Objective stress rate

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate (or the rate of deformation tensor). The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective.

Predictions from three objective stress rates under shear

There are numerous objective stress rates in continuum mechanics – all of which can be shown to be special forms of Lie derivatives. Some of the widely used objective stress rates are:

1. the Truesdell rate of the Cauchy stress tensor,
2. the Green–Naghdi rate of the Cauchy stress, and
3. the Zaremba-Jaumann rate of the Cauchy stress.

The adjacent figure shows the performance of various objective rates in a simple shear test where the material model is hypoelastic with constant elastic moduli. The ratio of the shear stress to the displacement is plotted as a function of time. The same moduli are used with the three objective stress rates. Clearly there are spurious oscillations observed for the Zaremba-Jaumann stress rate. This is not because one rate is better than another but because it is a misuse of material models to use the same constants with different objective rates. For this reason, a recent trend has been to avoid objective stress rates altogether where possible.

Non-objectivity of the time derivative of Cauchy stress

Under rigid body rotations (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Q}} ), the Cauchy stress tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}} transforms as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_r = \boldsymbol{Q}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{Q}^T ~;~~ \boldsymbol{Q}\cdot\boldsymbol{Q}^T = \boldsymbol{\mathit{1}}} Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}} is a spatial quantity and the transformation follows the rules of tensor transformations, ${\boldsymbol {\sigma }}$ is objective. However, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cfrac{d}{dt}(\boldsymbol{\sigma}_r) = \dot{\boldsymbol{\sigma}}_r = \dot{\boldsymbol{Q}}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{Q}^T + \boldsymbol{Q}\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{Q}^T + \boldsymbol{Q}\cdot\boldsymbol{\sigma}\cdot\dot{\boldsymbol{Q}}^T \ne \boldsymbol{Q}\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{Q}^T \,. } Therefore, the stress rate is not objective unless the rate of rotation is zero, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Q}} is constant.

Figure 1. Undeformed and deformed material element, and an elemental cube cut out from the deformed element.

For a physical understanding of the above, consider the situation shown in Figure 1. In the figure the components of the Cauchy (or true) stress tensor are denoted by the symbols Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij}} . This tensor, which describes the forces on a small material element imagined to be cut out from the material as currently deformed, is not objective at large deformations because it varies with rigid body rotations of the material. The material points must be characterized by their initial Lagrangian coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i} . Consequently, it is necessary to introduce the so-called objective stress rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{S}_{ij}} , or the corresponding increment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta S_{ij} = \overset{\circ}{S}_{ij} \Delta t} . The objectivity is necessary for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{S}_{ij}} to be functionally related to the element deformation. It means that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{S}_{ij}} must be invariant with respect to coordinate transformations, particularly the rigid-body rotations, and must characterize the state of the same material element as it deforms.

The objective stress rate can be derived in two ways:

• by tensorial coordinate transformations, which is the standard way in finite element textbooks
• variationally, from strain energy density in the material expressed in terms of the strain tensor (which is objective by definition)

While the former way is instructive and provides useful geometric insight, the latter way is mathematically shorter and has the additional advantage of automatically ensuring energy conservation, i.e., guaranteeing that the second-order work of the stress increment tensor on the strain increment tensor be correct (work conjugacy requirement).

Truesdell stress rate of the Cauchy stress

The relation between the Cauchy stress and the 2nd P-K stress is called the Piola transformation. This transformation can be written in terms of the pull-back of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}} or the push-forward of ${\boldsymbol {S}}$ as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S} = J~\phi^{*}[\boldsymbol{\sigma}] ~;~~ \boldsymbol{\sigma} = J^{-1}~\phi_{*}[\boldsymbol{S}] }

The Truesdell rate of the Cauchy stress is the Piola transformation of the material time derivative of the 2nd P-K stress. We thus define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~\phi_{*}[\dot{\boldsymbol{S}}] }

