Date of Award

December 2021

Document Type


Degree Name

Master of Science (MS)


Mechanical Engineering

Committee Member

Lonny Thompson

Committee Member

Gang Li

Committee Member

Garett Pataky


Pantographic lattices are cellular solids comprised of continuous beam fibers intersecting at periodically spaced pivots. The pivots are simulated in the discrete finite element beam model as torsional springs with varied torsional stiffness across orders of magnitude. An important functional performance feature of pantographic lattices is their ability to undergo large deformation without inducing large stresses at the pivots. There is a need for predictive models of this nonlinear behavior. In this study, parameter studies on the order of magnitude and nonlinear material behavior of the torsional stiffness at the pivots, combined with and without nonlinear geometric beam kinematic behavior is investigated. In this study, the mechanical response of pantographic lattice is analyzed for a series of elongation tests based on a set of kinematic assumptions. Finite element numerical results are presented for the axial bias test for in-plane stretch along the bisector of the beam fiber orientations. Strain energy distributions are used to analyze the stiffness of the deformed geometry behavior. Geometric nonlinearity is introduced to study the response for large deformation while a non-linear torsional spring expressed as a cubic polynomial function of relative beam rotations at connection nodes is utilized to model local pivot softening and stiffening effects. One use of the discrete frame model presented in the study is to serve as a validation tool for homogenized pantographic sheet models based on second gradient field theory in the case of light spring stiffness relative to the beam lengths and section properties, of order epsilon squared, where epsilon is a small-scale parameter measuring the ratio of a repeating unit cell size to the overall lattice size. For epsilon of order one, the discrete beam model serves to validate homogenized models based on the first gradient classical elasticity theory. The discrete beam model of an orthogonally oriented lattice is constituted by torsional springs at intersection points which dictate the internal moments. At a joint, displacements of the nodes are rigidly constrained while rotational degrees of freedom are proportional to a local torsional stiffness for the lattice pivots. The torsional stiffness of the spring is varied from (perfect) zero to infinite limits to replicate a free and rigid connection respectively between the intersecting beams joints. For the nonlinear torsional pivot spring model, the torsional stiffness is not constant, instead of being a function of the magnitude of pivot rotations. The linear torsional springs are generated at the lattice pivots in the discrete mathematical model using constraints setup using Lagrange multipliers. Geometric nonlinearity has been introduced using the large deformation beam kinematics implemented within ABAQUS finite element software. The pantographic lattice is constituted by Euler Bernoulli beams connecting nodes along the designated fibers. A nonlinear relationship between moment and angle of rotation is utilized in Abaqus python scripting to develop nonlinear torsional spring behavior at the pivots. The effective material non-linearity considered is a function of two parameters and is driven by the relative angle of rotation of two beams connecting at the pivot. A predictive model for the total energy of the lattice during a small stretch is also developed and verified in the case of the nonlinear material spring model at the pivots.In order to help understand the effects of different aspects of nonlinearity during lattice stretch, including, deformed shape, reaction force resultant, total strain, and energy distribution, several combinations are studied; small stretch, comparing linear vs. nonlinear torsional spring stiffness at pivots, with and without beam geometric nonlinear kinematics, and large stretch, comparing linear vs. nonlinear pivot stiffness, with and without geometric nonlinearity. The analysis has also been extended to a 3D pantographic lattice where each pivot is constituted by three torsional springs connecting the three combinations of beam fiber pairs at each intersection joint.



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