Strength of materials

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Strength of materials is materials science applied to the study of engineering materials and their mechanical behavior in general (such as stress, deformation, strain and stress-strain relations). 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.

Contents 1 Definitions 1.1 Stress terms 1.2 Strength terms 1.3 Strain - deformation terms 2 Stress-strain relations 3 Design terms 4 Suggested reading 5 Other fundamental engineering topics 6 External links

[edit] Definitions

[edit] Stress terms

Stress is the force applied over an area.

• 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

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 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).

Fatigue strength is a measure of the strength of a component under repeated or cyclical loading, and is usually more important for structures than the two previous measures of strength. It is always lower than those values, and is critical in assessing safety factors, for example. When components fail, they always fracture in a brittle manner, irrespective of whether or not the material normally behaves in a brittle or ductile fashion. Fatigue failure occurs by growth of one or more brittle cracks from stress concentrations in the component.

[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 English 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. FS = UTS / 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 = UTS / FS = 440/4 = 110 MPa, or R = 110×10⁶ N/m².

[edit] Suggested reading

• 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

[edit] Other fundamental engineering topics

• Analysis of resistive circuits
• Dynamics
  • Thermodynamics
  • Fluid dynamics
• Engineering economics
• Heat transfer
• Materials science
• Statics

[edit] External links

• Failure theories
• U. Wisconsin-Stout, Strength of Materials online lectures, problems, tests/solutions, links, software