Download Simplicity 1695079 Technical information

In terms of a cylindrical coordinate system ( r , θ , z ), with z as the axis of
material symmetry, and assuming an axially symmetric deformation field (i.e.,
ε zθ = ε rθ = 0 ), the constitutive law becomes:
ε r = a11 ⋅ σ r + a12 ⋅ σ θ + a13 ⋅ σ z ................................................................... (5.1.2a)
ε θ = a12 ⋅ σ r + a11 ⋅ σ θ + a13 ⋅ σ z ................................................................... (5.1.2b)
ε z = a13 ⋅ σ r + a13 ⋅ σ θ + a33 ⋅ σ z ................................................................... (5.1.2c)
ε rz = (a 44 / 2) ⋅ τ rz .......................................................................................... (5.1.2d)
in which a11 = 1 / E x , a33 = 1 / E z , a12 = −ν xy / E x , a13 = −ν zx / E z and a44 = 1 / Gxz .
Hence, five elastic constants are included, namely: two Young’s moduli E x (= E y ) and
E z ; two Poisson’s ratios ν xy (= ν yx ) and ν zx (= ν xz ⋅ E z / E x ) ; and one shear modulus
G xz (= G yz ) . The condition that the strain energy must be positive imposes the following
property restrictions (PRs) on the values of the elastic constants (e.g., Poulus and Davis,
1974): (PR1) E x , E z , G xz > 0 ; (PR2) 1 −ν xy − 2 ⋅ν xz ⋅ν zx > 0 ; and (PR3) 1 − ν xy > 0 .
Following Lekhnitskii (1963) and Singh (1986), the stresses ( σ r , σ θ , σ z ,τ rz ) and
displacements ( u, w in the r, z directions respectively) can be derived from a stress
function φ (r , z ) as follows:
σr = −
∂ ⎛ ∂φ 2 b ∂φ
∂φ 2
⎜⎜ 2 + ⋅
+a⋅ 2
r ∂r
∂z ⎝ ∂r
∂z
⎞
⎟⎟ ............................................................. (5.1.3a)
⎠
∂ ⎛ ∂φ 2 1 ∂φ
∂φ 2
⎜
σθ = − ⎜b ⋅ 2 + ⋅
+a⋅ 2
r ∂r
∂z ⎝ ∂r
∂z
σz =
τ rz
∂ ⎛ ∂φ 2 c ∂φ
∂φ 2
⎜⎜ c ⋅ 2 + ⋅
+d⋅ 2
r ∂r
∂z ⎝ ∂r
∂z
∂ ⎛ ∂φ 2 1 ∂φ
∂φ 2
⎜
= ⎜ 2 + ⋅
+a⋅ 2
r ∂r
∂r ⎝ ∂r
∂z
⎞
⎟⎟ ......................................................... (5.1.3b)
⎠
⎞
⎟⎟ ............................................................ (5.1.3c)
⎠
⎞
⎟⎟ ................................................................ (5.1.3d)
⎠
5-2