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Fix CS1 deprecated date parameter errors

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In [[continuum mechanics]], a '''hypoelastic material'''<ref>Truesdell (1963).</ref> is an [[elasticity (physics)|elastic]] material that has a [[constitutive model]] independent of [[finite strain theory|finite strain]] measures except in the linearized case. Hypoelastic material models are distinct from [[hyperelastic material]] models (or standard elasticity models) in that, except under special circumstances, they cannot be derived from a [[strain energy density function]].

Hypoelastic materials are described by a relation of the form
:<math>
\dot{\boldsymbol{T}} = \mathsf{D}:\dot{\boldsymbol{F}}
</math>
where <math> \boldsymbol{T}</math> is a [[stress measures|stress measure]], <math>\mathsf{D}</math> is the material property tensor, <math>\boldsymbol{F}</math> is the [[deformation gradient]], and superposed dots indicate time derivatives. In hypoelasticity, the quantity <math>\mathsf{D}</math> is a function of stress while, in hyperelasticity, the material tensor may also depend on strains and rotations.

== Hypoelasticity and objective stress rates ==
In many practical problems of solid mechanics, it is sufficient to characterize material deformation by the small (or linearized) strain
tensor
:<math>
\varepsilon_{ij} = \frac 1 2 (u_{i,j} + u_{j,i})
</math>
where <math>u_i</math> are the components of the displacements of continuum points, the subscripts refer to Cartesian coordinates <math>x_i</math> <math>(i=1,2,3)</math>, and the subscripts preceded by a comma denote partial derivatives (e.g.,
<math>u_{i,j} = \partial u_i /\partial x_j</math>). But there are also many problems where the finiteness of strain must be taken into account. These are of two kinds:
# large nonlinear elastic deformations possessing a potential energy, <math>W(\boldsymbol{F})</math> (exhibited, e.g., by rubber), in which the stress tensor components are obtained as the partial derivatives of <math>W</math> with respect to the finite strain tensor components; and
# inelastic deformations possessing no potential, in which the stress-strain relation is defined incrementally.

In the former kind, the total strain formulation described in the article on [[finite strain theory]] is appropriate. In the latter kind an incremental (or rate) formulation is necessary and must be used in every load or time step of a [[finite element]] computer program using updated Lagrangian procedure. The absence of a potential raises intricate questions due to the freedom in the choice of finite strain measure and characterization of the stress rate.

For a sufficiently small loading step (or increment), one may use the [[finite strain theory|deformation rate tensor]] (or velocity strain)
:<math>
d_{ij} = \dot \varepsilon_{ij} = \frac 1 2 (v_{i,j} + v_{j,i})
</math>
or increment
:<math>
\Delta \varepsilon_{ij} = \dot \varepsilon_{ij} \Delta t = d_{ij} \Delta t
</math>
representing the linearized strain increment from the initial (stressed and deformed) state in the step. Here the superior dot represents the time derivative <math>\partial /\partial t</math>, <math>\Delta</math> denotes a small increment over the step, <math>t</math> = time, and <math>v_i = \dot u_i</math> = material point velocity or displacement rate.

However, it would not be [[objectivity (frame invariance)|objective]] to use the time derivative of the [[cauchy stress tensor|Cauchy (or true) stress]] <math>\sigma_{ij}</math>. This stress, which describes the forces on a small material element imagined to be cut out from the material as currently deformed, is not objective because it varies with rigid body rotations of the material. The material points must be characterized by their initial coordinates <math>X_i</math> (called Lagrangian) because different material particles are contained in the element that is cut out (at the same location) before and after the incremental deformation.

Consequently, it is necessary to introduce the so-called [[objective stress rates|objective stress rate]] <math>\hat \sigma_{ij}</math>, or the corresponding increment <math>\Delta \sigma_{ij} = \hat \sigma_{ij} \Delta t</math>. The objectivity is necessary for <math>\hat \sigma_{ij}</math> to be functionally related to the element deformation. It means that that <math>\hat \sigma_{ij}</math> must be invariant with respect to coordinate transformations (particularly rotations) and must characterize the state of the same material element as it deforms.

== Notes ==
{{Reflist}}

== See also ==
* [[Stress measures]]
* [[Hyperelastic material]]
* [[Objective stress rates]]
* [[Objectivity (frame invariance)]]
* [[Principle of material objectivity]]
* [[Finite strain theory]]
* [[Infinitesimal strain theory]]

== Bibliography ==
* {{Citation|last=Truesdell|first=Clifford|year=1963|title=Remarks on hypo-elasticity|journal=Journal of research of the National Bureau of Standards - B. Mathematics and Mathematical Physics|volume=67B|number=3|pages=141–143}}

[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]

T-Distributed Stochastic Neighbor Embedding - Revision history

en>Monkbot: Fix CS1 deprecated date parameter errors