YIELDING%20AND%20RUPTURE%20CRITERIA%20(limit%20hypothesis)

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YIELDING AND RUPTURE CRITERIA (limit hypothesis)
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• The knowledge of stress and strain states and
  displacements in each point of a structure allows
  for design of its members. The dimensions of
  these members should assure functional and safe
  exploitation of a structure.

In the simplest case of uniaxial tension
(compression) it can be easily accomplished as
stress matrix is represented by one component ??1
only, and displacement along bar axis can be
easily measured to determine axial strain ?1
Measurements taken during the tensile test allow
also for determination of material
characteristics elastic and plastic limits as
well as ultimate strength. With these data one
can easily design tensile member of a structure
to assure its safety.
?
?expl
Ultimate strength
?explltltRm
Elastic limit
?explltRH
s-1
?expl ?1 RH /s
Safety coefficient
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• In the more complex states of stress (for example
  in combined bending and shear) the evaluation of
  safe dimensioning (related to elastic limit)
  becomes ambiguous.

Do we need to satisfy two independent conditions
?xlt RH
?xlt RH
where RHt i RHs denote elastic limits in tension
and shear, respectively?
Transformation to the principal axis of stress
matrix does not help either, as we do not know
whether the modulus of combined stresses is
smaller then RH
Thus, we need to formulate a hypothesis defining
which stress components should be taken as basis
for safe structure design.
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• In general case of 3D state of stress we
  introduce a function in 9-dimensional space of
  all stress components (or 3-dimensional in the
  case of principal axes) which are called the
  exertion function

In uniaxial sate of stress
We postulate that exertion function will take the
same value in given 3D state of stress as that in
uniaxial case.
The solution of this equation with respect to ?0
is called substitute stress according to the
adopted hypothesis defining function F and thus
function ?, as well.
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Let the exertion measure be
The ratio
gives the distance from unsafe state.
This distance can be dealt with as the exertion
in a given point.
SUCH A HYPOTHESIS DOES NOT EXIST !
A very similar one, which does exist
is called Gallieo-Clebsh-Rankine hypothesis
Associated ? function appears to bo
not-analytical one (derivatives on edges are
indefinable)
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GALIEO-CLEBSH-RANKINE hypothesis (GCR)
It is seen, that materials which obey this
hypothesis are isotopic with respect to their
strength.
They are also isonomic, as their strength
properties are identical for tension and
compression.
For plane stress state it reduces to a square.
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Exertion 100
Exertion 80
Exertion 60
Exertion 40
Exertion 0
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GALILEO hypothesis
Material isonomic and isotropic
Material insensitive to compression. (classical
Galileo hypothesis)
Material isotropic but not isonomic
where
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COULOMB-TRESCA-GUEST hypothesis CTG
For torsion
Many materials are sensitive to torsion
This hexagon represents Coulomb-Tresca hypothesis
(for plane stress state) the measure of exertion
is maximum shear stress
Uniaxial tension
In uniaxial state of stress
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HUBER-MISES-HENCKY hypothesis HMH
Small but important improvement has been made by
M.T. Huber followed by von Mises and Hencky
It is distortion energy only which decides on the
material exertion
For elastic materials (Hooke law obeys)
In uniaxial state of stress
In 3D space of principal stresses (Haigh space)
this hypothesis is represented by a cylinder with
open ends. In 2D plane stress state for
is an ellipse shown above.
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Hypothesis
GCR
CTG
HMH
Maximum normal stress
Maximum shear stress
Exertion measure
Deformation energy
Hexagonal prism with uniformly inclined axis
Circular cylinder with uniformly inclined axis
Cube with sides equal to 2R
3D image
2D image
Substitute stress
Substitute stress for beams
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