# Strength of materials

Strength of materials is the scientific area of applied mechanics for the study of the strength of engineering materials and their mechanical behaviour 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.

 Contents

## Definitions

### Stress terms

Stress is the internal distribution of forces within a body that balances and reacts to the loads applied to it. It is a complicated tensor quantity that can be broken down into simpler elements for engineering purposes;

• Compressive stress (or compression) is the stress state when the material tends to compact (volume decrease). A simple case of compression is the uniaxial compression induced by the action of opposite, pushing forces. Most materials can carry compressive stress, even the granulars such as sands.
• Tensile stress is a loading that tends to produce stretching on a material by the application of axially directed pulling forces. Materials can withstand some tensile loading, but if enough force is applied, they will eventually break into two parts. Steel is an example of a material 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 e.g. when cutting paper with scissors or a steel bolt with a bolt cutter.

### Strength terms

Compressive strength is a limit state of compressive stress that leads to compressive failure in the manner of ductile failure (infinite theoretically yield) or in the manner of brittle failure (rupture as the result of crack propagation, or sliding among 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).

### 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 loadings - displacements of a specimen (for example a bar element) it is expressed as the quotient of the displacement and the length of the speciment. For 3D displacement fields it is expressed as derivates of displacement functions in terms of a second order tensor (with 6 independent elements).

## Stress - strain relations

Elasticity is the linear response of materials in terms of stress and strain as described by Hooke's law (Sometimes elasticity has non-linear character as the recoverable stress-strain relation is a non linear-function). The simpler form of Hooke's law is the spring relation: F=k*Δx where [itex]k[itex] is the spring constant. Elasticity describes the state where the work offered by the application of external agents (forces), is stored in the material in form of elastic energy and it is recovered in form of displacement when external agents are removed (see Solid mechanics).

Plasticity is the non-linear response of materials in terms of stress and strain. Plastic behaviour includes the irrevesible transformation of work offered by the application of external agents (forces) to forms of energy such as thermal energy or crack propagation-growth. When the agents are removed, the deformation remains. Plastic behaviour is described by "Flow rules" such as the differential relationships between stress state, stress change and strain change.

Viscosity is the non-linear time dependent response of materials in terms of stress and strain. The most known form of viscosity in solid mechanics is creep. Viscosity in solids may include elastic deformation (Viscoelasticity) or/and plastic deformation (Viscoplasticity).

## 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 ([itex]N/mm^2[itex]). For example, Ultimate Tensile Strength (UTS) of mild steel is [itex]470 N/mm^2[itex]. It is useful to remember that [itex]1 MPa = 1 N/mm^2[itex].

Factor of safety is a design constraint that an engineered component or structure must achieve. [itex]FS = UTS/R[itex], where FS: the Factor of Safety, R: The acting force (or stress) and UTS: the Ultimate force (or stress).

For example to achieve a factor of safety of 4, the allowable stress in a mild steel component can be worked out as [itex]R = UTS/FS = 117.5 MPa[itex].

• Beer F.P., Johnston E.R., et al, Mechanics of Materials, 3rd edition, McGraw-Hill, 2001, ISBN 0072486732
• Timoshenko S., Strength of Materials, 3rd edition, Krieger Publishing Company, 1976, ISBN 0882754203
• 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 1560326867
• Den Hartog, Jacob P., Strength of Materials, Dover Publications, Inc., 1961, ISBN 0486607550
• Popov, Egor P., Engineering Mechanics of Solids,Prentice Hall, Englewood Cliffs, N. J., 1990, ISBN: 0132792583
• Groover, Mikell P., Fundamentals of Modern Manufacturing, John Wiley & Sons,Inc., 2002, 2nd Ed. ISBN 0-471-40051-3

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