Strength of materials
Overview
 
In materials science
Materials science
Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

, the strength of a material is its ability to withstand an applied stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

 without failure. The applied stress may be tensile
Tensile strength
Ultimate tensile strength , often shortened to tensile strength or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract...

, compressive
Compressive strength
Compressive strength is the capacity of a material or structure to withstand axially directed pushing forces. When the limit of compressive strength is reached, materials are crushed. Concrete can be made to have high compressive strength, e.g...

, or shear
Shear strength
Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is...

. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a mechanical member will induce internal forces within the member called stresses. The stresses acting on the material cause deformation of the material.
Encyclopedia
In materials science
Materials science
Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

, the strength of a material is its ability to withstand an applied stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

 without failure. The applied stress may be tensile
Tensile strength
Ultimate tensile strength , often shortened to tensile strength or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract...

, compressive
Compressive strength
Compressive strength is the capacity of a material or structure to withstand axially directed pushing forces. When the limit of compressive strength is reached, materials are crushed. Concrete can be made to have high compressive strength, e.g...

, or shear
Shear strength
Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is...

. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a mechanical member will induce internal forces within the member called stresses. The stresses acting on the material cause deformation of the material. Deformation of the material is called strain, while the intensity of the internal forces is called stress. The strength of any material relies on three different type of analytical method: strength, stiffness and stability, where strength refers to the load carrying capacity, stiffness refers to the deformation or elongation, and stability refers to the ability to maintain its initial configuration. Material yield strength refers to the point on the engineering stress-strain curve (as opposed to true stress-strain curve) beyond which the material experiences deformations that will not be completely reversed upon removal of the loading. The ultimate strength refers to the point on the engineering stress-strain curve corresponding to the stress that produces fracture.

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
Strengthening mechanisms of materials
Methods have been devised to modify the yield strength, ductility, and toughness of both crystalline and amorphous materials. These strengthening mechanisms give engineers the ability to tailor the mechanical properties of materials to suit a variety of different applications. For example, the...

 that alter the strength of a material includes work hardening
Work hardening
Work hardening, also known as strain hardening or cold working, is the strengthening of a metal by plastic deformation. This strengthening occurs because of dislocation movements within the crystal structure of the material. Any material with a reasonably high melting point such as metals and...

, solid solution strengthening
Solid solution strengthening
Solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal. The technique works by adding atoms of one element to the crystalline lattice of another element . The alloying element diffuses into the matrix, forming a solid solution...

, precipitation hardening and grain boundary strengthening
Grain boundary strengthening
Grain-boundary strengthening is a method of strengthening materials by changing their average crystallite size...

 and can be quantitatively and qualitatively explained. 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 expressed in terms of compressive strength
Compressive strength
Compressive strength is the capacity of a material or structure to withstand axially directed pushing forces. When the limit of compressive strength is reached, materials are crushed. Concrete can be made to have high compressive strength, e.g...

, tensile strength
Tensile strength
Ultimate tensile strength , often shortened to tensile strength or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract...

, and shear strength
Shear strength
Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is...

, namely the limit states of compressive stress, tensile stress and shear stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

, respectively. The effects of dynamic loading are probably the most important practical consideration of the strength of materials, especially the problem of fatigue
Fatigue (material)
'In materials science, fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The nominal maximum stress values are less than the ultimate tensile stress limit, and may be below the yield stress limit of the material.Fatigue occurs...

. Repeated loading often initiates brittle
Brittle
A material is brittle if, when subjected to stress, it breaks without significant deformation . Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Breaking is often accompanied by a snapping sound. Brittle materials include most ceramics and glasses ...

 cracks, which grow until failure occurs. The cracks always start at stress concentration
Stress concentration
A stress concentration is a location in an object where stress is concentrated. An object is strongest when force is evenly distributed over its area, so a reduction in area, e.g. caused by a crack, results in a localized increase in stress...

s, especially changes in cross-section of the product, near holes and corners.

The study of the subject of strength of materials often refers to various methods of calculating stresses in structural members, such as beams, columns and shafts. The methods 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).

Types of loadings

  • Transverse loading - Forces applied perpendicular 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 the change in curvature of the member. Transverse loading also induces shear forces that cause shear deformation of the material and increase the transverse deflection of the member.
  • Axial loading - The applied forces are collinear with the longitudinal axes of the member. The forces cause the member to either stretch or shorten.
  • Torsional loading - Twisting action caused by a pair of externally applied equal and oppositely directed force couples acting on parallel planes or by a single external couple applied to a member that has one end fixed against rotation.

