Lecture #2
Stress and strain
- Stress analysis-Triaxial in Cartesian coordinates
- Stress Tensor.
-
WHAT IS A TENSOR?
Dr.Haydar Al-Ethari
References:
1- Barber J.R., (2004), Elasticity, 2nd Edition, Kluwer academic publishers.
2- Robert J. A., Vlado A. L., (2006), Mechanics of solids and materials,
Cambridge university press.
3- Hearn E.j., 1977, Mechanics of Materials, Vol.1&2., Pergamon Press,
London.
Stress analysis-Triaxial in Cartesian coordinates:
Equations of equilibrium:
In many cases, a general system of direct stress, shear and body forces
(gravitational force due to own weight, centrifugal force……) will produce
stresses of variable magnitude throughout a component. The distribution of
these stresses must always be in equilibrium, and it is a consideration of the
conditions necessary to produce this equilibrium which produces the stress
equations of equilibrium.
Element with body force stresses with other stresses in the x- direction
Stress on one face = xx
Stress on opposite face =
=
+ change in stress
xx + rate of change × distance between faces
xx
Therefore
Stress on opposite face = σ xx +
∂σ xx
dx
∂x
Multiplying by the area dy dz of the face produces the force in X- direction,
thus for equilibrium of forces:


∂
∂
∂




σ xx + ∂x σ xx dx − σ xx  dydz + σ xy + ∂y σ xy dy − σ xy  dxdz + σ xz + ∂z σ xz dz − σ xz  dxdy + Fx dxdydz = 0






(The body force term being defined as a stress per unit volume is multiplied
by the volume dx dy dz to obtain the corresponding force)
Dividing by dx dy dz and simplifying,
∂σ xx ∂σ xy ∂σ xz
+
+
+ FX = 0
∂x
∂y
∂z
Similarly, for equilibrium in Y and in Z directions:
∂σ YX ∂σ YY ∂σ yz
+
+
+ Fy = 0
∂x
∂y
∂z
∂σ zx ∂σ zy ∂σ yz
+
+
+ Fz = 0
∂x
∂y
∂z
The symbol in above equations may be replaced by ; the mixed suffix
denotes that it is a shear stress.
The moment of forces must also be in equilibrium. Consider the element
below which shows only the stresses which produces moments about Y-axis.
For convenient and to eliminate some terms the origin of the coordinate has
been chosen to coincide with the centroid of the element.
dz 
dz 
dz 
dz
∂
∂

τ xz + ∂z (τ xz ) 2  dxdy 2 + τ xz − ∂z (τ xz ) 2  dxdy 2 −




∂
dx 
dx 
dx 
dx
∂

τ zx + ∂x (τ zx ) 2  dydz 2 − τ zx − ∂x (τ zx ) 2  dydz 2 = 0




Dividing by (dxdydz) and simplifying get that xz = zx .
Similarly, by considering the moments about the X&Z axes,
zy = yz and
xy = yx
The nine Cartesian stress components thus reduces to six independent
values,
 σ xx σ xy σ xz 


σ yx σ yy σ yz 

σ
 zx σ zy σ zz 
≡
σ xx σ xy σ xz 


σ yy σ yz 


σ zz 

Stress Tensor
The state of stress at a point can be defined in terms of six stresses:
σ xx σ xy σ xz 
i = 1,2,3
σ ij =  . σ yy σ yz  where
j = 1,2,3
 .
. σ zz 
WHAT IS A TENSOR?
Tensor is a group of numbers that represents a physical quantity. Tensors
have special properties that make it easy to transform and manipulate
physical quantities. The stress tensor is symmetric about its main diagonal.