration of dynamic fracture and wave propagation in heterogeneous media. For our analysis we leverage the advances in Molecular Dynamics (MD) integration methods for estimation of the trajectory of particles in the Lattice Element Method (LEM) and to simulate the dynamic response with a focus on structural (or building) systems. To this end we use Verlet-Velocity method for time integration, to estimate the location and momentum of each particle at every time step. To address the limitations of the explicit integration methods regarding small time-increments and to assure accuracy and the numerical stability, we also plan to explore implicit integration techniques such as Hilber-Hughes-Taylor method and midpoint method. Noting that the rotational degrees of freedom have minimal contribution to the kinetic energy of the system we develop an energy-based approach for static condensation to reduce the computational cost. Our approach relies on the Euler-Lagrange equation which manifests in the form of minimum potential energy theorem for mass-less degrees of freedom. Finally, shifting attention to another critical aspect of dynamic simulation the mass matrix, we adopt an energy-based approach and utilize the kinetic energy of the lattice elements to maintain consistency with the kinetic energy of their continuous bar counterpart. The framework is further extended to incorporate the nonlinear behavior of materials under various actions, including bending, torsion, and axial forces, through the introduction of novel potential functions inspired by the Force Analogy Method. These potential functions are calibrated using section properties that represent the nonlinear stress-strain responses of materials, such as nonlinear moment-curvature relationships. The proposed framework’s utility and accuracy are validated through its application in quasi-static linear and nonlinear simulations of large-scale buildings subjected to different loading conditions. These simulations evaluate the influence of structural parameters, such as the number of bays, number of stories, and bay lengths, on the overall behavior of structures in the nonlinear regime.