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.
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| - Objective stress rate (en)
- 客观应力率 (zh)
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| - 客观应力率（英語：objective stress rate）在连续介质力学中是指不依赖参考系的应力时间导数。一些本构关系可以表示为应力率与应变率之间的关系。由于物体的力学响应不应随参考系的变化而变化（即客观性），而由柯西应力张量求时间导数直接得到的应力率张量并不是客观的，故需要定义具有客观性的应力率。 在连续介质力学中有多种客观应力率的定义，它们都可以表示成李导数的形式。其中最常见的客观应力率包括
* 柯西应力张量的特鲁斯德尔（Truesdell）应力率 （其中为速度梯度张量）
* 柯西应力张量的格林－纳厄迪（Green-Naghdi）应力率 （其中，为转动张量）
* 柯西应力张量的耀曼（Jaumann）应力率 （其中为自旋张量，即速度梯度张量的反对称部分） (zh)
- 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. (en)
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| - 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. 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. (en)
- 客观应力率（英語：objective stress rate）在连续介质力学中是指不依赖参考系的应力时间导数。一些本构关系可以表示为应力率与应变率之间的关系。由于物体的力学响应不应随参考系的变化而变化（即客观性），而由柯西应力张量求时间导数直接得到的应力率张量并不是客观的，故需要定义具有客观性的应力率。 在连续介质力学中有多种客观应力率的定义，它们都可以表示成李导数的形式。其中最常见的客观应力率包括
* 柯西应力张量的特鲁斯德尔（Truesdell）应力率 （其中为速度梯度张量）
* 柯西应力张量的格林－纳厄迪（Green-Naghdi）应力率 （其中，为转动张量）
* 柯西应力张量的耀曼（Jaumann）应力率 （其中为自旋张量，即速度梯度张量的反对称部分） (zh)
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