Expanded out, this means that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~\boldsymbol{F}\cdot\dot{\boldsymbol{S}}\cdot\boldsymbol{F}^T = J^{-1}~\boldsymbol{F}\cdot \left[\cfrac{d}{dt}\left(J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}\right)\right] \cdot\boldsymbol{F}^T = J^{-1}~\mathcal{L}_\varphi[\boldsymbol{\tau}] } where the Kirchhoff stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau} = J~\boldsymbol{\sigma}} and the Lie derivative of the Kirchhoff stress is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}_\varphi[\boldsymbol{\tau}] = \boldsymbol{F}\cdot \left[\cfrac{d}{dt}\left(\boldsymbol{F}^{-1}\cdot\boldsymbol{\tau}\cdot\boldsymbol{F}^{-T}\right)\right] \cdot\boldsymbol{F}^T ~. }

This expression can be simplified to the well known expression for the Truesdell rate of the Cauchy stress

Truesdell rate of the Cauchy stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{l}\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma} \cdot \boldsymbol{l}^T + \text{tr}(\boldsymbol{l})~\boldsymbol{\sigma} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{l}} is the velocity gradient: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{l} = \dot{\boldsymbol{F}} \cdot \boldsymbol{F}^{-1}} . Proof We start with $${\overset {\circ }{\boldsymbol {\sigma }}}=J^{-1}~{\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left(J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}~.$$ Expanding the derivative inside the square brackets, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \overset{\circ}{\boldsymbol{\sigma}} & = J^{-1}~\boldsymbol{F}\cdot(\dot{J}~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T})\cdot\boldsymbol{F}^T + J^{-1}~\boldsymbol{F}\cdot(J~\dot{\boldsymbol{F}^{-1}}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T})\cdot\boldsymbol{F}^T \\ & + J^{-1}~\boldsymbol{F}\cdot(J~\boldsymbol{F}^{-1}\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{F}^{-T})\cdot\boldsymbol{F}^T + J^{-1}~\boldsymbol{F}\cdot(J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\dot{\boldsymbol{F}^{-T}})\cdot\boldsymbol{F}^T \end{align} } or, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~\dot{J}~\boldsymbol{\sigma} + \boldsymbol{F}\cdot\dot{\boldsymbol{F}^{-1}}\cdot\boldsymbol{\sigma} + \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\dot{\boldsymbol{F}^{-T}}\cdot\boldsymbol{F}^T } Now, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{F}\cdot\boldsymbol{F}^{-1} = \boldsymbol{\mathit{1}}} Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}\left(\boldsymbol{F}\cdot\boldsymbol{F}^{-1}\right) = 0 \quad \implies \quad \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} + \boldsymbol{F}\cdot\dot{\boldsymbol{F}^{-1}} = 0 } or, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\boldsymbol{F}^{-1}} = - \boldsymbol{F}^{-1}\cdot\boldsymbol{l} \quad \implies \quad \dot{\boldsymbol{F}^{-T}} = - \boldsymbol{l}^T\cdot\boldsymbol{F}^{-T} } where the velocity gradient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{l} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}} . Also, the rate of change of volume is given by $${\dot {J}}=J~{\text{tr}}({\boldsymbol {d}})=J~{\text{tr}}({\boldsymbol {l}})$$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{d}} is the rate of deformation tensor. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~J~\text{tr}(\boldsymbol{l})~\boldsymbol{\sigma} - \boldsymbol{F}\cdot\boldsymbol{F}^{-1}\cdot\boldsymbol{l}\cdot\boldsymbol{\sigma} + \dot{\boldsymbol{\sigma}} - \boldsymbol{\sigma}\cdot\boldsymbol{l}^T\cdot\boldsymbol{F}^{-T}\cdot\boldsymbol{F}^T } or, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{l}\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma}\cdot\boldsymbol{l}^T + \text{tr}(\boldsymbol{l})~\boldsymbol{\sigma} }

It can be shown that the Truesdell rate is objective.

Truesdell rate of the Kirchhoff stress

The Truesdell rate of the Kirchhoff stress can be obtained by noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S} = \phi^{*}[\boldsymbol{\tau}] ~;~~ \boldsymbol{\tau} = \phi_{*}[\boldsymbol{S}]} and defining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\tau}} = \phi_{*}[\dot{\boldsymbol{S}}]} Expanded out, this means that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\tau}} = \boldsymbol{F}\cdot\dot{\boldsymbol{S}}\cdot\boldsymbol{F}^T = \boldsymbol{F}\cdot \left[\cfrac{d}{dt}\left(\boldsymbol{F}^{-1}\cdot\boldsymbol{\tau}\cdot\boldsymbol{F}^{-T}\right)\right] \cdot\boldsymbol{F}^T = \mathcal{L}_\varphi[\boldsymbol{\tau}] } Therefore, the Lie derivative of ${\boldsymbol {\tau }}$ is the same as the Truesdell rate of the Kirchhoff stress.

