Mechanics of materials, also called strength of materials, is a subject which deals with
the behavior of solid objects subject to stresses and strains. The complete theory began with
the consideration of the behavior of one and two dimensional members of structures, whose
states of stress can be approximated as two dimensional, and was then generalized to three
dimensions to develop a more complete theory of the elastic and plastic behavior of materials.
An important founding pioneer in mechanics of materials was Stephen Timoshenko.
The study of strength of materials often refers to various methods of calculating the stresses
and strains 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 takes into account the properties of the materials such as its yield strength,
ultimate strength, Young's Modulus, and Poisson's ratio; in addition the mechanical element's
macroscopic properties, such as it length, width, thickness, boundary constraints and
abrupt changes in geometry such as holes are considered.
Definition In materials science, the strength of a material
is its ability to withstand an applied load without failure. The field of strength of
materials deals with forces and deformations that result from their acting on a material.
A load applied to a mechanical member will induce internal forces within the member called
stresses when those forces are expressed on a unit basis. The stresses acting on the material
cause deformation of the material. Deformation of the material is called strain when those
deformations too are placed on a unit basis. The applied loads may be axial, or shear.
The stresses and strains that develop within a mechanical member must be calculated in
order to assess the load capacity of that member. This requires a complete description
of the geometry of the member, its constraints, the loads applied to the member and the properties
of the material of which the member is composed. With a complete description of the loading
and the geometry of the member, the state of stress and of state of strain at any point
within the member can be calculated. Once the state of stress and strain within the
member is known, the strength of that member, its deformations, and its stability can be
calculated. The calculated stresses may then be compared to some measure of the strength
of the member such as its material yield or ultimate strength. The calculated deflection
of the member may be compared to a deflection criteria that is based on the member's use.
The calculated buckling load of the member may be compared to the applied load. The calculated
stiffness and mass distribution of the member may be used to calculate the member's dynamic
response and then compared to the acoustic environment in which it will be used.
Material strength refers to the point on the engineering stress-strain curve beyond which
the material experiences deformations that will not be completely reversed upon removal
of the loading and as a result the member will have a permanent deflection. The ultimate
strength refers to the point on the engineering stress-strain curve corresponding to the stress
that produces fracture. 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 axis 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 is the stress state caused by an applied load that acts to reduce the
length of the material along the axis of the applied load, it is in other words a stress
state that causes a squeezing of 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, such as buckling, that are dependent on the member's
geometry. Tensile stress is the stress state caused
by an applied load that tends to elongate the material along 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 can
tolerate some defects while brittle materials can fail well below their ultimate material
strength. Shear stress is the stress state caused by
the combined energy of a pair of 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 its the lowest stress that produces a permanent deformation in a material.
In some materials, like aluminium alloys, the point of yielding is difficult to identify,
thus it is usually defined 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 failure in a material in the manner of ductile failure or brittle
failure. Tensile strength or ultimate tensile strength
is a limit state of tensile stress that leads to tensile failure in the manner of ductile
failure or brittle failure. Tensile strength can be quoted as either true stress or engineering
stress, but engineering stress is the most commonly used.
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 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
or Charpy impact test, both of which measure the impact energy required to fracture a sample.
Volume, modulus of elasticity, distribution of forces, and yield strength affect the impact
strength of a material. In order for a material or object to have a high 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 terms Deformation of the material is the change
in geometry created when stress is applied. Deformation is expressed by the displacement
field of the material. Strain 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 the displacements of a
specimenlead to a calculation of strain expressed as the quotient of the displacement and the
original length of the specimen. For 3D displacement fields it is expressed as derivatives of displacement
functions in terms of a second order tensor. Deflection is a term to describe the magnitude
to which a structural element is displaced when subject to an applied 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, 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 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
occurs. Plasticity 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. The ultimate strength is the maximum stress that a material can withstand
before it breaks or weakens. For example, the ultimate tensile strength 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.
A Factor of safety 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
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,
it has been observed that the failure of the part occurs after a number of stress reversals
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:
maximum shear stress theory, maximum normal stress theory, maximum strain energy theory,
and 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. Of the latter 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. Maximum Shear stress Theory- This theory postulates
that failure will occur 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 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.
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. Fracture mechanics was established by Alan
Arnold Griffith and George Rankine Irwin. This important theory is also known as numeric
conversion of toughness of material in the case of crack existence.
Fractology was proposed by Takeo Yokobori because each fracure laws including creep
rupture criterion must be combined nonlinially. Microstructure
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 quantitatively and
qualitatively explained. Strengthening mechanisms are accompanied by the caveat that some other
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 the limiting values of the compressive
stress, tensile stress, and shear stresses that would cause failure. The effects of dynamic
loading are probably the most important practical consideration of the strength of materials,
especially the problem of fatigue. Repeated loading often initiates brittle cracks, which
grow until failure occurs. The cracks always start at stress concentrations, especially
changes in cross-section of the product, near holes and corners at nominal stress levels
far lower than those quoted for the strength of the material.
See also
References
Further reading
External links Failure theories
case studies in structural failure