Advanced Structural Analysis/Part I - Theory/General Properties of Materials/Measures/Elasticity

1. Introduction
2. Tensile Testing
3. Broad Implications
4. Additional Aspects

Introduction

All solid materials display an approximately linear response to moderate loads. This is manifested in Hooke's law:

$\epsilon _{x}={\frac {\sigma _{x}-\nu (\sigma _{y}+\sigma _{z})}{E}}$ 
$\epsilon _{y}={\frac {\sigma _{y}-\nu (\sigma _{x}+\sigma _{z})}{E}}$
$\epsilon _{z}={\frac {\sigma _{z}-\nu (\sigma _{x}+\sigma _{y})}{E}}$

Where:
$\sigma =$ the normal stress
$E=$ the elastic modulus (Young's modulus)
$\epsilon =$ the normal tension

The figure below illustrates the stress-strain curve of some material. Hooke's law is valid in the linear interval marked "Elastic region". The term elastic, as in elastic deformation, refers to the absence of energy dissipation and permanent deformation. Elastic deformations retract upon unloading.

Generic stress-strain graph for a ductile material.

Tensile Testing

Young's modulus can be measured by tensile testing of specimens such as the one in the picture below.

Tensile testing.

The basic idea of the test is to measure the length $l_{0}$ at rest, the response $\Delta l$ to the load difference $\Delta F$, and then estimate the elastic modulus from

$E={\frac {\Delta \sigma }{\Delta \epsilon }}={\frac {{\frac {F_{2}}{S_{2}}}-{\frac {F_{1}}{S_{1}}}}{\frac {l_{2}-l_{1}}{l_{1}}}}$

Although, the procedure may seem perfectly straight forward at a glimpse, there are certain things that should be noted. Firstly,...

Broad Implications

As is the case with many other elementary physical laws, Hooke's law spans the small and simple, and the big and complex. For instance, the theory is a good approximation of a simple test specimens subjected to axial loading. Naturally, it is also an accurate physical model for linear springs. And it is even valid for very complex structures, so long as they respond linearly. These are all trivial and self evident statements, yet they are powerful and important. Consider equation (1) and the relation below.

$\sigma ={\frac {F}{A}}$     (2)

If there is only one load affecting a linear structure, then according to equation (1), the strain at any point varies linearly with the stress at any other point in the structure. Furthermore, a linear stress measure at any point is a linear function of the applied load at that point, this is exemplified by equation (2). So, the strain and consequently the stress at any point is in fact a linear response to the load at any point. We conclude that

$\sigma _{i}=CF$     (3)

Where $C$ is a constant and $F$ is a load.

From equation (3) and the superposition principle we conclude that

$\sigma _{i}=c_{0}+c_{1}F_{1}+c_{2}F_{2}+...+c_{n-1}F_{n-1}+c_{n}F_{n}$     (4)

Equation (4) describes a stress measure of a system subjected to $n$ loads. The parameters of (4) can be obtained by performing $>=n$ tests/calculations. It follows that it is easy to, for instance, scale test results according to different safety factors.

It should also be noted, as is customary, that the stress-strain curve is identical during loading and unloading, if the system is linear.

It is important to know whether or not the structure is linear before drawing any critical conclusions from the theory of elasticity. Common sources of non-linear behavior in a structure are material, geometric, contact and dynamic effects.

Additional Aspects

The simplistic view on elasticity presented in this section is useful in many ways, and is an excellent introduction to the subject. It should be noted however, that it is incomplete. The chapter Continuum Mechanicspresents a theory of how 2D and 3D stress and strain fields interact.Moreover, the material property $E$, may be influenced by several factors, including: temperature and aging. If deformation time rates are very high, there might even be viscous forces present. Therefore, keep in mind that there is more to come on this subject later on in the book.