Plasticity General Formulation and Yield

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Plasticity- General Formulation and Yield

• MEEN 5330
• Sandeep Akula
• Bhargava Kunapareddi

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Introduction to Plasticity

• Definition Plasticity is a property of a
  material to undergo a non reversible change of
  shape in response to an applied force. Plastic
  deformation occurs under shear stress, as opposed
  to brittle fractures which occur under normal
  stresses
• Examples Clay and Mild steel
• In the theory of plasticity, the primary concern
  are with the mathematical formulation of stress-
  strain relationships suitable for description of
  plastic deformation and with the establishment of
  appropriate yield criteria for predicting the
  onset of plastic behavior.
• The basic concepts of plasticity can be explained
  in a elementary way by considering the
  stress-strain diagram for a simple one
  dimensional tension or compression test of some
  material.

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BASIC DEFINITIONS
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Generalized Hookes Law

• In the 1-D case, for a linear elastic material
  the stress s is proportional to the strain e,
  that is sEe, where the proportionality factor E
  is called modulus of elasticity, which is a
  property of the material.
• The relation sEe is known as Hookes law.
• Since we consider that the continuum material is
  a linear elastic material, we introduce the
  generalized Hookes Law in Cartesian coordinates.
• Where

Stiffness Tensor of the material of fourth order
Linear Strain
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Stress Strain Diagram

• Stress strain diagram is a plot between stress
  and strain .This shows the behavior of the
  material under a tensile or compressive load.
• The point P separates the graph in to elastic and
  plastic regions. Point P is called the yield
  point until which a one to one relation exists
  between stress and strain.
• J is know as the Johnson's apparent elastic limit
  where the slope of the curve is 50 of its
  initial value.
• In the plastic range unloading from point B
  results in path BC which is parallel to linear
  elastic portion. At C where stress is zero
  permanent plastic strain remains. The
  recoverable elastic strain from B is labeled as
  . A reloading from C back to B will follow
  path BC. From B a load increase will cause
  further deformation called work or strain
  hardening .
• In a plastic range the stress depends upon the
  entire loading, or strain history of the
  material. Plastic deformations are considered to
  be isothermal, time independent, separate from
  creep and relaxation.

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Idealized plastic behavior

• The theory for analyzing plastic behavior may be
  looked upon as generalization of certain
  idealizations of 1-D stress strain curve.
• The four most commonly used of these idealized
  stress strain diagrams are shown if fig. with a
  simple mechanical model of each.
• In these models the displacement of the mass
  depicts the plastic deformation and the force F
  plays the role of the stress.
• In fig a. elastic response and work hardening is
  absent and in fig b. elastic response prior to
  yield is included but not work hardening. In the
  absence of work hardening the plastic response is
  called perfectly plastic. Fig a and Fig b. are
  useful for studying contained plastic
  deformations.
• In Fig c. elastic response is omitted and the
  work hardening is assumed to be linear. This
  representation is useful for studying uncontained
  plastic flow.
• The stress strain curves considered here are
  tension curves. The compression curve for a
  previously unworked specimen is taken as
  reflection w.r.t origin of tension curve. If a
  stress reversal is carried out with a material
  that has been work hardened , a lowering of yield
  stress is observed in the 2nd type of
  loading.This is know as Bauschinger effect.

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Yield Condition

• Yield condition is a mathematical relationship
  among the stress components at a point that must
  be satisfied for the onset of plastic behavior at
  that point.
• In general, the yield condition may be expressed
  by the equation
• Where
• For an isotropic material the yield condition
  must be independent of direction and may
  therefore be expressed as a function of the
  stress invariants, or alternatively, as a
  symmetric function of the principal stresses.
• Thus it may appear as
• Where

Yield Function
Yield Constant
Principal Stresses
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Yield Conditions

• Of the numerous yield conditions which have been
  proposed, two are reasonably simple
  mathematically and yet accurate enough to be
  highly useful for the yield of isotropic
  materials.
• These are
• Tresca yield condition (Maximum Shear
  Theory)
• Von Mises yield condition (Distortion Energy
  Theory)

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Tresca Yield Condition

• This condition asserts that yielding occurs when
  the maximum shear stress reaches the prescribed
  value .
• This condition is expressed in its simplest form
  when given in terms of principal stresses.
• Thus for Tresa yield condition
  is given as

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Tresca Yield Conditions

• Tresca Yield Condition (Simple Tension)
• To relate the yield constant to the yield
  stress in simple tension the maximum shear in
  simple tension at yielding is observed to be
  .
• Therefore when referred to the yield stress in
  simple tension, Trescas yield condition becomes
  
