 # Shear rate tensor

It only takes a minute to sign up. I'd like to calculate the shear rate formula for CFD Non-newtonian Fluid and want to know if the following formula is the good one:. Viscious Stress General Equation Tensor :. So the magnitude of the shear rate is:. I didn't catch this the first time I read this thread, but your equation for the rate of deformation tensor D is incorrect; it should not have the dilatation terms along the diagonal.

The definition of the rate of deformation tensor is "the symmetric part of the velocity gradient tensor":. The linearized version of this is a Newtonian fluid, with "a" being a function only of the dilatation first invariantb being a constant, and c being zero. Sign up to join this community.

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It only takes a minute to sign up. Mike Stone is correct. There is no derivation from Newton's laws, and it is just geometry, but I will present it a little differently. Strain angles and rotation angles are how we parameterize all the 3x3 matrices that strain and rotate 3-vectors.

Rotations and strains form the group GL 3,R. This is the group of all invertible 3x3 matrices M of real numbers. All these elements are in radians. Now apply M to a vector x to get X. We have moved a piece of a body from x to X. Because lengths are invariant, the transformations are called rotations. That is, put your right thumb perpendicular to the plane formed by axis1 and axis2, such that you fingers would push axis1 into axis 2. For example, a square box with its sides initially along axis1 and axis2, becomes a parallelepiped with its sides tilted inward from axis1 and axis2 and its diagonal from the origin stretched.

The purpose of this rather too long answer was to show the strains and rotations are intimately related and are a group of transformations we can do with our fingers to any piece of material.

How the displacements u change as you move around in the body is just what the transformations cause, but is not the fundamental concept of strain or rotation. There is no derivation from Newton, because strain is purely geometric concept. It is measuring the deformation the change in the length and angles of the spacing between the atome of the body. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Strain rate tensor derivation Ask Question. Asked 3 years, 8 months ago.In continuum mechanicsthe strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time.

It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient derivative with respect to position of the flow velocity. In fluid mechanics it also can be described as the velocity gradienta measure of how the velocity of a fluid changes between different points within the fluid. The strain rate tensor is a purely kinematic concept that describes the macroscopic motion of the material.

Therefore, it does not depend on the nature of the material, or on the forces and stresses that may be acting on it; and it applies to any continuous mediumwhether solidliquid or gas. On the other hand, for any fluid except superfluidsany gradual change in its deformation i.

At any point in the fluid, these stresses can be described by a viscous stress tensor that is, almost always, completely determined by the strain rate tensor and by certain intrinsic properties of the fluid at that point.

Viscous stress also occur in solids, in addition to the elastic stress observed in static deformation; when it is too large to be ignored, the material is said to be viscoelastic. By performing dimensional analysisthe dimensions of velocity gradient can be determined. Therefore, the velocity gradient has the same dimensions as this ratio, i.

Sir Isaac Newton proposed that shear stress is directly proportional to the velocity gradient: . Note that J is a function of p and t. In this coordinate system, the Taylor approximation for the velocity near p is. Any matrix can be decomposed into the sum of a symmetric matrix and an antisymmetric matrix.

This decomposition is independent of coordinate system, and so has physical significance. Then the velocity field may be approximated as. The antisymmetric term R represents a rigid-like rotation of the fluid about the point p.

A rigid rotation does not change the relative positions of the fluid elements, so the antisymmetric term R of the velocity gradient does not contribute to the rate of change of the deformation. The actual strain rate is therefore described by the symmetric E term, which is the strain rate tensor. The symmetric term E of velocity gradient the rate-of-strain tensor can be broken down further as the sum of a scalar times the unit tensor, that represents a gradual isotropic expansion or contraction; and a traceless symmetric tensor which represents a gradual shearing deformation, with no change in volume: .

This decomposition is independent of the choice of coordinate system, and is therefore physically significant. This type of flow occurs, for example, when a rubber strip is stretched by pulling at the ends, or when honey falls from a spoon as a smooth unbroken stream.

For a two-dimensional flow, the divergence of v has only two terms and quantifies the change in area rather than volume. The study of velocity gradients is useful in analysing path dependent materials and in the subsequent study of stresses and strains; e. Consider the velocity field of a fluid flowing through a pipe. The layer of fluid in contact with the pipe tends to be at rest with respect to the pipe. This is called the no slip condition. This type of flow is called laminar flow.

From Wikipedia, the free encyclopedia. The third component of the velocity out of the screen is assumed to be zero everywhere. The symmetric part E pt r strain rate of the linear term of the example flow. The antisymmetric part R pt r rotation of the linear term. The scalar part D pt r uniform expansion, or compression, rate of the strain rate tensor E pt r.

The traceless part S pt r shear rate of the strain rate tensor E pt r. A Dictionary of Chemical Engineering. Oxford University Press.December 9,second invariant of rate-of-strain tensor.

The shear, or strain, rate is often calculated based on the square root of the second invariant of rate-of-strain tensor.

