ENG5312 – Mechanics of Solids II
25
Material Property Relationships
Generalized Hooke’s Law

Assumptions:
o Isotropic, homogeneous material;
o The principal of superposition applies, i.e. linear-elastic material
behaviour with small deflections.

Assume a material is exposed to three normal stresses  x ,  y and  z ,
which will result in three normal strains  x ,  y and  z .

Using Hooke’s Law, the strain in the
  x
 'x  
E

 ''x  





 will be normal strains in the
Due to Poisson’s ratio there
to  y and  z :


x -direction
due to  x is:


y
E
;  '''x  
x -direction due
z

E
Using superposition, the total normal strain in the x -direction is the sum
of the components due to stresses  x ,  y and  z :


1
 x   x    y   z 
E
 


 Similarly, in the y and z -directions:



1
 y    x   z 
E
(36b)
z 
1
 z    x   y 
E
(36c)
Note: A rectangular block will remain rectangular due to the application of
 x ,  y and  z since no shear strain exists.

 
y 


(36a)

ENG5312 – Mechanics of Solids II

From experiment, it has been found that the application of a shear stress
 xy will cause an element to deform due to shear strain  xy only.
Therefore, Hooke’s Law for shear stress and strain can be written as:
 xy 


26
 xy
G
;  yz 
 yz
G
;  xz 

 xz
(37)
G
The generalized form of Hooke’s Law Eqs. (EQREF) can be used to
determine three-dimensional deformations due to an applied stress field,
or stress field
 due to given
 strains. 
Relationships between E,  and G

The derivation requires use of the principal stresses, generalized Hooke’s
Law, and the strain transformation equations:

G
(38)

Dilation

E
21  
Consider an element ( dx,dy,dz ) exposed to a stress field (  x ,  y and
 z ). The deformed dimensions of the element will be (1  x )dx ,
(1  y )dy and (1  z )dz , and the change in volume ( V ) of the element
will be:





V  1  x 1  y 1  z dxdydz dxdydz




Since the strains are small, the strain products can be neglected, and V
can be written as:


V   x   y   z dxdydz

Division of V by the volume dV gives the dilation ( e ) or volumetric
strain, i.e. the change in volume per unit volume.



V
e
 x  y  z
 dV

(39)
ENG5312 – Mechanics of Solids II

Using the generalized Hooke’s Law the dilation can be written in terms of
the applied stresses:
e

27
1 2
 x   y   z 
E
(40)
The dilation can be used to determine changes in volume due to an
applied stress field.

Bulk Modulus

Consider an element subjected to uniform pressure on all sides (e.g. a
block submerged in water), then the normal stresses  x   y   z  p
(i.e. compressive stress due to applied pressure p), and using Eq. (40):
p
E

k
e 3(1 2 ) 

The LHS (normal stress/dilation) is similar to  /   E , therefore, the RHS
is called a bulk modulus, k .

Note: For most metals,  1/ 3, therefore, k  E . For a material that does
not change volume, k , therefore,
 the maximum theoretical value of 
is 0.5. During yielding
a material does not change volume, therefore,

  0.5 is used for plastic yielding.



(41)



ENG5312 – Mechanics of Solids II
28
Theories of Failure

To determine if a specific design will fail it is first necessary to determine
the maximum normal and shear stresses that occur in a member (using
mechanics fundamentals and stress concentration factors). Then the
principal stresses must be determined at these locations in the member.
Then an appropriate failure theory for biaxial or tri-axial stress field can be
applied.
Ductile Materials

For a uni-axial state of stress, the failure of a ductile material is usually
specified by the initiation of yielding.

Two failure theories exist for a ductile material exposed to multi-axial
stress fields: 1) maximum-shear-stress theory; and 2) maximum-distortionenergy theory.
1) Maximum-Shear Stress Theory

If a ductile material is subjected to a tension test it yields along slip
planes oriented at  45o to the axis of the applied load. Using
Mohr’s circle for an element in the tension test sample:


At 90 o to the applied normal stress (here the principal axis) on
Mohr’s circle we get the maximum shear stress. Therefore, the
material is failing in shear at 45 o .
  Henri Tresca used this result to determine a failure theory applied
to the failure of ductile materials subjected to any loading.

