# Shear stress

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

A shear stress, often denoted by τ (Greek: tau), is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician alphabet, and was the first alphabetic script to have distinct letters for vowels as well as consonants. In Archaic and early Classical times, the Greek alphabet existed in many different local variants, but, by the end of the fourth century BC, the Eucleidean alphabet, with twenty-four letters, ordered from alpha to omega, had become standard and it is this version that is still used to write Greek today. These twenty-four letters are: Α α, Β β, Γ γ, Δ δ, Ε ε, Ζ ζ, Η η, Θ θ, Ι ι, Κ κ, Λ λ, Μ μ, Ν ν, Ξ ξ, Ο ο, Π π, Ρ ρ, Σ σ/ς, Τ τ, Υ υ, Φ φ, Χ χ, Ψ ψ, and Ω ω.

Tau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.

## Contents

Shear stress arises from shear forces, which are pairs of equal and opposing forces acting on opposite sides of an object.

## General shear stress

The formula to calculate average shear stress is force per unit area.: [1]

${\displaystyle \tau ={F \over A},}$

where:

τ = the shear stress;
F = the force applied;
A = the cross-sectional area of material with area parallel to the applied force vector.

## Other forms

### Pure

Pure shear stress is related to pure shear strain, denoted γ, by the following equation: [2]

In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly. For soft materials, such as rubber, a strain state of pure shear is often used for characterizing hyperelastic and fracture mechanical behaviour. Pure shear is differentiated from simple shear in that pure shear involves no rigid body rotation.

${\displaystyle \tau =\gamma G\,}$

where G is the shear modulus of the isotropic material, given by

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain:

${\displaystyle G={\frac {E}{2(1+\nu )}}.}$

Here E is Young's modulus and ν is Poisson's ratio.

Young's modulus or Young modulus is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress and strain in a material in the linear elasticity regime of a uniaxial deformation.

Poisson's ratio, denoted by the Greek letter ('nu'), and named after the French mathematician and physicist Siméon Poisson, is the negative of the ratio of (signed) transverse strain to (signed) axial strain. For small values of these changes, is the amount of transversal expansion divided by the amount of axial compression.

### Beam shear

Beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam.

${\displaystyle \tau ={fQ \over Ib},}$

where

f = total shear force at the location in question;
Q = statical moment of area;
b = thickness (width) in the material perpendicular to the shear;
I = Moment of Inertia of the entire cross sectional area.

The beam shear formula is also known as Zhuravskii shear stress formula after Dmitrii Ivanovich Zhuravskii who derived it in 1855. [3] [4]

Dmitrii Ivanovich Zhuravskii (1821–1891) was a Russian engineer who was one of the pioneers of bridge construction and structural mechanics in Russia.

### Semi-monocoque shear

Shear stresses within a semi-monocoque structure may be calculated by idealizing the cross-section of the structure into a set of stringers (carrying only axial loads) and webs (carrying only shear 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

Also constructions in soil can fail due to shear; e.g., the weight of an earth-filled dam or dike may cause the subsoil to collapse, like a small landslide.

### Impact shear

The maximum shear stress created in a solid round bar subject to impact is given as the equation:

${\displaystyle \tau ={\sqrt {2UG \over V}},}$

where

U = change in kinetic energy;
G = shear modulus;
V = volume of rod;

and

U = Urotating + Uapplied;
Urotating = 1/22;
Uapplied = displaced;
I = mass moment of inertia;
ω = angular speed.

### Shear stress in fluids

Any real fluids (liquids and gases included) moving along a solid boundary will incur a shear stress at that boundary. The no-slip condition [5] dictates 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 the boundary layer. For all Newtonian fluids in laminar flow, the shear stress is proportional to the strain rate in the fluid, where the viscosity is the constant of proportionality. For non-Newtonian fluids, the viscosity is 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 point y is given by:

${\displaystyle \tau (y)=\mu {\frac {\partial u}{\partial y}}}$

where

μ is the dynamic viscosity of the flow;
u is the flow velocity along the boundary;
y is the height above the boundary.

Specifically, the wall shear stress is defined as:

${\displaystyle \tau _{\mathrm {w} }\equiv \tau (y=0)=\mu \left.{\frac {\partial u}{\partial y}}\right|_{y=0}~~.}$

The 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 velocity gradient (the velocity is a vector, so its gradient is a second-order tensor):

${\displaystyle \mathbf {\tau } ({\vec {u}})=\mu \nabla {\vec {u}}}$

and the constant of proportionality is named dynamic viscosity. For an isotropic Newtonian flow it is a scalar, while for anisotropic Newtonian flows it can be a second-order tensor too. The fundamental aspect is that for a Newtonian fluid the dynamic viscosity is independent on flow velocity (i.e., the shear stress constitutive law is linear), while non-Newtonian flows this is not true, and one should allow for the modification:

${\displaystyle \mathbf {\tau } ({\vec {u}})=\mu ({\vec {u}})\nabla {\vec {u}}}$

The above formula is no longer the Newton's law but a generic tensorial identity: one could 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:

${\displaystyle {\begin{pmatrix}\tau _{xx}&\tau _{xy}\\\tau _{yx}&\tau _{yy}\end{pmatrix}}={\begin{pmatrix}x{\frac {\partial u}{\partial x}}&0\\0&-t{\frac {\partial v}{\partial y}}\end{pmatrix}}}$

represents a Newtonian flow, in fact it can be expressed as:

${\displaystyle {\begin{pmatrix}\tau _{xx}&\tau _{xy}\\\tau _{yx}&\tau _{yy}\end{pmatrix}}={\begin{pmatrix}x&0\\0&-t\end{pmatrix}}\cdot {\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}}$,

i.e., an anisotropic flow with the viscosity tensor:

