Shear stress

Further information:Shear force

Shear stress(often denoted byτ,Greek:tau) is the component ofstresscoplanarwith amaterial cross section.It arises from theshear force,the component offorcevector parallel to the material cross section.Normal stress,on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

Shear stress
Common symbols	τ
SI unit	pascal
Derivations from other quantities	τ=⁠F/A⁠

A shearing force is applied to the top of the rectangle while the bottom is held in place. The resulting shear stress,τ,deforms the rectangle into aparallelogram.The area involved would be the top of the parallelogram.

General shear stress

The formula to calculate average shear stressτor force per unit area is: $$\tau ={F \over A},$$whereFis the force applied andAis the cross-sectional area.

The area involved corresponds to the materialfaceparallel to the applied force vector, i.e., with surfacenormal vectorperpendicular to the force.

Other forms

Wall shear stress

Wall shear stressexpresses the retarding force (per unit area) from a wall in the layers of a fluid flowing next to the wall. It is defined as:$$\tau _{w}:=\mu \left.{\frac {\partial u}{\partial y}}\right|_{y=0},$$whereμis thedynamic viscosity,uis the flow velocity, andyis the distance from the wall.

It is used, for example, in the description of arterialblood flow,where there is evidence that it affects theatherogenicprocess.

Pure

Pure shearstress is related to pureshear strain,denotedγ,by the equation$$\tau =\gamma G,$$whereGis theshear modulusof theisotropicmaterial, given by$$G={\frac {E}{2(1+\nu )}}.$$Here,EisYoung's modulusandνisPoisson's ratio.

Beam shear

Beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam:$$\tau:={\frac {fQ}{Ib}},$$where

• fis the total shear force at the location in question,
• Qis thestatical moment of area,
• bis the thickness (width) in the material perpendicular to the shear, and
• Iis themoment of inertiaof the entire cross-sectional area.

The beam shear formula is also known as Zhuravskii shear stress formula afterDmitrii Ivanovich Zhuravskii,who derived it in 1855.

Semi-monocoque shear

Further information:Shear flow

Shear stresses within asemi-monocoquestructure may be calculated by idealizing the cross-section of the structure into a set of stringers (carrying only axial loads) and webs (carrying onlyshear flows). Dividing the shear flow by the thickness of a given portion of the semi-monocoque structure yields the shear stress. Thus, the maximum shear stress will occur either in the web of maximum shear flow or minimum thickness.

Constructions in soil can also fail due to shear;e.g.,the weight of an earth-filleddamordikemay cause the subsoil to collapse, like a smalllandslide.

Impact shear

The maximum shear stress created in a solid round bar subject to impact is given by the equation$$\tau =2{\sqrt {\frac {UG}{V}}},$$where

• Uis the change in kinetic energy,
• Gis theshear modulus,and
• Vis the volume of the rod.

Furthermore,

U=U_rotating+U_applied,

where

• U_rotating=⁠1/2⁠Iω²,
• U_applied=Tθ_displaced,
• Iis the mass moment of inertia, and
• ωis the angular speed.

Shear stress in fluids

See also:Viscosity,Couette flow,Hagen-Poiseuille equation,Depth-slope product,Simple shear,andWind stress

Any realfluids(liquidsandgasesincluded) moving along a solid boundary will incur a shear stress at that boundary. Theno-slip conditiondictates that the speed of the fluid at the boundary (relative to the boundary) is zero; although at some height from the boundary, the flow speed must equal that of the fluid. The region between these two points is named theboundary layer.For allNewtonian fluidsinlaminar flow,the shear stress is proportional to thestrain ratein the fluid, where the viscosity is the constant of proportionality. Fornon-Newtonian fluids,theviscosityis not constant. The shear stress is imparted onto the boundary as a result of this loss of velocity.

For a Newtonian fluid, the shear stress at a surface element parallel to a flat plate at the pointyis given by$$\tau (y)=\mu {\frac {\partial u}{\partial y}},$$where

• μis thedynamic viscosityof the flow,
• uis theflow velocityalong the boundary, and
• yis the height above the boundary.

