The balance of forces within a material under stress

The drawing on the left is intended to represent a 3-dimensional solid body, acted upon by force PR on the right, and PL on the left. Within the body is a surface S.

Imagine that we have a fixed coordinate system (x,y,z) such that the positive x axis is normal to the surface S. We use the label SR for S when it represents a part of the surface of the volume VR, and SL for the part of the surface of the volume VL. (VR and VL together give the total volume).

It seems clear that each component of PR and PL in the x, y and z directions must be equal in magnitude, but with opposite sign. Otherwise, since the net force on a body is equal to the rate of change of its momentum with time, the body would accelerate in the direction of the greater force.

In elasticity theory, it is usually assumed that the material body is somehow held fixed in space, i.e. with balanced applied forces.

Now let's consider the surface S within the body: Here S, SL and SR all occupy the same region in space, but SR has normal vectors pointing into the region VL, and SL's normal vectors point into VR. At each point on S, the normal vectors for SR and SL have components in x, y and z which are equal in magnitude and opposite in direction.

Let's start with SL. We choose a point (x,y,z) on SL, and take the normal vector there to point in the positive x direction (into VR). Then the normal vector has only one non-zero component, as shown in the Cauchy formula on the left.

As an aside, note that we have used the symmetry of the stress matrix, which allows us to equate the off-diagonal components as shown.

Using matrix multiplication, we find that each component of the stress vector contains only one component of the stress tensor, as shown below.

The stress vector is determined by only three components of the stress tensor - the normal stress in the 11 or xx direction, and two shear stresses in the yx and zx directions.

Since the normal vector is of unit length, i.e. n = (1,0,0), that is n1 = 1, the equations on the left show that the x component of the stress vector (that is, of the force acting on the surface SL) is given by the xx component of the stress tensor.

Similarly the y component of the force t is given by the yx component of the stress tensor, and the z component of force is given by the zx component of stress.

Put another way, the yx component of the stress tensor is the y component of the force acting on the x plane. By"x plane," we mean the plane tangent to the surface SL at the point (x,y,z) whose normal vector points in the x direction. It's actually the yz plane, but calling it the x plane makes it easier to identify with the yx component of the stress tensor.

Similarly, the zx component of the stress tensor is the z component of the force acting on the x plane.

We now have a simple example of a particular stress vector acting on a surface within a body. The next step is to consider the stress vector acting on the same plane tangent to the same surface, but now imagining that surface to to belong to the remaining part of the body.

Referring again to the drawing at the top, we use the same coordinate system we had earlier, and consider the surface SR, which is part of the surface of the volume VR. We again choose the same point (x,y,z) that we chose before, with the same plane tangent to SL (and therefore to SR), only this time the normal vector has components (-1,0,0) in our coordinate system.

All we have to do to get stress vector components acting at that point is to replace the x component of n with -1. We get the equations shown below, and find that the components have the same magnitude as before, only they are in the opposite (-x) direction. Since this is true for any point (x,y,z) on the surface drawn anywhere in the body, it must be that the the stress vectors are balanced. Note that the stress components depend only on the location of (x,y,z) and not on the orientation of the surface through this point.

Therefore the stress vectors are balanced. If we choose a different surface through the same point, the components of n will change (the coordinate system is the same), but the stress vectors will still balance.

Truesdell pointed out that this result is actually a consequence of the conservation of momentum: The sum of all the external forces is zero (that PL and PR are equal and opposite). Otherwise, if there is a net force, the body will accelerate, and will no longer be in equilibrium.

Instead, the body will deform as a function of time. The result will be that the surface S will change over time, as will the normal vector: n=n(t).

Furthermore, if the applied forces PL and PR are unequal, then the integrals of t over the surfaces SL and SR will be unequal, since each of those integrals must balance the applied forces on their respective regions VL and VR. From this it is clear that the stress vectors cannot balance, even at a given instant of time. From this we conclude that when the body accelerates, there must be a discontinuity in the stress tensor across the surface S. That is, the value of the stress components will be different if we approach S from the right (VR) or the left (VL) side. This is the subject of the theory of shock waves.

We won't pause now (maybe later) to study shock waves, but suffice it to say that we would expect the acceleration necessary to cause a jump condition within the body of a material would have to be quite large - say of the magnitude that occurs when a massive object from space accelerates towards the earth.

In the next blog, we will return to our main subject - reversible, elastic deformations of homogeneous materials - and introduce some of the concepts needed to actually solve problems.

The Stress Principle of Euler and Cauchy

Near the end of an earlier blog - the one called "vectors," I almost inadvertently mentioned something so important that Truesdell referred to it as "the defining principle of continuum mechanics." He called it The Stress Principle of Euler and Cauchy. Note that it is a basic assumption about the nature of continuous media, rather than something that can be derived.

This principle states that there is a set of stress vectors acting on the surface of any region in a material which completely represent the forces which are exerted on the region by the material outside.

