Stress-Strain Stress-Strain Theory Theory Under action of applied forces, solid bodies Under action of applied forces, solid bodies undergo deformation, i.e., they change shape undergo deformation, i.e., they change shape and volume. The static mechanics of this and volume. The static mechanics of this deformations forms the theory of elasticity, deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic and dynamic mechanics forms elastodynamic theory. theory. Strain Tensor Strain Tensor x x x’ x’ dx dx dx’ dx’ u(x) u(x+dx) Displacement vector: Displacement vector: u(x) = x’- x Length squared: Length squared: dl = dx + dx + dx = dx dx 21 2 2 3 2 2 i i After deformation dl = dx’ dx’ = (du +dx ) i i i i 2 2 = du du + dx dx + 2 du dx i i i i i i dx’ dx’ dx dx Strain Tensor Strain Tensor x x x’ x’ dx dx dx’ dx’ u(x) u(x+dx) Length squared: Length squared: dl = dx + dx + dx = dx dx 21 2 2 3 2 2 i i After deformation dl = dx’ dx’ = (du +dx ) i i i i 2 2 = du du + dx dx + 2 du dx i i i i i i Length change: Length change: dl - dl = du du + 2du dx i 2 2 i i i du = du dx dx i j j i Substitute Substitute (1) into equation (1) dx’ dx’ dx dx Strain Tensor Strain Tensor x x x’ x’ dx dx dx’ dx’ u(x) u(x+dx) After deformation Length change: Length change: dl - dl = du du + 2du dx i 2 2 i i i du = du dx dx i j j i Substitute Substitute Length change: Length change: dl - dl = U U i 2 2 i (1) into equation (1) (du + du + du du )dx dx i j dx dx dx dx j j i i i j k k = Strain Tensor (2) dx’ dx’ dx dx Problem Problem 1 light year V > C V > C Problem Problem V > C V > C 1 light year V < C V < C Elastic Strain Theory Elastic Strain Theory Elastodynamics Elastodynamics Acoustics Acoustics L L L L’ L’ L’ = = ε ε xx xx = = dL L’-L dL L’-L L L L L = = Length Change Change Length Length Acoustics Acoustics L L L L’ L’ L’ = = ε ε xx xx = = dL L’-L dL L’-L L L L L = = Length Change Change Length Length Acoustics Acoustics L L L L’ L’ L’ = = ε ε xx xx = = dL L’-L dL L’-L L L L L = = Length Change Change Length Length No Shear Resistance = No Shear Strength No Shear Resistance = No Shear Strength Acoustics Acoustics dx dz du dw dw, du << dx, dz Tensional Tensional