The generalized Maxwell model also known as the Maxwell–Wiechert model (after James Clerk Maxwell and E Wiechert[1][2]) is the most general form of the linear model for viscoelasticity. In this model, several Maxwell elements are assembled in parallel. It takes into account that the relaxation does not occur at a single time, but in a set of times. Due to the presence of molecular segments of different lengths, with shorter ones contributing less than longer ones, there is a varying time distribution. The Wiechert model shows this by having as many spring–dashpot Maxwell elements as are necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.[3][4]
The generalized Maxwell model is widely applied to describe how materials deform under mechanical stress when both elastic and viscous effects are present. It assumes linear viscoelastic behavior and is suitable for cases involving small deformations.[5] Because of its ability to represent complex time-dependent responses, the model is commonly used in the study of polymers, soft tissues, and other viscoelastic solids.[6] The model can be expressed either in the time domain using a relaxation function or in the frequency domain through a complex modulus, making it adaptable for use in experimental and computational analyses. In engineering practice, it is often implemented using a Prony series to simulate viscoelastic behavior in finite element analysis.[7][8]
- ^ Wiechert, E (1889); "Ueber elastische Nachwirkung", Dissertation, Königsberg University, Germany
- ^ Wiechert, E (1893); "Gesetze der elastischen Nachwirkung für constante Temperatur", Annalen der Physik, Vol. 286, issue 10, p. 335–348 and issue 11, p. 546–570
- ^ Roylance, David (2001); "Engineering Viscoelasticity", 14-15
- ^ Tschoegl, Nicholas W. (1989); "The Phenomenological Theory of Linear Viscoelastic Behavior", 119-126
- ^ Petrie, Christopher J. S. (1977-05-01). "On stretching maxwell models". Journal of Non-Newtonian Fluid Mechanics. 2 (3): 221–253. doi:10.1016/0377-0257(77)80002-5. ISSN 0377-0257.
- ^ Song, Jake; Holten-Andersen, Niels; McKinley, Gareth H. (2023). "Non-Maxwellian viscoelastic stress relaxations in soft matter". Soft Matter. 19 (41): 7885–7906. doi:10.1039/D3SM00736G. ISSN 1744-683X.
- ^ Luk-Cyr, Jacques; Crochon, Thibaut; Li, Chun; Lévesque, Martin (2012-07-06). "Interconversion of linearly viscoelastic material functions expressed as Prony series: a closure". Mechanics of Time-Dependent Materials. 17 (1): 53–82. doi:10.1007/s11043-012-9176-y. ISSN 1385-2000.
- ^ Chen, T. "Determining a Prony Series for a viscoelastic material from time varying strain data". Technical Report, NASA (2000).
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