Strength of materials

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In materials science, the strength of a material refers to the material's ability to resist an applied force. A material's strength is a function of engineering processes, and scientists employ a variety of strengthening mechanisms to alter the strength of a material. These mechanisms include work hardening, solid solution strengthening, precipitation hardening and grain boundary strengthening and can be quantified and qualitatively explained. However, strengthening mechanisms are accompanied by the caveat that mechanical properties of the material may degenerate in an attempt to make the material stronger. For example, in grain boundary strengthening, although yield strength is maximized with decreasing grain size, ultimately, very small grain sizes make the material brittle. In general, the yield strength of a material is an adequate indicator of the material's mechanical strength. Considered in tandem with the fact that the yield strength is the parameter that predicts plastic deformation in the material, one can make informed decisions on how to increase the strength of a material depending its microstructural properties and the desired end effect. Strength is considered in terms of compressive strength, tensile strength, and shear strength, namely the limit states of compressive stress, tensile stress and shear stress, respectively. The effects of dynamic loading is probably the most important practical part of the strength of materials, especially the problem of fatigue. Repeated loading often initiates brittle cracks, which grow slowly until failure occurs.

However, the term strength of materials most often refers to various methods of calculating stresses in structural members, such as beams, columns and shafts, when the equations of equilibrium are not sufficient to solve the problem. In such problems, known as statically indeterminate problems, the elastic or plastic resistance of the material to deformation must be considered when calculating stresses. In this sense, the word ``strength" could well be replaced by ``stiffness", but the usage goes back to at least 1930 and is not likely to go away any time soon.

1 Definitions
2 Stress-strain relations
3 Design terms
4 Suggested reading
5 Other fundamental engineering topics
6 External links

[edit] Definitions

[edit] Stress terms

A material being loaded in a) compression, b) tension, c) shear.

Uniaxial stress is expressed by

$\sigma=\frac{F}{A},$

where F is the force (N) acting on an area A (m^2). The area can be the undeformed area or the deformed area, depending on whether engineering stress or true stress is used.

Compressive stress (or compression) is the stress state when the material (compression member) tends to compact. A simple case of compression is the uniaxial compression induced by the action of opposite, pushing forces. Compressive strength for materials is generally higher than that of tensile stress, but geometry is very important in the analysis, as compressive stress can lead to buckling.

Tensile stress is a loading that tends to produce stretching of a material by the application of axially directed pulling forces. Any material which falls into the "elastic" category can generally tolerate mild tensile stresses while materials such as ceramics and brittle alloys are very succeptable to failure under the same conditions. If a material is stressed beyond its limits, it will fail. The failure mode, either ductile or brittle, is based mostly on the microstructure of the material. Some Steel alloys are examples of materials with high tensile strength.

Shear stress is caused when a force is applied to produce a sliding failure of a material along a plane that is parallel to the direction of the applied force. An example is cutting paper with scissors.

[edit] Strength terms

Yield strength is the lowest stress that gives permanent deformation in a material. In some materials, like aluminium alloys, the point of yielding is hard to define, thus it is usually given as the stress required to cause 0.2% plastic strain.

Compressive strength is a limit state of compressive stress that leads to compressive failure in the manner of ductile failure (infinite theoretical yield) or in the manner of brittle failure (rupture as the result of crack propagation, or sliding along a weak plane - see shear strength).

Tensile strength or ultimate tensile strength is a limit state of tensile stress that leads to tensile failure in the manner of ductile failure (yield as the first stage of failure, some hardening in the second stage and break after a possible "neck" formation) or in the manner of brittle failure (sudden breaking in two or more pieces with a low stress state). Tensile strength can be given as either true stress or engineering stress.

Fatigue strength is a measure of the strength of a material or a component under cyclic loading, and is usually more difficult to assess than the static strength measures. Fatigue strength is given as stress amplitude or stress range ( $Δσ = σ max - σ min$ ), usually at zero mean stress, along with the number of cycles to failure.

Impact strength, it is the capability of the material in withstanding by the suddenly applied loads in terms of energy. Often measured with the Izod impact strength test or Charpy impact test, both of which measure the impact energy required to fracture a sample.

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[edit] Strain (deformation) terms

Deformation of the material is the change in geometry when stress is applied (in the form of force loading, gravitational field, acceleration, thermal expansion, etc.). Deformation is expressed by the displacement field of the material.

