1
Stress

1.1. Notion of stress

1.1.1. External forces

There are three types of external forces:

• - concentrated forces: this is a force exerted on a point (in Newton units, noted as N). In practice, this force does not actually exist. It is just a model. If we were to apply a force to a point that has zero surface, the contact pressure would be infinite and the deformation of the solid would therefore induce a non-zero contact surface. Nevertheless, it can still be imagined for studying problems with a very concentrated contact type load between balls. The results will thus yield an infinite stress and will need to be interpreted accordingly;
• - surface forces, which will be noted as Fext for the rest of this volume (in Pascal units, it is noted Pa). This type of force includes contact forces between two solids as well as the pressure of a fluid. Practically, any concentrated force can be seen as a surface force distributed onto a small contact surface;
• - volume forces, which will be noted as fv for the rest of this book (in N/m3). Examples of volume forces are forces of gravity, electromagnetic forces, etc.

Incidentally, in this book you will notice that vectors are underlined once and matrices (or tensors of rank 2), which you will come across further on, are underlined twice.

1.1.2. Internal cohesive forces

We wish to study the cohesive forces of the solid S, at point M and which is in equilibrium under the action of external forces. The solid is cut into two parts E1 and E2 by a plane with a normal vector n passing through M. The part E1 is in equilibrium under the action of the external forces on E1 and the cohesive force of E2 on E1.

Figure 1.1. Principle of internal cohesive forces. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

Let ?S be the surface around M and ?F be the cohesive force of 2 on 1 exerting on ?S, then the stress vector at the point M associated with the facet with a normal vector n is called:

The unit is N/m2 or Pa and we generally use MPa or N/mm2.

Physically, the stress notion is fairly close to the notion of pressure that can be found in everyday life (the unit is even the same!), but as we will see further on, pressure is but only one particular example of stress.

1.1.3. Normal stress, shear stress

We define the different stresses as:

• - normal stress, the projection of s (M, n) onto n, noted as s;
• - shear stress, the projection of s (M, n) onto the plane with normal n, noted as t.

Figure 1.2. Decomposition of a stress vector. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

Thus, s represents the cohesive forces perpendicular to the facet, meaning the traction/compression, and t the forces tangential to the facet, meaning the shear. In a physical sense, the pressure found in our everyday lives is simply a normal compression stress.

We then definitely have:

NOTE.- n and t must be unit vectors.

And conversely:

1.2. Properties of the stress vector

1.2.1. Boundary conditions

If n is an external normal, then:

Figure 1.3. External force and associated normal vector. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

NOTE.- Fext is in MPa, and a normal external vector is always moving from the matter towards the exterior.

So, Fext can be seen as a stress vector exerted on S, particularly if the surface is a free surface:

These relations are important as they translate the stress boundary conditions on the structure. In order for this to be the solution to the problem (see Chapter 3), these relations are part of a group of conditions that are needed to verify a stress field.

EXAMPLE: TANK UNDER PRESSURE.-

Figure 1.4. Tank under pressure

For every point on the internal wall of the tank, we find:

With the external normal vector moving towards the center of the circle, from where the normal and shear stresses are:

Given that the normal stress is negative and the shear stress is zero, the material is subjected to pure compression. The first relation shows that the physical notion of pressure is simply a normal stress of compression: hence the minus sign before the pressure!

1.2.2. Torsor of internal forces

Figure 1.5. Set of internal forces. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

The torsor of internal forces of 2 on 1 at G, the center of gravity of S, is:

At first sight, the torsor notion may seem primitive but it enables us to simply consolidate the force with the moment. Should the notion of torsor bother you, you may settle for referring to it in plainer language as force and moment. However, you should not forget that when speaking about internal forces between 2 parts of a solid, it needs to be remembered that there is a force (in N) and a moment (in N.mm). The ambiguity comes from the term "force", which is used for a force (in the common everyday sense of the word), and as a whole, force + moment!

Let us now seek to link this set of internal forces to the previously discussed stress vector. We then have:

therefore:

These relations are somewhat (or very) complex, but physically, they simply translate the fact that if we add up all of the stress vectors on section S, then we will obtain the force of part E2 on part E1. Lastly, we should not forget that when we add up the stress vectors, we will obtain not only a force, but also a moment (which obviously depends on the point at which it is calculated).

These relations can also be written on an external surface as:

These relations are important because in practice, although we know the resultant Rext/1 or Mext/1, we do not generally know Fext. In fact, an external force is practically applied via the intermediary of a beam, a screed, a jack, etc., and the applied resulting force (or the moment) is known, but the way in which it is divided is unknown.

EXAMPLE: TRACTION-

Figure 1.6. Tensile test. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

In a tensile test, we know that the resultant of the forces applied to Sy is worth F:

However, in order to deduce that:

we must add a homogeneity hypothesis of the force applied which remains to be verified. Incidentally, we can demonstrate that the two previous integrals are verified with this stress vector.

EXAMPLE: BENDING.-

Figure 1.7. Bending test. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

In a pure bending test, we know that the resultant of the forces applied to Sx is worth M.z:

However, by deducing that on Sx:

This formula is a classic example of the mechanics of material which we will discuss (and demonstrate) again when doing the exercises. Should you need to, you can read a more detailed publication, such as [AGA 08, BAM 08, CHE 08, DEL 08, DUP 09], etc.

Obviously, with the moment of inertia:

we must add a linear distribution hypothesis of the stress applied which remains to be verified. Incidentally, we can demonstrate that the two previous integrals are verified with this stress vector.

1.2.3. Reciprocal actions

Figure 1.8. Reciprocal actions. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical.zip

According to the Law of Reciprocal Action, we have:

Yet:

Hence:

This can be translated by the fact that a fine slice of matter of surface dS, which has a normal vector +n on one side and -n on the other, is at equilibrium under the action of the two opposing forces s(M, n). dS and s(M, -n). dS. Evidently, it is very much at equilibrium.

1.2.4. Cauchy reciprocal theorem

Figure 1.9. Stress vectors on the faces of a square. For a color version of this figure, see...