Post on 16-Apr-2017
05/03/2023 1
Anisotropy in sheet metals
ByDr. R. Narayanasamy, B.E.,M.Tech.,M.Engg.,Ph.D.,
(D.Sc.),Professor,
Department of Production Engineering,National Institute of Technology, Tiruchirappalli-
620 015 , Tamil Nadu, India.
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Yield Locus
• For a biaxial plane stress condition the Von Mises yield criterion can be expressed mathematically as: (Plot of this eqn is the Yield Locus)
is the equation of an ellipse. Major axis is and Minor axis is
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Yield Locus cont…
• For maximum shear stress condition – Yield locus falls inside of Von Mises Yield ellipse.
• (Uniaxial and balanced biaxial stress) predicts the same yield stress.
• For pure shear there occurs a greatest divergence
• Yield stress by Von Mises criterion 15.5% > Yield stress by maximum shear stress criterion.
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Comparison of yield criteria for plane stress (Fig 1)
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• Change in property values with direction of measurement.• Properties and amount of change depends on:
Structural origin of anisotropy Intensity with which it is developed
Properties are: Yield Strength (YS)Ultimate Tensile Strength (UTS)Percentage elongationReduction in areaUniform and total ductilityTrue stress at fractureNotched bar energy absorption.
Anisotropy in Yielding
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• Yield stress & ultimate tensile stress (under tension) are sensitive to structural anisotropy.
• Other properties depend on fibering of particles and weak interfaces.
• The intersection of (σ2 /σ1) = 1 (load path & yield locus) has physical significance.
Anisotropy in Yielding
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Imagine• A sheet is compressed through thickness
direction and yields at σy(3) . A hydrostatic tension (in all three directions) is applied to the yielding specimen.
• The magnitude of hydrostatic tension is σh = σy(3) (It does change the stress state but it will not yield).
Anisotropy in Yielding
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σ1
σ1 σ2
σ2σ3
σ3
-σ3
-σ3
σ1
σ1 σ2
σ2
Anisotropy in Yielding
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• By adding compressive & tensile hydrostatic stresses σh = σy(3) balanced biaxial tension can be produced (responsible for yielding of sheet metal).
• Yielding on (σ2 /σ1) = 1 load path is controlled by through thickness compression yield stress.
• Yield condition is given by: σ1 =σ2 = σy(3)
Anisotropy in Yielding
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• σy(3) changes independent of σy(1,2) (yield stress in direction 1 & 2 respectively).
• Anisotropy is described by – Normal anisotropy (&) Planar anisotropy
(which is related with crystallographic textures in rolled sheets)
– The texture is symmetrical around the normal sheet.
– Anisotropy has effect on yield locus.
Anisotropy in Yielding
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Anisotropy in Yielding
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• When σy(3) > σy(1,2) yielding is postponed to above uniaxial yield level. (This is texture hardening).
• When σy(3) < σy(1,2) the result is texture softening.• Empirical equation for anisotropic yield locus:
This equation follows from Hill’s general analysis of yielding in plastically anisotropic material.
Anisotropy in Yielding
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Anisotropy in Yielding
• Yielding criteria is considered if the material is isotropic.
• Material is no longer isotropic due to appreciable plastic deformation.
• Most fabricated metal shapes have anisotropic properties, so that it is likely that the tubular specimen used for basic studies of yield criteria incorporate some degree of anisotropy.
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Anisotropy in Yielding cont…
• Von-Mises’ criterion would not be valid for highly oriented cold-rolled sheet or a fiber-reinforced composite material.
• Hill’s formulated yield criterion for anisotropic material having orthotropic symmetry.
• where F,G,…..N are constants defining degree of anisotropy.
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Anisotropy in Yielding cont…
• For principal axes of orthotropic symmetry
• If X is yield stress in 1 direction, Y is yield stress in the 2 direction and Z is the yield stress in the 3 direction, then we can evaluate the constant as
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Anisotropy in Yielding cont…
• Lubahn and Felgar give detailed plasticity calculation for anisotropic behavoir
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Yield locus for textured titanium – alloy sheet (Fig 2)
ε = 0.002
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Anisotropy in Yielding cont…
• On a plane-stress yield locus as in fig 1, anisotropic yielding results in distortion of the yield locus.
• The yield locus is highly textured titanium alloy sheet (fig 2).
• The experimentally determined curve is non-symmetric when compared with ideal isotropic curve.
