1. Introduction

Before the advent of the personal computer, engineers would perform structural analysis using formulas that are simplified and suitable for hand calculation. Very elegant analytical methods of structural analysis were developed out of necessity and many such methods are still taught today in university and used in industry. Undergraduate students often take such courses in structural analysis around the same time that they take courses in mechanics of materials. This is where one learns about analytical theories relating to bending stress and buckling, phenomena originally investigated in the 18th century by Leonard Euler and Jacob Bernoulli, among others [see Timoshenko]. Advanced analytical theories relating to stability continued to be developed into the 20th century, and continue to this day to be useful for “design,” where one example of such a design process would be in selecting a safe yet economical I-beam to be used within the structural skeleton of a modern-day skyscraper. Thus, such methods are vital in order to do preliminary “design” calculations, and this author would argue that a deep understanding of classical methods of analysis also give the engineer an intuition about the way forces “flow” through a structure. However, a new field has emerged in recent years: simulation-based engineering.

Structural analysis work that is done in industry and academia is increasingly being accomplished via structural analysis software. This text considers one important category of structural analysis software: finite element analysis (FEA). In this text, FEA refers to high fidelity computer simulation, which is a type of structural analysis software that is used, for example, for performing car crash test simulations. This kind of software is also being increasingly used for the predictive modeling of other kinds of structures such as components of buildings, aircraft, biological tissue/biomechanics, etc. Whereas in classical mechanics and strength of materials, load-deformation behaviors are described by specialized analytical theories such as bending stress, torsion, buckling, etc., high fidelity computer models are able to simulate these same phenomena without the use of these analytical formulas. Instead, the software will discretize the structure into thousands of 8-noded bricks and the equations that govern the behavior of each brick are, in fact, conceptually quite simple, in particular if the structure remains elastic.

It was the first generation of French mathematicians, physicists, and engineers from École Polytechnique (early 19th century), who developed the framework that we now use as the governing equations for high fidelity computer simulation; a field called continuum mechanics (solid mechanics, to be more specific). Among this first generation of École Polytechnique students included Navier, Poisson, and Cauchy. Some of the expressions that we will see in this text are named after these continuum mechanics pioneers. Further specialization of the theory would not be needed until the late 20th century, with the advent of the computer. As we will see, for each of those 8-noded bricks that comprise the structure, the governing equations include a version of static equilibrium (or the equation of motion), along with some kind of three-dimensional version of Hooke’s Law. The governing equations for these 8-noded bricks are, thus, conceptually simple, but very powerful. The resulting computer model, which is capable of, naturally, simulating elastic buckling, bending, shear, and torsion, contains far fewer simplifications than the corresponding analytical theories from “mechanics of materials.”

Many undergraduate and graduate level courses that deal with finite element analysis methods for structural analysis aim to give the student the confidence (i.e. give the student all of the formulas) that they need so that they can write their own finite element analysis code. In this text, “theory” will be presented in a manner such that FEA itself is the “application.” This text will derive the governing formulas of “elastic isotropic” solid mechanics, whereas courses in FEA take such formulas as a given and focus, instead, on the ways that software implement the formulas, numerically.

note: In this text, FEA refers to “solid” elements (sometimes referred to as “continuum” elements). “Stick” elements (sometimes referred to as “beam” elements) or “shells” are simplifications of “solid” elements.

Consider, for example, the simplest case of isotropic linear infinitesimal elasticity, in FEA. If it’s 2D or 3D, then we need a constitutive equation analogous to \sigma=“E”\epsilon along with a strain-displacement equation analogous to \epsilon=\frac{\Delta L}{L}. The equations \boldsymbol{\sigma}=\boldsymbol{\mathcal{C}}:\boldsymbol{\epsilon} and \epsilon_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial X_j}+\frac{\partial u_j}{\partial X_i}) may look familiar, but in this text they will be derived.

Moreover, this text will cover large (“finite”) strain isotropy as well as provide insight into how more advanced FEA software handle large rigid body rotations and time-dependent problems. The topics covered in this text are essential for engineers in the field of materials research as well as those who use advanced FEA software to its full potential. We will start with “hyperelasticity”, which applies to large strains, large rigid body rotations, and nonlinear (though still elastic) stress-strain relationships. In general, this is the necessary starting point when one has a material with unknown properties. Rubbers actually behave nonlinearly (nonlinear isotropic-elastic behavior is a good starting point) and have large strains.

Then, “hypoelasticity” is considered, which relates stress rates (and therefore incremental stress) to strains and strain rates. Developing the constitutive relationship in hypoelastic form is the preferred method for the most advanced FEA software, such as ABAQUS and LS-DYNA, since they deal with rate and history dependent transient dynamic problems that involve time-stepping algorithms. Lastly, the aforementioned isotropic linear infinitesimal formulation will be discussed. Steel, in the elastic regime, can often be modeled in such a manner.

Before getting into these main ideas, we should introduce tensor notation. In doing so, we’ll also go through some math derivations that will be essential to our formulation of hyperelasticity, linear infinitesimal elasticity, and rate-form constitutive relationships. Tensor math is not a steep learning curve, and it is probably worthwhile to take a bit of time to make sure you understand it. While many of the derivations throughout this text will be complete and quite detailed, from the point of view of most readers, it is important to keep in mind that this text is written in a mechanics style rather than a mathematical style. While the author will do his best to avoid “sloppy” math, the overriding emphasis will always be on the physical interpretation and application of concepts.

This text is only intended to be an introduction to solid mechanics, and therefore only considers “elastic isotropic” materials. The information provided in this text serves as a prerequisite for topics such as anisotropy, viscoelasticity, and plasticity in solid mechanics.