Specifically, the wall shear stress is defined as$$\tau _{\mathrm {w} }:=\tau (y=0)=\mu \left.{\frac {\partial u}{\partial y}}\right|_{y=0}~.$$Newton's constitutive law,for any general geometry (including the flat plate above mentioned), states that shear tensor (a second-order tensor) is proportional to the flow velocitygradient(the velocity is a vector, so its gradient is a second-order tensor):$${\boldsymbol {\tau }}(\mathbf {u} )=\mu {\boldsymbol {\nabla }}\mathbf {u}.$$The constant of proportionality is nameddynamic viscosity.For an isotropic Newtonian flow, it is a scalar, while for anisotropic Newtonian flows, it can be a second-order tensor. The fundamental aspect is that for a Newtonian fluid, the dynamic viscosity is independent of flow velocity (i.e., the shear stress constitutive law islinear), while for non-Newtonian flows this is not true, and one should allow for the modification$${\boldsymbol {\tau }}(\mathbf {u} )=\mu (\mathbf {u} ){\boldsymbol {\nabla }}\mathbf {u}.$$This no longer Newton's law but a generic tensorial identity: one can always find an expression of the viscosity as function of the flow velocity given any expression of the shear stress as function of the flow velocity. On the other hand, given a shear stress as function of the flow velocity, it represents a Newtonian flow only if it can be expressed as a constant for the gradient of the flow velocity. The constant one finds in this case is the dynamic viscosity of the flow.

Example

Considering a 2D space in Cartesian coordinates(x,y)(the flow velocity components are respectively(u,v)), the shear stress matrix given by$${\begin{pmatrix}\tau _{xx}&\tau _{xy}\\\tau _{yx}&\tau _{yy}\end{pmatrix}}={\begin{pmatrix}x{\frac {\partial u}{\partial x}}&0\\0&-t{\frac {\partial v}{\partial y}}\end{pmatrix}}$$represents a Newtonian flow; in fact it can be expressed as$${\begin{pmatrix}\tau _{xx}&\tau _{xy}\\\tau _{yx}&\tau _{yy}\end{pmatrix}}={\begin{pmatrix}x&0\\0&-t\end{pmatrix}}\cdot {\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}},$$i.e., an anisotropic flow with the viscosity tensor$${\begin{pmatrix}\mu _{xx}&\mu _{xy}\\\mu _{yx}&\mu _{yy}\end{pmatrix}}={\begin{pmatrix}x&0\\0&-t\end{pmatrix}},$$which is nonuniform (depends on space coordinates) and transient, but is independent of the flow velocity:$${\boldsymbol {\mu }}(x,t)={\begin{pmatrix}x&0\\0&-t\end{pmatrix}}.$$This flow is therefore Newtonian. On the other hand, a flow in which the viscosity was$${\begin{pmatrix}\mu _{xx}&\mu _{xy}\\\mu _{yx}&\mu _{yy}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{u}}&0\\0&{\frac {1}{u}}\end{pmatrix}}$$is non-Newtonian since the viscosity depends on flow velocity. This non-Newtonian flow is isotropic (the matrix is proportional to the identity matrix), so the viscosity is simply a scalar:$$\mu (u)={\frac {1}{u}}.$$

Measurement with sensors

Diverging fringe shear stress sensor

This relationship can be exploited to measure the wall shear stress. If a sensor could directly measure the gradient of the velocity profile at the wall, then multiplying by the dynamic viscosity would yield the shear stress. Such a sensor was demonstrated by A. A. Naqwi and W. C. Reynolds.The interference pattern generated by sending a beam of light through two parallel slits forms a network of linearly diverging fringes that seem to originate from the plane of the two slits (seedouble-slit experiment). As a particle in a fluid passes through the fringes, a receiver detects the reflection of the fringe pattern. The signal can be processed, and from the fringe angle, the height and velocity of the particle can be extrapolated. The measured value of the wall velocity gradient is independent of the fluid properties, and as a result does not require calibration. Recentadvancements in the micro-optic fabrication technologies have made it possible to use integrated diffractive optical elements to fabricate diverging fringe shear stress sensors usable both in air and liquid.

Micro-pillar shear-stress sensor

A further measurement technique is that of slender wall-mounted micro-pillars made of the flexible polymerpolydimethylsiloxane,which bend in reaction to the applying drag forces in the vicinity of the wall. The sensor thereby belongs to the indirect measurement principles relying on the relationship between near-wall velocity gradients and the local wall-shear stress.

Electro-diffusional method

The electro-diffusional method measures the wall shear rate in the liquid phase from microelectrodes under limiting diffusion current conditions. A potential difference between an anode of a broad surface (usually located far from the measuring area) and the small working electrode acting as a cathode leads to a fast redox reaction. The ion disappearance occurs only on the microprobe active surface, causing the development of the diffusion boundary layer, in which the fast electro-diffusion reaction rate is controlled only by diffusion. The resolution of the convective-diffusive equation in the near-wall region of the microelectrode lead to analytical solutions relying the characteristics length of the microprobes, the diffusional properties of the electrochemical solution, and the wall shear rate.

See also

• Critical resolved shear stress
• Direct shear test
• Friction
• Shear and moment diagrams
• Shear rate
• Shear strain
• Shear strength
• Tensile stress
• Triaxial shear test