Stress terms

Uniaxial stress is expressed by
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 of interest.
  • Compressive stress (or compression) is the stress state caused by an applied load that acts to reduce the length of the material (compression member
    Compression member
    A compression member is a general class of structural elements of which a column is the most common specific example.-Description:For a compression member, such as a column, the principal stress comes mainly from axial forces, that is forces that fall along one line, usually the centerline.The...

    ) 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 their tensile strength. However, structures loaded in compression are subject to additional failure modes dependent on geometry, such as buckling
    Buckling
    In science, buckling is a mathematical instability, leading to a failure mode.Theoretically, buckling is caused by a bifurcation in the solution to the equations of static equilibrium...

    .

  • 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 shape of the cross section. Materials loaded in tension are susceptible to stress concentrations such as material defects or abrupt changes in geometry. However, materials exhibiting ductile behavior (most metals for example) can tolerate some defects while brittle materials (such as ceramics) can fail well below their ultimate material strength.

  • Shear stress
    Shear stress
    A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

    is the stress state caused by a pair of the built energy by opposing forces acting along parallel lines of action through the material, in other words the stress caused by faces of the material sliding relative to one another. An example is cutting paper with scissors or stresses due to torsional loading.

Strength terms

  • Yield strength
    Yield (engineering)
    The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...

    is the lowest stress that produces a permanent deformation in a material. In some materials, like aluminium alloy
    Aluminium alloy
    Aluminium alloys are alloys in which aluminium is the predominant metal. The typical alloying elements are copper, magnesium, manganese, silicon and zinc. There are two principal classifications, namely casting alloys and wrought alloys, both of which are further subdivided into the categories...

    s, the point of yielding is difficult to identify, thus it is usually defined as the stress required to cause 0.002% plastic strain. This is called a 0.002% proof stress.

  • Compressive strength
    Compressive strength
    Compressive strength is the capacity of a material or structure to withstand axially directed pushing forces. When the limit of compressive strength is reached, materials are crushed. Concrete can be made to have high compressive strength, e.g...

    is a limit state of compressive stress that leads to failure in the manner of ductile failure (infinite theoretical yield) or brittle failure (rupture as the result of crack propagation, or sliding along a weak plane - see shear strength
    Shear strength
    Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is...

    ).

  • Tensile strength
    Tensile strength
    Ultimate tensile strength , often shortened to tensile strength or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract...

    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 that failure, some hardening in the second stage and breakage after a possible "neck" formation) or brittle failure (sudden breaking in two or more pieces at a low stress state). Tensile strength can be quoted as either true stress or engineering stress.

  • Fatigue strength
    Fatigue (material)
    'In materials science, fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The nominal maximum stress values are less than the ultimate tensile stress limit, and may be below the yield stress limit of the material.Fatigue occurs...

    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 quoted as stress amplitude or stress range (), usually at zero mean stress, along with the number of cycles to failure under that condition of stress.

  • Impact strength, is the capability of the material to withstand a suddenly applied load and is expressed in terms of energy. Often measured with the Izod impact strength test
    Izod impact strength test
    Izod impact strength testing is an ASTM standard method of determining impact strength. A notched sample is generally used to determine impact strength....

     or Charpy impact test
    Charpy impact test
    The Charpy impact test, also known as the Charpy v-notch test, is a standardized high strain-rate test which determines the amount of energy absorbed by a material during fracture. This absorbed energy is a measure of a given material's toughness and acts as a tool to study temperature-dependent...

    , both of which measure the impact energy required to fracture a sample. Volume, modulus of elasticity, distribution of forces, and yield strength effect the impact strength of a material. In order for a material or object to have a higher impact strength the stresses must be distributed evenly throughout the object. It also must have a large volume with a low modulus of elasticity and a high material yield strength.


Strain (deformation) terms

  • Deformation of the material is the change in geometry created 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
    Strain (materials science)
    In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...

    or reduced deformation is a mathematical term that expresses the trend of the deformation change among the material field. Strain is the deformation per unit length. In the case of uniaxial loading - displacements of a specimen (for example a bar element) strain 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
    Tensor
    Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

     (with 6 independent elements).
  • Deflection
    Deflection (engineering)
    In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load and can be calculated by...

