Fact-checked by Grok 2 weeks ago

Strain-rate tensor

The strain-rate tensor, also known as the deformation-rate tensor or rate-of-strain tensor, is a second-order symmetric tensor in continuum mechanics that quantifies the instantaneous rate of deformation of a continuous medium, excluding rigid-body rotations.^[1] It is mathematically defined as the symmetric part of the velocity gradient tensor \mathbf{L} = \nabla \mathbf{v}, expressed in Cartesian coordinates as

D_{ij} = \frac{1}{2} \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right),

where \mathbf{v} is the velocity field and x_i are spatial coordinates; the diagonal components represent rates of extension or compression, while off-diagonal components capture shear rates.^[2]^[1] This tensor arises naturally from the decomposition of the velocity gradient into symmetric (deformation) and antisymmetric (rotation) parts, ensuring it has only six independent components due to symmetry (D_{ij} = D_{ji}) and real eigenvalues corresponding to principal strain rates.^[1] In solid mechanics, it describes the time derivative of the infinitesimal strain tensor, \dot{\varepsilon}_{ij}, linking applied stresses to deformation rates in viscoelastic materials via constitutive relations.^[3] For fluids, particularly Newtonian viscous fluids, the tensor plays a central role in the Cauchy stress tensor formulation, \mathbf{T} = -p \mathbf{I} + 2\mu \mathbf{D}, where p is pressure, \mu is dynamic viscosity, and \mathbf{I} is the identity tensor; the trace \operatorname{tr}(\mathbf{D}) governs volumetric changes, while the deviatoric part drives shear stresses.^[1] The strain-rate tensor's invariance under coordinate transformations makes it essential for analyzing material behavior in diverse applications, from structural engineering to fluid dynamics, including the Navier-Stokes equations where it determines viscous dissipation.^[1] Its properties, such as positive semi-definiteness in certain contexts, facilitate the study of energy dissipation rates, computed as \mathbf{T} : \mathbf{D} (double contraction), which quantifies mechanical work converted to heat in deforming continua.^[1]

Fundamentals

Dimensional Analysis

The strain-rate tensor serves as a measure of the rate of deformation in continuum media, where each of its components carries dimensions of inverse time, denoted as [T]^{-1}, which aligns with frequency units such as s^{-1} in the International System of Units (SI).^[4] This dimensional characteristic arises from the tensor's origin in velocity field gradients, where a velocity component (dimensions [L T^{-1}]) differentiated with respect to a spatial coordinate (dimensions [L]) results in [T^{-1}].^[5] Strain itself is dimensionless as a ratio of length changes, so its time derivative inherently scales with 1/time, emphasizing the tensor's role in capturing temporal rates of geometric distortion without inherent length or mass dependencies.^[4] The tensor's components further illustrate this uniformity in scaling. Normal strain-rate components along the principal axes, such as \dot{\epsilon}_{xx} = \frac{\partial u}{\partial x}, \dot{\epsilon}_{yy} = \frac{\partial v}{\partial y}, and \dot{\epsilon}_{zz} = \frac{\partial w}{\partial z}, each represent linear deformation rates and possess units of s^{-1}.^[5] Similarly, shear strain-rate components, exemplified by \dot{\epsilon}_{xy} = \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right), maintain the same [T]^{-1} dimensions, as the averaging does not alter the underlying velocity gradient scaling.^[4] This consistent dimensionality across all nine components in three-dimensional space ensures the tensor's invariance under coordinate transformations while facilitating direct comparisons of deformation rates in various directions. In engineering simulations and theoretical modeling, the [T]^{-1} scaling of the strain-rate tensor underpins non-dimensionalization techniques to isolate dominant physical effects. A key example is the Reynolds number, defined as \mathrm{Re} = \frac{\rho U L}{\mu}, which ratios inertial forces (\rho U^2 / L) to viscous forces (\mu U / L^2), where the viscous term implicitly involves strain rates on the order of U/L.^[6] This relation highlights how strain-rate magnitudes govern the balance between convective acceleration and diffusive momentum transport, enabling scaled analyses in computational fluid dynamics without loss of physical insight.^[6] The emphasis on dimensional consistency for the strain-rate tensor traces back to early continuum mechanics developments. In the 1820s, Claude-Louis Navier introduced viscous terms proportional to deformation rates in his equations of motion, using molecular arguments to dimensionally match stress ([M L^{-1} T^{-2}]) with dynamic viscosity times strain rate (\mu \dot{\epsilon}).^[7] George Stokes later refined this in 1845, providing a more rigorous formulation ensuring the full Navier-Stokes equations balanced across all terms, from pressure gradients to inertial accelerations.^[7] These foundational formulations established the tensor's dimensional framework, influencing subsequent derivations in viscous flow theory.^[8]

