Strength of materials

In materials science, the strength of a material refers to the material's ability to withstand an applied stress without failure. Yield strength refers to the point on the engineering stress-strain curve (as opposed to true stress-strain curve) beyond which the material begins deformation that cannot be reversed upon removal of the loading. Ultimate strength refers to the point on the engineering stress-strain curve corresponding to the maximum stress. The applied stress may be tensile, compressive, or shear.

A material's strength is dependent on its microstructure. The engineering processes to which a material is subjected can alter this microstrucure. The variety of strengthening mechanisms that alter the strength of a material includes 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 some 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. The methods that can be employed to predict the response of a structure under loading and its susceptibility to various failure modes may take into account various properties of the materials other than material (yield or ultimate) strength. For example failure in buckling is dependent on material stiffness (Young's Modulus).

Definitions

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 [m²]. 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 caused by an applied load that acts to reduce the length of the material (compression member) in the axis of the applied load, in other words the stress state caused by squeezing the material. 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. However, structures loaded in compression are subject to additional failure modes dependent on geometry, such as Euler buckling.

• Tensile stress is the stress state caused by an applied load that tends to elongate the material in the axis of the applied load, in other words the stress caused by pulling the material. The strength of structures of equal cross sectional area loaded in tension is independent of cross section geometry. Materials loaded in tension are susceptible to stress concentrations such as material defects or abrupt changes in geometry. However, materials exhibiting ductile behavior(metals for example) can tolerate some defects while brittle materials (such as ceramics) can fail well below their ultimate stress.

• Shear stress is the stress state caused by a opposing forces acting along parallel lines of action through the material, in other words the stress caused by sliding faces of the material relative to one another. An example is cutting paper with scissors.

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 thats 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 ($\Delta \sigma =\sigma _{\mathrm {max} }-\sigma _{\mathrm {min} }$), 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.

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.

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 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.

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. $FS=UTS/R$, 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.

Suggested reading

• Mechanics of Materials , E.J. Hearn
• 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.
• Ashby, M.F. Materials Selection in Design. Pergamon, 1992.
• Beer, F.P., E.R. Johnston, et al. Mechanics of Materials, 3rd edition. McGraw-Hill, 2001. ISBN 0-07-248673-2
• Cottrell, A.H. Mechanical Properties of Matter. Wiley, New York, 1964.
• Den Hartog, Jacob P. Strength of Materials. Dover Publications, Inc., 1961, ISBN 0-486-60755-0.
• Drucker, D.C. Introduction to Mechanics of Deformable Solids. McGraw-Hill, 1967.
• Gordon, J.E. The New Science of Strong Materials. Princeton, 1984.
• Groover, Mikell P. Fundamentals of Modern Manufacturing, 2nd edition. John Wiley & Sons,Inc., 2002. ISBN 0-471-40051-3.
• Hashemi, Javad and William F. Smith. Foundations of Materials Science and Engineering, 4th edition. McGraw-Hill, 2006. ISBN 007-125690-3.
• Hibbeler, R.C. Statics and Mechanics of Materials, SI Edition. Prentice-Hall, 2004. ISBN 013-129-011-8.
• Lebedev, Leonid P. and Michael J. Cloud. Approximating Perfection: A Mathematician's Journey into the World of Mechanics. Princeton University Press, 2004. ISBN 0-691-11726-8.
• Mott, Robert L. Applied Strength of Materials, 4th edition. Prentice-Hall, 2002. ISBN 0-13-088578-9.
• Popov, Egor P. Engineering Mechanics of Solids. Prentice Hall, Englewood Cliffs, N. J., 1990. ISBN 0-13-279258-3.
• Ramamrutham, S. Strength of Materials.
• Shames, I.H. and F.A. Cozzarelli. Elastic and inelastic stress analysis. Prentice-Hall, 1991. ISBN 1-56032-686-7.
• Timoshenko S. Strength of Materials, 3rd edition. Krieger Publishing Company, 1976, ISBN 0-88275-420-3.
• Timoshenko, S.P. and D.H. Young. Elements of Strength of Materials, 5th edition. (MKS System)

See also

• Dynamics
  • Thermodynamics
  • Fluid dynamics
• Forensic engineering
• Heat transfer
• Materials science
• Statics
• Strengthening mechanisms of materials
• Stress-strain relations
• Microstructures of materials
• Elasticity of materials
• Plasticity of materials
• Creep of materials
• Fracture toughness
• Fatigue of materials
• Diffusion in materials
• Deformation-mechanism maps
• Material selection
• Specific strength

External links

• Failure theories
• U. Wisconsin-Stout, Strength of Materials online lectures, problems, tests/solutions, links, software
• case studies in structural failure
• or http://en.sopromat.org Strength of Materials online organization. Beams calculation online.