• Tresca Yield Condition (Pure Shear)
• If the pure shear yield point value is k, the
  yield constant equals k .
• Then the yield condition becomes

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Mohrs Circle Simple tension Pure shear
(a) Simple Tension
(b) Pure Shear
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Von Mises Yield Condition (Distortion Energy
Theory)

• This condition asserts that yielding occurs when
  the second deviator stress invariant attains a
  specified value.
• Mathematically, the von Mises yield condition in
  terms of principal stresses is

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Von Mises Conditions in Simple Tension and Pure
Shear

• With reference to the yield stress in simple
  tension Von Mises condition becomes
• With respect to the pure shear yield value k, Von
  Mises condition becomes

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PLASTIC STRESS-STRAIN EQUATIONS.

• Once plastic deformation is initiated, the
  constitutive equations of elasticity are no
  longer valid. The plastic strains depend upon the
  entire loading history of the material.
• So these relations very often are given in terms
  of strain increments- the so called incremental
  theories.
• By neglecting the elastic portion and by assuming
  that the principal axes of strain increment
  coincide with the principal stress axes, the
  Levy-Mises equations relate the total strain
  increments to the deviatoric stress components
  through the equations.
• Where

Plastic Strain Increment
stress Deviator
Proportionality factor
NOTE Here appears in differential form
to emphasize that incremental strains are being
related to stress components. The factor
may change during loading and is therefore a
scalar multiplier and not a fixed constant. The
above equations represent the flow rule for a
rigid-perfectly plastic material.
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PLASTIC STRESS-STRAIN EQUATIONS

• The strain increment is split into elastic and
  plastic portions according to
• The plastic strain increments related to the
  stress deviator components is given by
• The resulting equations are know as
  Prandtl-Reuss equations. These equations
  represent the flow rule for an elastic-perfectly
  plastic material. They provide a relationship
  between the plastic strain increments and the
  current stress deviators but do not specify the
  strain increment magnitudes.

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EQUIVALENT STRESS AND EQUIVALENT PLASTIC STRAIN
INCREMENT

• With regard to the mathematical formulation of
  strain hardening rules, the equivalent or
  effective stress is given as
• In compact form it is written as
• The equivalent or effective plastic strain
  increment is defined by
• Which may be written in compact form as

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Example Problem 1

• For the state of stress
  Produced
  in a tension- torsion test of a thin-Walled tube,
  derive the yield curves in the plane
  for the Tresca and Von Mises Conditions if the
  yiled stress in Simple tension is ?
• A. For the given state of stress the principal
  stress values are
  and as shown
  by the Mohrs diagram in the below Fig. Thus from
  Tresca yield condition in simple tension the
  yield curve is or
  an ellipse in the
  plane. Like wise from von Mises stress in simple
  tension the Mises yield curve is the ellipse
  . The Tresca and Mises yield ellipse
  for this case are compared in the plot shown in
  Fig. below.

Plot
Mohrs Circle
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Example Problem 2

• 2. Show that the Prandtl-Ruess equations imply
  that principal axes of plastic strain increments
  coincide with principal stress axes and express
  the equations in terms of the principal stresses?
• A.
• From the form of Prandtl-Ruess equations,
  when referred to a coordinate system in which
  the shear stresses are zero, the plastic shear
  strain increments are seen to be zero also. In
  the principal axes system, the above equation
  becomes .
• Thus etc.,
  and by subtracting we get
• 
  Where

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HOME WORK PROBLEM

• Q. For the case of plastic plane strain with
  and ,show that the
  Levy-Mises equations lead to the conclusion that
  the Tresca and Mises yield conditions (when
  related to pure shear yield stress k) are
  identical?

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References

• 1. R.S.Khurmi,Machine Design, S.Chand and Co.
  Ltd, Chapter 2, Page 13(1996)
• 2. S.Ramamrutham and R.Narayanan, Strength of
  materials, Dhanpat Rai Sons, Chapter 1, Pages
  1-99(1995)
• 3. George E.Mase, Continuum Mechanics, Schaums
  outlines, Chapter 8, Pages 176-184(2004)
• 4. R.Hill, The mathematical theory of
  plasticity, Oxford at the Claredon Press, Page
  1(1971)
• 5. W.Johnson and P.B.Mellor, Engineering
  Plasticity, Van Nostrand Reinhold Company, page
  13(1973)

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