Wall shear stress Wall shear stress expresses 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},where is the
dynamic viscosity, is the flow velocity, and is the distance from the wall. It is used, for example, in the description of arterial
blood flow, where there is evidence that it affects the
atherogenic process.
Pure Pure shear stress is related to pure
shear strain, denoted , by the equation\tau = \gamma G,where is the
shear modulus of the
isotropic material, given by G = \frac{E}{2(1+\nu)}. Here, is
Young's modulus and is
Poisson'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 The beam shear formula is also known as Zhuravskii shear stress formula after
Dmitrii Ivanovich Zhuravskii, who derived it in 1855.
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. Constructions in soil can also 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 by the equation\tau = 2\sqrt {\frac{UG}{V}},where Furthermore, , where
Shear stress in fluids is frequently used to illustrate
shear-driven fluid motion. Any real
fluids (
liquids and
gases included) moving along a solid boundary will incur a shear stress at that boundary. The
no-slip condition 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 is given by\tau (y) = \mu \frac{\partial u}{\partial y},where {{unbulleted list | style = padding-left:1.5em; 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 velocity
gradient (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 named
dynamic 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 is
linear), 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. == Measurement with sensors ==