Strain or reduced deformation is a mathematical term to express the trend of the deformation change among the material field. For uniaxial loading - displacements of a specimen (for example a bar element) it is expressed as the quotient of the displacement and the length of the specimen. For 3D displacement fields it is expressed as derivatives of displacement functions in terms of a second order tensor (with 6 independent elements).

Deflection is a term to describe the magnitude to which a structural element bends under a load.

[edit] Stress-strain relations

Elasticity is the ability of a material to return to its previous shape after stress is released. In many materials, the relation between applied stress and the resulting strain is directly proportional (up to a certain limit), and a graph representing those two quantities is a straight line.

The slope of this line is known as Young's Modulus, or the "Modulus of Elasticity." The Modulus of Elasticity can be used to determine stress-strain relationships in the linear-elastic portion of the stress-strain curve. The linear-elastic region is taken to be between 0 and 0.2% strain, and is defined as the region of strain in which no yielding (permanent deformation) occurs.

Plasticity or plastic deformation is the opposite of elastic deformation and is accepted as unrecoverable strain. Plastic deformation is retained even after the relaxation of the applied stress. Most materials in the linear-elastic category are usually capable of plastic deformation. Brittle materials, like ceramics, do not experience any plastic deformation and will fracture under relatively low stress. Materials such as metals usually experience a small amount of plastic deformation before failure while soft or ductile polymers will plasticly deform much more.

Consider the difference between a fresh carrot and chewed bubble gum. The carrot will stretch very little before breaking, but nevertheless will still stretch. The chewed bubble gum, on the other hand, will plasticly deform enormously before finally breaking.

[edit] Design terms

Ultimate strength is an attribute directly related to a material, rather than just specific specimen of the material, and as such is quoted force per unit of cross section area (N/m²). For example, the ultimate tensile strength (UTS) of AISI 1018 Steel is 440 MN/m². In general, the SI unit of stress is the pascal, where 1 Pa = 1 N/m². In Imperial units, the unit of stress is given as lbf/in² or pounds-force per square inch. This unit is often abbreviated as psi. One thousand psi is abbreviated ksi.

Factor of safety is a design constraint that an engineered component or structure must achieve. $F S = U T S / R$ , where FS: the Factor of Safety, R: The applied stress, and UTS: the Ultimate force (or stress).

Margin of Safety is also sometimes used to as design constraint. It is defined MS=Factor of safety - 1

For example to achieve a factor of safety of 4, the allowable stress in an AISI 1018 steel component can be worked out as $R = U T S / F S$ = 440/4 = 110 MPa, or $R$ = 110×10⁶ N/m².

[edit] Suggested reading

Mott, Robert L, "Applied Strength of Materials", 4th edition, Prentice-Hall, 2002, ISBN 0-13-088578-9
Beer F.P., Johnston E.R., et al, Mechanics of Materials, 3rd edition, McGraw-Hill, 2001, ISBN 0-07-248673-2
Timoshenko S., Strength of Materials, 3rd edition, Krieger Publishing Company, 1976, ISBN 0-88275-420-3
Drucker D.C., Introduction to mechanics of deformable solids, McGraw-Hill, 1967.
Shames I.H., Cozzarelli F.A., Elastic and inelastic stress analysis, Prentice-Hall, 1991, ISBN 1-56032-686-7
Den Hartog, Jacob P., Strength of Materials, Dover Publications, Inc., 1961, ISBN 0-486-60755-0
Popov, Egor P., Engineering Mechanics of Solids, Prentice Hall, Englewood Cliffs, N. J., 1990, ISBN 0-13-279258-3
Groover, Mikell P., Fundamentals of Modern Manufacturing, John Wiley & Sons,Inc., 2002, 2nd Ed. ISBN 0-471-40051-3
Lebedev, Leonid P. and Cloud, Michael.J., Approximating Perfection: A Mathematician's Journey into the World of Mechanics, Princeton University Press, 2004, ISBN 0-691-11726-8
Alfirević, Ivo, Strength of Materials I, Tehnička knjiga, 1995, ISBN 953-172-010-X
Alfirević, Ivo, Strength of Materials II, Tehnička knjiga, 1999, ISBN 953-6168-85-5
Hashemi, Javad and Smith, William F., Foundations of Materials Science and Engineering,McGraw-Hill, 2006, 4th Ed. ISBN 007-125690-3