Viscoelastic models revisited: characteristics and interconversion formulas for generalized Kelvin–Voigt and Maxwell models

Abstract

Generalized Kelvin–Voigt and Maxwell models using Prony series are some of the most well-known models to characterize the behavior of polymers. The simulation software for viscoelastic materials generally implement only some material models. Therefore, for the practice of the engineer, it is very useful to have formulas that establish the equivalence between different models. Although the existence of these relationships is a well-established fact, moving from one model to another involves a relatively long process. This article presents a development of the relationships between generalized Kelvin–Voigt and Maxwell models using the aforementioned series and their respective relaxation and creep coefficients for one and two summations. The relationship between the singular points (maximums, minimums and inflexion points) is also included.

Appendices

Appendix A

1.1 A.1 Constants of differential equations

1.1.1 A.1.1 Constants of Eq. (27) in the text

$$A = \mathop \sum \limits_{i = 0}^{n} \mathop {\mathop \prod \limits_{j = 0}^{n} }\limits_{j \ne i} E_{{i_{K} }} = \varDelta \cdot \mathop \sum \limits_{i = 0}^{n} \frac{1}{{E_{{i_{K} }} }},$$

with $\varDelta = \mathop \prod \limits_{i = 0}^{n} E_{{i_{K} }} ,$

$$B = \frac{\varDelta }{{E_{{0_{K} }} }} \cdot \mathop \sum \limits_{j = 1}^{n} \frac{{\eta_{{j_{K} }} }}{{E_{{j_{K} }} }} + \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n} }\limits_{j \ne i} \frac{{\eta_{{j_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} }},$$

$$C = \frac{\varDelta }{{E_{{0_{K} }} }} \cdot \mathop \sum \limits_{i = 1}^{n - 1} \mathop {\mathop \sum \limits_{j = 2}^{n} }\limits_{j > i} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} }} + \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n - 1} }\limits_{j \ne i} \mathop {\mathop {\mathop \sum \limits_{k = 2}^{n} }\limits_{k > j} }\limits_{k \ne i} \frac{{\eta_{{j_{K} }} \eta_{{k_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} }},$$

$$D = \frac{\varDelta }{{E_{{0_{K} }} }} \cdot \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} \eta_{{k_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} }} + \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n - 2} }\limits_{j \ne i} \mathop {\mathop {\mathop \sum \limits_{k = 2}^{n - 1} }\limits_{k > j} }\limits_{k \ne i} \mathop {\mathop {\mathop \sum \limits_{l = 3}^{n} }\limits_{l > k} }\limits_{l \ne i} \frac{{\eta_{{j_{K} }} \eta_{{k_{K} }} \eta_{{l_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} E_{{k_{K} }} }},$$

$$\vdots$$

$$Y = \mathop \sum \limits_{i = 1}^{n} \left[ {\left( {E_{{0_{K} }} + E_{{i_{K} }} } \right)\mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{{j_{K} }} } \right],$$

$$Z = \mathop \prod \limits_{i = 1}^{n} \eta_{{i_{K} }} ,$$

$$a = \varDelta ,$$

$$b = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \frac{{\eta_{{i_{K} }} }}{{E_{{i_{K} }} }},$$

$$c = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 1} \mathop {\mathop \sum \limits_{j = 2}^{n} }\limits_{j > i} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} }},$$

$$d = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} \eta_{{k_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} }},$$

$$\vdots$$

$$y = E_{{0_{K} }} \cdot \mathop \sum \limits_{i = 1}^{n} E_{{i_{K} }} \mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{{j_{K} }} ,$$

$$z = E_{{0_{K} }} \cdot\mathop \prod \limits_{i = 1}^{n} \eta_{{i_{K} }} .$$

1.1.2 A.1.2 Constants of Eq. (56) in the text

$$A = \mathop \prod \limits_{i = 1}^{n} E_{{i_{M} }} \equiv \varDelta ,$$

$$B = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \frac{{\eta_{{i_{M} }} }}{{E_{{i_{M} }} }},$$

$$C = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 1} \mathop {\mathop \sum \limits_{j = 2}^{n} }\limits_{j > i} \frac{{\eta_{{i_{M} }} \eta_{{j_{M} }} }}{{E_{{i_{M} }} E_{{j_{M} }} }},$$

