06 - concept of stress 06 - concept of stress holzapfel nonlinear solid mechanics [2000], chapter 3, pages 109-129 holzapfel nonlinear solid mechanics [2000], chapter 3, pages 109-129 06 - concept of stress 1 06 - concept of stress 2 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. 06 - concept of stress 3 06 - concept of stress 4 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 augustin louis caucy newton‘s third law actio = reactio augustin louis caucy xP [1789-1857] [1789-1857] n t 06 - concept of stress 5 06 - concept of stress 6 cauchy‘s theorem cauchy X3 , x3 • cauchy‘s postulate t1 stress vector t to a plane with normal n at t2 position x only depends on plane‘s normal n t X , x 2 2 • cauchy‘s lemma augustin louis caucy augustin louis caucy X , x [1789-1857] newton‘s third law actio = reactio [1789-1857] 1 1 t3 • cauchy‘s theorem • cauchy‘s theorem existence of second order tensor field σ is inde- existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n pendent of n, such that t is a linear function of n 06 - concept of stress 7 06 - concept of stress 8 illustration of stress components normal and tangential stress x3 tn e3 n • stress vector t tt interpretation of 3x3 components • normal stress x2 e2 • tangential stress e1 x1 06 - concept of stress 9 06 - concept of stress 10 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! • a) project stress tensor onto plane with normal n • b) what is the magnitude of the normal and the shear stress? • c) is the normal stress tensile or compressive? 06 - concept of stress 11 06 - concept of stress 12 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? • c) since the normal stress is tensile or 06 - concept of stress 13 06 - concept of stress 14 minimum/maximium principal stress concept of stress - example 2 • principal normal stresses include the maximum and minimum normal stress among all possible directions • given the following stress tensor • they follow from the characteristic equation • where are the stress invariants • a) what are the maximal stress values? • b) what are the principal directions? • principal directions are the directions associated with the principal values and follow from • c) what is their significance? with (no summation) 06 - concept of stress 15 06 - concept of stress 16 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? 06 - concept of stress 17 06 - concept of stress 18 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 06 - concept of stress 19 06 - concept of stress 20 stress tensors stress tensors cauchy / true stress relates spatial force to spatial area / nominal stress first piola kirchhoff first piola kirchhoff relates spatial force to material area gustav robert kirchhoff augustin louis caucy [1824-1887] [1789-1857] second piola kirchhoff stress relates material force to material area second piola kirchhoff cauchy 06 - concept of stress 21 06 - concept of stress 22 pull back / push forward covariant / strains pull back push forward contravariant / stresses pull back push forward 06 - concept of stress 23 .