Velocity Gradient Tensor

In continuum mechanics, the velocity gradient tensor, denoted \mathbf{L}, is defined as the gradient of the velocity vector field \mathbf{v}, expressed as \mathbf{L} = \nabla \mathbf{v}.^[1] In index notation, this is generally written as \mathbf{L} = \nabla \mathbf{v}, where the gradient operator \nabla acts on the spatial coordinates.^[9] In Cartesian coordinates, the components of \mathbf{L} are given by

L_{ij} = \frac{\partial v_i}{\partial x_j},

where v_i are the components of the velocity field and x_j are the spatial coordinates.^[1] This tensorial representation captures the spatial variation of velocity in a second-order tensor form.^[9] Physically, the velocity gradient tensor describes how the velocity field changes over infinitesimal distances in space, thereby encoding the local kinematics of motion in a continuum.^[10] It encompasses both the deformative aspects, such as stretching and shearing, and the rigid rotational components inherent in the flow or deformation of material elements.^[1] For an infinitesimal line element \mathrm{d}\mathbf{x}, the relative velocity across it is \mathrm{d}\mathbf{v} = \mathbf{L} \cdot \mathrm{d}\mathbf{x}, illustrating how \mathbf{L} governs the differential motion within the continuum.^[9] As a foundational kinematic quantity in continuum mechanics, the velocity gradient tensor provides the basis for analyzing infinitesimal changes in velocity across space, essential for deriving rates of deformation in both fluids and solids.^[11] The components of \mathbf{L} possess dimensions of inverse time, [T]^{-1}, reflecting the scaling of velocity differences per unit length.^[12]

Formal Definition and Properties

Strain-Rate Tensor Components

The strain-rate tensor, often denoted as \mathbf{D}, is formally defined as the symmetric portion of the velocity gradient tensor \mathbf{L} = \nabla \mathbf{v}, where \mathbf{v} is the velocity field of the continuum. It is expressed in coordinate-free notation as

\mathbf{D} = \frac{1}{2} \left( \mathbf{L} + \mathbf{L}^T \right).

This definition captures the rate at which neighboring material elements deform relative to one another, excluding rigid-body rotation.^[13]^[9] In three-dimensional Cartesian coordinates, with velocity components u, v, and w corresponding to the x, y, and z directions, the nine components of \mathbf{D} take the explicit form

\begin{aligned} D_{xx} &= \frac{\partial u}{\partial x}, \\ D_{yy} &= \frac{\partial v}{\partial y}, \\ D_{zz} &= \frac{\partial w}{\partial z}, \\ D_{xy} = D_{yx} &= \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right), \\ D_{xz} = D_{zx} &= \frac{1}{2} \left( \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right), \\ D_{yz} = D_{zy} &= \frac{1}{2} \left( \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \right). \end{aligned}

The symmetry \mathbf{D} = \mathbf{D}^T implies only six independent components, corresponding to three normal strain rates and three shear strain rates.^[13] The trace of the strain-rate tensor, \operatorname{tr}(\mathbf{D}) = D_{xx} + D_{yy} + D_{zz} = \nabla \cdot \mathbf{v}, quantifies the dilatation or volumetric strain rate, which indicates the rate of local volume change in the material.^[13] As a second-order tensor, \mathbf{D} transforms under rotation of the coordinate system via \mathbf{D}' = \mathbf{R} \mathbf{D} \mathbf{R}^T, where \mathbf{R} is the proper orthogonal rotation tensor; this ensures that the tensor's symmetry and key physical properties remain invariant across coordinate frames.^[9]

