13.2 Tensors in Plasticity Theory
167
T σ =
⎛
⎝
σ x τ xy τ xz
τ yx σ y τ yz
τ zx τ zy σ z
⎞
⎠ , T ε =
⎛
⎜
⎜
⎜
⎜
⎝
ε x
1
2
γ xy
1
2
γ xz
1
2
γ yx ε y
1
2
γ yz
1
2
γ zx
1
2
γ zy ε z
⎞
⎟
⎟
⎟
⎟
⎠
.
(13.6)
To designate stress and strain components, we will use the following designations:
σ ij , ε ij , (i,j ∼ x, y, z),
whereas
σ ij =
σ i at i = j,
τ ij at i = j ;
ε ij =
ε i at i = j,
1
2
γ ij at i = j.
For reduction purposes, we will write tensors (13.6) as follows:
T σ = {σ ij }, T ε = {ε ij }.
Below, deformations mean low deformations unless agreed otherwise. The
components of low deformations are expressed through the components u i , (i ∼
x, y, z) of the displacement vector #» u using formulas (2.2), (2.3) that can be
represented as follows:
ε ij =
1
2
u i,j + u j,i
,
(13.7)
where the comma after the first index means differentiation by variable corresponding to the second index; for example,
u x,y =
∂u x
∂y
; u z,x =
∂u z
∂x
;
ε xx =
1
2
∂u x
∂x
+
∂u x
∂x
=
∂u x
∂x
;
ε zx =
1
2
γ zx =
1
2
∂u z
∂x
+
∂u x
∂z
.
Apart from stress and strain tensors (13.6), plasticity theory also uses stress and
strain rate tensors:
T ˙
ε = {˙ ε ij }, T ˙
σ = { ˙
σ ij },
(13.8)