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Anisotropy in Yielding cont…
• An important aspect of yield anisotropy is texture hardening.
• Consider a highly textured sheet that is fabricated into a thin-wall pressure vessel, so that the thickness stress is negligible.
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Anisotropy in Yielding cont…
or • We can assume that the yield stress in the
plane of the sheet are equal, i.e. X=Y.•
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Anisotropy in Yielding cont…
• However the yield stress in the thickness direction of the sheet, Z, which is a difficult property to measure.
• This problem can be circumvented by measuring R value, (R = the ratio of the width strain to thickness strain.)
The yield locus equation is written as:
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Anisotropy in Yielding cont…
• The yield locus can be written as
• High through thickness yield stress Z results in low-thickness strain and high value of R.
• The extent of strength is seen in fig. 1 from the texture effect for spherical vessel
• The resistance to yielding (plastic deformation) increases with increased R.
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Anisotropy in Yielding cont…
Normal anisotropy
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Anisotropy – Dr.R.Sowerby Notes
• One of the simplest functions capable of description of initial anisotropy is the quadratic form
f= 0 ………….(1)• If for example, , is expressed by the following
fourth order isotropic tensor
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Anisotropy cont…
• = - , where the etc. are the Kvoneeker delta, then (1) reduces to the Von-Mises yield criterion.NB: The represents 81 coefficients but condition of symmetry reduces the independent coefficients.
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Anisotropy cont…
• Note from (1) by flow rule
since === ……….(b)• Plastic work increment =
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Anisotropy cont…
• Since is a scalar quantity the ijkl all dummy suffices and are interchangeable.
• So =………..(c)• From (a)-(c), ,• Then the 81 independent coefficients, can be
reduced to 21.
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Anisotropy cont…
• If it is now assumed that there exists three orthogonal principal axes of anisotropy such that:
i. The direct strain increments are independent of shear stress,
ii. The shear strain increment are independent of direct stresses,
iii. The shear strain increment depend on the corresponding shear stresses only;
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Anisotropy cont…
• Then by invoking the plastic incompressibility assumption the number of independent coefficients are reduced to 6.
• With 6 independent coefficient the yield condition can be expressed in the following form, due to Hill
++ + …………..(2)
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Anisotropy cont…
• The coefficients F, G, H etc. characterize the current state of anisotropy.
when L=M=N=3F=3G=3H,• The expression (2) reduces to the Von-Mises
yield criterion if F is equated to with y the yield stress in uniaxial tension.
• If f in (2) is taken as plastic potential then the plastic strain increments, referred to the plastic axes of anisotropy are
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Anisotropy cont…
• -+-• -+-• -+-
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Some theoretical consideration of the effect of Anisotropy
• We have already met Hill’s mode of an Anisotropy Yield criterion :-
++ + ………….…(18)
• If we consider only principal stresses acting in the principal direction of Anisotropy then (18) reduces to
=1 …………........…(19)
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Some theoretical consideration of the effect of Anisotropy cont…
• Applying the flow rule to either (19) or (18) (The form is unchanged for the direct strain increments)We obtain • [H(-)+G(-)] ……(a)• [F(-)+H(-)] ……(b)• [G(-)+F(-)] ……(c)
………………………………….(20)
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Some theoretical consideration of the effect of Anisotropy cont…
• Imagine preforming and tensile test in the ① direction (i.e. in the X- direction of anisotropy)
• Then = =0 Hence :=(G+H):-H:-G ………(21)• The ratio is referred to as the value . (the suffix ‘x’ implying the x- direction of anisotropy)
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Some theoretical consideration of the effect of Anisotropy cont…
• It is the ratio of the width strain increment to the thickness strain increment.
• In an isotropic material it would have the value of unity.
• It is usual to refer to the integrated form of (21) so that
=
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Some theoretical consideration of the effect of Anisotropy cont…
• (in practice however the R value could change during continuous loading)
• Similarly for tensile test in the Y-direction of anisotropy.
• :=(F+H):-H:-F ………(22)And = = ( if the integrated form of the strain is employed)• In general ≠ …………..(23)
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Plane Stress System
• This is appropriate to sheet metal forming operation and (say) is taken as zero.
• Equation (19) and (20) still hold of course with =0 .• The form of equation (21) and (22) are unchanged.• It is customary to identify the “Rolling Direction“
“Transverse Direction” and “Through Thickness Direction” of a sheet metal of material, as the principal direction of Anisotropy.