Decomposition into Symmetric and Antisymmetric Parts

The velocity gradient tensor \mathbf{L} = \nabla \mathbf{v} can be uniquely decomposed into a symmetric part, known as the strain-rate tensor \mathbf{D}, and an antisymmetric part, known as the rotation tensor \mathbf{W}, such that \mathbf{L} = \mathbf{D} + \mathbf{W}.^[14] The strain-rate tensor is defined as \mathbf{D} = \frac{1}{2} (\mathbf{L} + \mathbf{L}^T), while the rotation tensor is \mathbf{W} = \frac{1}{2} (\mathbf{L} - \mathbf{L}^T).^[15] This decomposition separates the contributions of deformation and rotation in the local fluid motion.^[16] The components of the rotation tensor \mathbf{W} are given by W_{ij} = \frac{1}{2} \left( \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i} \right).^[14] For example, in the xy-plane, W_{xy} = \frac{1}{2} \left( \frac{\partial u}{\partial y} - \frac{\partial v}{\partial x} \right).^[15] This tensor is closely related to the vorticity vector \boldsymbol{\omega} = \nabla \times \mathbf{v}, which quantifies the local rotation of fluid elements; in three dimensions, the vorticity components are extracted from \mathbf{W} via \omega_k = -\epsilon_{kij} W_{ij}, where \epsilon_{kij} is the Levi-Civita symbol.^[14] Specifically, for the z-component in two dimensions, \omega_z = -2 W_{xy}.^[16] Physically, the strain-rate tensor \mathbf{D} describes the rate of change in shape (through shearing) and volume (through dilatation) of a fluid element, representing pure deformation without rotation.^[16] In contrast, the rotation tensor \mathbf{W} captures the rigid-body rotation of the fluid element, which does not involve any deformation or change in shape.^[15] This distinction arises because symmetric tensors like \mathbf{D} affect relative positions through stretching and compression, while antisymmetric tensors like \mathbf{W} correspond to infinitesimal rotations equivalent to a skew-symmetric operator.^[16] The decomposition \mathbf{L} = \mathbf{D} + \mathbf{W} is invariant under orthogonal coordinate transformations, as \mathbf{L}, \mathbf{D}, and \mathbf{W} all transform as second-order tensors, thereby preserving the symmetry properties and the separation into deformation and rotation components regardless of the chosen frame.^[17]

