# Hypo-elasticity

Hypo-elasticity (hypo-, ancient Greek “under”) is a material model for elasticity proposed by Clifford Truesdell , in which the change in stresses is a linear function of the change in strains determined exclusively by the stresses . Only by following this relationship over time does the stress - strain relationship arise for the present material law .

There are material models that belong to both hypo and Cauchy elasticity . However, both theories also have members that are not covered by the other theory . The formulation of hypo-elasticity is so general that it even has connections to plasticity theory .

Originally, Truesdell intended to formulate a new concept for elastic behavior that, except in the special linear case, should be exclusive for large deformations . Truesdell himself said in 1963: “Hypo-elasticity seems to offer a convenient summary of certain aspects of response common to many materials, rather than a theory of any particular material” (in German, for example: “Hypo-elasticity seems to be more of a suitable option to simulate certain aspects of behavior that is common to many materials, as being a theory for a concrete material. ")

## Historical summary

C. Truesdell describes the development of his theory as follows:

The basic concept was proposed by Augustin-Louis Cauchy himself in his theory about elastic media with residual stresses , whereby he did not mention rates directly, but implied it in infinitesimal static deformations. Special cases of hypo-elasticity with constant rates were given by C. Jaumann and Erwin Lohr . A theory of viscoelasticity that included a special case of hypoelasticity was proposed by Stanisław Zaremba without mentioning the implications for an elasticity theory . The general theory was formulated by F. Fromm, but not investigated further. Ultimately, the theory was then proposed by C. Truesdell. Barry Bernstein discovered the connection with plasticity and formulated integrability conditions. Research into the theory continues well into the twenty-first century.

## definition

A hypo-elastic material obeys a constitutive law of form

${\ displaystyle {\ stackrel {\ bullet} {\ boldsymbol {\ sigma}}} = {\ stackrel {4} {\ mathbf {H}}} ({\ boldsymbol {\ sigma}}): \ mathbf {d} }$.

In it is

• ${\ displaystyle {\ stackrel {\ bullet} {\ boldsymbol {\ sigma}}}}$a suitable rate
• of Cauchy's stress tensor ${\ displaystyle {\ boldsymbol {\ sigma}}}$
• ${\ displaystyle {\ stackrel {4} {\ mathbf {H}}}}$a constitutive fourth order tensor that only depends on the stresses and not on the strains or their rates
• ":" The Frobenius scalar product of tensors
• ${\ displaystyle \ mathbf {d} = {\ frac {1} {2}} (\ mathbf {l + l} ^ {\ mathrm {T}}) = {\ stackrel {\ triangle} {\ mathbf {e} }}}$ the spatial distortion rate
• ${\ displaystyle \ mathbf {l} = {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ {- 1}}$the spatial velocity gradient
• ${\ displaystyle {\ stackrel {\ triangle} {\ mathbf {e}}}: = {\ dot {\ mathbf {e}}} + \ mathbf {e \ cdot l} + \ mathbf {l} ^ {\ mathrm {T}} \ cdot \ mathbf {e}}$the objective covariant Oldroyd derivative
• of the Euler-Almansi tensor .${\ displaystyle \ mathbf {e} = {\ frac {1} {2}} \ left (\ mathbf {1} - (\ mathbf {F \ cdot F} ^ {\ mathrm {T}}) ^ {- 1 } \ right)}$

The time derivatives of the stress tensor can be derivatives that are invariant under a Euclidean transformation of the observer , e.g. B. the derivations from the following table: ${\ displaystyle {\ stackrel {\ bullet} {\ boldsymbol {\ sigma}}}}$

