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 =      σxx τxy τxz τyx σyy τyz τzx τzy σzz     

τzx τzy σzz σxx τxy τxz τyx σyy τyz

Positive Sign Convention

σx σy τxy (+)

Nominal vs true stress

Strain 1D Strain

l0 ∆l

ǫ = ∆l l0

u0 u1 x1 x0

ǫx = u1 − u0 x1 − x0 → ∂u ∂x Normal strain can occur in all three directions. ǫx = ∂u ∂x ǫy = ∂v ∂y ǫz = ∂w ∂z

Analogous to shear stress, there are three independent shear strain components. γxy = θ1 + θ2

θ2 θ1

γxy = γyx = ∂v ∂x + ∂u ∂y γyz = γzy = ∂w ∂y + ∂v ∂z γxz = γzx = ∂u ∂z + ∂w ∂x

Plane Stress and Plane Strain Plane Stress

τzx τzy σzz σxx τxy τxz τyx σyy τyz

σx σy τxy (+)

Plane Strain Materials

Homogeneous: Isotropic:

Linear Stress-Strain Relationships

Most general material (in the elastic region):               ǫxx ǫyy ǫzz γxy γxz γyz               =               S11 S12 S13 S14 S15 S16 S21 S22 S23 S24 S25 S26 S31 S32 S33 S34 S35 S36 S41 S42 S43 S44 S45 S46 S51 S52 S53 S54 S55 S56 S61 S62 S63 S64 S65 S66                             σxx σyy σzz τxy τxz τyz              

ǫ = Sσ S: compliance matrix Inverse: σ = Kǫ K: stiffness matrix.

Orthotropic Materials

Usually, a material contain certain symmetries.               ǫxx ǫyy ǫzz γxy γxz γyz               =               S11 S12 S13 S21 S22 S23 S31 S32 S33 S44 S55 S66                             σxx σyy σzz τxy τxz τyz              

Isotropic

A special case of orthotropy is isotropy, in which the elastic properties are the same in every direction.

              ǫx ǫy ǫz γxy γxz γyz               = 1 E               1 −ν −ν −ν 1 −ν −ν −ν 1 2(1 + ν) 2(1 + ν) 2(1 + ν)                             σx σy σz τxy τxz τyz              

Inverse (stiffness matrix)

              σx σy σz τxy τxz τyz               = E (1 + ν)(1 − 2ν)               1 − ν ν ν ν 1 − ν ν ν ν 1 − ν

1−2ν 2 1−2ν 2 1−2ν 2

                            ǫx ǫy ǫz γxy γxz γyz              

G = E 2(1 + ν) Strain in x-direction ǫx = 1 E [σx − ν(σy + σz)]