Stress (physics)

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Image:Stress tensor.png

In physics, stress is a measure of the internal distribution of force per unit area within a body that balances and reacts to the loads applied to it. Stress is a tensor quantity with nine terms, but can be described fully by six terms due to symmetry. Simplifying assumptions are often used to represent stress as a vector for engineering calculations.

The stress tensor <math>\sigma_{ij}</math> is defined by the equation:

<math>dF_i=\sum_{j=1}^3 \sigma_{ij}\,dA_j</math>

where <math>[dF_1,dF_2,dF_3]</math> is the force on a small area element <math>[dA_1,dA_2,dA_3]</math> where the subscripts 1,2,3 refer to the x,y, and z axes respectively and the area vector is a vector perpendicular to the area element, with length equal to the area of the element.

The stress in an axially loaded bar is equal to the applied force divided by the bar's area (see also pressure). Stresses in a 2-D or 3-D solid are more complex and need to be defined more rigorously. The internal force acting on a small area dA of a plane that passes through a point P can be resolved into three components: one normal to the plane and two parallel to the plane. The normal component divided by dA gives the normal stress (usually denoted by σ), and the parallel components divided by the area dA give the shear stress (usually denoted by τ). These stresses are average stresses, as the area dA is finite; but when the area dA is allowed to approach zero, the stresses become stresses at the point P. In general, the stress may vary from point to point, but for simple cases, such as circular cylinders with pure axial loading, the stress normal to the cross section is constant.

Since stresses are defined in relation to the plane that passes through the point under consideration, and the number of such planes is infinite, there appear an infinite set of stresses at a point P. Fortunately, it can be proven by equilibrium that the stresses on any plane can be computed from the stresses on three orthogonal planes passing through the point. The three planes are normally chosen to be the x-, y-, and z-planes. As each plane has three stresses, the stress tensor has nine stress components, which completely describe the state of stress at a point. By using Mohr's circle method or stress tensor transformation, the stresses on an arbitrary plane through P can be computed from the stress tensor at P.

Stress can occur in liquids, gases, and solids. Liquids and gases support normal stress (pressure), but flow under shear stress (see viscosity). Solids support both shear and normal stress, with brittle materials failing under normal stress, and plastic or ductile materials failing under shear stress.

Contents 1 Stress in one-dimensional bodies 2 Cauchy's principle 3 Plane stress 3.1 Principal stresses 3.2 Mohr's circle 4 Stress in three dimensions 5 Stress tensor 5.1 Generalized notation 5.2 Why is Newtonian stress a symmetric tensor? 5.3 Equilibrium conditions 6 Stress measurement 7 Units 8 Residual stress 9 See also 10 Books 11 External links

Stress in one-dimensional bodies

The idea of stress originates in two simple, but important, observations of the loading (in tension) of a one-dimensional body, for example, a steel wire.

1. When a wire is pulled tight, it stretches (undergoes strain). Up to a certain limit, the amount it stretches is proportional to the load divided by the cross-sectional area of the wire, σ = F/A.
2. Failure occurs when the load exceeds a critical value for the material, the tensile strength multiplied by the cross-sectional area of the wire, F_c = σ_t A.

These observations suggest that the fundamental characteristic that affects the deformation and failure of materials is stress, force divided by the area over which it is applied.

This definition of stress, σ = F/A, is sometimes called engineering stress and is used for rating the strength of materials loaded in one dimension. The cross-sectional area is measured prior to applying strain for testing. Poisson's ratio, however, reveals that any applied strain will produce a change in the area, A. Engineering stress neglects this change in area. Stress-strain diagrams are usually presented as engineering stress, even though the sample may undergo a substantial change in cross-sectional area during testing.

True stress is a definition of stress that includes the change in cross-sectional area. This is true in the sense that, once you stretch a material it tends to contract in the transverse direction (Poisson contraction) so the actual force per unit area is over that new (usually smaller) area. For engineering applications, the initial geometry is known, so calculations are generally easier in terms of the initial area, hence engineering stress. The distinction between engineering and true stresses is especially important for rubber-like substances and for plasticity, since in these cases the changes in cross-sectional areas can be significant. In the case of small strains, cross-sectional area effectively does not change in which case true and engineering stresses are identical. Both engineering stress and true stress are evaluated as tensors for three-dimensional cases. When considering true stress in one dimension, it can be calculated using the formula σ_true = σ(1 + ε) where ε is engineering strain and σ is engineering stress.