$$D = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{{i_{M} }} \eta_{{j_{M} }} \eta_{{k_{M} }} }}{{E_{{i_{M} }} E_{{j_{M} }} E_{{k_{M} }} }},$$

$$\vdots$$

$$Y = \mathop \sum \limits_{i = 1}^{n} E_{{i_{M} }} \mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{{j_{M} }} ,$$

$$Z = \mathop \prod \limits_{i = 1}^{n} \eta_{{i_{M} }} ,$$

$$a = E_{{\infty_{M} }} \cdot \varDelta ,$$

$$b = \varDelta \cdot \left( {E_{{\infty_{M} }} \cdot\mathop \sum \limits_{i = 1}^{n} \frac{{\eta_{{i_{M} }} }}{{E_{{i_{M} }} }} + \mathop \sum \limits_{i = 1}^{n} \eta_{{i_{M} }} } \right),$$

$$c = {\varDelta} \cdot \left( {E_{{\infty _{M} }} \sum\limits_{{i = 1}}^{{n - 1}} {\sum\limits_{\begin{subarray}{l} j = 2 \\ j > 1 \end{subarray} }^{n} {\frac{{\eta _{{I_{M} }} \eta _{{j_{M} }} }}{{E_{{i_{M} }} E_{{j_{M} }} }}} + \sum\limits_{{i = 1}}^{n} {\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{n} {\frac{{\eta _{{i_{M} }} \eta _{{j_{M} }} }}{{E_{{j_{M} }} }}} } } } \right),$$

$$d = \varDelta \cdot \left( {E_{\infty _M} \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{i_M} \eta_{j_M} \eta_{k_M} }}{{E_{{i_{M} }} E_{{j_{M} }} E_{{k_{M} }} }} + \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n} }\limits_{j \ne i} \mathop {\mathop {\mathop \sum \limits_{k = 1}^{n} }\limits_{k \ne i} }\limits_{k > j} \frac{{\eta_{i_M} \eta_{j_M} \eta_{k_M} }}{{E_{{j_{M} }} E_{{k_{M} }} }}} \right),$$

$$\vdots$$

$$y = E_{\infty _M} \mathop \sum \limits_{i = 1}^{n} E_{i_M} \mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{j_M} + \mathop \sum \limits_{i = 1}^{n} E_{i_M} \eta_{i_M} \mathop {\mathop \sum \limits_{j = 1}^{n} }\limits_{j \ne i} E_{j_M} \mathop {\mathop {\mathop \prod \limits_{k = 1}^{n} }\limits_{k \ne i} }\limits_{k \ne j} \eta_{k_M} ,$$

$$z = \left(E_{\infty _M} + \mathop \sum \limits_{i = 1}^{n} E_{i_M} \right) \mathop \prod \limits_{j = 1}^{n} \eta_{j_M}.$$

1.2 A.2 Derivatives of Prony series

In this Appendix, the mathematical expressions of the first and second order derivatives of Prony series are shown.

1.2.1 A.2.1 Derivatives as a function of time

A.2.1.1 First order derivatives Mathematical expressions to calculate maximums or minimums in relaxation modulus and creep compliance as a function of time, using Prony series, are shown in Eqs. (A.1) and (A.2), respectively

$$\frac{{{\text{d}}Y}}{{{\text{d}}t}} = - Y_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} }}{{\tau_{i} }}{\text{e}}^{{ - \frac{t}{{\tau_{i} }}}} ,$$

(A.1)

$$\frac{{{\text{d}}J}}{{{\text{d}}t}} = J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} }}{{\lambda_{i} }}{\text{e}}^{{ - \frac{t}{{\lambda_{i} }}}} .$$

(A.2)

A.2.1.2 Second order derivatives Mathematical expressions to calculate inflexion points in relaxation modulus and creep compliance as a function of time, using Prony series, are shown in equation Eqs. (A.3) and (A.4), respectively

$$\frac{{{\text{d}}^{2} Y}}{{{\text{d}}t^{2} }} = Y_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} }}{{\tau_{i}^{2} }}{\text{e}}^{{ - \frac{t}{{\tau_{i} }}}} ,$$