Applications in Mechanics

Role in Fluid Dynamics

In fluid dynamics, the strain-rate tensor, denoted as \mathbf{D}, quantifies the rate of deformation in the velocity field and directly influences the viscous stresses that govern flow behavior, particularly in viscous flows where momentum transport occurs through molecular interactions. This tensor emerges as the symmetric part of the velocity gradient, capturing the stretching and shearing motions essential for modeling energy dissipation in fluids. Historically, Sir George Gabriel Stokes incorporated the strain-rate tensor into the equations of motion for viscous fluids in 1845, deriving what are now known as the Navier-Stokes equations by adding rate-dependent viscous terms to Euler's inviscid equations, thereby establishing viscosity as a fundamental property dependent on deformation rates.^[18]^[19] For Newtonian fluids, the constitutive relation links the deviatoric part of the Cauchy stress tensor \boldsymbol{\tau} linearly to the strain-rate tensor via \boldsymbol{\tau} = 2\mu \mathbf{D}, where \mu is the dynamic viscosity coefficient, assumed constant and independent of the deformation rate. This relation implies that viscous stresses arise solely from the symmetric deformation, with no contribution from rigid-body rotation. The complete Cauchy stress tensor \boldsymbol{\sigma} for such fluids is then \boldsymbol{\sigma} = -p \mathbf{I} + 2\mu \mathbf{D} in the incompressible case, or more generally for compressible flows, \boldsymbol{\sigma} = -p \mathbf{I} + 2\mu \mathbf{D} + \lambda (\operatorname{tr} \mathbf{D}) \mathbf{I}, where p is the pressure, \mathbf{I} is the identity tensor, \lambda is the second viscosity coefficient, and bulk viscosity effects account for volumetric changes.^[20]^[21]^[22] In non-Newtonian fluids, where viscosity varies with deformation rate, the strain-rate tensor's role extends to generalized models that capture shear-thinning or shear-thickening behaviors. A common example is the power-law fluid model, where the effective viscosity depends on the shear rate magnitude \dot{\gamma} = \sqrt{2 \mathbf{D} : \mathbf{D}}, defined using the double contraction (or Frobenius inner product) of \mathbf{D} with itself; the deviatoric stress is then \boldsymbol{\tau} = 2 K \dot{\gamma}^{n-1} \mathbf{D}, with K as the consistency index and n as the power-law index (n < 1 for shear-thinning, n > 1 for shear-thickening). This formulation allows the strain-rate tensor to dictate nonlinear stress responses in complex flows like polymer melts or slurries.^[23]^[24] The trace of the strain-rate tensor, \operatorname{tr} \mathbf{D} = \nabla \cdot \mathbf{u}, represents the rate of volumetric compression or expansion and is directly tied to fluid compressibility; in incompressible flows, such as those of liquids at low speeds, \operatorname{tr} \mathbf{D} = 0, enforcing constant density and simplifying the constitutive relations by eliminating bulk viscosity terms. This condition ensures that deformations are purely deviatoric, focusing viscous effects on shear and extension without volume change.^[25]^[26]

Role in Solid Mechanics

In solid mechanics, the strain-rate tensor, often denoted as \mathbf{D} or \dot{\boldsymbol{\varepsilon}}, plays a crucial role in describing the time-dependent deformation of materials under dynamic loading conditions, particularly for rate-sensitive behaviors beyond the quasi-static infinitesimal strain tensor \boldsymbol{\varepsilon}. For small deformations, the strain-rate tensor approximates the material time derivative of the infinitesimal strain tensor, such that \dot{\boldsymbol{\varepsilon}} = \mathbf{D} = \frac{1}{2} (\nabla \mathbf{v} + (\nabla \mathbf{v})^T), where \mathbf{v} is the velocity field, and the total strain accumulates via time integration \boldsymbol{\varepsilon} = \int \dot{\boldsymbol{\varepsilon}} \, dt.^[27] This linkage enables the modeling of incremental deformation histories in solids, distinguishing it from static analyses where rate effects are negligible.^[28] A primary application of the strain-rate tensor in solid mechanics is within viscoplasticity models, which capture the rate-dependent inelastic flow in materials like metals under high strain rates or elevated temperatures. In these frameworks, the plastic component of the strain-rate tensor \dot{\boldsymbol{\varepsilon}}^p follows an associated flow rule derived from a yield potential, typically the von Mises criterion, given by

\dot{\boldsymbol{\varepsilon}}^p = \frac{3}{2} \frac{\boldsymbol{\sigma}'}{\|\boldsymbol{\sigma}'\|} \dot{\varepsilon}_{eq},