    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 is directly proportional to the resulting strain (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
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

, or the "Modulus of Elasticity." The Modulus of Elasticity can be used to determine the stress-strain relationship 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 easily identified on the stress-strain plot it is defined to be between 0 and 0.2% strain, and is defined as the region of strain in which no yielding (permanent deformation) occurs.
  • Plasticity
    Plasticity (physics)
    In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...

    or plastic deformation is the opposite of elastic deformation and is defined as unrecoverable strain. Plastic deformation is retained after the release 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 ductile metals such as copper and lead or polymers will plasticly deform much more.


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

Design terms

Ultimate strength is an attribute related to a material, rather than just a specific specimen made of the material, and as such it is quoted as the force per unit of cross section area (N/m²). The ultimate strength is the maximum stress that a material can withstand before it breaks or weakens. 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
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

, 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.

A Factor of safety
Factor of safety
Factor of safety , also known as safety factor , is a term describing the structural capacity of a system beyond the expected loads or actual loads. Essentially, how much stronger the system is than it usually needs to be for an intended load...

 is a design criteria that an engineered component or structure must achieve. , where FS: the factor of safety, R: The applied stress, and UTS: ultimate stress (psi or N/m^2)

Margin of Safety is also sometimes used to as design criteria. It is defined MS = Failure Load/(Factor of Safety * Predicted Load) - 1

For example to achieve a factor of safety of 4, the allowable stress in an AISI 1018 steel component can be calculated to be = 440/4 = 110 MPa, or = 110×106 N/m². Such allowable stresses are also known as "design stresses" or "working stresses."

Design stresses that have been determined from the ultimate or yield point values of the materials give safe and reliable results only for the case of static loading. Many machine parts fail when subjected to a non steady and continuously varying loads even though the developed stresses are below the yield point. Such failures are called fatigue failure. The failure is by a fracture that appears to be brittle with little or no visible evidence of yielding. However, when the stress is kept below "fatigue stress" or "endurance limit stress", the part will endure indefinitely. A purely reversing or cyclic stress is one that alternates between equal positive and negative peak stresses during each cycle of operation. In a purely cyclic stress, the average stress is zero. When a part is subjected to a cyclic stress, also known as stress range (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 stress range is below the material’s yield strength. Generally, higher the range stress, the fewer the number of reversals needed for failure.

Failure theories

There are four important failure theories, namely (1) maximum shear stress theory, (2) maximum normal stress theory, (3) maximum strain energy theory, and (4) maximum distortion energy theory. Out of these four theories of failure, the maximum normal stress theory is only applicable for brittle materials, and the remaining three theories are applicable for ductile
materials.
  • Maximum Shear stress Theory- This theory postulates that failure will occur in a machine part if the magnitude of the maximum shear stress (tmax) in the part exceeds the shear strength (typ) of the material determined from uniaxial testing.

This theory postulates, that failure will occur when,
tmax = typ
or max of [|S1-S2|/2 , |S2-S3|/2 , and |S3-S1|/2] = Syp/2
Dividing both side by 2,
max of [|S1-S2| , |S2-S3|, and |S3-S1|] = Syp
Using a design factor of safety Nfs, the theory formulates the design equation as,
max of [|S1-S2| , |S2-S3|, and |S3-S1|] should be less than or equal to Syp/Nfs
  • Maximum normal stress theory- this theory postulates, that failure will occur in machine part if the maximum normal stress in the part exceeds the ultimate tensile stress 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.

As the three principal stresses at a point in the part S1, S2, or S3 may be both tensile and compressive stresses, when this theory is applied, we need to check for failures both from tension and compression. The method of application of this theory is to find the maximum tensile stress, and the maximum compressive stress from the given values of S1, S2, and S3. The largest positive value among S1, S2, and S3 is the maximum tensile stress and the smallest negative value is the maximum compressive stress. For example if S1 = 80 MPa, S2 = -100 MPa, and S3 = -150 MPa, then the maximum tensile stress = 80 MPa, and the maximum compressive stress = -150 MPa (smallest negative value!).Thus according to this theory, the safe design condition for brittle material can be given by:
The maximum tensile stress should be less than or equal to Sut/Nfs and The magnitude of the maximum compressive stress should less than Suc/Nfs
  • 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.

