On the oscillatority of viscoelastic functions

Interconversions between linear viscoelastic functions with a time-dependent bulk modulus

“…In addition, the two special cases of equation (72) are the model used by Leadermann [28] and the model used by Rabotnov [29] Equations (71) and (72) are linear equations respective to Σ ( t , σ ) and E ( t , ε ), although they are nonlinear ones respective to σ ( t ) and ε ( t ). By taking the Sumudu transform to equations (71) and (72), and multiplying the projections E (1) and E (2) given by equations (11), respectively and it can be seen that the inverse relations are hold One may use equations (14) to define E ( t ) and ν ( t ), and the interconversion relations that discussed in Section 2 still hold. It should be noted that a stress- (strain-) controlled characterization [11] in which σ 11 ( t ) ≠ 0 ( ε 11 ( t ) ≠ 0 ) and σ ij ≠ 11 ( t ) = 0 ( ε ij ≠ 11 ( t ) = 0 ) may not lead to σ 11 ( …”

“…It should be noted that a stress- (strain-) controlled characterization [11] in which σ 11 ( t ) ≠ 0 ( ε 11 ( t ) ≠ 0 ) and σ ij ≠ 11 ( t ) = 0 ( ε ij ≠ 11 ( t ) = 0 ) may not lead to σ 11 ( t , σ 11 ) ≠ 0 ( E 11 ( t , ε 11 ) ≠ 0 ) and Σ ij ≠ 11 ( t , σ 11 ) = 0 ( …”

“…As shown in equations (11) and (13), the stress responses of volumetric change, represented by 3K(t), and deviatoric change, represented by 2G(t), of a viscoelastic material can be expressed independently. These functions and their Stieltjes inverses, which are the eigen-functions of the relaxation or creep tensors, can be recognized as the fundamental viscoelastic functions (analogic to the fundamental elastic constants described by Tschoegl et al [6]).…”

“…It should be noted that A ( t ) in equation (15) is monotonic if all 0 ≤ a n A ≤ 1 , a constant if A 0 = A ∞ , a relaxation modulus if A 0 > A ∞ , a creep compliance if A 0 < A ∞ [2], non-monotonic if some of a n A < 0 or a n A > 1 [10], and oscillatory if some of τ n A are complex-valued [11].…”

“…A ' , a creep compliance if A 0 \A ' [2], non-monotonic if some of a A n \0 or a A n . 1 [10], and oscillatory if some of t A n are complex-valued [11]. In this paper, K(t) is considered as time-dependent, and the interconversion relations between viscoelastic functions are derived.…”