Mathematical Problems in Engineering

Research Article

Spectral Classification of Non-Coaxiality for Two-Dimensional Incremental Stress-Strain Response

Table 2

Coupled spectral classification of incremental non-coaxiality.
πœ† 2 β‰₯ πœ† 3 β‰₯ 0 0 > πœ† 2 β‰₯ πœ† 3 πœ† 2 > 0
& πœ† 3 < 0 		πœ† 2 , πœ† 3 complex
Properties of πœ† 𝑖 ( πœ† 1 = 0 ) and πœ† 𝑖 πœ† 2 β‰  πœ† 3 πœ† 2 = πœ† 3 πœ† 2 β‰  πœ† 3 πœ† 2 = πœ† 3
𝐢 𝑑 𝑖 𝑗 π‘˜ 𝑙 𝛿 π‘˜ 𝑙 = 0
( πœ† 1 = 0 )		 πœ† 2 β‰₯ πœ† 3
& πœ† 3 β‰₯ 0 		πœ† 2 β‰  πœ† 3 SIC = 4
stable		SIC = 2
stable		No possibilitySIC = UIC = 0
stable
πœ† 2 = πœ† 3
& πœ† 2 β‰  0 		SIC = 4
stable		SIC
stable		No possibilitySIC = UIC = 0
stable
πœ† 3 ≀ πœ† 2
& πœ† 2 ≀ 0 		πœ† 2 β‰  πœ† 3 No possibilityUIC = 4
unstable		UIC = 2
unstable		No possibilitySIC = UIC = 0
unstable
πœ† 2 = πœ† 3
& πœ† 2 β‰  0 		No possibilityUIC = 4
unstable		UIC
unstable		No possibilitySIC = UIC = 0
unstable
πœ† 2 β‰₯ 0 &
πœ† 3 ≀ 0 		πœ† 2 β‰  πœ† 3 SIC = 4
partially stable		SIC = 2
partially stable		UIC = 4
partially stable		UIC = 2
partially stable		SIC = UIC = 2
partially stable		SIC = UIC = 0
partially stable
πœ† 2 = πœ† 3 No possibilitySIC = UIC = 0
Stable-Unstable Transition
𝐢 𝑑 𝑖 𝑗 π‘˜ 𝑙 𝛿 π‘˜ 𝑙 β‰  0 SIC = UIC = 0
partially stable		SIC = UIC = 0
partially stable
SIC = 𝑛 : number of directions with identical direction of incremental stress and deviator strain, c o s 2 πœƒ i n c ≑ 1 (SIC alone means that incremental stress and deviator strain have identical direction). UIC = 𝑛 : number of directions with opposite direction of incremental stress and deviator strain, c o s 2 πœƒ i n c ≑ βˆ’ 1 (UIC alone means that all incremental stress and deviator strain have opposite direction).
Stable-Unstable Transition: c o s 2 πœƒ i n c ≑ 0 ; stable: 0 ≀ c o s 2 πœƒ i n c ≀ 1 ; unstable: 0 > c o s 2 πœƒ i n c β‰₯ βˆ’ 1 ; partially stable: βˆ’ 1 ≀ c o s 2 πœƒ i n c ≀ 1 .