(A.3)

$$\frac{{{\text{d}}^{2} J}}{{{\text{d}}t^{2} }} = - J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} }}{{\lambda_{i}^{2} }}{\text{e}}^{{ - \frac{t}{{\lambda_{i} }}}} .$$

(A.4)

1.2.2 A.2.2 Derivatives as a function of frequency

A.2.2.1 First order derivatives Mathematical expressions to calculate maximums or minimums in storage modulus, loss modulus, storage compliance, loss compliances and tangents of the phase angle as a function of frequency, using Prony series, are shown in equations Eqs. (A.5)–(A.9), respectively

$$\frac{{{\text{d}}G^{\prime}}}{{{\text{d}}\omega }} = 2G_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} \tau_{i}^{2} \omega }}{{\left( {1 + \tau_{i}^{2} \omega^{2} } \right)^{2} }} ,$$

(A.5)

$$\frac{{{\text{d}}G^{\prime\prime}}}{{{\text{d}}\omega }} = G_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} \tau_{i} \left( {1 - \tau_{i}^{2} \omega^{2} } \right)}}{{\left( {1 + \tau_{i}^{2} \omega^{2} } \right)^{2} }},$$

(A.6)

$$\frac{{{\text{d}}J^{\prime}}}{{{\text{d}}\omega }} = - \,2J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} \lambda_{i}^{2} \omega }}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{2} }} ,$$

(A.7)

$$\frac{{{\text{d}}J^{\prime\prime}}}{{{\text{d}}\omega }} = J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} \lambda_{i} \left( {1 - \lambda_{i}^{2} \omega^{2} } \right)}}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{2} }},$$

(A.8)

$$\frac{{{\text{d}}\tan \delta }}{{{\text{d}}\omega }} = \frac{{\frac{{{\text{d}}G^{\prime\prime}}}{{{\text{d}}\omega }} \cdot G^{\prime} - G^{\prime\prime} \cdot \frac{{{\text{d}}G^{\prime}}}{{{\text{d}}\omega }}}}{{ {G^{\prime}}^{2} }}.$$

(A.9)

A.2.2.2 Second order derivatives Mathematical expressions to calculate inflexion points in storage modulus, loss modulus, storage compliance, loss compliances and tangents of the phase angle as a function of frequency, using Prony series, are shown in equations Eqs. (A.10)–(A.14), respectively

$$\frac{{{\text{d}}^{2} G '}}{{{\text{d}}\omega^{2} }} = 2G_{0} \mathop \sum \limits_{i = 1}^{n} p_{i} \tau_{i}^{2} \frac{{1 - 3\tau_{i}^{2} \omega^{2} }}{{\left( {1 + \tau_{i}^{2} \omega^{2} } \right)^{3} }} ,$$

(A.10)

$$\frac{{{{\text{d}}^{2}} {G ^{\prime\prime}}}}{{{{\text{d}}\omega^{2}} }} = 2G_{0} \mathop \sum \limits_{i = 1}^{n} {p_{i}} {\tau_{i}^{3}} \frac{{\omega \left( {{\tau_{i}^{2}} {\omega^{2}} - 3} \right)}}{{\left( {1 + {\tau_{i}^{2}} {\omega^{2}} } \right)^{3} }} ,$$

(A.11)

$$\frac{{{\text{d}}^{2} J '}}{{{\text{d}}\omega^{2} }} = - 2J_{0} \mathop \sum \limits_{i = 1}^{n} q_{i} \lambda_{i}^{2} \frac{{1 - 3\lambda_{i}^{2} \omega^{2} }}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{3} }} ,$$

(A.12)

$$\frac{{{\text{d}}^{2} J ^{\prime\prime}}}{{{\text{d}}\omega^{2} }} = 2J_{0} \mathop \sum \limits_{i = 1}^{n} q_{i} \lambda_{i}^{3} \frac{{\omega \left( {\lambda_{i}^{2} \omega^{2} - 3} \right)}}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{3} }} ,$$