Surname formula
Zaremba-Jaumann derivation ${\ displaystyle {\ stackrel {\ circ} {\ varvec {\ sigma}}}: = {\ dot {\ varvec {\ varvec {\ sigma}}} + {\ varvec {\ sigma}} \ cdot \ mathbf {w} - \ mathbf {w} \ cdot {\ boldsymbol {\ sigma}}}$
Convective contravariant Oldroyd derivative ${\ displaystyle {\ stackrel {\ nabla} {\ varvec {\ sigma}}}: = {\ dot {\ varvec {\ sigma}}} - \ mathbf {l} \ cdot {\ varvec {\ sigma}} - {\ boldsymbol {\ sigma}} \ cdot \ mathbf {l} ^ {\ mathrm {T}}}$
Cauchy derivative ${\ displaystyle {\ stackrel {\ diamond} {\ varvec {\ sigma}}}: = {\ dot {\ varvec {\ varvec {\ sigma}}} + \ operatorname {Sp} (\ mathbf {l}) {\ varvec { \ sigma}} - \ mathbf {l} \ cdot {\ boldsymbol {\ sigma}} - {\ boldsymbol {\ sigma}} \ cdot \ mathbf {l} ^ {\ mathrm {T}}}$

The tensor

${\ displaystyle \ mathbf {w} = \ mathbf {ld} = {\ frac {1} {2}} (\ mathbf {l} - \ mathbf {l} ^ {\ mathrm {T}})}$

is called the spin or vortex tensor .

## objectivity

In the definition of hypo-elasticity, the time derivatives have already been taken to ensure that they do not depend on a Euclidean transformation of the observer. However, this does not guarantee that the product of the constitutive tensor and the deformation speed is an objective tensor. For this to be the case, it is necessary and sufficient that ${\ displaystyle {\ stackrel {4} {\ mathbf {H}}} ({\ boldsymbol {\ sigma}}): \ mathbf {d}}$

${\ displaystyle {\ stackrel {4} {\ mathbf {H}}} (\ mathbf {Q} \ cdot {\ boldsymbol {\ sigma}} \ cdot \ mathbf {Q} ^ {\ mathrm {T}}): (\ mathbf {Q \ cdot d \ cdot Q} ^ {\ mathrm {T}}) = \ mathbf {Q} \ cdot \ left ({\ stackrel {4} {\ mathbf {H}}} ({\ varvec {\ sigma}}): \ mathbf {d} \ right) \ cdot \ mathbf {Q} ^ {\ mathrm {T}}}$

holds for any orthogonal tensors , ${\ displaystyle \ mathbf {Q}}$

${\ displaystyle {\ stackrel {4} {\ mathbf {H}}} ({\ boldsymbol {\ sigma}}): \ mathbf {d}}$

so is an isotropic tensor function of . ${\ displaystyle {\ boldsymbol {\ sigma}} \; {\ textsf {and}} \; \ mathbf {d}}$

## Anisotropy

In an unstressed particle of a hypo-elastic material free of residual stresses . If there is a load in this state, the result is: ${\ displaystyle {\ boldsymbol {\ sigma}} = \ mathbf {0}}$

${\ displaystyle {\ dot {\ boldsymbol {\ sigma}}} = {\ stackrel {4} {\ mathbf {H}}} (\ mathbf {0}) \ cdot \ mathbf {d}}$

regardless of which objective time derivative is used for the voltages. Because there is an isotropic tensor function on the right-hand side of this equation, the stresses develop from a stress-free state with small distortions initially as in an isotropic material. ${\ displaystyle {\ dot {\ boldsymbol {\ sigma}}}}$

Conversely, a material that does not react isotropically - but anisotropically - from a stress-free state in the event of small deformations is not hypo-elastic.

## Isotropic Cauchy elasticity

In the case of an isotropic Cauchy elastic material, the Cauchy stress tensor results as an isotropic tensor function of the left Cauchy-Green tensor , e.g. B .: ${\ displaystyle \ mathbf {b}: = \ mathbf {F \ cdot F} ^ {\ mathrm {T}}}$

${\ displaystyle {\ boldsymbol {\ sigma}} = \ phi _ {0} \ mathbf {I} + \ phi _ {1} \ mathbf {b} + \ phi _ {2} \ mathbf {b \ cdot b} }$.

