1 06 - concept of stress
06 - concept of stress
holzapfel �nonlinear solid mechanics� [2000], chapter 3, pages 109-129
2 06 - concept of stress
06 - concept of stress
holzapfel �nonlinear solid mechanics� [2000], chapter 3, pages 109-129
3 06 - concept of stress
me338 - syllabus
4 06 - concept of stress
definition of stress stress [‘stres] is a measure of the internal forces acting within a deformable body. quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. these internal forces arise as a reaction to external forces applied to the body. because the loaded deformable body is assumed to behave as a continuum, these internal forces are distributed continuously within the volume of the material body, and result in deformation of the body's shape.
• cauchy‘s postulate
5 06 - concept of stress
cauchy‘s postulate
stress vector t to a plane with normal n at position x only depends on plane‘s normal n
augustin louis caucy [1789-1857] Px
6 06 - concept of stress
cauchy‘s lemma
augustin louis caucy [1789-1857]
• cauchy‘s lemma
newton‘s third law actio = reactio
t n
-t -n
7 06 - concept of stress
cauchy‘s theorem
augustin louis caucy [1789-1857]
existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n
X2 , x2
X3 , x3
X1 , x1
t2
t1
t3
t
• cauchy‘s theorem
• cauchy‘s postulate
8 06 - concept of stress
cauchy
augustin louis caucy [1789-1857]
• cauchy‘s lemma
• cauchy‘s theorem
stress vector t to a plane with normal n at position x only depends on plane‘s normal n
newton‘s third law actio = reactio
existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n
9 06 - concept of stress
x2
x3
x1
e3
e2
e1
illustration of stress components
interpretation of 3x3 components
10 06 - concept of stress
normal and tangential stress
t n tn
tt
• stress vector
• normal stress
• tangential stress
11 06 - concept of stress
concept of stress - example 1 • consider the cauchy stress tensor as given below
• a) find the traction vector corresponding to the plane! • b) what is the magnitude of the normal and the shear stress? • c) is the normal stress tensile or compressive?
12 06 - concept of stress
concept of stress - example 1
• a) find the traction vector corresponding to the plane!
don’t forget to normalize the normal vector
• a) project stress tensor onto plane with normal n
• b) what is the magnitude of the normal stress?
13 06 - concept of stress
concept of stress - example 1
• b) what is the magnitude of the shear stress? or
14 06 - concept of stress
concept of stress - example 1
• c) is the normal stress tensile or compressive?
• c) since the normal stress is tensile
• they follow from the characteristic equation
15 06 - concept of stress
minimum/maximium principal stress • principal normal stresses include the maximum and minimum normal stress among all possible directions
• where are the stress invariants
• principal directions are the directions associated with the principal values and follow from
with (no summation) 16 06 - concept of stress
concept of stress - example 2
• given the following stress tensor
• a) what are the maximal stress values? • b) what are the principal directions? • c) what is their significance?
17 06 - concept of stress
concept of stress - example 2 • a) what are the maximal stress values?
• first we derive the characteristic equation (cubic eqn)
• and solve for the eigenvalues
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concept of stress - example 2 • b) what are the principal directions?
• solve three evp‘s obtain the three principal directions
• c) what is their significance?
06 - concept of stress 19
concept of stress - research example
20 06 - concept of stress
stress tensors
cauchy / true stress relates spatial force to spatial area
first piola kirchhoff / nominal stress relates spatial force to material area
second piola kirchhoff stress relates material force to material area
cauchy / true stress relates spatial force to spatial area
first piola kirchhoff / nominal stress relates spatial force to material area
second piola kirchhoff stress relates material force to material area
21 06 - concept of stress
stress tensors
22 06 - concept of stress
stress tensors
gustav robert kirchhoff [1824-1887]
augustin louis caucy
[1789-1857]
first piola kirchhoff
second piola kirchhoff cauchy
23 06 - concept of stress
pull back / push forward
covariant / strains
contravariant / stresses
pull back push forward
pull back push forward