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

Table 3

(a) Model parameters for stable INC (𝐶^{𝑑𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0,𝜆₁=𝜆₁=0)

Classification Domains in Figure 6	𝐶𝑖𝑗𝑘𝑙 coefficients	Spectral properties	Computation results in Figure 7
I	^1.5−0.50−0.51.50000.1	𝜆2=2.0,𝜆3=0.2,	(a)
(𝐵′=𝐵)		𝜆2=2.0,𝜆3=0.2
𝑎^+𝐴^+	^1.0−0.51.5−1.01.5−1.5001.0	𝜆2=3.5,𝜆3=0.5,	(b)
(𝐵′>𝐵)		𝜆2=𝜆3=2.0
𝑜𝐴^+	^1.5−0.50−0.51.50001.0	𝜆2=𝜆3=𝜆2=𝜆3=2.0	(c)
(𝐵′=𝐵)
VI	^1.0−0.51.0−1.01.5−1.0−2.02.01.0	𝜆2=3.0,𝜆3=1.0	(d)
(𝐵′>𝐵)		𝜆2=2.0+2.8𝑖 (complex) 𝜆3=2.0−2.8𝑖 (complex)

(b) Model parameters for instable INC (𝐶^{𝑑𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0,𝜆₁=𝜆₁=0)

Classification Domains in Figure 6	𝐶𝑖𝑗𝑘𝑙 coefficients	Spectral properties	Computation results in Figure 8
II	^0.51.501.50.5000−0.1	𝜆2=−0.2,𝜆3=−1.0,	(a)
(𝐵′=𝐵)		𝜆2=−0.2,𝜆3=−1.0
𝑎^−𝐴^−	^0.52.001.51.000.5−0.5−0.5	𝜆2=−0.5,𝜆3=−1.5,	(b)
(𝐵′>𝐵)		𝜆2=𝜆3=−1.0
𝑜𝐴^−	^0.51.501.50.5000−0.5	𝜆2=𝜆3=𝜆2=	(c)
(𝐵′=𝐵)		𝜆3=−1.0
VII	^0.52.01.01.51.0−1.0−1.51.5−0.5	𝜆2=−0.5,𝜆3=−1.5	(d)
		𝜆2=−1.0+2.45𝑖 (complex)
(𝐵′>𝐵)		𝜆3=−1.0−2.45𝑖 (complex)

(c) Model parameters for stable-instable INC transition (𝐶^{𝑑𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0,𝜆₁=𝜆₁=0)

Classification Domains in Figure 6	𝐶𝑖𝑗𝑘𝑙 coefficient	Spectral properties	Computation results in Figure 9
Point 𝑜′	^1.01.5−0.11.01.50.10.1−0.10.0	𝜆2=𝜆3=0
(𝐵′>𝐵)		𝜆2=0.2𝑖 (complex) 𝜆3=−0.2𝑖 (complex)

(d) Model parameters for partially stable INC (𝐶^{𝑑𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0,𝜆₁=𝜆₁=0)

Classification Domains in Figure 6	𝐶𝑖𝑗𝑘𝑙 coefficients	Spectral properties	Computation results in Figure 10
IV	^1.0−0.50.6−1.01.5−0.61.5−1.51.0	𝜆2=4.1,𝜆3=−0.1	(a)
(𝐵′>𝐵)		𝜆2=3.9,𝜆3=0.1
𝑜𝑎^+	^1.0−0.5−4.0−1.01.54.00.00.01.0	𝜆2=6.0,𝜆3=−2.0	(b)
(𝐵′>𝐵)		𝜆2=𝜆3=2.0
V	^0.52.00.251.51.0−0.251.0−1.0−0.6	𝜆2=0.15,𝜆3=−2.35	(c)
(𝐵′>𝐵)		𝜆2=−0.095,𝜆3=−2.1
𝑜𝑎^−	^0.52.001.51.002.0−2.0−0.5	𝜆2=1.0,𝜆3=−3.0	(d)
(𝐵′>𝐵)		𝜆2=𝜆3=−1.0
III	^1.5−0.51.0−0.51.5−1.01.0−1.00.6	𝜆2=2.96,𝜆3=−2.16	(e)
(𝐵′=𝐵)		𝜆2=2.96,𝜆3=−2.16
VIII	^1.0−0.50.4−1.01.5−0.4−5.05.00.5	𝜆2=6.13,𝜆3=−3.13	(f)
(𝐵′>𝐵)		𝜆2=1.5+2.78𝑖 (complex) 𝜆3=1.5−2.78𝑖 (complex)

(e) Model parameters for partially stable INC (𝐶^{𝑑𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙≠0,𝜆₁=0)

𝐶𝑖𝑗𝑘𝑙 coefficient	Spectral properties	Computation Results in Figure 11
^1.0−0.50−0.55.00001.0	𝜆2=3.5,𝜆3=2.0	(a)
	𝜆1=3.77,𝜆2=2.0,𝜆3=−0.27
^1.00.5−1.01.02.01.01.0−1.00.75	𝜆2=1.125+1.96𝑖 (complex)	(b)
	𝜆3=1.125−1.96𝑖 (complex)
	𝜆1=1.5,𝜆2=0.9,𝜆3=−0.155