where \boldsymbol{\sigma}' is the deviatoric stress tensor, \|\boldsymbol{\sigma}'\| = \sqrt{\frac{3}{2} \boldsymbol{\sigma}' : \boldsymbol{\sigma}'} is its magnitude, and the equivalent plastic strain rate is \dot{\varepsilon}_{eq} = \sqrt{\frac{2}{3} \dot{\boldsymbol{\varepsilon}}^p : \dot{\boldsymbol{\varepsilon}}^p}.^[29] This rule ensures that plastic straining aligns with the direction of maximum shear stress, with the magnitude \dot{\varepsilon}_{eq} governing the rate sensitivity through viscoplastic evolution laws, such as Perzyna-type overstress formulations.^[30] Such models are essential for predicting creep, recovery, and strain-rate hardening in polycrystalline solids.^[31] The strain-rate tensor also informs rate-dependent yield criteria, extending classical plasticity to scenarios involving high-speed impacts and ballistic loading. A seminal example is the Johnson-Cook constitutive model, which incorporates strain-rate sensitivity into the flow stress via a logarithmic term \sigma = \left( A + B \varepsilon^n \right) \left( 1 + C \ln \frac{\dot{\varepsilon}}{\dot{\varepsilon}_0} \right) \left( 1 - T^{*m} \right), where C quantifies the rate sensitivity, \dot{\varepsilon} is the equivalent strain rate, and other terms account for strain hardening, thermal softening, and reference values. This model is widely adopted for simulating dynamic fracture and penetration in metals, as the strain-rate term captures increased yield strength at elevated rates (e.g., $10^3 to $10^6 s^{-1}) observed in split-Hopkinson bar tests.^[32] In finite element simulations of dynamic loading on solids, the strain-rate tensor must be handled with objective rates to ensure frame-invariance in large-deformation contexts, such as rotating or convected material frames. The Jaumann corotational derivative, defined as \overset{\circ}{\mathbf{D}}^J = \mathbf{D} - \mathbf{W} \cdot \mathbf{D} + \mathbf{D} \cdot \mathbf{W}, where \mathbf{W} is the spin tensor (antisymmetric part of the velocity gradient), provides a commonly used objective measure that accounts for rigid-body rotations without altering the deformation physics.^[33] This formulation is integrated into constitutive updates within explicit or implicit time-stepping schemes, enabling accurate prediction of wave propagation, impact responses, and localization in viscoplastic materials.^[34]

Examples

Flow in a Pipe

In steady, laminar flow through a straight circular pipe of radius R and length L, known as Hagen-Poiseuille flow, the motion is axisymmetric, fully developed, and directed along the pipe's axial z-direction, driven by a constant pressure gradient \Delta P / L.^[35] The velocity field in cylindrical coordinates (r, \theta, z) takes the form \mathbf{v} = (v_r, v_\theta, v_z) = (0, 0, v_z(r)), where the axial velocity component follows a parabolic profile given by

v_z(r) = \frac{\Delta P}{4 \mu L} (R^2 - r^2),

with \mu denoting the constant dynamic viscosity of the fluid and r the radial coordinate from the pipe centerline.^[36] This profile arises from solving the Navier-Stokes equations under no-slip boundary conditions at the wall (v_z(R) = 0) and symmetry at the center (dv_z/dr|_{r=0} = 0).^[37] The velocity gradient tensor in this flow has a single non-zero derivative, \partial v_z / \partial r, leading to a strain-rate tensor \mathbf{D} with only off-diagonal components non-zero: D_{rz} = D_{zr} = \frac{1}{2} \frac{d v_z}{dr}. Substituting the velocity profile yields

\frac{d v_z}{dr} = -\frac{\Delta P \, r}{2 \mu L}, \quad D_{rz} = D_{zr} = -\frac{\Delta P \, r}{4 \mu L},

while all other components vanish, including the diagonal elements.^[36] The trace of \mathbf{D} is zero (\operatorname{tr}(\mathbf{D}) = 0), confirming the flow's incompressibility as required for typical liquids.^[35] For a Newtonian fluid, the deviatoric part of the Cauchy stress tensor relates linearly to the strain-rate tensor via \boldsymbol{\tau} = 2 \mu \mathbf{D}, resulting in a single non-zero shear stress component \tau_{rz} = \mu \frac{d v_z}{dr} = -\frac{\Delta P \, r}{2 L}.^[37] This stress distribution, independent of viscosity, follows directly from axial momentum balance and provides the wall shear stress \tau_w = -\tau_{rz}|_{r=R} = \frac{\Delta P \, R}{2 L} responsible for frictional pressure losses.^[35] The magnitude of the velocity gradient at the wall defines the wall shear rate \dot{\gamma}_w = \left| \frac{d v_z}{dr} \right|_{r=R} = \frac{\Delta P \, R}{2 \mu L}, a key parameter for characterizing viscous effects and linking to the laminar friction factor f = 64 / \operatorname{Re} via \tau_w = f (\rho V^2 / 8), where V is the average velocity and \operatorname{Re} the Reynolds number.^[37] This example demonstrates how the strain-rate tensor quantifies the deformation in pressure-driven channel flows, essential for engineering designs like pipelines and blood vessels.^[36]