Following the same process as for the Cauchy stress above, we can show that

Truesdell rate of the Kirchhoff stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\tau}} = \dot{\boldsymbol{\tau}} - \boldsymbol{l}\cdot\boldsymbol{\tau} - \boldsymbol{\tau}\cdot\boldsymbol{l}^T }

Green-Naghdi rate of the Cauchy stress

This is a special form of the Lie derivative (or the Truesdell rate of the Cauchy stress). Recall that the Truesdell rate of the Cauchy stress is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~\boldsymbol{F}\cdot \left[\cfrac{d}{dt}\left(J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}\right)\right] \cdot\boldsymbol{F}^T ~. } From the polar decomposition theorem we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{R}} is the orthogonal rotation tensor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{R}^{-1} = \boldsymbol{R}^T} ) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{U}} is the symmetric, positive definite, right stretch.

If we assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{U} = \boldsymbol{\mathit{1}}} we get ${\boldsymbol {F}}={\boldsymbol {R}}$. Also since there is no stretch Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = 1} and we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau} = \boldsymbol{\sigma}} . Note that this doesn't mean that there is not stretch in the actual body - this simplification is just for the purposes of defining an objective stress rate. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = \boldsymbol{R}\cdot \left[\cfrac{d}{dt}\left(\boldsymbol{R}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{R}^{-T}\right)\right] \cdot\boldsymbol{R}^T = \boldsymbol{R}\cdot\left[\cfrac{d}{dt}\left(\boldsymbol{R}^T\cdot\boldsymbol{\sigma}\cdot\boldsymbol{R}\right)\right] \cdot\boldsymbol{R}^T } We can show that this expression can be simplified to the commonly used form of the Green-Naghdi rate

Green-Naghdi rate of the Cauchy stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\square}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{\Omega} - \boldsymbol{\Omega}\cdot\boldsymbol{\sigma} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\Omega} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T} . Proof Expanding out the derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = \boldsymbol{R}\cdot\dot{\boldsymbol{R}^T}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{R}\cdot\boldsymbol{R}^T + \boldsymbol{R}\cdot\boldsymbol{R}^T\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{R}\cdot\boldsymbol{R}^T + \boldsymbol{R}\cdot\boldsymbol{R}^T\cdot\boldsymbol{\sigma}\cdot\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T } or, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = \boldsymbol{R}\cdot\dot{\boldsymbol{R}^T}\cdot\boldsymbol{\sigma} + \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T } Now, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{R}\cdot\boldsymbol{R}^T = \boldsymbol{\mathit{1}} \quad \implies \quad \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T = - \boldsymbol{R}\cdot\dot{\boldsymbol{R}^T} } Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T - \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T\cdot\boldsymbol{\sigma} } If we define the angular velocity as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\Omega} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T} we get the commonly used form of the Green–Naghdi rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\square}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{\Omega} - \boldsymbol{\Omega}\cdot\boldsymbol{\sigma} }

The Green–Naghdi rate of the Kirchhoff stress also has the form since the stretch is not taken into consideration, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\square}{\boldsymbol{\tau}} = \dot{\boldsymbol{\tau}} + \boldsymbol{\tau}\cdot\boldsymbol{\Omega} - \boldsymbol{\Omega}\cdot\boldsymbol{\tau} }

Zaremba-Jaumann rate of the Cauchy stress

The Zaremba-Jaumann rate of the Cauchy stress is a further specialization of the Lie derivative (Truesdell rate). This rate has the form

Zaremba-Jaumann rate of the Cauchy stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\triangle}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{w} - \boldsymbol{w}\cdot\boldsymbol{\sigma} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{w}} is the spin tensor.