The coefficients are scalar , isotropic functions of the main invariants or other invariants of the left Cauchy-Green tensor. Because of ${\ displaystyle \ phi _ {0,1,2}}$

${\ displaystyle {\ dot {\ mathbf {b}}} = {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ {- 1} \ cdot \ mathbf {F} \ cdot \ mathbf { F} ^ {\ mathrm {T}} + \ mathbf {F} \ cdot \ mathbf {F} ^ {\ mathrm {T}} \ cdot \ mathbf {F} ^ {\ mathrm {T} -1} \ cdot {\ dot {\ mathbf {F}}} ^ {\ mathrm {T}} = \ mathbf {l} \ cdot \ mathbf {b} + \ mathbf {b} \ cdot \ mathbf {l} ^ {\ mathrm { T}}}$

is calculated

${\ displaystyle {\ dot {\ phi _ {i}}} = {\ frac {\ mathrm {d} \ phi _ {i}} {\ mathrm {d} \ mathbf {b}}}: {\ dot { \ mathbf {b}}} = {\ frac {\ mathrm {d} \ phi _ {i}} {\ mathrm {d} \ mathbf {b}}}: (\ mathbf {l} \ cdot \ mathbf {b } + \ mathbf {b} \ cdot \ mathbf {l} ^ {\ mathrm {T}}) = 2 \ left ({\ frac {\ mathrm {d} \ phi _ {i}} {\ mathrm {d} \ mathbf {b}}} \ cdot \ mathbf {b} \ right): \ mathbf {d}}$

The derivative of the stress tensor with respect to time yields:

${\ displaystyle {\ begin {array} {rcl} {\ dot {\ boldsymbol {\ sigma}}} & = & {\ dot {\ phi}} _ {0} \ mathbf {I} + {\ dot {\ phi}} _ {1} \ mathbf {b} + \ phi _ {1} {\ dot {\ mathbf {b}}} + {\ dot {\ phi}} _ {2} \ mathbf {b \ cdot b } + \ phi _ {2} {\ dot {\ mathbf {b}}} \ cdot \ mathbf {b} + \ phi _ {2} \ mathbf {b} \ cdot {\ dot {\ mathbf {b}} } \\ & = & 2 \ left [\ mathbf {I} \ otimes \ left ({\ frac {\ mathrm {d} \ phi _ {0}} {\ mathrm {d} \ mathbf {b}}} \ cdot \ mathbf {b} \ right) + \ mathbf {b} \ otimes \ left ({\ frac {\ mathrm {d} \ phi _ {1}} {\ mathrm {d} \ mathbf {b}}} \ cdot \ mathbf {b} \ right) + (\ mathbf {b \ cdot b}) \ otimes \ left ({\ frac {\ mathrm {d} \ phi _ {2}} {\ mathrm {d} \ mathbf {b }}} \ cdot \ mathbf {b} \ right) \ right]: \ mathbf {d} \\ && + \ mathbf {l} \ cdot {\ varvec {\ sigma}} + {\ varvec {\ sigma}} \ cdot \ mathbf {l} ^ {\ mathrm {T}} +2 \ phi _ {2} \ mathbf {b \ cdot d \ cdot b} -2 \ phi _ {0} \ mathbf {d} \\\ rightarrow {\ stackrel {\ nabla} {\ boldsymbol {\ sigma}}} & = & 2 \ left [\ mathbf {I} \ otimes \ left ({\ frac {\ mathrm {d} \ phi _ {0}} { \ mathrm {d} \ mathbf {b}}} \ cdot \ mathbf {b} \ right ) + \ mathbf {b} \ otimes \ left ({\ frac {\ mathrm {d} \ phi _ {1}} {\ mathrm {d} \ mathbf {b}}} \ cdot \ mathbf {b} \ right ) + (\ mathbf {b \ cdot b}) \ otimes \ left ({\ frac {\ mathrm {d} \ phi _ {2}} {\ mathrm {d} \ mathbf {b}}} \ cdot \ mathbf {b} \ right) + \ phi _ {2} (\ mathbf {b \ otimes b}) ^ {\ stackrel {23} {\ mathrm {T}}} - \ phi _ {0} {\ stackrel {4 } {\ mathbf {I}}} \ right]: \ mathbf {d} \\ & = & {\ stackrel {4} {\ mathbf {H}}} (\ mathbf {b}): \ mathbf {d} . \ end {array}}}$

