Lect07 Failure Modes

MCEN90029 Advanced Mechanics of Solids Lecture L07 - 1

Lecture L07 Failure modes

MCEN90029

Advanced Mechanics of Solids

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Summary Over the next two lectures we will cover yield criteria for

brittle and ductile materials These include yield criterion such as Tresca, Von Mises

and Colomb-Mohr theories We will demonstrate that different yield criteria lead to

different failure modes


Strength theories Stress-strain curve for a material obtained by subjecting a

test specimen to axial tensile force Initial stress-strain relationship linear. After elastic limit,

material acquires inelastic or permanent deformation


Strength theories For example, if a tensile specimen made of a ductile

material is loaded to failure in tension, fractures will be at an angle of 45 to the load axis Thus, shear stress contributes more to failure, and a maximum

shear stress criterion may be suitable


Yield criteria Failure: occurs at initiation of inelastic material behaviour Plastic theory

Yield criterion (yield initiation) Flow rule (stress/strain after yield) Hardening (change in yield strength due to plastic

strain) Yield criterion: predicting initiation of yielding using a given

criterion Yield criterion often a mathematical function:

f (ij ,Y )

ij = state of stressY = yield strength

where

when

f (ij ,Y ) = 0 yield occurs

f (ij ,Y ) < 0 stress state is elastic

f (ij ,Y ) > 0 post-yield deformation


Yield criteria There are many theories of static failure which

can be postulated for which the consequences can be seen in a tensile test. 1. Maximum-principal-stress theory 2. Maximum-principal strain theory (St. Venants

criterion) 3. Strain-energy density criterion 4. Maximum-shear-stress theory (Tresca theory) 5. Maximum distortion-energy theory (Von Mises

theory) 6. Colomb-Mohr theory for brittle materials


1. Maximum principal stress criterion Often called Rankines criterion Yielding begins when the maximum principal stress

is equal to the uniaxial tensile (or compressive) yield stress Y. Occurs when 1 reaches Y

If two principal stresses 1 and 2 (|1| > |2|) both act at a point, yielding is predicted when 1 = Y, regardless of 2

If 1 = -2, the shear stress is equal in magnitude to and occurs on 45 diagonal planes (e.g torsion).

Thus, if this criterion (1 = = Y) is to be valid for a given material, the shear yield stress Y of the material must be equal to the tensile yield stress

For ductile materials, Y is much less than the tensile yield stress Y!


1. Maximum principal stress criterion For brittle materials that fail by brittle fracture rather

than yielding, the maximum principal stress criterion may adequately predict tension fracture

The maximum principal stress criterion can be expressed by the yield function:

f =max(1 ,2 ,3 ) YThe effective stress is:

e =max(1 ,2 ,3 )The corresponding yield surface is defined by the stress states that satisfy the yield criterion (f = 0), hence:

1 = Y, 2 = Y, 3 = Y, The yield surface consists of 6 planes, perpendicular to the principal stress coordinate axes

1

1

-1

-1

2 /u

1 /u


2. Maximum principal strain criterion Often called St. Venants criterion Yielding begins when the maximum principal strain

is equal to the yield strain of a material in tension

For the simple block, yielding occurs when which corresponds to

Under biaxial stress, max principal strain is:

Y =Y /E

1 = Y

=Y

1 = (1 /E) v( 2 /E)For this stress state, yielding will begin when 1 > Y. If 2 is negative (compressive), max value of 1 that can be applied without yield will be less than Y


2. Maximum principal strain criterion For an isotropic material, the principal strain (in

terms of the principal stresses), in the 1 direction, is, from Hookes law:

1 =1E (1 v 2 v 3)

Assuming that 1 is the principal strain of largest magnitude, we equate |1| with Y to obtain the yield function:

f1 = 1 v2 v3 Y = 0 or 1 v2 v3 = Y

If we dont know the magnitudes of the principal strains, the other possibilities are:

f2 = 2 v1 v3 Y = 0 or 2 v1 v3 = Yf3 = 3 v1 v2 Y = 0 or 3 v1 v2 = Y

1 =1E (1 v 2 v 3)


2. Maximum principal strain criterion Hence, the effective stress e may be defined as:

and the yield function as:

e = max i jk i v j vk

f1 =e Y

Hence, the effective stress e may be defined as:

and the yield function as:

The yield surface for the maximum principal strain for biaxial stress state (1 = 0) is as follows. For the yield surface ABCD, individual stresses greater than Y can occur without causing yielding


3. Strain-energy density criterion

For normal stress x applied to an element, the element increases length in x-dir, and decreases length in the y- and z-dir. A new length in any direction, in terms of normal strains is:

# x = x +xx # y = y +yy # z = z +zz

From Hookes law:

x = xE

From Poissons ratio for isotropic materials, contractions in y and z directions are equal:

y = z = vx = v xE

Strains caused by y and z are:

Yielding at a point begins when the strain-energy density at the point equals the strain energy density at yield in uniaxial tension (or compression).

