Rheol Acta (2008) 47:159–167 DOI 10.1007/s00397-007-0223-6 ORIGINAL CONTRIBUTION On the sensitivity of interconversion between relaxation and creep R. S. Anderssen · A. R. Davies · F. R. de Hoog Received: 24 March 2007 / Accepted: 1 July 2007 / Published online: 31 October 2007 © Springer-Verlag 2007 Abstract The interconversion equation of linear vis- coelasticity defines implicitly the interrelations between the relaxation and creep functions G(t) and J (t). It is widely utilised in rheology to estimate J (t) from mea- surements of G(t) and conversely. Because different molecular details can be recovered from G(t) and J (t), it is necessary to work with both. This leads naturally to the need to identify whether it is better to first mea- sure G(t) and then determine J (t) or conversely. This requires an understanding of the stability (sensitivity) of the recovery of J (t) from G(t) compared with that of G(t) from J (t). Although algorithms are available that work adequately in both directions, numerical exper- imentation strongly suggests that the recovery of J (t) from G(t) measurements is the more stable. An ele- mentary theoretical rationale has been given recently by Anderssen et al. (ANZIAM J 48:C346–C363, 2007) for single exponential models of G(t) and J (t). It ex- plicitly exploits the simple algebra of such functions. In this paper, corresponding bounds are derived that hold for arbitrary sums of exponentials. The paper concludes with a discussion, from a practical rheological perspec- tive, about the implications and implementations of the results. R. S. Anderssen (B ) · F. R. de Hoog CSIRO Mathematical and Information Sciences, GPO Box 664, Canberra ACT 2601, Australia e-mail: Bob.Anderssen@csiro.au A. R. Davies School of Mathematics, Cardiff University, Wales, UK Keywords Integral constitutive equation · Laplace transformation · Relaxation modulus · Creep · Relaxation · Linear viscoelasticity Introduction For the characterisation of the linear viscoelastic behaviour of a given material, the key functions are the relaxation and creep (retardation) moduli, G(t) and J (t). They correspond to the kernels in the Boltzmann causal integral equation models of that material’s stress–strain response in relaxation and creep experi- ments (Ferry 1980) σ(t) =  t 0 G(t − s) ˙ γ(s)ds ˙ γ(t) = dγ(t) dt , (1) and γ(t) =  t 0 J (t − s) ˙ σ(s)ds ˙ σ(t) = dσ(t) dt , (2) where σ(t) and γ(t) denote, respectively, the stress and strain with σ(t) = γ(t) = 0 for −∞ < t ≤ 0. Normally, for t > 0, the functions G(t) and J (t) are defined, respectively, in terms of a relaxation spectrum H(τ) and a retardation spectrum L(τ) G(t) = G(∞) +  ∞ 0 exp(−t/τ) H(τ) τ dτ, H(τ) ≥ 0, (3)