The Zaremba-Jaumann rate is used widely in computations primarily for two reasons

1. it is relatively easy to implement.
2. it leads to symmetric tangent moduli.

Recall that the spin tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{w}} (the skew part of the velocity gradient) can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T + \frac{1}{2}~\boldsymbol{R}\cdot(\dot{\boldsymbol{U}}\cdot\boldsymbol{U}^{-1} - \boldsymbol{U}^{-1}\cdot\dot{\boldsymbol{U}})\cdot\boldsymbol{R}^T } Thus for pure rigid body motion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T = \boldsymbol{\Omega} } Alternatively, we can consider the case of proportional loading when the principal directions of strain remain constant. An example of this situation is the axial loading of a cylindrical bar. In that situation, since

$${\boldsymbol {U}}={\begin{bmatrix}\lambda _{X}\\&\lambda _{Y}\\&&\lambda _{Z}\end{bmatrix}}$$

we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\boldsymbol{U}} = \begin{bmatrix} \dot{\lambda}_{X}\\ & \dot{\lambda}_{Y}\\ & & \dot{\lambda}_{Z} \end{bmatrix}} Also,

$${\boldsymbol {U}}^{-1}={\begin{bmatrix}1/\lambda _{X}\\&1/\lambda _{Y}\\&&1/\lambda _{Z}\end{bmatrix}}$$

of the Cauchy stress. Therefore,

$${\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}={\begin{bmatrix}{\dot {\lambda }}_{X}/\lambda _{X}\\&{\dot {\lambda }}_{Y}/\lambda _{Y}\\&&{\dot {\lambda }}_{Z}/\lambda _{Z}\end{bmatrix}}=U^{-1}{\dot {U}}$$

This once again gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T = \boldsymbol{\Omega}} In general, if we approximate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{w} \approx \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T } the Green–Naghdi rate becomes the Zaremba-Jaumann rate of the Cauchy stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\triangle}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{w} - \boldsymbol{w}\cdot\boldsymbol{\sigma} }

Other objective stress rates

There can be an infinite variety of objective stress rates. One of these is the Oldroyd stress rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\triangledown}{\boldsymbol{\sigma}} = \mathcal{L}_\varphi[\boldsymbol{\sigma}] = \boldsymbol{F}\cdot\left[\cfrac{d}{dt}\left(\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}\right) \right]\cdot\boldsymbol{F}^T } In simpler form, the Oldroyd rate is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\triangledown}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{l}\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma}\cdot\boldsymbol{l}^T }

If the current configuration is assumed to be the reference configuration then the pull back and push forward operations can be conducted using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{F}^T} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{F}^{-T}} respectively. The Lie derivative of the Cauchy stress is then called the convective stress rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\diamond}{\boldsymbol{\sigma}} = \boldsymbol{F}^{-T}\cdot\left[\cfrac{d}{dt}\left(\boldsymbol{F}^T\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}\right) \right]\cdot\boldsymbol{F}^{-1} } In simpler form, the convective rate is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\diamond}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{l}\cdot\boldsymbol{\sigma} + \boldsymbol{\sigma}\cdot\boldsymbol{l}^T }

Objective stress rates in finite strain inelasticity

Many materials undergo inelastic deformations caused by plasticity and damage. These material behaviors cannot be described in terms of a potential. It is also often the case that no memory of the initial virgin state exists, particularly when large deformations are involved. The constitutive relation is typically defined in incremental form in such cases to make the computation of stresses and deformations easier.

The incremental loading procedure

For a small enough load step, the material deformation can be characterized by the small (or linearized) strain increment tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{e} = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^T\right] \quad \equiv \quad e_{ij} = \tfrac{1}{2}(u_{i,j} + u_{j,i}) } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} is the displacement increment of the continuum points. The time derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial\boldsymbol{e}}{\partial t} = \dot{\boldsymbol{e}} = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{v} + (\boldsymbol{\nabla}\mathbf{v})^T\right] \quad \equiv \quad\dot{e}_{ij} = \tfrac{1}{2} (v_{i,j} + v_{j,i}) } is the strain rate tensor (also called the velocity strain) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} = \dot{\mathbf{u}}} is the material point velocity or displacement rate. For finite strains, measures from the Seth–Hill family (also called Doyle–Ericksen tensors) can be used: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf E_{(m)}=\frac{1}{2m}(\mathbf U^{2m}- \mathbf I) } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}} is the right stretch. A second-order approximation of these tensors is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E}_{(m)} \approx \boldsymbol{e} + {\tfrac 1 2}(\nabla\mathbf{u})^T\cdot\nabla\mathbf{u} - (1 - m) \boldsymbol{e} \cdot \boldsymbol{e}}