ENG5312 – Mechanics of Solids II

29
The maximum-shear-stress theory (or Tresca yield criterion) states:
yielding of a ductile material begins when the maximum shear
stress in the material reaches the shear stress in the material that
causes yielding in a simple axial tension test, or:
max
 abs


Y
2
Where  Y is determined from a tension test.
max

The maximum shear-stress-theory
requires  abs
for the applied
state of stress, therefore, the principal stresses are required.
 Remember, if the principal stresses have the same sign then  max
abs
occurs out of plane:



max
 abs
 max
2

max
And if the principal stresses have opposite signs  abs
occurs in
plane:

max
 abs


 max   min

2
The maximum-shear-stress theory can be stated as follows when
 1 and  2 have the same sign:



 1   Y or  1   Y
(42a)
 or as follows when  1 and  2 have opposite signs:
 1 
 2  Y




(42b)
ENG5312 – Mechanics of Solids II

30
Plot the point (  1 , 2 ). If it falls within the hexagon the material will
not fail. If the point falls on the boundary or outside the hexagon,
the material will fail.

2) Maximum-distortion-energy theory

A material stores energy internally as it is deformed by an external
loading. The strain-energy density ( u) is the energy stored pre unit
volume, and when a material is subjected to uni-axial stress:


1
u  
2
Consider an element subjected to the three principal stresses  1,  2
and  3 , then using superposition the total strain-energy density is:

1
1
1
u   11   2 2   3 3  
2
2
2


And using Hooke’s Law, Eq. (36):
1
u    12   12   12  2  1 2   1 3   2 3 
2E


(43)
The strain-energy density can be perceived as being composed of a
part due to volume change, and another part due to change of shape.

Defining
 avg  ( 1   2   3 ) / 3, then  avg which acts on all faces of
an element will cause the same principal strains in all directions (i.e.
 avg causes a change in volume but no change in shape). The




ENG5312 – Mechanics of Solids II
31
remaining portion of the stress will cause the change in shape, i.e.
( 1   avg ) , ( 2   avg ) and ( 3   avg ).

Experiments have shown that materials do not yield when subjected to
uniform stress (e.g.  avg above). Therefore, it was proposed that a
 material willyield when the distortion energy per unit volume of
ductile
a material subjected to simple tension, i.e. maximum-distortion-energy
theory.

To determine the distortion energy, substitute ( 1   avg ) , ( 2   avg )
and ( 3   avg ) into Eq. (43):


ud 



1 
2
2
2
 1   2    2  3    3  1 
6E
ud 




ud Y


1  2
Y
3E
For the maximum-distortion-energy theory, ud  u d Y , therefore:


1  2
 1   1 2   22
3E
For a uni-axial tension test  1   Y ,  2   3  0 :


(44)
For plane stress  3  0 and:



 12   1 2   22   Y2
Which is the equation of an ellipseon the  1 , 2 axes:


(45)
ENG5312 – Mechanics of Solids II
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Brittle Materials


In a uni-axial stress field the failure of brittle materials is specified by
fracture.

In tension tests, brittle materials fail when the normal stress reaches the
ultimate stress (  ult ).

In torsion tests, brittle materials fail due to a maximum tensile stress (at
45 o ) to the applied shear, and the maximum tensile stress is

approximately
the same as the normal stress required for failure in a
tension test.
1) Maximum-normal stress theory

This theory states that a brittle material will fail when the maximum
principal stress (sigma1) reaches the ultimate normal stress the
material can endure in a pure tension test.
 1   ult
 2   ult

Graphically


2) Mohr’s failure criterion
(46)
ENG5312 – Mechanics of Solids II
33

Applies when the tension and compression properties of a brittle
material differ.

Tests performed to determine: 1)  ult t in a tension test; 2)  ult c in
a compression test; and 3)  ult in a torsion test.

Three Mohr’s circles are drawn: 1)  1   2  0 ,  3   ult c ;

2)  1   ult t ,  
2   3  0 ; and 3)  ult .







Construct a failure envelope, i.e. a curve tangent to the three circles.

To check if a material will fail: draw the Mohr’s circle for the state of
stress. If the circle is contained within the failure envelop there will be
no failure. If the circle is tangent to, or crosses the failure envelope
then failure will occur.
ENG5312 – Mechanics of Solids II
34