1. Analysis of a cracked plate
Casa Day 16/11/05

2. Motivation Aloha Airlines Flight 243
Honolulu International Airport, Hawaii
No visual exterior inspection (nor required)
At feet… loud clap
Flight attendant swept out of the cabin
1 dead, 65 injured
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
$5 million damage

3. Peel Test Force Displacement (constant speed) Nickel layer 45º
Substrate
Peel-Test Support
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress

4. Uniaxial Stress Stress Strain
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material
Tomemos uma barra que é carregada com uma força F. Suponhamos que esta força é uniforme ao longo da face do cilindro, e perpendicular à mesma. Entãp define-se o stress=F/A. como consequência desta força F, o cilindro soferá uma elongação, vai crescer Delta l. à deformação causada pelo stress sigma chama-se strain, que denotamos por epsilon. Temos epsilon=Delta l / l.
Consideremos agora uma barra em que a secção no meio da barra é mais pequena. Considermos que a barra é carregada pela mesma força F. Então no meio da barra o stress será maior. A maneira como definimos o strain no ponto que dista x do ponto de apoio da barra é epsilon(x)=…
Como definir o strain no caso da força não ser uniforme? Strain = lim …

5. Displacement Gradient
Approximation
Temos o ponto P, na reference configuration. Consideremos um ponto na vizinhança de P. Chamemos-lhe Q. No instante t, P e Q ter-se-ão deslocado. Displacement de P e de Q u(x) e u(x+dx). Podemos escrever u(x+dx)=u(x)+du.dx+…. Se dx é pequeno (o que não é o caso da figura, hehe), então podemos ignorar o “…”. A du chamamos displacement gradient matrix. É importante, vamos explicar porquê.
Displacement Gradient

6. Strain Tensor Strain Tensor symmetric skew-symmetric deformations
rigid body motions
Strain Tensor
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material
symmetric

7. Physical Interpretation
diagonal
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material
off-diagonal

8. Stress Tensor Traction Cauchy Stress
Stress- a tensorial quantity that characterizes the state of internal forces acting in a body

9. Equation of motion Balance of linear momentum
body force
density
Balance of angular momentum
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material
Equation of motion

10. Summary Equations Unknowns 3 6 6 3 6 6 + constitutive equations + 15 9
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material 15 9 15

11. Constitutive Equations
Linear Elasticity:
Young’s Modulus
Poisson’s Ratio
Lamé Moduli
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material

12. Problem Description
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material

13. Continuum Mechanics Plane Stress Boundary Conditions 1º plane stress
2º equações
3º boundary conditions

14. Continuum Mechanics
1º plane stress
2º equações
3º boundary conditions

15. Domain decomposition
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material

16. Macro equations, Micro equations
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material

17. Schwarz
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material

18. Example Difference between successive iterations for u u1 u2-u1 u1-u0
5.13
u2-u1
1.16x10-1
u3-u2
6.56x10-2
u4-u3
5.86x10-2
u2-u1
Stress- an applied force or system of forces concentrated over an area that tends to strain or deform a body
Strain- a deformation produced by stress
É abusivo usar a letra epsilon?
Stress-strain diagram obtained by tensile test, it is characteristic of each material

19. Future Work - Grain structure - Time - Convergence analysis E=2x10e-11
Nu=0.33