Maxwell model#Effect of a sudden deformation
{{Short description|Model of viscoelastic material}}
{{Use dmy dates|date=November 2017}}
A Maxwell model is the most simple model viscoelastic material showing properties of a typical liquid.{{cite book |doi=10.1016/bs.aams.2022.09.003 |quote=The Maxwell model is a type of simplest and basic mathematical model to describe the mechanics characteristic of viscoelastic solid material. |chapter=Mechanics constitutive models for viscoelastic solid materials: Development and a critical review |title=Advances in Applied Mechanics |date=2023 |last1=Zhou |first1=Xiaoqiang |last2=Yu |first2=Daoyuan |last3=Barrera |first3=Olga |volume=56 |pages=189–321 |isbn=978-0-323-99248-0 }} It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations.{{cite report |last1=Roylance |first1=David |date=24 October 2001 |title=Engineering Viscoelasticity |pages=8–11 |url=http://web.mit.edu/course/3/3.11/www/modules/visco.pdf }}{{self-published inline|date=May 2025}} It is named for James Clerk Maxwell who proposed the model in 1867.{{cite journal |last1=Boyaval |first1=Sébastien |title=Viscoelastic flows of Maxwell fluids with conservation laws |journal=ESAIM: Mathematical Modelling and Numerical Analysis |date=May 2021 |volume=55 |issue=3 |pages=807–831 |doi=10.1051/m2an/2020076 |arxiv=2007.16075 }}{{cite journal |title=IV. On the dynamical theory of gases |journal=Philosophical Transactions of the Royal Society of London |date=31 December 1867 |volume=157 |pages=49–88 |doi=10.1098/rstl.1867.0004 }} It is also known as a Maxwell fluid. A generalization of the scalar relation to a tensor equation lacks motivation from more microscopic models and does not comply with the concept of material objectivity. However, these criteria are fulfilled by the Upper-convected Maxwell model.{{fact|date=May 2025}}
Definition
The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,{{cite book |doi=10.1016/B978-0-12-174252-2.50005-3 |chapter=Viscoelastic Stress Strain Constitutive Relations |title=Theory of Viscoelasticity |date=1982 |last1=Christensen |first1=R.M. |pages=1–34 |isbn=978-0-12-174252-2 }} as shown in the diagram. If, instead, we connect these two elements in parallel, we get the generalized model of a solid Kelvin–Voigt material.
In Maxwell configuration, under an applied axial stress, the total stress, and the total strain, can be defined as follows:
:
:
where the subscript D indicates the stress–strain in the damper and the subscript S indicates the stress–strain in the spring. Taking the derivative of strain with respect to time, we obtain:
:
where E is the elastic modulus and η is the material coefficient of viscosity. This model describes the damper as a Newtonian fluid and models the spring with Hooke's law.
In a Maxwell material, stress σ, strain ε and their rates of change with respect to time t are governed by equations of the form:
:
or, in dot notation:
:
The equation can be applied either to the shear stress or to the uniform tension in a material. In the former case, the viscosity corresponds to that for a Newtonian fluid. In the latter case, it has a slightly different meaning relating stress and rate of strain.
The model is usually applied to the case of small deformations. For the large deformations we should include some geometrical non-linearity. For the simplest way of generalizing the Maxwell model, refer to the upper-convected Maxwell model.
Effect of a sudden deformation
If a Maxwell material is suddenly deformed and held to a strain of , then the stress decays on a characteristic timescale of , known as the relaxation time. The phenomenon is known as stress relaxation.
The picture shows dependence of dimensionless stress
upon dimensionless time :
If we free the material at time , then the elastic element will spring back by the value of
:
Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:
:
Effect of a sudden stress
If a Maxwell material is suddenly subjected to a stress , then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
:
If at some time we released the material, then the deformation of the elastic element would be the spring-back deformation and the deformation of the viscous element would not change:
:
:
The Maxwell model does not exhibit creep since it models strain as linear function of time.
If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Thus, Maxwell material is a type of liquid.
Effect of a constant strain rate
If a Maxwell material is subject to a constant strain rate then the stress increases, reaching a constant value of
In general
Dynamic modulus
Image:Maxwell relax spectra.PNG
The complex dynamic modulus of a Maxwell material would be:
:
Thus, the components of the dynamic modulus are :
:
and
:
The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is .
border="1" cellspacing="0"
| Blue curve | dimensionless elastic modulus |
Pink curve | dimensionless modulus of losses |
Yellow curve | dimensionless apparent viscosity |
X-axis | dimensionless frequency . |
See also
References
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{{DEFAULTSORT:Maxwell Material}}