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<p>The interaction between the slow and fast eigenmodes of a viscoelastic material gives rise to “dampened elasticity,” playing a crucial role in both technical applications and natural processes. This phenomenon occurs during the flow of a viscoelastic liquid or the deformation of a viscoelastic solid, as slow eigenmodes work to elastically restore the material’s previous shapes, while fast relaxation modes resist any such elastic recovery by attempting to preserve the material’s current shape. In this way, fast modes create a viscous background that dampens elastic recovery. Elastic-dominated and viscous-dominated stress components act across a sequence of process times, 0 &lt; <i>s</i> &lt; ∞, when probing eigenmodes with a spectrum of relaxation times spanning 0 &lt; <i>τ</i> &lt; <i>τ</i><sub>max</sub>. Classification of an eigenmode as fast or slow depends on the respective value of the Deborah function, <i>D</i>(<i>τ</i>;<i>s</i>) = <i>τ</i>/<i>s</i>, a key parameter introduced here. The critical value <i>D</i> = 1 separates the eigenmodes into two groups, those dominated by elasticity (<i>D</i> &gt; 1) and those dominated by viscosity (<i>D</i> &lt; 1). The resulting ratio of elastic to viscous stress, referred to as elastic-to-viscous ratio EVR, defines the viscoelastic state of soft matter under specific flow or strain conditions. The EVR(<Emphasis Type="BoldItalic">x</Emphasis>,<i>t</i>) can vary with position (<Emphasis Type="BoldItalic">x</Emphasis>) and/or evolve over time. Additionally, the damping potential, <i>P</i><sub><i>d</i></sub>, defines the ratio of transient to permanent elasticity in solids. Illustrating examples involve gels and the process of gelation with its characteristic evolution of relaxation times. The distinction between elasticity-dominated and viscosity-dominated stress is equally applicable to linear and non-linear viscoelasticity.</p> Graphical Abstract <p></p>

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Dampened elasticity of gels and the Deborah function

  • 期刊论文

  • 发表日期：

    2025年5月6日

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  • H. Henning Winter

摘要

The interaction between the slow and fast eigenmodes of a viscoelastic material gives rise to “dampened elasticity,” playing a crucial role in both technical applications and natural processes. This phenomenon occurs during the flow of a viscoelastic liquid or the deformation of a viscoelastic solid, as slow eigenmodes work to elastically restore the material’s previous shapes, while fast relaxation modes resist any such elastic recovery by attempting to preserve the material’s current shape. In this way, fast modes create a viscous background that dampens elastic recovery. Elastic-dominated and viscous-dominated stress components act across a sequence of process times, 0 < s < ∞, when probing eigenmodes with a spectrum of relaxation times spanning 0 < τ < τmax. Classification of an eigenmode as fast or slow depends on the respective value of the Deborah function, D(τ;s) = τ/s, a key parameter introduced here. The critical value D = 1 separates the eigenmodes into two groups, those dominated by elasticity (D > 1) and those dominated by viscosity (D < 1). The resulting ratio of elastic to viscous stress, referred to as elastic-to-viscous ratio EVR, defines the viscoelastic state of soft matter under specific flow or strain conditions. The EVR(x,t) can vary with position (x) and/or evolve over time. Additionally, the damping potential, Pd, defines the ratio of transient to permanent elasticity in solids. Illustrating examples involve gels and the process of gelation with its characteristic evolution of relaxation times. The distinction between elasticity-dominated and viscosity-dominated stress is equally applicable to linear and non-linear viscoelasticity.

Graphical Abstract