## 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^^].

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

$$\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 small^{1}^{}, 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:

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

- $\sigma$ – 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 forming^{2}^{} 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.

- 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. - For example during the forming of a trunk for a car.