Elastičnost (fizika) — разлика између измена
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Ред 40:
Tehnička granica elastičnosti je naprezanje pri kojem osetljivi instrumenti za merenje osete prvo primetno trajno produženje materijala (pri još nepromenjenom preseku A<sub>o</sub>). Nakon te granice (obično na kraju linearnog rastezanja) materijal se rasteže plastično i nakon prestanka delovanja sile ne vraća se više na početnu dužinu -{''L<sub>0</sub>''}-, već ostaje određeno trajno produženje, uz suženje preseka, ''A < A<sub>0</sub>)''.
== Konačna elastičnost ==
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The elastic behavior of objects that undergo finite deformations has been described using a number of models, such as [[Cauchy elastic material]] models, [[Hypoelastic material]] models, and [[Hyperelastic material]] models. The [[deformation gradient]] ('''''F''''') is the primary deformation measure used in [[finite strain theory]].
=== Košijevi elastični materijali ===
{{main|Košijevi elastični materijal}}
A material is said to be Cauchy-elastic if the [[Cauchy stress tensor]] '''''σ''''' is a function of the [[deformation gradient]] '''''F''''' alone:
:<math>\ \boldsymbol{\sigma} = \mathcal{G}(\boldsymbol{F}) </math>
It is generally incorrect to state that Cauchy stress is a function of merely a [[strain tensor]], as such a model lacks crucial information about material rotation needed to produce correct results for an anisotropic medium subjected to vertical extension in comparison to the same extension applied horizontally and then subjected to a 90-degree rotation; both these deformations have the same spatial strain tensors yet must produce different values of the Cauchy stress tensor.
Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses might depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "[[hyperelastic material]]" models (for which stress can be derived from a scalar "elastic potential" function).
=== Hipoelastični materijali ===
{{main-lat|Hipoelastični materijal}}
Hipoelastični materijali can be rigorously defined as one that is modeled using a [[constitutive equation]] satisfying the following two criteria:<ref>{{cite book|last1=Truesdell|first1=Clifford|last2=Noll|first2=Walter|title=The Non-linear Field Theories of Mechanics|year=2004|publisher=Springer-Verlag|location=Berlin Heidelberg New York|edition=3rd|isbn=978-3-540-02779-9|page=401}}</ref>
1. The Cauchy stress <math>\boldsymbol{\sigma}</math> at time <math>t</math> depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a [[Cauchy elastic material]], for which the current stress depends only on the current configuration rather than the history of past configurations.
2. There is a tensor-valued function <math>G</math> such that
<math>
\dot{\boldsymbol{\sigma}} = G(\boldsymbol{\sigma},\boldsymbol{L}) \,,
</math>
in which <math>\dot{\boldsymbol{\sigma}}</math> is the material rate of the Cauchy stress tensor, and <math>\boldsymbol{L}</math> is the spatial [[velocity gradient]] tensor.
If only these two original criteria are used to define hypoelasticity, then [[hyperelasticity]] would be included as a special case, which prompts some constitutive modelers to append a third criterion that specifically requires a hypoelastic model to ''not'' be hyperelastic (i.e., hypoelasticity implies that stress is not derivable from an energy potential). If this third criterion is adopted, it follows that a hypoelastic material might admit nonconservative adiabatic loading paths that start and end with the same [[deformation gradient]] but do ''not'' start and end at the same internal energy.
Note that the second criterion requires only that the function <math>G</math> ''exists''. As detailed in the main [[Hypoelastic material]] article, specific formulations of hypoelastic models typically employ so-called objective rates so that the <math>G</math> function exists only implicitly and is typically needed explicitly only for numerical stress updates performed via direct integration of the actual (not objective) stress rate.
=== Hiperelastični materijali ===
{{main-lat|Hiperelastični materijal}}
Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a [[strain energy density function]] (''W''). A model is hyperelastic if and only if it is possible to express the [[Cauchy stress tensor]] as a function of the [[deformation gradient]] via a relationship of the form
:<math>
\boldsymbol{\sigma} = \cfrac{1}{J}~ \cfrac{\partial W}{\partial \boldsymbol{F}}\boldsymbol{F}^T \quad \text{where} \quad J := \det\boldsymbol{F} \,.
</math>
This formulation takes the energy potential (''W'') as a function of the [[deformation gradient]] (<math>\boldsymbol{F}</math>). By also requiring satisfaction of [[material objectivity]], the energy potential may be alternatively regarded as a function of the [[Cauchy-Green deformation tensor]] (<math>\boldsymbol{C}:=\boldsymbol{F}^T\boldsymbol{F}</math>), in which case the hyperelastic model may be written alternatively as
:<math>
\boldsymbol{\sigma} = \cfrac{2}{J}~ \boldsymbol{F}\cfrac{\partial W}{\partial \boldsymbol{C}}\boldsymbol{F}^T \quad \text{where} \quad J := \det\boldsymbol{F} \,.
</math>
== Reference ==
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