# Objective stress rate

**stress increment**

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference.wikipedia

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### Hypoelastic material

**hypoelastic**

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.

As explained below, specific formulations of hypoelastic models typically employ a so-called objective stress rate so that the G function exists only implicitly.

### Continuum mechanics

**continuumcontinuous mediacontinuum physics**

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. If the stress and strain measures are material quantities then objectivity is automatically satisfied.

### Derivative

**differentiationdifferentiablefirst derivative**

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference.

### Stress (mechanics)

**stressstressestensile stress**

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference.

### Frame of reference

**reference frameframes of referencereference frames**

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference.

### Constitutive equation

**constitutive relationconstitutive equationsconstitutive model**

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).

### Finite strain theory

**deformation gradientfinite strainfinite deformations**

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).

### Stress measures

**Biot stress tensorsecond Piola-Kirchhoff stressstress**

If the stress and strain measures are material quantities then objectivity is automatically satisfied.

### Lie derivative

**Lie bracketcommuting vector fieldsLie bracket of vector fields**

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

### Simple shear

**shearplane shearshearing**

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.

### Elastic modulus

**modulus of elasticityelastic modulimodulus**

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.

### Shear stress

**shearshearingwall shear stress**

The ratio of the shear stress to the displacement is plotted as a function of time.

### Displacement field (mechanics)

**displacementdisplacement fielddisplacements**

The ratio of the shear stress to the displacement is plotted as a function of time.

### Cauchy stress tensor

**stress tensorprincipal stressstress**

1) the Truesdell rate of the Cauchy stress tensor,

### Covariant transformation

**covariancecovariantcovariantly**

Since is a spatial quantity and the transformation follows the rules of tensor transformations, is objective.

### Infinitesimal strain theory

**strain tensorstraininfinitesimal strain tensor**

For a small enough load step, the material deformation can be characterized by the small (or linearized) strain increment tensor

### Strain-rate tensor

**strain rate tensordeformation rate tensorrate of change of deformation**

is the strain rate tensor (also called the velocity strain) and is the material point velocity or displacement rate.

### Walter Noll

**material objectivityobjectiveframe-indifferent (objective)**

In other words, material constitutive equations should be frame-indifferent (objective).

### Force gauge

**Force gauge and Force Sensor**

Force Gauges usually measure pressure in stress increments and other dependant human factors.