Simple Shear Flow

Simple shear flow provides a fundamental example of the strain-rate tensor in action, particularly in planar Couette flow between two infinite parallel plates separated by a distance h. The lower plate remains stationary, while the upper plate moves at a constant velocity U in the x-direction, yielding a unidirectional velocity field \mathbf{v} = \left( u(y) = \frac{U}{h} y, \, 0, \, 0 \right). This configuration produces a uniform shear rate \dot{\gamma} = \frac{U}{h}, representing the velocity gradient perpendicular to the flow direction.^[38]^[39] The strain-rate tensor \mathbf{D} for this flow is symmetric and traceless, with non-zero components limited to the off-diagonal shear terms: D_{xy} = D_{yx} = \frac{\dot{\gamma}}{2}, while all other components, including the normal strains, are zero. The trace \operatorname{tr}(\mathbf{D}) = 0 confirms the flow's incompressibility, as there are no volumetric changes. This setup isolates pure shear deformation without extension or compression in the principal directions.^[40] For a Newtonian fluid, the deviatoric stress tensor \boldsymbol{\tau} relates directly to the strain-rate tensor via \boldsymbol{\tau} = 2\mu \mathbf{D}, where \mu is the dynamic viscosity. Consequently, the shear stress component is \tau_{xy} = \mu \dot{\gamma}, with vanishing normal stresses (\tau_{xx} = \tau_{yy} = \tau_{zz} = 0). This linear relationship highlights the tensor's role in linking kinematics to viscous forces in simple flows.^[41]^[42] In non-Newtonian fluids, such as polymer melts or suspensions, the response deviates from linearity, and the apparent viscosity \eta(\dot{\gamma}) = \frac{\tau_{xy}}{\dot{\gamma}} becomes shear-rate dependent. Shear-thinning behavior is common, where \eta decreases with increasing \dot{\gamma}, often modeled by power-law relations like \tau_{xy} = K \dot{\gamma}^n with n < 1. This phenomenon is critical for rheological characterization in Couette viscometers, where simple shear isolates viscosity variations without geometric complexities.^[43]^[44] The velocity gradient tensor in this flow decomposes into the symmetric strain-rate tensor and an antisymmetric rotation tensor, with the rotation component W_{xy} = -\frac{\dot{\gamma}}{2} accounting for the local rigid-body rotation that accompanies but does not contribute to deformation.^[45]

