Strength of materials

In materials science, the strength of a material is its 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 microstructure. 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 are 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).

1 Types of loadings
2 Definitions
3 Stress-strain relations
4 Design terms
- 4.1 Failure theories
5 See also
6 References
7 Further reading
8 External links

Types of loadings

Transverse loading - Forces applied perpendicularly to the longitudinal axis of a member. Transverse loading causes the member to bend and deflect from its original position, with internal tensile and compressive strains accompanying change in curvature.
Axial loading - The applied forces are collinear with the longitudinal axes of the members. The forces causes the member to either stretch or shortens.
Torsional loading - Twisting action caused by a pair of externally applied equal and oppositely directed couples acting in a parallel planes or by a single external couple applied to a member that has one end fixed against rotation.

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 pair of 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. This is called a 0.2% proof stress.

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 ( $\Delta\sigma= \sigma_\mathrm{max} - \sigma_\mathrm{min}$ ), 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 either below the yield point, or if a yield point is not well defined for the material, taken to be between 0 and 0.2% strain, and is defined as the region of strain in which no yielding (permanent deformation) occurs.^[1]

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 plastically 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: ultimate tensile stress (psi or N/m^2)

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.

Design for cyclic loading working stress that have been determined from the ultimate or yield point values of the materials divided by a factor of safety give safe and reliable results only for static loading. Many machine parts are subjected to a non steady and continuously varying stress. Failures in such machine parts are usually caused by repeated loadings and at stresses that are below the yield point. When the part is below fatigue or endurance limit value, it will last indefinitely. However, the fatigue failures due to bending are the most common. These failures usually take place across the crystals. Failure is by fracture or separation without visible yielding or distortion. A purely reversing or cyclic stress means when the stress alternates between equal positive and negative peak stresses during each cycle of operation. Cyclic stress over time can be represented b a sinusoidal curve. In pure cyclic stress, the average stress is zero. When a part is subjected to cyclic stress, also known as range stress (Sr), it has been observed that the failure of the part occurs after a number of stress reversals (N) even if the magnitude of the range stress is below the material’s yield strength. Generally, higher the range stress, lesser number of reversals is needed for failure.

Failure theories

Maximum Shear stress Theory- this theory postulates, that failure will occur in machine part if the magnitude of the maximum shear stress in the part exceeds the shear strength of the material determined from uniaxial testing.

Maximum normal stress theory- this theory postulates, that failure will occur in machine part if the maximum normal stress in the part exceeds the shear strength of the material as determined from uniaxial testing. This theory deals with brittle materials only. The maximum tensile stress should be less than or equal to ultimate tensile stress divided by factor of safety. The magnitude of the maximum compressive stress should be less than ultimate compressive stress divided by factor of safety.

Maximum strain energy theory-this theory postulates that failure will occur when the strain energy per unit volume due to the applied stresses in a part equals the strain energy per unit volume at the yield point in uniaxial testing.

Maximum distortion energy theory- this theory is also known as shear energy theory or von Mises-Hencky theory. This theory postulates that failure will occur when the distortion energy per unit volume due to the applied stresses in a part equals the distortion energy per unit volume at the yield point in uniaxial testing. The total elastic energy due to strain can be divided into two parts. One part causes change in volume, and the other part causes change in shape. Distortion energy is the amount of energy that is needed to change the shape.

References

↑ Beer & Johnston (2006). Mechanics of Materials (5th ed.). McGraw Hill. pp. 53–56. ISBN 978-0-07-352938-7.