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

review: 04 - kinematics 4

configurations

• displacement field in the spatial description

review: 04 - kinematics 5

displacement fields

• displacement field in the material description • acceleration field in material and spatial description

review: 04 - kinematics 6

velocity and acceleration field

• velocity field in material and spatial description

and

and

7

material derivative of a spatial field

• material derivative of a spatial field

• under application of the chain rule

convective term • easier to remember

review: 04 - kinematics 8

deformation gradient

• by using the chain rule we can relate the line elements

• the deformation gradient relates vectors

and

and

review: 04 - kinematics

9 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. • the body is in equilibrium under the external forces

10 06 – concept of stress

traction vector

• we cut the body along a plane through point

11 06 – concept of stress

traction vector

• and establish a force equilibrium again, where

12 06 – concept of stress

traction vector

13 06 – concept of stress

• we call the traction vector or surface traction or more specifically the cauchy traction vector

traction vector

• the traction vector follows newton‘s third law of action and reaction

14 06 – concept of stress

traction vector • lets look at the traction vector in more detail

• the vectors m and n are perpendicular to each other or mathematical speaking

15 06 – concept of stress

traction vector

16 06 – concept of stress

cauchy‘s stress theorem • for every possible plane cutting through the point x at time t there exists a traction vector t

• the infinite number of traction vectors at point x define the stress state of at that point

17 06 – concept of stress

• cauchy‘s stress theorem states further that

cauchy‘s tetrahedron

• where is the cauchy stress tensor

18 06 – concept of stress

stress components

keeping in mind:

19 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?

20 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) it follows that

• b) what is the magnitude of the normal stress?

21 06 – concept of stress

concept of stress – example 1

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

or

22 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

23 06 – concept of stress

extremal stress values • principal normal stresses include the maximum and minimum normal stress among all possible directions

• where are denoted the stress invariants

24 06 – concept of stress

extremal stress values • principal directions are the directions associated with the principal values and follow from

with (no summation)

25 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?

26 06 – concept of stress

concept of stress – example 2 • a) what are the maximal stress values?

• first we derive the characteristic equation (cubic)

• and solve for the eigenvalues

27 06 – concept of stress

concept of stress – example 2 • b) what are the principal directions?

• and so forth to obtain the three principal directions

• c) what is their significance?

06 – concept of stress 28

concept of stress – research example

• first piola-kirchhoff stress tensor from follows that

06 – concept of stress 29

alternative stress tensors

• cauchy stress tensor from follows that

• kirchhoff stress tensor different from only by

• second piola-kirchhoff stress tensor from follows