${\displaystyle {\begin{pmatrix}\mu _{xx}&\mu _{xy}\\\mu _{yx}&\mu _{yy}\end{pmatrix}}={\begin{pmatrix}x&0\\0&-t\end{pmatrix}}}$

which is nonuniform (depends on space coordinates) and transient, but relevantly it is independent on the flow velocity:

${\displaystyle \mathbf {\mu } (x,t)={\begin{pmatrix}x&0\\0&-t\end{pmatrix}}}$

This flow is therefore newtonian. On the other hand, a flow in which the viscosity were:

${\displaystyle {\begin{pmatrix}\mu _{xx}&\mu _{xy}\\\mu _{yx}&\mu _{yy}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{u}}&0\\0&{\frac {1}{u}}\end{pmatrix}}}$

is Nonnewtonian since the viscosity depends on flow velocity. This nonnewtonian flow is isotropic (the matrix is proportional to the identity matrix), so the viscosity is simply a scalar:

${\displaystyle \mu (u)={\frac {1}{u}}}$

## 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. [6] 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 (see double-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 knowing the fringe angle, the height and velocity of the particle can be extrapolated. The measured value of wall velocity gradient is independent of the fluid properties and as a result does not require calibration. Recent advancements in the micro-optic fabrication technologies have made it possible to use integrated diffractive optical element to fabricate diverging fringe shear stress sensors usable both in air and liquid. [7]

### Micro-pillar shear-stress sensor

A further measurement technique is that of slender wall-mounted micro-pillars made of the flexible polymer PDMS, 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. [8] [9]

## Related Research Articles

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

In physics and fluid mechanics, a boundary layer is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.

A Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly proportional to the local strain rate—the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.

A power-law fluid, or the Ostwald–de Waele relationship, is a type of generalized Newtonian fluid for which the shear stress, τ, is given by

A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after Eugene C. Bingham who proposed its mathematical form.

Hemorheology, also spelled haemorheology, or blood rheology, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The configuration often takes the form of two parallel plates or the gap between two concentric cylinders. The flow is driven by virtue of viscous drag force acting on the fluid, but may additionally be motivated by an applied pressure gradient in the flow direction. The Couette configuration models certain practical problems, like flow in lightly loaded journal bearings, and is often employed in viscometry and to demonstrate approximations of reversibility. This type of flow is named in honor of Maurice Couette, a Professor of Physics at the French University of Angers in the late 19th century.

Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology.

In physics and fluid mechanics, a Blasius boundary layer describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow, i.e. flows in which the plate is not parallel to the flow.

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.

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Apparent viscosity is the shear stress applied to a fluid divided by the shear rate. For a Newtonian fluid, the apparent viscosity is constant, and equal to the Newtonian viscosity of the fluid, but for non-Newtonian fluids, the apparent viscosity depends on the shear rate. Apparent viscosity has the SI derived unit Pa·s (Pascal-second, but the centipoise is frequently used in practice:.

The Herschel–Bulkley fluid is a generalized model of a non-Newtonian fluid, in which the strain experienced by the fluid is related to the stress in a complicated, non-linear way. Three parameters characterize this relationship: the consistency k, the flow index n, and the yield shear stress . The consistency is a simple constant of proportionality, while the flow index measures the degree to which the fluid is shear-thinning or shear-thickening. Ordinary paint is one example of a shear-thinning fluid, while oobleck provides one realization of a shear-thickening fluid. Finally, the yield stress quantifies the amount of stress that the fluid may experience before it yields and begins to flow.

The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.

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Particle-laden flows refers to a class of two-phase fluid flow, in which one of the phases is continuously connected and the other phase is made up of small, immiscible, and typically dilute particles. Fine aerosol particles in air is an example of a particle-laden flow; the aerosols are the dispersed phase, and the air is the carrier phase.

The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.

Skin friction drag is a component of profile drag, which is resistant force exerted on an object moving in a fluid. Skin friction drag is caused by the viscosity of fluids and is developed from laminar drag to turbulent drag as a fluid moves on the surface of an object. Skin friction drag is generally expressed in term of the Reynolds number, which is the ratio between inertial force and viscous force.

An important class of non-Newtonian fluids presents a yield stress limit which must be exceeded before significant deformation can occur – the so-called viscoplastic fluids or Bingham plastics. In order to model the stress-strain relation in these fluids, some fitting have been proposed such as the linear Bingham equation and the non-linear Herschel-Bulkley and Casson models.

## References

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4. "Flexure of Beams" (PDF). Mechanical Engineering Lectures. McMaster University.
5. Day, Michael A. (2004), The no-slip condition of fluid dynamics, Springer Netherlands, pp. 285–296, ISSN   0165-0106 .
6. Naqwi, A. A.; Reynolds, W. C. (Jan 1987), "Dual cylindrical wave laser-Doppler method for measurement of skin friction in fluid flow", NASA STI/Recon Technical Report N, 87
7. {microS Shear Stress Sensor, MSE}
8. Große, S.; Schröder, W. (2009), "Two-Dimensional Visualization of Turbulent Wall Shear Stress Using Micropillars", AIAA Journal, 47 (2): 314–321, Bibcode:2009AIAAJ..47..314G, doi:10.2514/1.36892
9. Große, S.; Schröder, W. (2008), "Dynamic Wall-Shear Stress Measurements in Turbulent Pipe Flow using the Micro-Pillar Sensor MPS3", International Journal of Heat and Fluid Flow, 29 (3): 830–840, doi:10.1016/j.ijheatfluidflow.2008.01.008