Recall, strain energy density:

x = z = vy = v yE

x = y = vz = v zE

y = yE

z = zE


3. Strain-energy density criterion Thus, for an element undergoing x, y, z simultaneously, the effect of each stress can be added using the concept of linear superposition:

x =1E x v( y + z)[ ]

y =1E y v( z + x )[ ]

z =1E z v( x + y )[ ]

Now, for a uniaxial stress x, the force due to x is Fx = xyz, which displaces in a linear manner the amount x=xx. The total work on the element is:

W = 12 Fxx =12 x x (xyz)

The work per unit volume w is determined by dividing the work by the volume xyz

w = 12 x x Nm/m3


3. Strain-energy density criterion If the material is perfectly elastic, total work increases the potential energy (strain energy) of the volume. Thus:

u = w = 12x xIn the elastic region, . The resulting strain energy per unit vol is:

x =x /E

u = 12E x2

If the normal stresses y, z are also present,

u = 12x x +12y y +

12z z

Substituting the normal strain relationship derived previously, the strain energy per unit volume is:

x =1E x v( y + z)[ ]

y =1E y v( z + x )[ ]

z =1E z v( x + y )[ ]

u = 12E x2 + y

2 + z2 2v( x y + y z + z x )[ ] (1)



The strain-energy density at yield in uniaxial tension where 1 = Y, 2 = 3 = 0 is:

U0 =12E 1

2 +22 +3

2 2v(12 +13 +23)[ ] > 0

Yielding at a point begins when the strain-energy density at the point equals the strain energy density at yield in uniaxial tension (or compression). Written in terms of principal stresses, from equation (1):

U0Y =Y 22E

Thus, the strain-energy density criterion states that yield is initiated when the strain energy density U0=U0Y.

For uniaxial tension, yielding is predicted to occur when 1 = Y

For a biaxial stress state, when 1 = 2 = , yielding is predicted to occur when

22(1 v) =Y 2

(2)



12 +2

2 +32 2v(12 +13 +23) Y 2 = 0

The yield function for the strain-energy density criterion is obtained by setting U0 from equation (2) equal to U0Y (the strain-energy density at yield):

f =e2 Y 2Hence the yield function has the form

Where the effective stress is

e = 12 +2

2 +32 2v(12 +13 +23)

In general, the yield surface for the strain-energy density criterion is an ellipsoid in principal stress space. The specific shape depends on Poissons ratio v. (Left, a bi-axial stress state, 3 = 0 )


4. Tresca (maximum shear stress) theory for ductile materials

Yielding begins when the maximum shear stress at a point equals the maximum shear stress at yield in uniaxial tension (or compression).

For a multiaxial stress state, the maximum shear stress is max = (max - min)/2, where max and min denote the maximum and minimum order principal stresses

In uniaxial tension, 1 = , 2 = 3 = 0, the maximum shear stress is

Since yield in uniaxial tension must begin when = Y, the shear stress associated with yielding is predicted to be

max =2

Y =Y2

Thus the yield function for max shear stress criterion is:

f = e Y2 Where the effective stress is

e = max



From the 3-dimensional Mohrs circle, the magnitudes of the extreme values of the shear stresses in the principal coordinate system (the radius of the three Mohrs circles) are

The maximum shear stress max is the largest of (1, 2 , 3). If the principal stresses are unordered, yielding under multiaxial stress can occur for any one of the following conditions

The Tresca criterion exhibits good agreement with experimental results for certain ductile metals. For pure shear (e.g. torsion), the shear yield stress of some ductile metals is found to be 15% higher than that predicted by Tresca criterion (Tresca criterion is conservative)

1 = 2 3

2 ; 2 = 3 1

2 ; 3 =1 2

2

2 3 = Y; 3 1 = Y; 1 2 = Y

Yield surface



EXAMPLE: Determine the yield stress using the Tresca criterion of an arbitrary three dimensional state of stress at a point given by. Assume plane-stress

[ ] =xx xy zxxy yy yzzx yz zz

#

$

% % %

&

'

( ( (

Let N be the unit normal to the principal plane, N = li + mj + nk where (l,m,n) are directional cosines of the unit normal N

SOLUTION

The projection of this stress vector on principal planes is P = N

Px = lxx +mxy + nxzPy = lxy +myy + nyzPz = lxz +myz + nzz

l(xx ) +mxy + nxz = 0lxy +m(yy ) + nyz = 0lxz +myz + n(zz ) = 0

Thus, (1)



Find the solution to equation (1). To avoid zero solution to directional cosines (as l2 + m2 + n2 = 1), the determinant of the coefficients must be zero. Thus,

xx xy xzxy yy yzxz yz zz

= 0

Evaluating the determinant:

3 (xx +yy +zz)2 + (xxyy +yyzz +zzxx yz2 zx2 xy2 ) (xxyyzz + 2yzzxxy xxyz2 yyzx2 zzxy2 ) = 0

The three solutions to this cubic equation are the three principal stresses, 1, 2, 3

If we assume plane stress, zz= yz= zx= 0 this equation reduces to:

3 (xx +yy )2 + (xxyy xy2 ) = 0



The three roots to the equation are = 0 and

3 (xx +yy )2 + (xy xy2 ) = 0

=xx +yy2

xx yy2

$

% &

'

( )

2

+xy2

With the principal stresses determined, order the principal stresses such that 1> 2> 3, The yield criterion for the Tresca criterion is

max =max min

2 = 3 12


Lecture summary

Today we covered: Maximum-principal-stress theory Maximum-principal strain-energy theory (St. Venants

criterion) Strain-energy density criterion Maximum-shear-stress theory (Tresca theory)

In the next lecture we will discuss Von Mises theory for ductile materials, and the Colomb-Mohr theory for brittle materials

Lect07 Failure Modes

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