As an example: a steel bolt of diameter 5 mm, has a cross-sectional area of 19.6 mm². A load of 50 N induces a stress (force distributed over the cross section) of σ = 50/19.6 = 2.55 MPa (N/mm²). This can be thought of as each square millimeter of the bolt supporting 2.55 N of the total load. In another bolt with half the diameter, and hence a quarter the cross-sectional area, carrying the same 50 N load, the stress will be quadrupled (10.2 MPa).

The ultimate tensile strength is a property of a material loaded in one dimension. It allows the calculation of the load that would cause fracture. The compressive strength is a similar property for compressive loads. The yield tensile strength is the value of stress causing plastic deformation. These values are determined experimentally using the measurement procedure known as the tensile test.

Cauchy's principle

Augustin Louis Cauchy enunciated the principle that, within a body, the forces that an enclosed volume imposes on the remainder of the material must be in equilibrium with the forces upon it from the remainder of the body.

This intuition provides a route to characterizing and calculating complicated patterns of stress. To be exact, the stress at a point may be determined by considering a small element of the body that has an area ΔA, over which a force ΔF acts. By making the element infinitesimally small, the stress vector σ is defined as the limit:

<math>

\sigma = \lim_{\Delta A \to 0} \frac {\Delta F} {\Delta A} = {dF \over dA} </math>

Being a tensor, the stress has two directional components: one for force and one for plane orientation; in three dimensions these can be two forces within the plane of the area A, the shear components, and one force perpendicular to A, the normal component. Therefore the shear stress can be further decomposed into two orthogonal force components within the plane. This gives rise to three total stress components acting on this plane. For example in a plane orthogonal to the x axis, there can be a normal force applied in the x direction and a combination of y and z in plane force components.

Plane stress

Plane stress is a two-dimensional state of stress (Figure 2). This 2-D state models well the state of stresses in a flat, thin plate loaded in the plane of the plate. Figure 2 shows the stresses on the x- and y-faces of a differential element. Not shown in the figure are the stresses in the opposite faces and the external forces acting on the material. Since moment equilibrium of the differential element shows that the shear stresses on the perpendicular faces are equal, the 2-D state of stresses is characterized by three independent stress components (σ_x, σ_y, τ_xy). Note that forces perpendicular to the plane can be abbreviated. For example, σ_x is an abbreviation for σ_xx. This notation is described further below. Image:Stress 3.PNG

Principal stresses

Cauchy was the first to demonstrate that at a given point, it is always possible to locate two orthogonal planes in which the shear stress vanishes. These planes are called the principal planes, while the normal stresses on these planes are the principal stresses. The common technique for doing this is by use of Mohr's circle.

Principal stresses are the maximum and minimum values of the normal stresses. Eigenvalues of a stress tensor show the principal stresses, and the eigenvectors show the direction of the principal stresses.

Mohr's circle

Image:Mohrs circle fig 1.PNGImage:Mohrs circle fig 3.PNG A graphical representation of any 2-D stress state was proposed by Christian Otto Mohr in 1882. Consider the state of stress at a point P in a body (Figure 2). The Mohr's circle may be constructed as follows.

1. Draw two perpendicular axes with the horizontal axis representing normal stress, while the vertical axis the shear stress.

2. Plot the state of stress on the x-plane as the point A, whose abscissa (x value) is the magnitude of the normal stress, σ_x (tension is positive), and whose ordinate (y value) is the shear stress (clockwise shear is positive).

3. Mark the magnitude of the normal stress σ_y on the horizontal axis (tension being positive).

4. Mark the midpoint of the two normal stresses, O (Figure 3).

Image:Mohrs circle fig 2.PNG

5. Draw the circle with radius OA, centered at O (Figure 4).

6. A point on the Mohr's circle represents the state of stresses on a particular plane at the point P. Of special interest are the points where the circle crosses the horizontal axis, for they represent the magnitudes of the principal stresses (Figure 5).

Mohr's circle may also be applied to three-dimensional stress. In this case, the diagram has three circles, two within a third.

Engineers use Mohr's circle to find the planes of maximum normal and shear stresses, as well as the stresses on known weak planes. For example, if the material is brittle, the engineer might use Mohr's circle to find the maximum component of normal stress (tension or compression); and for ductile materials, the engineer might look for the maximum shear stress.

Stress in three dimensions

The considerations above can be generalized to three dimensions. However, this is very complicated, since each shear loading produces shear stresses in one orientation and normal stresses in other orientations, and vice versa. Often, only certain components of stress will be important, depending on the material in question.

The von Mises stress is derived from the distortion energy theory and is a simple way to combine stresses in three dimensions to calculate failure criteria of ductile materials. In this way, the strength of material in a 3-D state of stress can be compared to a test sample that was loaded in one dimension.

