Leonard Euler and Daniel Bernoulli developed a model for linear elastic bending of beams around 1750, shortly after Isaac Newton and Gottfried Leibnitz developed calculus in the mid 1600’s. Without any concept of stress and strain, the Euler-Bernoulli model related bending moment to bending curvature. Only in the 1820’s did Claude-Louis Navier and Augustin-Louis Cauchy develop a linear elastic model for the deformation of three-dimensional solid bodies. Cauchy conceived the concepts of stress and strain, both analytical calculus concepts, which have guided our thinking about solid mechanics ever since. 

 

During the 1900’s, scholars extended the linear elastic Navier-Cauchy model to include the concepts of plasticity, creep, fracture, and damage. Concurrently, mathematicians Clifford Truesdell and Walter Noll developed a model called “continuum mechanics”, which generalized the concepts of stress and strain to allow for large deformations.

 

By developing the finite element method, John Argyris, Ray Clough, and Olgierd Zienkiewicz first implemented computational elasticity on digital computers in the 1950s.  Since that time, scholars have extended the finite element method to simulate the nonlinear behavior of solids, but their methods fall short in modeling damage and fracture.

 

Therefore, in 2000 Stewart Silling developed the continuum peridynamic theory, which allows the deformation field to be discontinuous while assuming the material reference space is a continuum. Thus, engineers must first discretize continuum peridynamics to implement it on a digital computer. 

 

The question arises: can we directly develop a discrete, rather than a continuum, model for the deformation of solid bodies that is immediately implementable on a digital computer? The state-based peridynamic lattice model, presented here, is such a model.

 

We describe the state-based peridynamic lattice model, present some simulation examples, and describe its benefits and shortcomings. 

 

Have we committed a grave error by departing from an analytical (calculus) model for solid mechanics? Let us discuss this in the seminar!