Strain energy is the energy stored in a material due elastic deformation, which is, work done during elastic deformation. Work done per unit volume = strain x average stress. During tensile test, stress increases from zero to Syp, that is average stress = Syp/2.Elastic strain at yield point = Syp/E, where E is the elastic modulus of elasticity.Strain energy per unit volume during uniaxial tension = average stress x strain = Syp2/2E
  • 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.

Application of failure theory

Out of the four theories, only the maximum normal stress theory predicts failure for brittle materials.
The rest of the three theories are applicable for ductile materials. Out of these three, the distortion energy theory provides most accurate results in majority of the stress conditions.The strain energy theory needs the value of Poisson’s ratio of the part material, which is often not readily available. The maximum shear stress theory is conservative.
For simple unidirectional normal stresses all theories are equivalent, which means all theories will give the same result.

See also

  • Creep of materials
  • Deformation-mechanism maps
  • Diffusion in materials
  • Dynamics
    Dynamics (mechanics)
    In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...

    • Thermodynamics
      Thermodynamics
      Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

    • Fluid dynamics
      Fluid dynamics
      In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

  • Elasticity of materials
  • Fatigue of materials
  • Forensic engineering
    Forensic engineering
    Forensic engineering is the investigation of materials, products, structures or components that fail or do not operate or function as intended, causing personal injury or damage to property. The consequences of failure are dealt with by the law of product liability. The field also deals with...

  • Fracture mechanics
    Fracture mechanics
    Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.In...

  • Fracture toughness
    Fracture toughness
    In materials science, fracture toughness is a property which describes the ability of a material containing a crack to resist fracture, and is one of the most important properties of any material for virtually all design applications. The fracture toughness of a material is determined from the...

  • Heat transfer
    Heat transfer
    Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

  • Materials science
    Materials science
    Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

  • Material selection
    Material selection
    Material selection is a step in the process of designing any physical object. In the context of product design, the main goal of material selection is to minimize cost while meeting product performance goals. Systematic selection of the best material for a given application begins with properties...

  • Microstructures of materials
  • Plastics deformation in solids
  • Plasticity of materials
  • Schmidt hammer
    Schmidt hammer
    A Schmidt hammer, also known as a Swiss hammer or a rebound hammer, is a device to measure the elastic properties or strength of concrete or rock, mainly surface hardness and penetration resistance.It was invented by Ernst Schmidt, a Swiss engineer....

  • Specific strength
    Specific strength
    The specific strength is a material's strength divided by its density. It is also known as the strength-to-weight ratio or strength/weight ratio. In fiber or textile applications, tenacity is the usual measure of specific strength...

  • Statics
    Statics
    Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

  • Stress concentration
    Stress concentration
    A stress concentration is a location in an object where stress is concentrated. An object is strongest when force is evenly distributed over its area, so a reduction in area, e.g. caused by a crack, results in a localized increase in stress...

  • Strengthening mechanisms of materials
    Strengthening mechanisms of materials
    Methods have been devised to modify the yield strength, ductility, and toughness of both crystalline and amorphous materials. These strengthening mechanisms give engineers the ability to tailor the mechanical properties of materials to suit a variety of different applications. For example, the...



Further reading

  • Fa-Hwa Cheng, Initials. (1997). Strength of material. Ohio: McGraw-Hill
  • 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.
    J.E. Gordon
    James Edward Gordon was one of the founders of materials science and biomechanics, and a well known author of three books on structures and materials, which have been translated in many languages and are still widely used in schools and universities.-Biography:Gordon graduated in naval...

     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.
    Stephen Timoshenko
    Stanford University:* Bergman, E. O., * Kurzweil, A. C., * , * Huang, Y. S., * Wang, T. K., * Weber, H. S., * , * , * , -Publications:...

    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)
  • Davidge, R.W., Mechanical Behavior of Ceramics, Cambridge Solid State Science Series, (1979)
  • Lawn, B.R., Fracture of Brittle Solids, Cambridge Solid State Science Series, 2nd Edn. (1993)
  • Green, D., An Introduction to the Mechanical Properties of Ceramics, Cambridge Solid State Science Series, Eds. Clarke, D.R., Suresh, S., Ward, I.M. (1998)


External links

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