### second invariant of rate-of-strain tensor

The tensor itself is made up of all the possible deformation of a fluid element, which includes volumetric and shear deformation. I would like to invite comments from everyone concerning: 1 if this second invariant, in its general definition, includes both the volumetric-rate of deformation and shear-rate of deformation? Or simply shear-rate of deformation alone? I take it as a way to 'average' all the strain components in the tensor, thus an 'effective' strain rate.

I look forward for your comments. December 9,Re: second invariant of rate-of-strain tensor. December 10,Correction Here is the corrected version. December 10,Re: Correction So, if the tensor A is decomposed into its deviatoric and volumetric tensors, the 2nd invariant of the deviatoric tensor would not contain the volumetric part; while the 2nd invariant for the volumetric tensor would of course contain the voluemtric part.

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13.0. The Stress Tensor

November 20, June 3, July 31,Shear Strain Rate. Hi all, I have been reading about non-Newtonian fluids recently and I have a question about the scalar shear strain rate.

Any insight would be greatly appreciated. Thanks, Dave. August 1, So I think I may have figured this out. So thats three out of four definitions accounted for.

## Viscous stress tensor

Still not sure of the last one. Cant understand Why the 0. Any ideas? Nucleophobe likes this. October 24, Hi, Dave. Multiply Dij by Dij using Matrix multiplication. Sum up the diagonal elements of the Result Matrix and Multiply by 2. Kind Regards, Dave. Hi Dave, I think that the deformation tensor Dij is defined for compatibility with the definition of gamma.

October 25, I'm sure that there is probably a very good reason for dropping the 0. I just never figured it out. I can't imagine that a different definition of Dij would be used just to satisfy a different definition of gamma?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 ratethe rate at which it is deforming around that point.

The viscous stress tensor is formally similar to the elastic stress tensor Cauchy tensor that describes internal forces in an elastic material due to its deformation. Both tensors map the normal vector of a surface element to the density and direction of the stress acting on that surface element. However, elastic stress is due to the amount of deformation strainwhile viscous stress is due to the rate of change of deformation over time strain rate.

In viscoelastic materials, whose behavior is intermediate between those of liquids and solids, the total stress tensor comprises both viscous and elastic "static" components. For a completely fluid material, the elastic term reduces to the hydrostatic pressure.

Internal mechanical stresses in a continuous medium are generally related to deformation of the material from some "relaxed" unstressed state. These stresses generally include an elastic "static" stress component, that is related to the current amount of deformation and acts to restore the material to its rest state; and a viscous stress component, that depends on the rate at which the deformation is changing with time and opposes that change.

Like the total and elastic stresses, the viscous stress around a certain point in the material, at any time, can be modeled by a stress tensor, a linear relationship between the normal direction vector of an ideal plane through the point and the local stress density on that plane at that point. Note that these numbers usually change with the point p and time t.

Consider an infinitesimal flat surface element centered on the point prepresented by a vector dA whose length is the area of the element and whose direction is perpendicular to it. Let dF be the infinitesimal force due to viscous stress that is applied across that surface element to the material on the side opposite to dA.

The components of dF along each coordinate axis are then given by. In a perfectly fluid material, that by definition cannot have static shear stress, the elastic stress tensor is zero:.

Ignoring the torque on an element due to the flow "extrinsic" torquethe viscous "intrinsic" torque per unit volume on a fluid element is written as an antisymmetric tensor as. If the particles have rotational degrees of freedom, this will imply an intrinsic angular momentum and if this angular momentum can be changed by collisions, it is possible that this intrinsic angular momentum can change in time, resulting in an intrinsic torque that is not zero, which will imply that the viscous stress tensor will have an antisymmetric component with a corresponding rotational viscosity coefficient.

External forces can result in an asymmetric component to the stress tensor e. In a solid material, the elastic component of the stress can be ascribed to the deformation of the bonds between the atoms and molecules of the material, and may include shear stresses. In a fluid, elastic stress can be attributed to the increase or decrease in the mean spacing of the particles, that affects their collision or interaction rate and hence the transfer of momentum across the fluid; it is therefore related to the microscopic thermal random component of the particles' motion, and manifests itself as an isotropic hydrostatic pressure stress.

The viscous component of the stress, on the other hand, arises from the macroscopic mean velocity of the particles.The most obvious first step is to take the time derivative of the deformation gradient,as follows:. Since is a function ofwe can use the chain rule to arrive at the following important expression:.

We can see from eq. For our purposes, consider that if we were to have analytical functions of time,for each term in that describes how and are changing with time, for example, then we could find by noting that and. This is quite academic and potentially quite difficult compared to the way that would be found in FEA, so such an example will not be given.

However, is a very important quantity as we will see in the chapter on rate-form constitutive relationships. The velocity gradient,is a very important quantity and can be decomposed as follows:. A complete proof of this can be found in [Asaro]. It can also be easily shown that is, in fact, skew. This proof can be found in Appendix A. Consider the time rate of change of length of a particular element:. Additionally, if then we must have rigid body motions only.

The way that eq. Similar to the shear strain,we can again consider how the angle between two vectors,andchanges under deformation see Fig below. In general, though. Similarly, in general. Your email address will not be published. Save my name, email, and website in this browser for the next time I comment. Notify me of new posts by email.