# Basic concepts

## Normal stress — equilibrium

A tensile specimen is loaded by the external loads $P$ according to the figure below. The external loads are directed in opposite directions and are equal in magnitude — this implies the bar is in equilibrium.

Now we wish to investigate how stressed the material in the specimen is. To this end, we make a (virtual) section according to Figure @fig:1d-bar-section below and study what happens within the tensile specimen.

Considering equilibrium of the left figure, as well as the right, shows that the internal normal force $N$ must equal the external load $P$, i.e. $N=P$.

The measure used to characterize “the amount of load a piece of material is subjected to” is called stress and is denoted $\sigma$. Stress is defined as force per cross-sectional area $A$ and has the unit Pascal [Pa] = [N/m^^2^^].

Tension and compression

Depending on the sign of the normal force and stress, respectively, they are called tensile or compressive:

• $N > 0$ — tensile force
• $N < 0$ — compressive force
• $\sigma > 0$ — tensile stress
• $\sigma < 0$ — compressive stress

If we load a tensile specimen, the main part of the specimen will experience a constant stress, both along the specimen and across the cross-section, see the yellow area in the figure below.

If the studied section is taken at a large enough distance from the loaded ends, then one can assume the stress is constant over the cross-section giving

$$\sigma=\frac{N}{A}$$

this type of stress is called a normal stress since it is directed along the normal of the cross-section.

## Normal strain — kinematics

Assume we have a bar with length $L$ before loading according to the figure below. Then, the bar is loaded with external forces such that the length is changed to $L+\delta$.

Introduce the concept of deformation as the change in length $\delta$. If $\delta > 0$ it is called an elongation and if $\delta < 0$ it is called a shortening.

The concept of strain is defined as the “change in length per unit length” and is denoted $\epsilon$. For a bar of length $L$ undergoing a deformation $\delta$, the strain becomes

$$\epsilon=\frac{\delta}{L}$$

Now, introduce a coordinate $x$ along the bar according to the figure below. The deformed bar is shown below the undeformed bar.

The point which was located at position $x$ before deformation has now moved to $x+u$, where $u$ is called the displacement at $x$. Generally, $u$ varies with each point $x$, i.e. $u=u(x)$, and is called the displacement field.

The change in length of the bar — the deformation — is obtained as $\delta = u(L)-u(0)$ and the strain, therefore, becomes

$$\epsilon=\frac{u(L)-u(0)}{L}$$

If the strain is sought at a specific coordinate $x$ one needs to know the displacement field $u(x)$. To determine the strain at a point $x$, we define the following limit

Normal strain

$$\epsilon(x)=\lim_{\Delta x \rightarrow 0} \frac{u(x+\Delta x)-u(x)}{\Delta x}=\frac{{\rm d} u}{{\rm d} x}=u'(x)$$

In this book, we only consider what is called small-strain theory which implies that the strains are small1, usually below 5$\%$ ($|\epsilon| < 0.05$). By assuming small strains, we can make geometric simplifications as we will see several examples of later.

## Material model — Hooke’s law

To characterize the behavior of a material one can perform different experiments where the so-called tensile test is the most common. This experiment determines the relationship between applied tensile force and specimen elongation $\Delta L$.

In the experiment, the applied force is $P=\sigma \, A$ and the measured deformation is $\delta =\epsilon \, L$. However, since we are interested in obtaining the material behavior without the influence of the dimensions of the test specimen, we divide with these ($A$ and $L$). We then see that the material behavior becomes a relationship between stress $\sigma$ and strain $\epsilon$. This material relationship is also called a constitutive relationship and for a construction steel, it may usually look something like the figure below.

When the strains are small, the figure above can be approximated by a linear relationship between stress and strain as

$$\sigma = E \, \epsilon$$

where $E$ is a proportionality constant called Young’s modulus or modulus of elasticity and have the unit [N/m${}^2$ = Pa]. This relation between stress and strain is called Hooke’s law after Robert Hooke and is shown below. This equation is a fundamental material relation, within the field of strength of materials. As an example, the value of $E$ for steel is approximately $2.1 \cdot 10^{12} \Pa = 210 \GPa$ and for concrete $30\GPa$.

We summarize:

Hooke's law

$$\sigma = E \, \epsilon$$

• $\sigma$ – Normal stres [Pa]
• $E$ – Young’s modulus or modulus of elasticity [Pa]
• $\epsilon$ – Normal strain [-]

Yield limit

As long as the stress is below the yield limit $\sigma_{\rm y}$, the material behaves elastically. This means, upon unloading to zero stress (force) the strain goes to zero (deformation). However, if the material is loaded with a stress higher than $\sigma_{\rm y}$ there will be a permanent strain after unloading — we say that the material yields or flow. This behavior is desired in some application such as metal forming2 but should otherwise be avoided. The maximum stress the material can withstand, before failing, is called ultimate strength and is denoted $\sigma_{\rm u}$.

Most (solid) materials can be described by a linear relationship such as Hooke’s law within a certain strain range. But outside this range, materials generally behave quite different. There is a large difference between a metal, a polymer or a bread-dow.

Every relation between stress and strain is called a material relationship or a constitutive relationship and it is important to realize that these are mathematical models. They describe how materials behave during deformation but there is no material that is linear elastic. The study of constitutive relations is called material mechanics (or mechanics of materials) and is a large research field.

## Lateral contraction — Poisson’s effect

If one pulls in a bar it becomes elongated but it also becomes thinner — an illustrative example is when you pull on an elastic band. If the strain along the bar is denoted $\eps_{\parallel}$ and the lateral strain (in the thickness direction) is denoted $\eps_{\perp}$, experiments have shown the following relationship:

$$\eps_{\perp} = – \nu \cdot \eps_{\parallel} = – \nu \cdot \frac{\sigma}{E}$$

where $\nu$ is a constant called coefficient of lateral contraction or more commonly as Poisson’s ratio [-], after Siméon Denis Poisson. Typical values for $\nu$ is $0.3$ for metals and $0.2$ for concrete. If $\nu=0.5$ the material will not change volume during deformation and such materials are called incompressible, for example, rubber and many fluids. The fact that $\nu=0.5$ does not lead to a change in volume is shown in the section about Hooke’s generalized law.

1. A suitable strain measure for large deformations is the logarithmic strain (or true strain) $\epsilon=\int_0^{\epsilon} {\rm d} \epsilon=\int_{L_0}^{L_0+\delta }{ \rm d} L/L={\rm ln}(1+\delta/L_0)$. Using this gives a non-linear relation between deformation and strain. Also, the change in cross-section should be accounted for when determining the stress. A suitable stress measure is then the true stress, defined as the normal force divided by the deformed cross-sectional area.
2. For example during the forming of a trunk for a car.