References

[1]
[PDF] Introduction to Continuum Mechanics
This is a set of notes for a senior-year course on continuum mechanics, covering solids and fluids as continuous media.
[2]
[PDF] Chapter 1 Continuum mechanics review
1.3 Strain and strain-rate tensors A matrix, two indexed variables, tensor of rank two, like the stress matrix. • Meaning of the elements: Diagonal elements ...
[3]
[PDF] Continuum Mechanics Continuum Mechanics and Constitutive ...
For viscoelastic and viscous materials the rate of strain is the dominant response to an applied stress. The rate of strain tensor, ˙ γ , is the derivative of ...
[4]
[PDF] Strains, strain rate, stresses - Docenti.unina.it
The strain rate is the strain achieved per unit time, and is particularly important in fluid materials. Dimension is inverse time, typically s−1. The strain ...
[5]
Rates of Strain - an overview | ScienceDirect Topics
Strain rate is defined as the rate at which strain occurs in a workpiece, measured in units of 1/s or s − 1, and is calculated as the product of the drawing ...
[6]
[PDF] Dimensional Analysis, Scaling, and Similarity - UC Davis Math
The inertial term in the Navier-Stokes equation has the order of magnitude ρ0u · ∇u = O ρ0U2 L , while the viscous term has the order of magnitude ต∆u = O µU L ...
[7]
200 years of the Navier–Stokes equation - SciELO
The Navier-Stokes equation, introduced in 1822, is a nonlinear partial differential equation governing the motion of real viscous fluids, like Newton's second ...
[8]
A revisit of Navier–Stokes equation - ScienceDirect.com
The first partial differential equation for fluid dynamics was much later formulated by Euler in 1752 when he considered only for inviscid fluids (that is why ...
[9]
[PDF] Fundamentals of Continuum Mechanics
Nov 3, 2011 · Continuum mechanics is a mathematical framework for studying the transmis- sion of force through and deformation of materials of all types.
[10]
[PDF] Chapter 2 Kinematics: Deformation and Flow
velocity gradient tensor composed with the deformation gradient. A physical interpretation is that it is gradients (differences) in velocity that lead to ...
[11]
[PDF] 2.6 Material and Spatial Velocity Gradients
Having the spatial velocity gradient l defined, we are now in a position to define the the rate of deformation tensor d and the spin tensor ...
[12]
Dimensional Formula of Velocity Gradient - BYJU'S
Dimensional Formula of Velocity Gradient. The dimensional formula of velocity gradient is given by,. [M0 L0 T-1]. Where,. M = Mass; L = Length; T = Time ...
[13]
[PDF] Chapter 7 Kinematics
Thus, the velocity gradient tensor contains infor- mation of the variation of all components of the velocity in all coordinate directions, as shown in figure ...<|control11|><|separator|>
[14]
[PDF] Lecture 5
The velocity gradient tensor, G, provides the ... strain rate of 3 rad/sec. Rigid rotation rate: 1 ... ˜r ≡ rotation tensor. If a circle gets ...
[15]
Velocity Gradients - Continuum Mechanics
Introduction. Velocity gradients are absolutely essential to analyses involving path dependent materials, such as the plastic deformation of metals.Missing: seminal | Show results with:seminal
[16]
[PDF] Chapter 1 Fluid equations
A Newtonian fluid is one in which the components of the stress tensor are related linearly to the components of the strain rate tensor. This relationship.<|control11|><|separator|>
[17]
[PDF] A Simple Example of 2D Tensor
Tensors are independent of specific reference frames, i.e. they are invariant under coordinate transformations. • Invariance qualifies tensors to describe ...
[18]
[PDF] Stokes's Fundamental Contributions to Fluid Dynamics
Jul 12, 2019 · The basic equations for- mulated by him, the Navier-Stokes equations, are capable of describing fluid flows over a vast range of magnitudes.
[19]
[PDF] Modeling turbulence with the Navier-Stokes equations
Aug 6, 2019 · The general equations of motion for a viscous fluid were obtained by Sir George Stokes in 1845. ... Navier-Stokes equations with viscosity= 0 ...
[20]
Newtonian Fluid - an overview | ScienceDirect Topics
A Newtonian fluid is defined as a type of fluid that follows a constitutive equation where the stress tensor is a linear function of the rate of deformation ...
[21]
[PDF] 1 Governing equations of fluid motion - University of Bristol
In general a constitutive relationship provides the connection between the deviatoric stress tensor and the rate of strain tensor. The latter provides a ...
[22]
[PDF] CHAPTER 6 Viscosity & The Stress Tensor