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Plane Stress System cont…
• Consequently a Tensile test conducted on a specimen cut from the plane of the sheet.
• But in the Rolling Direction is usually designated as “R” value of .
• The suffix 0 indicates to the rolling direction.• Similarly the would be the “R” value for a tensile
specimen cut at 90° to the rolling direction.• Again, in general,
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Planar Anisotropy
• If by performing experiments on the sheet it is found that the measured R value is unchanged with orientation , the sheet is said to have Planar Anisotropy.
• (if R=1 then the sheet is of course isotropic)
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Planar Anisotropy cont…
• Digression • It can be shown that the R value at any angle α
(measured anticlockwise from rolling direction) is:-
• ……….(24)• Planar isotropy occurs (according to this
theory)• When N=F+2H=G+2H (NB: F=G)
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Planar Anisotropy cont…
• In which case • R(α) always equals (or) …….(25)• Technological definition of planar anisotropy• You will some times see this defined as:-• ………………………..(26)
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Normal Anisotropy
• Already mentioned that a constant R value, independent of rotation, implies Planar Anisotropy.
• However for R≠1 the material is still Anisotropic.• A high R value(R>1) implies that the material has
a high resistance to thinning (A high through thickness strength)
• The reverse is true for R<1.
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Normal Anisotropy
• Consequently the R value(for a sheet with planar isotropy) is regarded as a measure of the Normal Anisotropy.
• A technological measure of Normal Anisotropy is often given as
• ͞b= …………………….(27)
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Normal Anisotropy cont…
• The recognition of the role that the R value can play in certain metal forming processes has lead to an increased activity in the role of “Texture Hardening”.
• The problem lies in first identifying the role ‘R’ might play and then suitably processing the sheet to produce the desired Texture.
• For example an increased in ‘R’ value tends to increase the deep drawing capabilities of sheet.
• All evidence up to now would suggest a similar trend for Stretch Forming Operation.
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Influence of the Anisotropic Parameters in the Shape of the Yield Surface for the Plane Stress Case
• With equation (19) reduces to
……………(23)• By altering the ratio of the parameters H,G& F
you can easily recognize influence.• This has on the shape of the yield surface ,
plotted in space.
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Representative Stress and Representative Strain Increment
• As with the isotropic case expression for representative stress and representative strain increment can be formulated.
• Written in terms of principal components and directed in the principal axes of Anisotropy, they are:-
• …(29)
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Representative Stress and Representative Strain Increment cont…
………..(30)
Where Q= (FG+GH+HF) When F=G=H (29)and (30) reduces to the Isotropic expression.
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy
• From (18) and the flow rule ……(a)…(b)……(c) ………… ….. (d) ……………....(e)d ……………….(f)……………………...(31)
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• From (a),(b) and (c) of (31) we may obtain• …….(a)• H…….(b)• …….(c) ………………………….(32) where Q=FG+GH+HF
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Square the equation (32)(a),(b) and (c) in turn and multiply by F,G and H respectively add to the sum
On L.H.S. Note(the addition to R.H.S through 31(d),(e)&(f))
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Verification of the and d ϵ̅ expression for Hill’s Model of Anisotropy cont…
• Using equation (18) we obtain
=…………….(33)
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Now choose a generalized (representative) yield stress, σ̅ , and generalized (representative) plastic strain increment dϵ̅ such that an increment of plastic work is given by
• And hence from (18) =…………………(34)
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• The choice of one representative quantity is Arbitrary.
• Select
……………(35)
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Hence from (33),(34) and (35)
…………..(36)
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Note the assumption for σ ̅~ reducing to the expression for the isotropic case when F=G=H.
It then follows that, so to will dϵ̅ .• Note also that the equation (29) and (30) are a
particular form of (35) and (36) where etc. where principal stresses in the principal direction of
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• We have already seen that conducting tensile test and measuring the resulting strain increments (or total strain) from the number of tensile tests.
• We can obtain a ratio of the anisotropic coefficient.
• See for example equation (21),(22) and (25).
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Similarly the coefficient are also related to the tensile stress properties for example using equation(18) or (19).
• And measure of the yield strength in the X(1),Y(2)and Z(3) direction we have
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• ……(a) • ……(b)• ……(c) ……………………(37)• Here etc. are the measured yield strengths.
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Hence if we know the coefficient(or their ratio) then we are a liberty to predict the yield strength in other direction if we know it in one direction.
