Shear Stress

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

Normal Shear Stress

The formula for calculating average shear stress is force per unit area.

\tau = {F \over A},

Where from:

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

Other forms

Pure

The net shears stress is related to the net shears stress , denoted , by the following equation:

\tau = \gamma G \,

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

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

Here e is Young’s modulus and is Poisson ‘s ratio .

Beam shear

Beam shear is defined as the internal shear stress of the beam due to the shear force applied to the beam.

\tau ~=~\frac{fQ}{Ib}

Where from

• f = total shear force at the site under consideration;
• Q = stationary moment of the field ;
• 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 the Zhuravsky shear stress formula after Dmitry Ivanovich Zhuravsky , who derived it in 1855.

semi-monocoque shear

The shear stress within a quasi-monocoque structure can be calculated by idealizing the cross-section of the structure into a set of stringers (carrying only axial load) and webs (carrying only shear flow ). Shear stress is generated by dividing the shear flow by the thickness of a given part of a semi-monocoque structure. Thus, the maximum shear stress will be either at maximum shear flow or in a web of minimum thickness

Shear can also cause construction in the soil to fail; As such, the weight of an earth-filled dam or embankment may cause the subsoil to collapse, like a minor landslide .

impact shear

The maximum shear stress created in a solid round bar under impact is given by the equation:

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

Where from

• U = change in kinetic energy;
• g = shear modulus ;
• V = volume of rods;
u = u_{rotate} + u_{apply} ;\\
u = rotation~=~\frac{1}{2}I^2\\
U_{applied} = Tθ~~ displaced ;\\
I = mass moment of inertia;\\
= angular momentum.

Shear stress in liquids

Any real fluid ( liquids and gases included) moving along a solid boundary will have a shear stress at that boundary. The no-slip condition [5] stipulates that the speed of the fluid at the limit (relative to the limit) is zero; However at some height above the limit the speed of flow must be equal to that of the fluid. The area 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 viscosity is a constant of proportionality. For non-Newtonian fluids , viscosityis not stable. This loss of velocity results in a shear stress imposed on the boundary.

For a Newtonian fluid, the shear stress on a surface element parallel to a flat plate at point y is given by:

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

Where from

μ is the dynamic viscosity of the flow ;
U is with the flow velocity limit;
y is the height above the limit.
Specifically, the wall shear stress is defined as:

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

Newton’s constitutional law, for any general geometry (including the flat plate above), states that the shear tensor (a second-order tensor) is proportional to the flow velocity gradient (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 called dynamic viscosity . For an isotropic Newtonian flow it is a scalar, while for an anisotropic Newtonian flow it can be a second order tensor. The fundamental aspect is that for Newtonian fluids the dynamical viscosity is independent on flow velocity (i.e., the shear stress constitutional law is linear ), whereas for non-Newtonian flows this is not true, and one must allow a modification:

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

The above formula is no longer Newton’s law but a general tensorial identity: given any expression of shear stress as a function of flow velocity, one can always find the expression of viscosity as a function of flow velocity. On the other hand, given a shear stress as a function of 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 we get 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 (u,v) respectively), the shear stress matrix is ​​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 flux, 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 a 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 non-uniform (depending on space coordinates) and transient, but contextually independent on flow velocity:

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

Hence this flow is Newtonian. On the other hand, a flow which had viscosity:

{\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 because viscosity depends on flow velocity. This non-Newtonian flow is isotropic (the matrix identity is proportional to the matrix), so viscosity is only a scalar quantity:

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

Measurement with sensor

Diverging Fringe Shear Stress Sensor

This relationship can be exploited to measure wall shear stress. If a sensor could directly measure the gradient of the velocity profile on the wall, multiplying by the dynamic viscosity would generate the shear stress. Such sensors were demonstrated by AA Naqwi and WC Reynolds. [6]The interference pattern produced by sending a beam of light through two parallel membranes creates a network of linearly diverging fringes that appear to originate from the plane of the two slits (see the double-slit experiment). As a particle in a fluid passes through a fringe, a receiver detects the reflection of the fringe pattern. The signal can be processed, and by knowing 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 consequently does not require calibration. Recent advances in micro-optic fabrication technologies have made it possible to use an integrated diffractive optical element to create diverging fringe shear stress sensors usable in both air and liquid.

Micro-pillar shear-stress sensor

Another measurement technique is that of thin wall-mounted micro-columns made of flexible polymer PDMS, which bend in response to applied drag forces in the vicinity of the wall. The sensor thus deals with indirect measurement principles that rely on the relationship between the near-wall velocity gradient and the local wall-shear stress.

Electro-diffusional method

The electro-diffusional method measures the wall shear rate from the microelectrode to the liquid phase under limiting diffusion current conditions. A potential difference between the anode of a wider surface (usually located away from the measuring area) and the smaller working electrode serving as the cathode leads to a faster redox reaction. Ion disappearance only occurs at the microprobe active surface, leading to the development of a diffusion boundary layer, in which the rapid electro-diffusion reaction rate is controlled only by diffusion. The resolution of the convective-disruptive equation in the near-wall region of the microelectrode leads to analytical solutions dependent on the length of the microscopic probe’s characteristics, the diffusion properties of the electrochemical solution, and the wall shear rate.