Lec 16 - YIELD CRITERION

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YIELD CRITERIA : 

YIELD CRITERIA

INTRODUCTION : 

INTRODUCTION 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. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible.

INTRODUCTION : 

INTRODUCTION Knowledge of the yield point is vital when designing a component since it generally represents an upper limit to the load that can be applied. It is also important for the control of many materials production techniques such as forging, rolling, or pressing. In structural engineering, this is a soft failure mode which does not normally cause catastrophic failure or ultimate failure unless it accelerates buckling.

YIELD POINT : 

YIELD POINT It is often difficult to precisely define yielding due to the wide variety of stress–strain curves exhibited by real materials. In addition, there are several possible ways to define yielding True elastic limit The lowest stress at which dislocations move. This definition is rarely used, since dislocations move at very low stresses, and detecting such movement is very difficult. Proportionality limit Up to this amount of stress, stress is proportional to strain (Hooke's law), so the stress-strain graph is a straight line, and the gradient will be equal to the elastic modulus of the material.

YIELD POINT : 

YIELD POINT Elastic limit (yield strength) Beyond the elastic limit, permanent deformation will occur. The lowest stress at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on equipment and operator skill. For elastomers, such as rubber, the elastic limit is much larger than the proportionality limit. Also, precise strain measurements have shown that plastic strain begins at low stresses. Offset yield point (proof stress) This is the most widely used strength measure of metals, and is found from the stress-strain curve as shown in the figure in next slide. A plastic strain of 0.2% is usually used to define the offset yield stress, although other values may be used depending on the material and the application. The offset value is given as a subscript, e.g. Rp0.2=310 MPa. In some materials there is essentially no linear region and so a certain value of strain is defined instead. Although somewhat arbitrary, this method does allow for a consistent comparison of materials.

YIELD POINT : 

YIELD POINT

YIELD POINT : 

YIELD POINT Upper yield point and lower yield point Some metals, such as mild steel, reach an upper yield point before dropping rapidly to a lower yield point. The material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value.

Yield Criteria : 

Yield Criteria 9 Parts / structure if uni-axially loaded are easy to be predicted for failure or otherwise if subjected to specific load e.g. elastic limit, yield point etc. Problem becomes complex when the parts are loaded bi – axially or tri – axially Here one has 2, 3 stresses to compare with single stress i.e. elastic limit, yield point etc etc. Decision? yield criteria help engineers

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12 So far, no universal method has been established that correlates failure in a uniaxial test with failure due to multiaxial loading Yield criteria, failure criteria or theories of failure attempt to answer the question! Can data obtained from a uniaxial tension or compression test be used to predict failure under more complex loadings? Yield Criteria

Yield Criteria : 

Yield Criteria 13 A yield criteria is a hypothesis defining the limit of elasticity in a material and the onset of plastic deformation under any possible combination of stresses. A yield criteria can be any descriptive statement that defines conditions under which yielding will occur. It may be ex-pressed in terms of specific quantities, such as the stress state, the strain state, a strain energy quantity or others.

Yield Criteria : 

Yield Criteria 14 A yield criteria is usually expressed in mathematical form by means of A yield function f(σij , Y) Where σij defines the state of stress Y ( also σY or σf )is the yield strength in uni-axial tension or compression The yield function is defined such that the yield criteria is satisfied when f(σij , Y) = 0 When f(σij , Y) < 0 , the stress state is elastic When f(σij , Y) >0 , the stress state is un-defined

YIELD SURFACE : 

YIELD SURFACE A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the surface itself may change shape and size as the plastic deformation evolves, this is because stress states that lie outside the yield surface are non-permissible.

YIELD SURFACE : 

YIELD SURFACE The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (σ1,σ2,σ3), a two- or three-dimensional space spanned by stress invariants (I1,J2,J3).

YIELD SURFACE : 

YIELD SURFACE Thus we may write the equation of the yield surface (that is, the yield function) in the forms Where σ is the Cauchy stress and σ1,σ2,σ3 are its principal values, is the deviatoric part of the Cauchy stress and s1,s2,s3 are its principal values.

YIELD SURFACE : 

YIELD SURFACE There are several different yield surfaces known in engineering, and those most popular are listed below. Tresca yield surface von Mises yield surface Mohr–Coulomb yield surface Drucker–Prager yield surface Bresler–Pister yield surface Willam–Warnke yield surface

Yield Surface : 

Yield Surface 19 Yield surface is a graphical representation of a yield function. The yield surface is plotted using a three dimensional stresses (principal stresses) σ1, σ2 , σ3 , as coordinates of the mutually perpendicular axes. In “Max Principle Stress Yield Criterion”, a right rectangular prism encloses all safe value of any combination of stress components. The compressive strength σc (also Sc or Yc) need not be equal to tensile strength σt (also St or Yt)

Yield Criteria : 

Yield Criteria 20 There are five hypotheses to help in deciding, whether the part is going to yield or not Maximum principal stress criteria (W Rankine’s criteria) Maximum principal strain criteria (Saint Venant’s Theory) Strain energy density criteria Maximum shear stress criteria (Tresca Criteria) Distorsion Energy Density criteria (Von Mises Criteria)

YIELD CRITERION : 

YIELD CRITERION A yield criterion, often expressed as yield surface, or yield locus, is an hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles.