• For example planar isotropy (with F=G) requires
• Thus knowing then is given by• …………………….(38)
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• Already mentioned that altering the coefficient s H,G and F will alter the shape of the yield locus .
• You can verify this for yourselves using equations (19) or (29), since one can establish a Geometric representation of the yield surface when plotted in principal stresses space.
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Verification of the and dϵ̅ expression for Hill’s Model of Anisotropy cont…
• The plot can be further simplified by considering the plane stress case (e.g. =0)
• For example using equation (29)(with =0) and setting H=0, F=G . The yield surface when plotted in is a circle.
• Putting F=G, but H>F elongates the yield surface along the axis bisecting the axis (i.e. along a 45° line) .
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Verification of the and dϵ̅expression for Hill’s Model of Anisotropy cont…
• The yield surface is still symmetric about this 45° line with F=G .
• If F≠G then the surface is not symmetric about this 45° line.
• Note also the influence of F,G and H on a Plane Strain loading line.
• Consider the case of , from (20) or (31 (b)) (with=0), then is given by
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Verification of the and dϵ̅expression for Hill’s Model of Anisotropy cont…
• With H=0 then along a line=0• If H>>F then along a line≃• i.e. almost balanced biaxial tension.• A high R value has already mentioned
indicates a high through thickness strength in theory one should be able to obtain a measure of by conducting a balanced biaxial test(i.e. =).
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Verification of the and dϵ̅expression for Hill’s Model of Anisotropy cont…
• For Planar Anisotropy• So ……….(39)• Where for example would be the R values in
the Rolling and Transverse directions of a rolled sheet and and are the rolling strengths of tensile specimen cut in the rolling and transverse directions.
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Verification of the and dϵ̅expression for Hill’s Model of Anisotropy cont…
• The expression for the yield strength at any angle measured α anticlockwise to the rolling direction is given by
…………………………..(40)• For the derivation of equation(40)(likewise
equation 24)~ which defines the R value at any angle α to the rolling direction.
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Verification of the and dϵ̅expression for Hill’s Model of Anisotropy cont…
• Therefore equal biaxial stresses σ in the plane of the sheet is the same as doing a compression (or tension) test (i.e.in the 3 directions).
• Hence looking at the ellipse on diagram the intersection the ellipse makes with the 45° direction (distance OC) measures the through thickness strength.
• This can be greater or less than OA or OB depending on the values of H,G and F.
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Analytical approach to show improvement in drawability with R value
• Assumption• Material has planar
isotropy i.e. F=G• Material does not strain
harden.• Flange does not deform
under condition of plane strain; no thickening in Z direction i.e.
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Analytical approach to show improvement in drawability with R value cont..
• From assumption (iii) using equation (20) with Z=3, 𝜃=2, r=1; then
• N.B. F=G ………(1)
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Analytical approach to show improvement in drawability with R value cont..
• Using the yield function and inserting this value of from (1)
• (say)……..(2)• Now the radial equilibrium condition for an
element in the flange is • …………………(3)
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Analytical approach to show improvement in drawability with R value cont..
• Using the value (2) in (3) and noting the material does not harden (assumption(ii)) gives
• …………..(4)• Now at i.e. outside of Flange
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Analytical approach to show improvement in drawability with R value cont..
• At inside of flange at then • ………………….(5)• In the absence of friction the punch load (P)
must be very nearly equal to
(t= material thickness)• From equation (5) • ………………..(6)
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Analytical approach to show improvement in drawability with R value cont..
• Now consider the element in the cup wall
• 0
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Analytical approach to show improvement in drawability with R value cont..
• Substitute this in the yield function with gives• …………………(7)
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Analytical approach to show improvement in drawability with R value cont..
• The drawing load (P) must also equal to ………………………..(8)
• Using (7) and (8)• ………………(9)• Equation (9) and (6) (the assumption that t is
the same in each of these equations neglects the possibility of the material thinning as it behaves over the profile radius).
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Analytical approach to show improvement in drawability with R value cont..
• Hence the equation(9) and(6) ...(10)• Using the value of A from (2) in (10)
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Analytical approach to show improvement in drawability with R value cont..
• Defining R as • ………………(11)• Equation (11) states that the bigger the R
value the bigger the ratio (which is the L.D.R)• There is a limit, as found in practice to how
increasing R effects the L.D.R. in actual fact the relationship is not
Limiting Draw Ratio in deep drawing.