Energy-consistent objective stress rates

Consider a material element of unit initial volume, starting from an initial state under initial Cauchy (or true) stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_0} and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}} be the Cauchy stress in the final configuration. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} be the work done (per unit initial volume) by the internal forces during an incremental deformation from this initial state. Then the variation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W} corresponds to the variation in the work done due to a variation in the displacement Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \mathbf{u}} . The displacement variation has to satisfy the displacement boundary conditions.

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}_{(m)}} be an objective stress tensor in the initial configuration. Define the stress increment with respect to the initial configuration as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S} = \boldsymbol{S}_{(m)} - \boldsymbol{\sigma}_0} . Alternatively, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{P}} is the unsymmetric first Piola–Kirchhoff stress referred to the initial configuration, the increment in stress can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{T} = \boldsymbol{P} - \boldsymbol{\sigma}_0} .

Variation of work done

Then the variation in work done can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W = \boldsymbol{S}_{(m)}:\delta\boldsymbol{E}_{(m)} = \boldsymbol{P}:\delta\nabla\mathbf{u} } where the finite strain measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{E}_{(m)}} is energy conjugate to the stress measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}^{(m)}} . Expanded out, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W = \left(\boldsymbol{S}+\boldsymbol{\sigma}_0\right):\delta\boldsymbol{E}_{(m)} = \left(\boldsymbol{T}+\boldsymbol{\sigma}_0\right):\delta\nabla\mathbf{u} \,. } The objectivity of stress tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}_{(m)}} is ensured by its transformation as a second-order tensor under coordinate rotations (which causes the principal stresses to be independent from coordinate rotations) and by the correctness of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}_{(m)}:\delta\boldsymbol{E}_{(m)}} as a second-order energy expression.

From the symmetry of the Cauchy stress, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_0:\delta\nabla\mathbf{u} = \boldsymbol{\sigma}_0:\delta\boldsymbol{e} \,. } For small variations in strain, using the approximation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}:\delta\boldsymbol{E}_{(m)} \approx \boldsymbol{S}:\delta\nabla\mathbf{u} } and the expansions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_0:\delta\boldsymbol{E}_{(m)} = \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{E}_{(m)}}{\partial \nabla\mathbf{u}}:\delta\nabla\mathbf{u}\right] ~,~~ \boldsymbol{\sigma}_0:\delta\boldsymbol{e} = \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{e}}{\partial \nabla\mathbf{u}}:\delta\nabla\mathbf{u}\right] } we get the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{E}_{(m)}}{\partial \nabla\mathbf{u}}:\delta\nabla\mathbf{u}\right] + \boldsymbol{S}:\delta\nabla\mathbf{u} = \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{e}}{\partial \nabla\mathbf{u}}:\delta\nabla\mathbf{u}\right] + \boldsymbol{T}:\delta\nabla\mathbf{u} \,. } Imposing the variational condition that the resulting equation must be valid for any strain gradient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\nabla\mathbf{u}} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S} = \boldsymbol{T} - \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{E}_{(m)}}{\partial \nabla\mathbf{u}}-\frac{\partial \boldsymbol{e}}{\partial \nabla\mathbf{u}}\right] } (1)

We can also write the above equation as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}_{(m)} = \boldsymbol{P} - \boldsymbol{\sigma}_0:\frac{\partial}{\partial \nabla\mathbf{u}}\left[\boldsymbol{E}_{(m)} - \boldsymbol{e}\right] \,. } (2)

