Stress/Strain Lecture 1 ME EN 372 Andrew Ning aning@byu.edu - PDF document

Stress/Strain Lecture 1 ME EN 372 Andrew Ning aning@byu.edu Outline Stress Strain Plane Stress and Plane Strain Materials

Stress 1D Stress P P σ = P A

3D Stress State σ yy τ yx   τ yz τ xy σ xx τ xy τ xz   τ zy σ = τ yx σ yy τ yz   σ xx     τ xz τ zx τ zy σ zz τ zx σ zz

Positive Sign Convention σ y τ xy (+) σ x Nominal vs true stress

Strain 1D Strain ∆ l l 0 ǫ = ∆ l l 0

u 1 u 0 x 0 x 1 ǫ x = u 1 − u 0 → ∂u x 1 − x 0 ∂x Normal strain can occur in all three directions. ǫ x = ∂u ∂x ǫ y = ∂v ∂y ǫ z = ∂w ∂z

Analogous to shear stress, there are three independent shear strain components. γ xy = θ 1 + θ 2 θ 2 θ 1 γ xy = γ yx = ∂v ∂x + ∂u ∂y γ yz = γ zy = ∂w ∂y + ∂v ∂z γ xz = γ zx = ∂u ∂z + ∂w ∂x

Plane Stress and Plane Strain Plane Stress σ yy σ y τ yx τ xy τ yz τ xy τ zy (+) σ x σ xx τ xz τ zx σ zz

Plane Strain Materials

Homogeneous: Isotropic: Linear Stress-Strain Relationships Most general material (in the elastic region):       ǫ xx S 11 S 12 S 13 S 14 S 15 S 16 σ xx        ǫ yy   S 21 S 22 S 23 S 24 S 25 S 26   σ yy               ǫ zz   S 31 S 32 S 33 S 34 S 35 S 36   σ zz        =       γ xy S 41 S 42 S 43 S 44 S 45 S 46 τ xy                   γ xz S 51 S 52 S 53 S 54 S 55 S 56 τ xz                   γ yz S 61 S 62 S 63 S 64 S 65 S 66 τ yz

ǫ = Sσ S : compliance matrix Inverse: σ = Kǫ K : stiffness matrix. Orthotropic Materials Usually, a material contain certain symmetries.       ǫ xx S 11 S 12 S 13 0 0 0 σ xx        ǫ yy   S 21 S 22 S 23 0 0 0   σ yy               ǫ zz   S 31 S 32 S 33 0 0 0   σ zz        =       γ xy 0 0 0 S 44 0 0 τ xy                   γ xz 0 0 0 0 S 55 0 τ xz                   γ yz 0 0 0 0 0 S 66 τ yz

Isotropic A special case of orthotropy is isotropy, in which the elastic properties are the same in every direction.       ǫ x 1 − ν − ν 0 0 0 σ x       1 0 0 0  ǫ y   − ν − ν   σ y              1 0 0 0  ǫ z   − ν − ν   σ z  = 1             E  γ xy   0 0 0 2(1 + ν ) 0 0   τ xy               γ xz   0 0 0 0 2(1 + ν ) 0   τ xz              0 0 0 0 0 2(1 + ν ) γ yz τ yz Inverse (stiffness matrix)       σ x 1 − ν ν ν 0 0 0 ǫ x        σ y   ν 1 − ν ν 0 0 0   ǫ y              1 − ν 0 0 0  σ z   ν ν   ǫ z  E       =       (1 + ν )(1 − 2 ν ) 1 − 2 ν 0 0 0 0 0  τ xy     γ xy     2          1 − 2 ν  τ xz   0 0 0 0 0   γ xz     2          1 − 2 ν τ yz 0 0 0 0 0 γ yz 2

E G = 2(1 + ν ) Strain in x-direction ǫ x = 1 E [ σ x − ν ( σ y + σ z )]