Stress (physics)

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Stress tensor

In physics, stress is the internal distribution of forces within a body that balance and react to the loads applied to it. Stress is a tensor quantity with nine terms, but which 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 in an axially loaded bar is equal to the applied force divided by the bar's area (see also pressure (physics)). Stresses in a 2-D or 3-D solid are more complex and need be defined more rigorously. The internal force acting on a small area dA of a plane which passes 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 <math> \boldsymbol{\sigma} <math>), and parallel components divided by the area dA give the shear stress (usually denoted by <math> \boldsymbol{\tau} <math>). 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 9 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 gasses 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

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). 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, Fc = σt x A

These observations suggest that the fundamental characteristics 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 simplified definition of stress that includes the change in cross-sectional area. It should be noted, however, that True stress is still a simplified version of the stress tensor.

An example; a steel bolt of diameter 5 mm, has a cross-sectional area of 19.63 mm2. A load of 50 N induces a stress (force distributed across the cross-section) of σ = 50/19.63 = 2.55 N/mm2 (MPa). This can be thought of as each m2 of bolt supporting 2.55 MN 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 rest 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 vector, the stress has two components, one in the plane of the area, A, the shear stress, and one perpendicular, the normal stress. The shear stress can be further decomposed into two orthogonal components within the plane, thus giving rise to three stress components acting on this plane.

Plane stress

Plane stress is a two-dimensional state of stress (Figure 1). This 2-D state models well the state of stresses in a flat thin plate loaded in the plane of the plate. Figure 1 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 characterised by three independent stress components (σx, σy, τxy).

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Figure 1: Stresses normal and tangential to faces

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 the use of the Mohr's circle.

Mohr's circle

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Figure 4: Mohr's circle, stage 1
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Figure 6: Mohr's circle, stage 3

A graphical representation of any 2-D stress state was proposed by Otto Mohr in 1882. Consider the state of stress at a point P in a body (Figure 1). 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 is the magnitude of the normal stress (tension is positive), and whose ordinate is the shear stress (counter-clockwise shear is negative).

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

4. Mark the mid-point of the two normal stresses, O. (Figure 4)

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Figure 5: Mohr's circle, stage 2

5. Draw the circle with radius OA, centered at O.

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 6)

Mohr's circle may also be applied to three-dimension 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 orientation, and vice versa. Often, only certain components of stress will be important, depending on the material in question.

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 3D 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 the third of the trace of the tensor, i.e. the mean of the diagonal terms

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

Generalized notation

In the generalized stress tensor notation, the tensor component are written "σij" 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> ares called "j" and "-j" ;
  • point from the center of the cube, <math>\vec{e_j}<math> points towards the j side, the -j side is at the opposite.

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Image:Numerotation_faces_cube.png

σij is then the component along the i axis that applies on the j side of the cube.

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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 σ21
τxz σ31
...

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.

Units

The SI unit for stress is the pascal (symbol Pa); in US Customary units, stress is expressed in pounds per square inch (PSI). See also: Pressure (physics)

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, which is taken up during welding by either the molten metal or the placement of parts being welded. 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 un-controlled residual stresses are undesirable, many designs rely on them. For example, toughened glass and pre-stressed concrete rely on them 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 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. Parts 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

Books

  • Dieter, G. E. (1988) Mechanical Metallurgy, ISBN 0071004068
  • Love, A. E. H. (1944) A Treatise on the Mathematical Theory of Elasticity ISBN 0486601749
  • Marsden, J. E. & Hughes, T. J. R. (1983) Mathematical Foundations of Elasticity ISBN 0486678652

External links

ko:변형력 it:Stress (fisica) nl:Mechanische spanning ja:ストレス (物理学) sl:Mehanska napetost

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