We can put this statement in the form of the equation below. The left hand side (LHS) represents the volume integral of the i th component of the force density acting on each point inside the volume, and the right hand side (RHS) is the integral of the i th component of the stress vector acting on the surface of V.

The mass density of the material at each point is given by the Greek letter rho, the force density f is the force per unit mass, and the stress vector is the force per unit area acting on the surface of V.

Since the region of of interest in the material can be any shape or size, including arbitrarily small, it follows from this equation that we don't need to know the forces inside the region in order to calculate anything. We only need to know the forces on its surface.

The integral on the LHS represents the total force due to the exterior material which acts upon the interior material. Similarly, the interior material acts on the exterior, with the same total force, because we are considering bodies which are in equilibrium.

By equilibrium, we mean the following. When we first apply some combination of constant forces to a body, it moves - it changes its shape. When the movement stops, we say the body is in equilibrium. The vector sum of the external forces applied to the body are exactly balanced by the reactive forces of the body onto whatever is applying the external forces, and a similar statement applies to each region of the body we choose.

Note that this force balancing only applies to elastic materials, like rubber. For example, if you pulled hard enough on the ends of a length of aluminum rod, it would be permanently deformed. You could hold the rod without pulling on it at all, and it would still be deformed.

On the other hand, a long piece of aluminum (or copper, or steel) wire can be held horizontally at one end, and moved up and down vertically at the other end without any permanent deformation, as long as the distance traveled by the free end is small enough compared with the wire's length. So if you let go of the free end, the wire will return to its initial position.

This observation about the balance of interior and exterior forces also applies to the RHS of the equation, but it requires some further discussion, along with a drawing, so I'll save that for the next blog.

Cauchy formula in matrix form

The stress vector t is now written as a 3-component column vector. The equation tells how to write three equations for the components of t, by multiplying the stress matrix by the unit vector n.

The multiplication is done as follows (see equations below): the first component of t is given by multiplying the "11" component of sigma by the first component of n, etc.

Note how the first subscript of sigma corresponds to the t subscript, and the second subscript of sigma matches the n subscript.

Each component of the stress vector t depends on three components of the stress tensor, and on the three components of the surface normal vector n. Recall that t is the force applied at each point on the surface of a chosen region of the body by the surrounding material, and that n is the unit vector normal to that surface at the same point.

Put another way, here's the remarkable thing about Cauchy's theorem: it means that for every possible surface through a given point in the body - and their are an infinite number of such surfaces - all you need are the components of the stress tensor at that point. Then you can calculate the stress vector for any surface through that point, converting its normal vector into the stress vector with the Cauchy formula above.

The first, or x, component of t represents the force in the x-direction. If positive, the x component points out from the surface at that point; i.e., it is a tensile force. If negative, the x component points into the region enclosed by the surface, and is therefore compressive.

Similarly, the y and z components of t are in the y and z directions.

Note that the x component of t depends not only on the component of the normal vector n in the x direction, but also on the y and z components of n, and similarly for the y and z components of t.

The stress matrix element that corresponds to the x component of t and the x component of n is the -11- or -xx- element, and similarly for the y and z components. Later we will see why these components of the stress tensor are called normal stresses. The off-diagonal components are the shear stresses.

Stress tensor in matrix form

The 9 components of the stress tensor may be written in matrix form as shown above. In the lower of the two matrices above, the first number in the subscripts to the sigmas gives the row of the matrix, and the second number gives the column. So, for example the subscript "11" is used to represent row 1, column 1, and so on.

The subscripts also have another meaning: the three x, y, and z axes of our Cartesian coordinate system can be labelled 1, 2, and 3 respectively. Then the subscript "12" stands for "xy," and so on, allowing us to write the stress matrix as it's shown in the upper matrix.

An important simplification to elasticity theory is made possible by the fact that the stress tensor for most materials is symmetric: That is, the "xy" and "yx" components are equal, and similarly for "xz" and "zx," and for "yz" and "zy. Therefore the stress tensor has only six distinct components, or "unknowns" to be determined. They are the three diagonal components, and the three (different) components off-diagonal.

Vectors can also be written as matrices, with the three vector components as either a column or row of three numbers. I'll give an example in the next post, by writing the Cauchy formula in matrix form, followed by the use of matrix multiplication to write an equation for each component of the stress vector t.

Cauchy's stress theorem

Here we have Cauchy's theorem: The stress vector t can be written as the vector dot product of the stress tensor, sigma, and the unit vector normal to the surface, n. Whereas t depends on n, sigma does not. Sigma is a function only of position in the elastic body.

However, being a symmetric tensor (defined in the next blog), sigma also requires six numbers to specify - six of the nine components of the tensor.

So while a vector has only three components, a tensor has as many as nine (some of the components may be the same, as we'll see in the next blog).

This formula for Cauchy's theorem is in vector notation, a shorthand form which can be understood by writing it out in matrix form, which I'll add to the next post.

more on the stress vector, and the stress tensor

Here is the imaginary cube inside the rubber band, oriented so that the vertical faces are perpendicular to the pulling direction. The stress is perpendicular to the two vertical faces of the cube on the left and right sides, and parallel to all the other faces.

