06 - concept of stress concept of stress concept of stress concept of stress. me338 - syllabus. definition of stress

Transcription

1 holzapfel nonlinear solid mechanics [2000], chapter 3, pages holzapfel nonlinear solid mechanics [2000], chapter 3, pages 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. 3 4

2 cauchy s postulate cauchy s lemma cauchy s postulate stress vector t to a plane with normal n at position x only depends on plane s normal n cauchy s lemma -t -n Px newton s third law actio = reactio n t 5 6 cauchy s theorem cauchy X 3, x 3 cauchy s postulate t 2 t t 1 X 2, x 2 stress vector t to a plane with normal n at position x only depends on plane s normal n cauchy s lemma X 1, x 1 t 3 newton s third law actio = reactio cauchy s theorem cauchy s theorem existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n 7 8

3 illustration of stress components x 3 normal and tangential stress e 3 stress vector t n n t interpretation of 3x3 components normal stress t t e 2 x 2 tangential stress x 1 e concept of stress - example 1 concept of stress - example 1 consider the cauchy stress tensor as given below a) find the traction vector corresponding to the plane! don t forget to normalize the normal vector 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? 11 a) project stress tensor onto plane with normal n 12

4 concept of stress - example 1 concept of stress - example 1 b) what is the magnitude of the normal stress? c) is the normal stress tensile or compressive? b) what is the magnitude of the shear stress? or c) since the normal stress is tensile minimum/maximium principal stress principal normal stresses include the maximum and minimum normal stress among all possible directions they follow from the characteristic equation where are the stress invariants principal directions are the directions associated with the principal values and follow from with (no summation) 15 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? 16

5 concept of stress - example 2 concept of stress - example 2 a) what are the maximal stress values? b) what are the principal directions? first we derive the characteristic equation (cubic eqn) solve three evp s obtain the three principal directions and solve for the eigenvalues c) what is their significance? 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 19 20

6 stress tensors stress tensors cauchy / true stress relates spatial force to spatial area first piola kirchhoff / nominal stress relates spatial force to material area first piola kirchhoff second piola kirchhoff stress relates material force to material area gustav robert kirchhoff [ ] second piola kirchhoff cauchy pull back / push forward covariant / strains pull back push forward contravariant / stresses pull back push forward 23