So if the stress-deformation relationship is invertible

${\ displaystyle {\ boldsymbol {\ sigma}} = \ mathbf {f} (\ mathbf {b}) \ quad \ leftrightarrow \ quad \ mathbf {b} = \ mathbf {f} ^ {- 1} ({\ boldsymbol {\ sigma}})}$,

then an isotropic, Cauchy-elastic material is also hypo-elastic, which Walter Noll first demonstrated .

## plasticity

Hypo-elasticity also includes models that are incompatible with elasticity. For example, if the constitutive tensor vanishes in a certain state of stress , the stresses remain constant as the strain progresses. This behavior is known from ideal plasticity. ${\ displaystyle {\ stackrel {4} {\ mathbf {H}}} ({\ boldsymbol {\ sigma}})}$

Barry Bernstein formulated conditions that must be fulfilled for hypo-elasticity to be consistent with a certain stress-strain relationship.

## References and footnotes

1. C. Truesdell: Remarks on Hypo-Elasticity , Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics, Vol. 67B, No. July 3-September 1963
2. AL Cauchy: Sur l'equilibre et le mouvement intérieur des corps considérés comme des masses continues , Oeuvres 9, 243-269 (1829)
3. ^ C. Jaumann: System of physical and chemical differential laws , session reports of the Academy of Sciences in Vienna (IIa) 120 , 385-530 (1911), Chapter IX.
4. ^ S. Zaremba: Sur une forme perfectionnée de la théorie de la relaxation , Bulletin International de l'Academie des Sciences de Cracovie, 1903 , 534–614.
5. ^ F. Fromm: Material laws of the isotropic continuum , Ingenieur-Archiv 4 , pp. 432-466 ​​(1933), equation 53a
6. ^ C. Truesdell: Hypo-elasticity , Journal of Rational Mech. Anal. 4 (1955), pp. 83-183 and 1019-1020
7. after James G. Oldroyd (1921 - 1982)
8. This derivation is also named after C. Truesdell. He himself named the derivation after Cauchy and wrote that this derivation of time was baptized after him for no inventive reason ("came to be named, for no good reason, after [...] me") (see C. Truesdell [1963, P. 141])
9. The Fréchet derivative of a scalar function with respect to a tensor is the tensor for which - if it exists - applies: ${\ displaystyle f (\ mathbf {T})}$${\ displaystyle \ mathbf {T}}$${\ displaystyle \ mathbf {A}}$
${\ displaystyle \ mathbf {A}: \ mathbf {H} = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} s}} f (\ mathbf {T} + s \ mathbf {H }) \ right | _ {s = 0} = \ lim _ {s \ rightarrow 0} {\ frac {f (\ mathbf {T} + s \ mathbf {H}) -f (\ mathbf {T})} {s}} \ quad \ forall \; \ mathbf {H}}$
There is and “:” the
Frobenius scalar product . Then will too ${\ displaystyle s \ in \ mathbb {R}}$
${\ displaystyle {\ frac {\ partial f} {\ partial \ mathbf {T}}} = \ mathbf {A}}$
written.
10. W. Noll: On the continuity of the solid and fluid states , Journal of Rational Mechanics and Analysis 4 , 3-81 (1955)
11. ^ B. Bernstein: Hypo-elasticity and elasticity , Archive for Rational Mechanics and Analysis 6 (1960), pp. 89-104 and B. Bernstein: Relation between hypo-elasticity and elasticity , Transaction of the Society of Rheology 4 (1960) , Pp. 23-28.