YIELD CRITERION : 

YIELD CRITERION Since stress and strain are tensor quantities they can be described on the basis of three principal directions, in the case of stress these are denoted by . The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions). Other equations have been proposed or are used in specialist situations.

Max. Principal Stress Criteria(Max. Normal Stress theory) : 

Max. Principal Stress Criteria(Max. Normal Stress theory) 23 Maximum normal stress theory predicts failure of a specimen subjected to any combination of loads when maximum normal stress at any point in the specimen reaches the axial failure stress as determined by the axial tensile or compressive test for the same material Maximum normal stress theory states that failure occurs when ever one of the three principal stress equal the strength Maximum principal stress criteria states that yielding begins at a point in a member when the maximum principal stress reaches a value equal to the tensile or compressive yield stress Y (also SY or σY )

Max. Principal Stress Criteria(Max. Normal Stress theory) : 

Max. Principal Stress Criteria(Max. Normal Stress theory) 24 According to this theory, for the principal stresses, , when the criterion of failure is the yielding stress, the failure occurs whenever is equal to the tensile yield strength, or is equal to the negative value of the compressive yield strength. The yielding failure criterion was used mostly for ductile materials. For brittle materials, the ultimate strength is used as the criterion of failure, and this theory predicts that the failure occurs whenever is equal to the ultimate tensile strength, or is equal to the negative value of the ultimate compressive strength.

Yield Surface : 

Yield Surface 25

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26 Max. Principal Stress Criteria(Max. Normal Stress theory) By maximum principal stress criteria, if a point is subjected to both principal stress i.e. σ1 , σ2 and |σ1|> |σ2|, then yielding shall occur when σ1 = σY regardless of the fact that σ2 also acts at that point For 3D, suppose we arrange three principal stresses for any stress state in ordered form σ1> σ2> σ3 , then this theory predicts that failure occurs whenever σ1= yield strength in tensile = σt Or σ3= yield strength in compression = σc

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27 This can be expressed as yield function f = max (|σ1|, |σ2|, |σ3|)-Y Yield occurs when f = 0 We get following 6 equations for this criteria σ1 = ±Y , σ2 = ±Y , σ3 = ±Y Max. Principal Stress Criteria(Max. Normal Stress theory)

Max Principal Strain Criteria( Max Normal Strain Theory)(Saint Venant’s theory) : 

Max Principal Strain Criteria( Max Normal Strain Theory)(Saint Venant’s theory) 28 The maximum normal strain theory predicts failure of a specimen subjected to any combination of loads when the maximum normal strain at any point in the specimen reaches the failure strain (εf) σf / E at the proportional limit, as determined by an axial tensile or compressive test of the same material The maximum normal strain theory, also called Saint – Venant theory, applies only in the elastic range of stresses. This theory states that yielding occurs when the largest of the three principal strains become equal to the strain corresponding to the yield strength The maximum principal strain criteria, also known as saint- Venant criteria, states that yielding begins when the maximum principal strain at a point reaches a value equal to the yield strain εY = Y/E

Elastic Stress – Strain Relations : 

Elastic Stress – Strain Relations 29

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Incase of uni-axial strains Incase of bi-axial strains Eq. 2 & 4 gives four equations to generate the yield surface for max. principal strain criteria 30

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32 For tri-axial stress

Strain Energy Density Criteria(Max Strain Energy Theory) : 

Strain Energy Density Criteria(Max Strain Energy Theory) 33 The maximum strain energy theory predicts failure of a specimen subjected to any combination of loads when the strain energy per unit volume of any portion of the stressed member reaches the failure value of the strain energy per unit volume as determined by an axial tensile or compression test of the same material The strain energy density criteria, propose by Beltrami, states that yielding at a point begins when strain energy density at the point equals the strain energy density at yield in uniaxial tension or compression

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34 Strain energy is given by

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Maximum Shear Stress Criteria(Maximum Shear Stress theory) : 

Maximum Shear Stress Criteria(Maximum Shear Stress theory) 37 The maximum shearing stress theory predicts failure of a specimen subjected to any combination of loads when the maximum shearing stress at any point in the specimen reaches the failure shear stress ςf equal to σf/2 as determined by an axial tensile or compressive test of the same material The maximum shear stress theory states that yielding begins whenever the maximum shear stress in any element becomes equal to the maximum shear stress in a tension test specimen of same material when the specimen begins to yield The maximum shear stress criteria, also known as Tresca Criteria , states that yielding begins when the maximum shear stress at a point equals the maximum shear stress at yield in uniaxial tension or compression

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38 Max shear stress at a point is given by In case of uniaxial loading, max shear stress is given by Thus , the mathematical model for max shear stress criteria is

TRESCA YIELD SURFACE : 

TRESCA YIELD SURFACE Figure 1 shows the Tresca–Guest yield surface in the three-dimensional space of principal stresses. It is a prism of six sides and having infinite length. This means that the material remains elastic when all three principal stresses are roughly equivalent (a hydrostatic pressure), no matter how much it is compressed or stretched. However, when one of principal stresses becomes smaller (or larger) than the others the material is subject to shearing. In such situations, if the shear stress reaches the yield limit then the material enters the plastic domain.

