Theories of failure
Namita Karna
Assistance Professor
Theories of failure
• Failure: Every material has certain strength, expressed in terms of stress or strain,
beyond which it fractures or fails to carry the load
• Material strength are determined from uni-axial tests.
• Thus, the strength obtained from those tension test cannot be directly used for
component design since, in actual scenarios component undergo multi-axial stress
conditions.
• Failure criterion: A criterion (standard/principle/measure/gauge/norm) is used to
hypothesize (imagine/assume/theory/visualize) the failure.
Why we need failure theories?
Failure Mode
• Yielding: a process of global permanent plastic deformation. Change in the
geometry of the object.
• Low stiffness: excessive elastic deflection.
• Fracture: a process in which cracks grow to the extent that the component breaks
apart.
• Buckling: the loss of stable equilibrium. Compressive loading can lead to bucking in
columns.
• Creep: a high-temperature effect. Load carrying capacity drops.
Theories of Failure
• Maximum principle stress theory (Rankine)
• Maximum shear stress theory (Tresca-guest )
• Maximum principle strain theory (St.Venants)
• Total strain Energy theory (Haigh’s Theory).
• Maximum distortion energy theory (Von-Mises and Hencky’s theory)
Maximum principle stress theory
• Designed for brittle material
• Can be used for ductile material in special occasion:
• Uni-axial or bi-axial loading(if principle stresses are of similar nature)
• Hydrostatic stress condition (no shear stress)
• Condition for safe design:
• Maximum principle stress (σ1) <= Permissible stress (σper)
• σ1 <= Synt/N or Sut/N
N- is the safety factor
Synt and Sut are yield strength and ultimate strength
Maximum principle stress theory
• Safe zone: Area inside the square
Maximum shear stress theory
• Suitable for ductile material as these are weak in shear.
• Gives over safe design which could be uneconomic sometimes
• Condition for safe design:
Maximum shear stress theory
• Safe zone: Area inside the square
Maximum principle strain theory (St.Venants)
Maximum principle strain theory (St.Venants)
Total strain Energy theory (Haigh’s Theory)
Total strain Energy theory (Haigh’s Theory)
Maximum distortion energy theory (VonMises and Hencky’s theory)
Maximum distortion energy theory (VonMises and Hencky’s theory)
Imperfection in solid
• A crystalline defect refers to a lattice irregularity having one or more of its
dimensions on the order of an atomic diameter.
• Classification of crystalline imperfections is frequently made according to
the geometry or dimensionality of the defect.
POINT DEFECT:VACANCIES AND SELFINTERSTITIALS
• The simplest of the point defects is a vacancy, or
vacant lattice site, one normally occupied but
from which an atom is missing.
• The necessity of the existence of vacancies is
explained using principles of thermodynamics;
in essence, the presence of vacancies increases the
entropy (i.e., the randomness) of the crystal.
• The equilibrium number of vacancies N𝜐 for a
given quantity of material (usually per meter
cubed) depends on and increases with
temperature according to
POINT DEFECT:VACANCIES AND SELFINTERSTITIALS
• N is the total number of atomic sites (most commonly
per cubic meter),
• Q𝜐 is the energy required for the formation of a
vacancy (J/mol or eV/atom),
• T is the absolute temperature in kelvins
• k is the gas or Boltzmann’s constant
• For most metals, the fraction of vacancies N𝜐/N just below the melting temperature
is on the order of 10−4 that is, one lattice site out of 10,000 will be empty.
POINT DEFECT:VACANCIES AND SELFINTERSTITIALS
• A self-interstitial is an atom from the crystal that is crowded into
an interstitial site—a small void space that under ordinary
circumstances is not occupied.
• In metals, a self-interstitial introduces relatively large distortions
in the surrounding lattice because the atom is substantially
larger than the interstitial position in which it is situated.
• Substitutional impurity atom, in which an atom of the parent
lattice lying in a lattice site is replaced by the impurity atom.
• Interstitial impurity atom, in which the impurity atom is at a
non-lattice site similar to the self-interstitial atoms referred to
above.
Miscellaneous Imperfections: Dislocation
• A dislocation is a linear or onedimensional defect around which some
of the atoms are misaligned.
Plastic Deformation
• The transition from elastic to plastic is a gradual one for most
metals; some curvature results at the onset of plastic
deformation, which increases more rapidly with rising stress.
• Plastic deformation is permanent, and strength and hardness
are measures of an material’s resistance to this deformation.
• On a microscopic scale, plastic deformation corresponds to the net
movement of large numbers of atoms in response to an applied
stress.
• In crystalline solids, plastic deformation most often involves the
motion of dislocations, linear crystalline defects.
Failure of material
• Simple fracture is the separation of a body into two or more pieces in response to an imposed
stress that is static (i.e., constant or slowly changing with time) and at temperatures that are low
relative to the melting temperature of the material.
• Fracture can also occur from fatigue (when cyclic stresses are imposed) and creep (time-dependent
deformation, normally at elevated temperatures).
• Although applied stresses may be tensile, compressive, shear, or torsional (or combinations of these).
• For metals, two fracture modes are possible: ductile and brittle.
• Ductile metals typically exhibit substantial plastic deformation with high energy absorption before
fracture. However, there is normally little or no plastic deformation with low energy absorption
accompanying a brittle fracture.
• Ductile fracture is almost always preferred to brittle fracture for two reasons:
Ductile
• Ductile fracture is almost always preferred to brittle fracture
for two reasons:
• First, brittle fracture occurs suddenly and catastrophically
without any warning; this is a consequence of the
spontaneous and rapid crack propagation. By contrast, in
ductile fracture, the presence of plastic deformation gives
warning that failure is imminent, allowing preventive
measures to be taken
• Second, more strain energy is required to induce ductile
fracture inasmuch as these materials are generally tougher.
Under the action of an applied tensile stress, many metal
alloys are ductile, whereas ceramics are typically brittle, and
polymers may exhibit a range of behaviors..
Ductile fracture
• found for extremely soft metals, such as pure gold and
lead at room temperature, and other metals, polymers,
and inorganic glasses at elevated temperatures
• fracture is preceded by only a moderate amount of
necking, small cavities, or micro-voids, form in the
interior of the cross section
• these micro-voids enlarge, come together, and coalesce
to form an elliptical crack, crack continues to grow in a
direction parallel to its major axis by this micro-void
coalescence process
• fracture ensues by the rapid propagation of a crack
around the outer perimeter of the neck
Ductile fracture
• Sometimes a fracture having this characteristic
surface contour is termed a cup-and-cone fracture
because one of the mating surfaces is in the form
of a cup and the other like a cone.
Ductile fracture
• Brittle fracture takes place without any appreciable deformation and by rapid crack propagation. The direction
of crack motion is very nearly perpendicular to the direction of the applied tensile stress and yields a
relatively flat fracture surface
• crack propagation corresponds to the successive and repeated breaking of atomic bonds along specific
crystallographic planes
• This type of fracture is said to be transgranular (or transcrystalline) because the fracture cracks pass through
the grains.
such a process is termed cleavage
Ductile fracture
• In some alloys, crack propagation is
along grain boundaries this fracture
is termed intergranular.
• This type of fracture normally
results subsequent to the occurrence
of processes that weaken or
embrittle grain boundary regions.
Brittle fracture
• The direction of crack motion is very nearly perpendicular to the direction of the applied tensile
stress and yields a relatively flat fracture surface,
Summary
• Discuss topics covered
• Reiterate welcome
• Wrap-up