(I keep forgetting to mention, you can enlarge the drawing by pressing the ctrl and + keys, and then shrink back with crtl and -).

Imagine that the cube shrinks to an infinitesimal size. The normal and shear stresses will be associated with almost the same point in the material - and yet, they will be different because they represent the force on different surfaces, with different normal vectors n.

So we say that t is a function of both the position in the body (x), and the unit vector normal to the surface (n) under consideration. Thus the stress vector t is very different from most vectors in continuum mechanics, which are functions only of their position in space (and of time, in the case of time-dependent phenomena).

Putting it another way, it takes 6 numbers to specify the stress vector: 3 for the position x, and 3 for the surface with unit normal n, for which we want to calculate the force.

Thus each component of the stress vector (each of the x, y and z components) is a mathematical function of 6 variables. In order to develop a theory of stress, a new concept was invented which allowed an important simplification: a way of representing the stress vector with a new function, which depends only on the position x. This new idea has been called Cauchy's theorem,* and is stated in the beginning of the next post.

*According to Truesdell (The Classical Field Theories of Mechanics, by Clifford Truesdell and Richard Toupin, published in 1960 in Volume III of the Handbuch der Physik), it was the French mathematician Augustin-Louis Cauchy (1789–1857) who invented the concept of the stress tensor. As we get further into the theory, I think you will begin to appreciate the importance of this fundamental theorem, and the genius of the man who came up with it.

vectors

We all live in a three-dimensional space, in which objects can be located by their distances from an arbitrary point in the space - call it the "origin." To facilitate this, we imagine a rectilinear, orthogonal coordinate system. "Rectilinear," because the space can be defined with straight lines (rather than curves, as, say, on the surface of a sphere), and "orthogonal" because a point in the space can be uniquely determined by specifying its location on three mutually perpendicular directions.

In the drawing, we have marked off the three perpendicular axes, labelled x, y and z. The location of the tip of the arrow (or vector) in the drawing can be represented in standard notation by (a,b,c), where a is the distance from the origin along the x axis, and similarly for b and c along the y and z axes. We say that the vector x = (a,b,c).

In the previous blog I showed a drawing with a point x on the surface of an arbitrary volume within a solid body. Now I can say more precisely, that x is the vector from the origin of the coordinate system to the point on the surface, but to keep the drawing simpler I omitted the coordinate system. Anyhow, I wanted to explain this so you'd know why x is a vector.

Now, to continue that earlier discussion: We have the unit vector n perpendicular to the arbitrary surface drawn through the point located by the vector x. It's easier to talk about n if we choose a coordinate system so that the surface through x is perpendicular to one of the axes. For example, if this surface is perpendicular to the x axis, then n = (1,0,0). This notation means that n has a value of 1 in the x direction, and zero in the y and z directions. (Remember, this value 1 has whatever unit of length you choose, as long as you are consistent, and use the same unit for every length).

Now we can talk about the stress vector, t. What makes it so interesting to me is that t depends not only on the location in the solid defined by x, but also on the orientation of the surface we've chosen to examine, that contains the point defined by x.

Before we go any further, let's define stress: stress results from the force of a solid material acting on itself.

Imagine that we cut a rubber band, so that we have a single piece of material, and we pull on it from each end. We are then exerting a force on the ends (it has to be the same force on each end, for the material to hold still), but what about in the interior of the body?

Imagine a very small cube, somewhere inside the rubber, with its surfaces lined up so one pair is perpendicular to the direction we are pulling, and the others are therefore parallel to that direction. The forces acting on those surfaces are exactly balanced by forces in the the opposite directions, since we are holding the rubber band still - i.e., the elastic body is stationary. The material is actually pulling on itself.

From a microscopic point of view, we are stretching the bonds between molecules of rubber (or even stretching the long chain molecules themselves), and the magnitude of the stress depends on the strength of the molecular bonds. These molecular forces act only within short distances, so the force on each molecule is due only to other molecules nearby. Electrical shielding by the nearby molecules prevents long range effects, except in special cases - piezoelectric materials, for example - when long range forces are important.

But here we are confining ourselves to a continuum model, in which the stress and the properties of the material can be described by mathematically continuous functions in space. That's why we just say that the material "acts on itself." Note also that this means for any region of the elastic body we want to consider, the deformation of that region is entirely determined by the forces acting on its surface. Also, the sum of these forces, taking their directions into account, must add up to zero - otherwise the body would move.

The forces on the perpendicular surfaces mentioned above are perpendicular, or "normal" to the surfaces (they are equal and in opposite directions), whereas the forces on the parallel surfaces are parallel to them, or "shear," (and they are also equal and in opposite directions). Since we are pulling on the band, the normal forces are called "tensile." If we had a material we could squeeze, like a sponge, the normal forces would be "compressive."

I'll add a drawing of this cube next.