Time derivatives

The Cauchy stress and the first Piola-Kirchhoff stress are related by (see Stress measures) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma} = \boldsymbol{P}\cdot\boldsymbol{F}^T J^{-1} = (\boldsymbol{P} + \boldsymbol{P}\cdot\nabla\mathbf{u}^T) J^{-1} \,. } For small incremental deformations, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^{-1} \approx 1 - \nabla\cdot\mathbf{u} \,. } Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\boldsymbol{\sigma} = \boldsymbol{\sigma} - \boldsymbol{\sigma}_0 \approx (\boldsymbol{P} + \boldsymbol{P}\cdot\nabla\mathbf{u}^T) (1 - \nabla\cdot\mathbf{u}) - \boldsymbol{\sigma}_0 \,. } Substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{T} + \boldsymbol{\sigma}_0 = \boldsymbol{P}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\boldsymbol{\sigma} \approx [\boldsymbol{T} + \boldsymbol{\sigma}_0 + (\boldsymbol{T} + \boldsymbol{\sigma}_0)\cdot\nabla\mathbf{u}^T] (1 - \nabla\cdot\mathbf{u}) - \boldsymbol{\sigma}_0 \,. } For small increments of stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{T}} relative to the initial stress Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_0} , the above reduces to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\boldsymbol{\sigma} \approx \boldsymbol{T} - \boldsymbol{\sigma}_0 (\nabla \cdot \mathbf{u}) + \boldsymbol{\sigma}_0 \cdot \nabla \mathbf{u}^T \,. } (3)

From equations (1) and (3) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S} = \Delta\boldsymbol{\sigma} + \boldsymbol{\sigma}_0 (\nabla \cdot \mathbf{u}) - \boldsymbol{\sigma}_0 \cdot \nabla \mathbf{u}^T- \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{E}_{(m)}}{\partial \nabla\mathbf{u}}-\frac{\partial \boldsymbol{e}}{\partial \nabla\mathbf{u}}\right] } (4)

Recall that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}} is an increment of the stress tensor measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S}_{(m)}} . Defining the stress rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{S} =: \overset{\circ}{\boldsymbol{S}}_{(m)} \Delta t } and noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\boldsymbol{\sigma} = \dot{\boldsymbol{\sigma}} \Delta t } we can write equation (4) as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{S}}_{(m)} \Delta t = \dot{\boldsymbol{\sigma}} \Delta t + \boldsymbol{\sigma}_0 (\nabla \cdot \mathbf{v}) \Delta t - \boldsymbol{\sigma}_0 \cdot \nabla \mathbf{v}^T \Delta t - \boldsymbol{\sigma}_0:\left[\frac{\partial \boldsymbol{E}_{(m)}}{\partial \nabla\mathbf{u}}-\frac{\partial \boldsymbol{e}}{\partial \nabla\mathbf{u}}\right]} (5)

Taking the limit at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t \rightarrow 0} , and noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_0 = \boldsymbol{\sigma}} at this limit, one gets the following expression for the objective stress rate associated with the strain measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{E}_{(m)}} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{S}}_{(m)} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}(\nabla \cdot \mathbf{v}) - \boldsymbol{\sigma} \cdot \nabla \mathbf{v}^T - \boldsymbol{\sigma} : \frac{\partial}{\partial t} \left[\frac{\partial}{\partial \nabla\mathbf{u}}\left(\boldsymbol{E}_{(m)} - \boldsymbol{e}\right)\right] \,. } (6)

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot \sigma_{ij} = \partial \sigma_{ij} /\partial t} = material rate of Cauchy stress (i.e., the rate in Lagrangian coordinates of the initial stressed state).

Work-conjugate stress rates

A rate for which there exists no legitimate finite strain tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{E}_{(m)}} associated according to Eq. (6) is energetically inconsistent, i.e., its use violates energy balance (i.e., the first law of thermodynamics).

Evaluating Eq. (6) for general Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=2} , one gets a general expression for the objective stress rate:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{S}}_{(m)} = \overset{\circ}{\boldsymbol{S}}_{(2)} + \tfrac{1}{2}(2 - m) [\boldsymbol{\sigma}\cdot \dot \boldsymbol{e} + (\boldsymbol{\sigma} \cdot \dot \boldsymbol{e})^T] } (7)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{\circ}{\boldsymbol{S}}_{(2)}} is the objective stress rate associated with the Green-Lagrangian strain (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=2} ).

