63 lines
3 KiB
Markdown
63 lines
3 KiB
Markdown
([livre](https://musee-stendhal.bm-grenoble.fr/detailstatic.aspx?RSC_BASE=SYRACUSE&RSC_DOCID=1066937&TITLE=initiation-a-la-rheologie-g-couarraze-j-l-grossiord&_lg=fr-FR) ISBN: 2-85206-660-2)
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The study of the material properties that govern the flow and deformation of matter.
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[[Continuum Mechanics]] is a foundation of rheology
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The fundamental [[dynamics vs kinematics|dynamic physical quantity]] is the shear stress $\tau$, and the fundamentals kinematic physical quantities are shear strain $\varepsilon$ and the shear rate $\dot{\varepsilon}$ .
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## Laminar shear displacement
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The material presents a laminate structure, with infinity thin adjacent layers;
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The strain occurs as a slipping of the different layers to each other, without transfer of matter between layers;
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It is an ordered and layered movement, without mixage of the material nor variation of its volume.
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-> friction between layers -> shear forces / unit of volume -> shear stress
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## Boundary layer hypothesis
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The layer of a material touching a solid surface is bound to it and moves with the same speed.
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So between a moving plate and a static plate, the viscous materials follows a laminar shear displacement, and a gradient of speed appears between the boundary layers.
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## Shear strain
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$$\varepsilon(x,t)=\frac{du(x,t)}{dx}$$
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With the x axis being the normal to the stacked layers. So $\varepsilon$ does not depends on the displacement $u$ but the variation of the displacement between two adjacent layers.
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Usually the strain depends on the position within the material, but in some cases the variation of displacement between layers is constant, and depends only on time.
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![[shear strain]]
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## Shear rate
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$$\dot{\varepsilon}=\frac{d\varepsilon}{dt}=\frac{d}{dt}\frac{du}{dx}=\frac{d}{dx}\frac{du}{dt}$$
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so $\dot{\varepsilon}$ is often called the speed gradient, and denoted with the letter $D$
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$$\dot{\varepsilon}(x,t)=\frac{dv(x,t)}{dx}$$
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## Rheologic state equation
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$$\varepsilon=f[\tau]$$
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The goal of rheology is to find such an equation for a given material, through experiments. (also, $\varepsilon=f[\tau,P,T]$ would be more exact)
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## Dynamic viscosity
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$$\mu=\frac{\tau}{\dot{\varepsilon}}$$
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Newtonian fluids have a $\mu$ than does not depend on shear stress, their dynamic viscosity is then called absolute viscosity and denoted $\eta$ . The rheologic state function of a Newtonian body is then :
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$$\dot{\varepsilon}=\frac{\tau}{\eta}$$
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## Linear viscosity
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$\varepsilon=f[\tau]$ is linear
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definitions :
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- the creep function $f(t)$ is the strain when a constant unit stress is applied
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- the relaxation function $g(t)$ is the stress when a constant unit strain is applied
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$f$ and $g$ depends on the material and need to be determined experimentally
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![[creep and relaxation]]
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$$\varepsilon(t)=\tau(t)f(0)+\int_{0}^{t}\dot{f}(t-t')\tau(t')dt'$$
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where
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$$\dot{f}(u)\equiv \frac{df}{du}(u)$$
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$$\tau(t)=\varepsilon(t)g(0)+\int_{0}^{t}\dot{g}(t-t')\varepsilon(t')dt'$$
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where
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$$\dot{g}(u)\equiv \frac{dg}{du}(u)$$
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## Viscoelastic models
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(see [[Viscoelasticity#Models]]) |