06 - concept of stress

1 06 - concept of stress

06 - concept of stress

holzapfel �nonlinear solid mechanics� [2000], chapter 3, pages 109-129


06 - concept of stress

holzapfel �nonlinear solid mechanics� [2000], chapter 3, pages 109-129


me338 - syllabus


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


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


cauchy‘s lemma

augustin louis caucy [1789-1857]

• cauchy‘s lemma

newton‘s third law actio = reactio

t n

-t -n


cauchy‘s theorem


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


cauchy


• 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


x2

x3

x1

e3

e2

e1

illustration of stress components

interpretation of 3x3 components


normal and tangential stress

t n tn

tt

• stress vector

• normal stress

• tangential 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?


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?



• b) what is the magnitude of the shear stress? or



• c) is the normal stress tensile or compressive?

• c) since the normal stress is tensile

• they follow from the characteristic equation


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


• given the following stress tensor

• a) what are the maximal stress values? • b) what are the principal directions? • c) what is their significance?


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


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


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


stress tensors


stress tensors

gustav robert kirchhoff [1824-1887]

augustin louis caucy

[1789-1857]

first piola kirchhoff

second piola kirchhoff cauchy


pull back / push forward

covariant / strains

contravariant / stresses

pull back push forward

pull back push forward

06 - concept of stress

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