Stress tensor

Because the behavior of a body does not depend on the coordinates used to measure it, stress can be described by a tensor. The stress tensor is symmetric and can always be resolved into the sum of two symmetric tensors:

• a mean or hydrostatic stress tensor, involving only pure tension and compression; and
• a shear stress tensor, involving only shear stress.

In the case of a fluid, Pascal's law shows that the hydrostatic stress is the same in all directions, at least to a first approximation, so can be captured by the scalar quantity pressure. Thus, in the case of a solid, the hydrostatic (or isostatic) pressure p is defined as one third of the trace of the tensor, i.e., the mean of the diagonal terms.

<math>p = \frac{\mathrm{tr}(T)}{3} = \frac{\sigma_{11} + \sigma_{22} + \sigma_{33}}{3} </math>

Generalized notation

In the generalized stress tensor notation, the tensor components are written σ_ij, where i and j are in {1;2;3}.

The first step is to number the sides of the cube. When the lines are parallel to a vector base <math>(\vec{e_1},\vec{e_2},\vec{e_3})</math>, then:

• the sides perpendicular to <math>\vec{e_j}</math> are called j and -j; and
• from the center of the cube, <math>\vec{e_j}</math> points toward the j side, while the -j side is at the opposite.

Image:Numerotation faces cube.png

σ_ij is then the component along the i axis that applies on the j side of the cube. (Or in books in the English language, σ_ij is the stress on the i face acting in the j direction -- the transpose of the subscript notation herein. But transposing the subscript notation produces the same stress tensor, since a symmetric matrix is equal to its transpose.)

Image:Tenseur des contraintes generalise.png

This generalized notation allows an easy writing of equations of the continuum mechanics, such as the generalized Hooke's law:

<math>\sigma_{ij} = \sum_{kl} C_{ijkl} \cdot \varepsilon_{kl}</math>

The correspondence with the former notation is thus:

x	→	1
y	→	2
z	→	3
σxx	→	σ11
τxy	→	σ12
τxz	→	σ13
...

Why is Newtonian stress a symmetric tensor?

The fact that the Newtonian stress is a symmetric tensor follows from some simple considerations. The force on a small volume element will be the sum of all the stress forces over the surface area of that element. Suppose we have a volume element in the form of a long bar with a triangular cross section, where the triangle is a right triangle. We can neglect the forces on the ends of the bar, because they are small compared to the faces of the bar. Let <math>\vec{A}</math> be the vector area of one face of the bar, <math>\vec{B}</math> be the area of the other, and <math>\vec{C}</math> be the area of the "hypotenuse face" of the bar. It can be seen that

<math>\vec{C}=-\vec{A}-\vec{B}</math>

Let's say <math>\sigma(\vec{A})</math> is the force on area <math>\vec{A}</math> and likewise for the other faces. Since the stress is by definition the force per unit area, it is clear that

<math>\sigma(k\vec{A})=k\sigma(\vec{A})</math>

The total force on the volume element will be:

<math>\vec{F}=\sigma(\vec{A})+\sigma(\vec{B})-\sigma(\vec{A}+\vec{B})</math>

Let's suppose that the volume element contains mass, at a constant density. The important point is that if we make the volume smaller, say by halving all lengths, the area will decrease by a factor of four, while the volume will decrease by a factor of eight. As the size of the volume element goes to zero, the ratio of area to volume will become infinite. The total stress force on the element is proportional to its area, and so as the volume of the element goes to zero, the force/mass (i.e. acceleration) will also become infinite, unless the total force is zero. In other words:

<math>\sigma(\vec{A}+\vec{B})=\sigma(\vec{A})+\sigma(\vec{B})</math>

This, along with the second equation above, proves that the <math>\sigma</math> function is a linear vector operator (i.e. a tensor). By an entirely analogous argument, we can show that the total torque on the volume element (due to stress forces) must be zero, and that it follows from this restriction that the stress tensor must be symmetric.

However, there are two fundamental ways in which this mode of thinking can be misleading. First, when applying this argument in tandem with the underlying assumption from continuum mechanics that the Knudsen number is strictly less than one, then in the limit <math>K_{n}\rightarrow 1</math>, the symmetry assumptions in the stress tensor may break down. This is the case of Non-Newtonian fluid, and can lead to rotationally non-invariant fluids, such as polymers. The other case is when the system is operating on a purely finite scale, such as is the case in mechanics where Finite deformation tensors are used.