(A.13)

$$\frac{{{\text{d}}^{2} \tan \delta }}{{{\text{d}}\omega^{2} }} = \frac{{G^{\prime}\left( {\frac{{{{\rm d}^{2}} {{{G}}^{\prime\prime}}}}{{{\rm d}\omega^{2} }} \cdot G^{\prime} - G^{\prime\prime} \cdot \frac{{{{\rm d}^{2}} {G^{\prime}}}}{{{\rm d}\omega^{2} }}} \right) - 2\frac{{{\rm d}G^{\prime}}}{{\rm d}\omega } \left( {\frac{{{\rm d}G^{\prime\prime}}}{{\rm d}\omega } \cdot G^{\prime} - G^{\prime\prime} \cdot \frac{{{\rm d}G^{\prime}}}{{\rm d}\omega }} \right)}}{{ {G^{\prime}}^{3} }}.$$

(A.14)

1.3 A.3 Roots of polynomial equations

In this Appendix, the resolution of cubic and quartic polynomial equations is presented.

1.3.1 A.3.1 Roots of the cubic equation $x^{3} + ax^{2} + bx + c = 0$

$$x_{1} = \frac{1}{12}\left[ {\frac{{ - 2^{4/3} \left( {1 + \sqrt 3 {\text{i}}} \right)\left( {a^{2} - 3b} \right) + 2^{2/3} \left( { - 1 + \sqrt 3 i} \right)H^{2} - 4Ha}}{H}} \right],$$

(A.15)

$$x_{2} = \frac{1}{12}\left[ {\frac{{2^{4/3} \left( { - 1 + \sqrt 3 {\text{i}}} \right)\left( {a^{2} - 3b} \right) - 2^{2/3} \left( {1 + \sqrt 3 i} \right)H^{2} - 4Ha}}{H}} \right],$$

(A.16)

$$x_{3} = \frac{1}{6}\left[ {\frac{{2^{4/3} \left( {a^{2} - 3b} \right) + 2^{2/3} H^{2} - 2Ha}}{H}} \right],$$

(A.17)

with

$$H = \sqrt[3]{{ - 2a^{3} + 9ab - 27c + 3\sqrt 3 \sqrt { - a^{2} b^{2} + 4b^{3} + 4a^{3} c - 18abc + 27c^{2} } }}.$$

(A.18)

1.3.2 A.3.2 Roots of the quartic equation $x^{4} + ax^{3} + bx^{2} + cx + d = 0$

$$x_{1} = - \frac{{3a + \sqrt 3 H_{4} }}{12} - \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }},}$$

(A.19)

$$x_{2} = - \frac{{3a + \sqrt 3 H_{4} }}{12} + \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }}} ,$$

(A.20)

$$x_{3} = - \frac{{3a - \sqrt 3 H_{4} }}{12} - \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }}} ,$$

(A.21)

$$x_{4} = - \frac{{3a - \sqrt 3 H_{4} }}{12} + \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }}} ,$$

(A.22)

with

$$H_{1} = - 4\left( {b^{2} - 3ac + 12d} \right)^{3} + \left[ {2b^{3} - 9b\left( {ac + 8d} \right) + 27\left( {c^{2} + a^{2} d} \right)} \right]^{2} ,$$

(A.23)

$$H_{2} = 2b^{3} - 9abc + 27c^{2} + 27a^{2} d - 72bd + \sqrt {H_{1} } ,$$

(A.24)

$$H_{3} = \sqrt[3]{2} \left( {b^{2} - 3ac + 12d} \right),$$

(A.25)

$$H_{4} = \sqrt {3a^{2} - 8b + \frac{{4H_{3} }}{{\sqrt[3]{{H_{2} }}}} + \sqrt[3]{{32H_{2} }}} .$$

(A.26)

Appendix B

2.1 B.1 Relationships with n = 1

See Table 1.

Table 1 Relationships for models with n = 1

2.2 B.2 Relationships with n = 2

See Table 2.

Table 2 Relationships for models with n = 2

With the auxiliary constants: (a) Case “GM parameters known” (1st row in the table) (Eq. (54) in the text)

$$C_{1_M} = \frac{1}{{E_{\infty_M} }},$$

$$C_{2_M} = - \frac{{E_{1_M} + E_{2_M} }}{{E_{{\infty_{M} }} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)}},$$

$$C_{3_M} = \frac{\alpha }{{2\eta_{1_M} \eta_{2_M} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)}},$$

$$C_{4_M} = \frac{{\sqrt {\alpha^{2} - 4E_{{\infty_{M} }} E_{1_M} E_{2_M} \eta_{1_M} \eta_{2_M} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)} }}{{2\eta_{1_M} \eta_{2_M} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)}},$$