TRESCA YIELD SURFACE : 

TRESCA YIELD SURFACE

TRESCA YIELD SURFACE : 

TRESCA YIELD SURFACE Figure 2 shows the Tresca–Guest yield surface in two-dimensional stress space, it is a cross section of the prism along the σ1,σ2 plane.

Distortion Energy Density Criteria(Maximum Distortion Energy Theory) : 

Distortion Energy Density Criteria(Maximum Distortion Energy Theory) 42 This theory differs from the maximum strain energy theory in that the portion of the strain energy producing volume change is considered in effective in causing failure by yielding. Supporting evidence comes from experiments showing that homogenous materials can withstand very high hydrostatic stresses without yielding. Therefore only the portion of the starin energy producing a change of shape is assumed to be responsible for the failure of the material by inelastic action

Hydrostatic and Deviatoric Stresses : 

Hydrostatic and Deviatoric Stresses

INTRODUCTION : 

INTRODUCTION Consider a block lying in 3-D space. For a given load, the faces of this block experience tensile stresses normal to the faces and shear stresses tangent to the faces. See Figure 1

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These stresses can be expressed as a matrix known as a stress tensor:

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The tensor can be broken down into two components: stresses that act on the block to change the volume and stresses that act to distort the volume. The former stresses are referred to as hydrostatic stresses and the latter as deviatoric stresses.

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Since the hydrostatic stresses maintain the original proportions of the volume, it makes sense that they would be equal. Thus the hydrostatic stresses are simply the mean of the principal stresses

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The deviatoric stresses consist simply of the hydrostatic stresses subtracted from the original stress tensor. The resulting matrix includes tensile stresses that elongate the volume as well as shear stresses that cause angular distortion—in other words, stresses that cause the volume to deviate from its original proportions.

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For a simple example, consider the cube in Figure, subject to an evenly distributed compressive load of 210 lb in the y-direction. Assume each face has an area of 1 in2. Assume the loading as uni-axial.

HYDROSTATIC STRESSES : 

HYDROSTATIC STRESSES It is apparent that the cube will compress in the y-direction. It is also apparent that the cube will expand in the x- and z-directions due to Poisson’s effect. The hydrostatic stresses on each face may be calculated as: Thus the hydrostatic stresses here are represented as 70 psi stresses on each face that act to compress the volume.

DEVIATORIC STRESSES : 

DEVIATORIC STRESSES Deviatoric stresses are simply Thus the contraction/expansion of the volume is represented as a 140 psi compressive stress in the y-direction and 70 psi tensile stress in the x and z-directions.

VON MISES STRESS : 

VON MISES STRESS The Von Mises Stress provides a measure of the shear, or distortional, stress in the material. This type of stress tends to cause yielding in metals. It is independent of the amount of hydrostatic stress (σ1= σ2= σ 3) action on the material. The Von Mises Stress is identified in terms of the principal stresses as σvm= √1/2[(σ1- σ2)2+(σ1- σ3)2+(σ2- σ3)2]

VON MISES STRESS : 

VON MISES STRESS In a state of pure tension, say σ11= σ and all other stresses are zero, then σ vm= σ. In a state of pure shear, say σ12= t and all other stresses are zero, then σvm=√3 t For materials, initial yielding can be expected when σvm= σy , where σy is the tensile yield stress, or when σvm=√3ty, where ty is the yield stress in shear. For other materials, particularly frictional materials such as soil and concrete, the Von Mises Stress may have no value in predicting yield or failure.

Distortion Energy Density Criteria(Maximum Distortion Energy Theory) : 

Distortion Energy Density Criteria(Maximum Distortion Energy Theory) 54 The distortion energy theory predicts that yielding will occur whenever the distortion energy in a unit volume equals the distortion energy in same volume when uniaxially stressed to the yielding strength The distortion energy density criteria, often attributed to Von Mises, states that yielding begins when the distortional strain energy density at a point equals the distortional strain energy density at yield in uniaxial tension or compression. Distortion strain energy density is that energy associated with a change in the shape of a body

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56 σ1 σ2 σ3 σ1-p σ2-p σ3-p p p p

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Yield Criteria : Mathematical Models : 

59 Yield Criteria : Mathematical Models Maximum principal stress criteria Maximum principal strain criteria (Saint Venant’s Theory) Strain energy density criteria Maximum shear stress criteria (Tresca Criteria) Distorsion Energy Density criteria (Von Mises Criteria) Lec5