In particular,

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=2} gives the Truesdell stress rate
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=0} gives the Zaremba-Jaumann rate of Kirchhoff stress
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=1} gives the Biot stress rate

(Note that m = 2 leads to Engesser's formula for critical load in shear buckling, while m = -2 leads to Haringx's formula which can give critical loads differing by >100%).

Non work-conjugate stress rates

Other rates, used in most commercial codes, which are not work-conjugate to any finite strain tensor are:

• the Zaremba-Jaumann, or corotational, rate of Cauchy stress: It differs from Zaremba-Jaumann rate of Kirchhoff stress by missing the rate of relative volume change of material. The lack of work-conjugacy is usually not a serious problem since that term is negligibly small for many materials and zero for incompressible materials (but in indentation of a sandwich plate with foam core, this rate can give an error of >30% in the indentation force).
• the Cotter–Rivlin rate corresponds to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = -2} but it again misses the volumetric term.
• the Green–Naghdi rate: This objective stress rate is not work-conjugate to any finite strain tensor, not only because of the missing volumetric term but also because the material rotation velocity is not exactly equal to the spin tensor. In the vast majority of applications, the errors in the energy calculation, caused by these differences, are negligible. However, it must be pointed out that a large energy error was already demonstrated for a case with shear strains and rotations exceeding about 0.25.
• the Oldroyd rate.

Objective rates and Lie derivatives

The objective stress rates could also be regarded as the Lie derivatives of various types of stress tensor (i.e., the associated covariant, contravariant and mixed components of Cauchy stress) and their linear combinations. The Lie derivative does not include the concept of work-conjugacy.

Tangential stiffness moduli and their transformations to achieve energy consistency

The tangential stress-strain relation has generally the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot S_{ij}^{(m)} = C_{ijkl}^{(m)} \dot e_{kl}} (6)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ijkl}^{(m)}} are the tangential moduli (components of a 4th-order tensor) associated with strain tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ij}^{(m)}} . They are different for different choices of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} , and are related as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[C_{ijkl}^{(m)} - C^{(2)}_{ijkl} - {\tfrac 1 4}(2-m)(S_{ik}\delta_{jl} + S_{jk}\delta_{il} + S_{il}\delta_{jk} + S_{jl}\delta_{ik}) \right] v_{k,l}= 0} (7)

From the fact that Eq. (7) must hold true for any velocity gradient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{k,l}} , it follows that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ijkl}^{(m)} = C^{(2)}_{ijkl} + (2-m)[S_{ik}\delta_{jl}]_\mathrm{sym},~~[S_{ik}\delta_{jl}]_\mathrm{sym} = {\tfrac 1 4} (S_{ik}\delta_{jl} + S_{jk}\delta_{il} + S_{il}\delta_{jk} + S_{jl}\delta_{ik})} (8)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ijkl}^{(2)}} are the tangential moduli associated with the Green–Lagrangian strain (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=2} ), taken as a reference, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij}} = current Cauchy stress, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{ij}} = Kronecker delta (or unit tensor).

Eq. (8) can be used to convert one objective stress rate to another. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij} \dot e_{kk} = (S_{ij} \delta_{kl}) \delta e_{kl}} , the transformation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ijkl}^{\mathrm{conj}} = C_{ijkl}^{\mathrm{nonconj}} + S_{ij}\delta_{kl}} (9)

can further correct for the absence of the term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij} v_{k,k}} (note that the term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij}\delta_{km}} does not allow interchanging subscripts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ij} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle kl} , which means that its absence breaks the major symmetry of the tangential moduli tensor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ijkl}^{\mathrm{nonconj}}} ).

Large strain often develops when the material behavior becomes nonlinear, due to plasticity or damage. Then the primary cause of stress dependence of the tangential moduli is the physical behavior of material. What Eq. (8) means that the nonlinear dependence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ijkl}} on the stress must be different for different objective stress rates. Yet none of them is fundamentally preferable, except if there exists one stress rate, one Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} , for which the moduli can be considered constant.

See also

• Cauchy stress tensor
• Stress measures
• Principle of material objectivity
• Hypoelastic material

External links

• Objectivity in Classical Mechanics