$$C_{5_M} = \frac{{E_{1_M} E_{2_M} \left( {\eta_{1_M} + \eta_{2_M} } \right)}}{{\eta_{1_M} \eta_{2_M} \left( {E_{1_M} + E_{2_M} } \right)}},$$

with

$$\alpha = E_{{\infty_{M} }} \left( {E_{1_M} \eta_{2_M} + E_{2_M} \eta_{1_M} } \right) + E_{1_M} E_{2_M} \left( {\eta_{1_M} + \eta_{2_M} } \right).$$

(b) Case “relaxation coefficients known” (2nd row in the table)

$$C_{1_R} = \frac{1}{{Y_{0} \left( {1 - p_{1} - p_{2} } \right)}},$$

$$C_{2_R} = - \frac{{p_{1} + p_{2} }}{{Y_{0} \left( {1 - p_{1} - p_{2} } \right)}},$$

$$C_{3_R} = \frac{\beta }{{2\tau_{1} \tau_{2} }},$$

$$C_{4_R} = \frac{{\sqrt {\beta^{2} - 4\tau_{1} \tau_{2} \left( {1 - p_{1} - p_{2} } \right)} }}{{2\tau_{1} \tau_{2} }},$$

$$C_{5_R} = \frac{{p_{1} \tau_{1} + p_{2} \tau_{2} }}{{\tau_{1} \tau_{2} \left( {p_{1} + p_{2} } \right)}},$$

with

$$\beta = \tau_{1} \left( {1 - p_{2} } \right) + \tau_{2} \left( {1 - p_{1} } \right).$$

(c) Case “creep coefficients known” (3rd row in the table)

$$C_{1_C} = \frac{1}{{J_{0} \left( {1 + q_{1} + q_{2} } \right)}},$$

$$C_{2_C} = \frac{{q_{1} + q_{2} }}{{J_{0} \left( {1 + q_{1} + q_{2} } \right)}},$$

$$C_{3_C} = \frac{\delta }{{2\lambda_{1} \lambda_{2} }},$$

$$C_{4_C} = \frac{{\sqrt {\delta^{2} - 4\lambda_{1} \lambda_{2} \left( {1 + q_{1} + q_{2} } \right)} }}{{2\lambda_{1} \lambda_{2} }},$$

$$C_{5_C} = \frac{{q_{1} \lambda_{1} + q_{2} \lambda_{2} }}{{\lambda_{1} \lambda_{2} \left( {q_{1} + q_{2} } \right) }},$$

with

$$\delta = \lambda_{1} \left( {1 + q_{2} } \right) + \lambda_{2} \left( {1 + q_{1} } \right).$$

(d) Case “GKV parameters known” (4th row in the table) (Eq. (17) in the text)

$$C_{1_K} = \frac{{E_{0_K} E_{1_K} E_{2_K} }}{{E_{0_K} E_{1_K} + E_{0_K} E_{2_K} + E_{1_K} E_{2_K} }},$$

$$C_{2_K} = \frac{{E_{0_K}^{2} \left( {E_{1_K} + E_{2_K} } \right)}}{{E_{0_K} E_{1_K} + E_{0_K} E_{2_K} + E_{1_K} E_{2_K} }},$$

$$C_{3_K} = \frac{\gamma }{{2\eta_{1_K} \eta_{2_K} }},$$

$$C_{4_K} = \frac{{\sqrt {\gamma^{2} - 4\eta_{1_K} \eta_{2_K} \left( {E_{0_K} E_{1_K} + E_{0_K} E_{2_K} + E_{1_K} E_{2_K} } \right)} }}{{2\eta_{1_K} \eta_{2_K} }},$$

$$C_{5_K} = \frac{{E_{1_K}^{2} \eta_{2_K} + E_{2_K}^{2} \eta_{1_K} }}{{\eta_{1_K} \eta_{2_K} \left( {E_{1_K} + E_{2_K} } \right)}},$$

with

$$\gamma = E_{0_K} \left( {\eta_{1_K} + \eta_{2_K} } \right) + E_{1_K} \eta_{2_K} + E_{2_K} \eta_{1_K} .$$