Stress is a force per unit area, with normal and shear components. The stress tensor (σij) is defined as Σi(n) = σij nj. Viscosity is a fluid's resistance to ...Missing: continuum | Show results with:continuum
[23]
Lattice Boltzmann method for non-Newtonian (power-law) fluids
The magnitude of the local shear rate γ ̇ is related to the second invariant of the rate-of-strain tensor, γ ̇ = D i j D i j , where the components of the rate- ...
[24]
[PDF] The Generalized Newtonian Fluid - MIT OpenCourseWare
For the power-law fluid. ○. From Table B.3 in DPL, we get the rate-of-strain tensor in cylindrical coordinates from which the shear rate is found. ( ). 1. 1. 2.
[25]
[PDF] ON INVARIANTS OF FLUID MECHANICS TENSORS
where the velocity gradient tensor ... To grant a physical interpretation to third invariants one should first find the physical interpretation of the strain rate ...<|control11|><|separator|>
[26]
[PDF] On the velocity gradient tensor
The components of the rate-of-strain tensor e describe motions that are resisted by viscosity; shearing, compression and expansion. The com- ponents of the ...
[27]
3.7 Small strain, rate independent plasticity
... strain rate tensor are parallel to the components of ... For small 𝛾 , we can approximate the Lagrange strain tensor by the infinitesimal strain tensor.
[28]
[PDF] Chapter 11: Strain and Strain Tensors - TU Delft OpenCourseWare
strain rates for local examples. 11-1 Strain ellipsoid. The state of strain in three dimensions can be graphically represented by a strain ellipsoid.
[29]
[PDF] Flow Theory of Plasticity
ε , called equivalent plastic strain, is a scalar measure of plastic strain tensor defined through time integration of the equivalent plastic strain rate. 0.
[30]
Classes of flow rules for finite viscoplasticity - ScienceDirect.com
Classes of flow rules for finite viscoplasticity are defined by assuming that certain measures for plastic strain rate and plastic spin depend on the state ...
[31]
A dynamic flow rule for viscoplasticity in polycrystalline solids under ...
Aug 6, 2025 · For visco-plasticity in polycrystalline solids under high strain rates, we introduce a dynamic flow rule (also called the micro-force ...
[32]
A review on Johnson Cook material model - ScienceDirect.com
Flow behaviors, high strain rates, temperature of metals, deformation of diverse materials are predicted by the parameters acquired by the Johnson cook model.
[33]
Objective Rate - an overview | ScienceDirect Topics
Here, elastic Zaremba-Jaumann corotational derivative is used to express objective rate constitutive relations of SMA materials in view of the possibility of ...
[34]
(PDF) Nonlinear, finite deformation, finite element analysis
Aug 10, 2025 · Its derivation should be based on the Jaumann ... The objective stress rates used in most commercial finite element programs are the Jaumann rate ...
[35]
[PDF] Lecture 3: Simple Flows
For the pipe flow, the strain rate can be expressed as: ˙γ = − dw dr ... component of the strain rate tensor: ˙γ = r d dr uθ r . Therefore, we could ...
[36]
3 9/27 W
In Poiseuille flow (laminar flow through a pipe), the velocity field ... Evaluate the strain rate tensor at $r=0$ , $\frac12R$ and $R$ . What can you ...
[37]
[PDF] viscous fluid flow
Jun 15, 1992 · Viscous fluid flow/Frank M. White.-2nd ed. p. cm. Includes index ... strain-rate tensor; Reynolds stress dissipation, Eq. (6-111).
[38]
Simple Shear Flow - an overview | ScienceDirect Topics
The gradient of the velocity in the direction at right angles to the flow is called the shear rate (sometimes called the velocity gradient or strain rate), and ...
[39]
https://www.sciencedirect.com/science/article/pii/B9781855731981500108
[40]
https://www.sciencedirect.com/science/article/pii/B9780750661652500369
[41]
Newtonian Fluid - an overview | ScienceDirect Topics
A Newtonian fluid is defined as a fluid in which the shear stress is directly proportional to the shear rate at a constant temperature and pressure, ...
[42]
https://www.sciencedirect.com/science/article/pii/B9780444828774500226
[43]
Shear Viscosity - an overview | ScienceDirect Topics
For non-Newtonian fluids, the shear viscosity or apparent viscosity is determined as follows: (6.4) η = τ γ. For shear-thinning and shear-thickening fluids ...
[44]
https://www.sciencedirect.com/science/article/pii/S016931079780007X
[45]
3.4 Strain rates and rates of rotation - Cardiovascular biomechanics
... tensors are defined as a consequence: the velocity gradient tensor L, the symmetric rate of deformation tensor D, and the antisymmetric rate of rotation tensor ...