Equilibrium conditions

The state of stress as defined by the stress tensor is an equilibrium state if the following conditions are satisfied:

<math> \frac {\partial {\sigma_{11}}} {\partial {x_{1}}} + \frac {\partial {\sigma_{12}}} {\partial {x_{2}}} + \frac {\partial {\sigma_{13}}} {\partial {x_{3}}} = f_{1} </math>
<math> \frac {\partial {\sigma_{21}}} {\partial {x_{1}}} + \frac {\partial {\sigma_{22}}} {\partial {x_{2}}} + \frac {\partial {\sigma_{23}}} {\partial {x_{3}}} = f_{2} </math>
<math> \frac {\partial {\sigma_{31}}} {\partial {x_{1}}} + \frac {\partial {\sigma_{32}}} {\partial {x_{2}}} + \frac {\partial {\sigma_{33}}} {\partial {x_{3}}} = f_{3} </math>

<math> \sigma_{ij} </math> are the components of the tensor, and f ₁ , f ₂ , and f ₃ are the body forces (force per unit volume).

These equations can be compactly written using Einstein notation in which repeated indices are summed. Defining <math>\partial_i</math> as <math>\partial/\partial x_i</math> the equilibrium conditions are written:

<math>\partial_j\sigma_{ij}=f_i</math>

The equilibrium conditions may be derived from the condition that the net force on an infinitesimal volume element must be zero. Consider an infinitesimal cube aligned with the <math>x_1</math>, <math>x_2</math>, and <math>x_3</math> axes, with one corner at <math>x_i</math> and the opposite corner at <math>x_i+dx_i</math> and having each face of area <math>dA</math>. Consider just the faces of the cube which are perpendicular to the <math>x_1</math> axis. The area vector for the near face is <math>[-dA,0,0]</math> and for the far face it is <math>[dA,0,0]</math>. The net stress force on these two opposite faces is

<math>dF_i=\sigma_{i1}([x_1+dx_1,x_2,x_3])\,dA-\sigma_{i1}([x_1,x_2,x_3])\,dA \approx

\partial_1\sigma_{i1}\,dA</math>

A similar calculation can be carried out for the other pairs of faces. The sum of all the stress forces on the infinitesimal cube will then be

<math>dF_i=\partial_j\sigma_{ij}\,dA</math>

Since the net force on the cube must be zero, it follows that this stress force must be balanced by the force per unit volume <math>f_i</math> on the cube (e.g., due to gravitation, electromagnetic forces, etc.) which yields the equilibrium conditions written above. Note that the equilibrium conditions are sometimes written with the indices of the stress tensor transposed, but this is of no consequence since the stress tensor is symmetric (<math>\sigma_{ij}=\sigma_{ji}</math>).

Stress measurement

As with force, stress cannot be measured directly but is usually inferred from measurements of strain and knowledge of elastic properties of the material. Devices capable of measuring stress indirectly in this way are strain gages and piezoresistors.

Units

The SI unit for stress is the pascal (symbol Pa), the same as that of pressure. In US Customary units, stress is expressed in pounds-force per square inch (psi). See also pressure.

Residual stress

Residual stresses are stresses that remain after the original cause of the stresses has been removed. Residual stresses occur for a variety of reasons, including inelastic deformations and heat treatment. Heat from welding may cause localized expansion. When the finished weldment cools, some areas cool and contract more than others, leaving residual stresses. Castings may also have large residual stresses due to uneven cooling.

While uncontrolled residual stresses are undesirable, many designs rely on them. For example, toughened glass and prestressed concrete rely on residual stress to prevent brittle failure. Similarly, a gradient in martensite formation leaves residual stress in some swords with particularly hard edges (notably the katana), which can prevent the opening of edge cracks. In certain types of gun barrels made with two telescoping tubes forced together, the inner tube is compressed while the outer tube stretches, preventing cracks from opening in the rifling when the gun is fired. These tubes are often heated or dunked in liquid nitrogen to aid assembly.

Press fits are the most common intentional use of residual stress. Automotive wheel studs, for example, are pressed into holes on the wheel hub. The holes are smaller than the studs, requiring force to drive the studs into place. The residual stresses fasten the parts together. Nails are another example.

See also

• Strain tensor
• Stress-energy tensor
• Stress concentration

Books

• Dieter, G. E. (3 ed.). (1989). Mechanical Metallurgy. New York: McGraw-Hill. ISBN 0071004068.
• Love, A. E. H. (4 ed.). (1944). Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications. ISBN 0486601749.
• Marsden, J. E., & Hughes, T. J. R. (1994). Mathematical Foundations of Elasticity. New York: Dover Publications. ISBN 0486678652.

External links

• Stress analysis, Wolfram Research

• ESDU Stress Analysis Methodsde:Spannung (Mechanik)

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