Hypoelasticity
Hypoelasticity (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 hypoelasticity 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: “Hypoelasticity 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: “Hypoelasticity 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 AugustinLouis 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 hypoelasticity 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 twentyfirst century.
definition
A hypoelastic material obeys a constitutive law of form
 .
In it is

a suitable rate
 of Cauchy's stress tensor
 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

the spatial distortion rate
 the spatial velocity gradient

the objective covariant Oldroyd derivative
 of the EulerAlmansi tensor .
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:
Surname  formula 

ZarembaJaumann derivation  
Convective contravariant Oldroyd derivative  
Cauchy derivative 
The tensor
is called the spin or vortex tensor .
objectivity
In the definition of hypoelasticity, 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
holds for any orthogonal tensors ,
so is an isotropic tensor function of .
Anisotropy
In an unstressed particle of a hypoelastic material free of residual stresses . If there is a load in this state, the result is:
regardless of which objective time derivative is used for the voltages. Because there is an isotropic tensor function on the righthand side of this equation, the stresses develop from a stressfree state with small distortions initially as in an isotropic material.
Conversely, a material that does not react isotropically  but anisotropically  from a stressfree state in the event of small deformations is not hypoelastic.
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 CauchyGreen tensor , e.g. B .:
 .
The coefficients are scalar , isotropic functions of the main invariants or other invariants of the left CauchyGreen tensor. Because of
is calculated
The derivative of the stress tensor with respect to time yields:
So if the stressdeformation relationship is invertible
 ,
then an isotropic, Cauchyelastic material is also hypoelastic, which Walter Noll first demonstrated .
plasticity
Hypoelasticity 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.
Barry Bernstein formulated conditions that must be fulfilled for hypoelasticity to be consistent with a certain stressstrain relationship.
References and footnotes
 ↑ C. Truesdell: Remarks on HypoElasticity , Journal of Research of the National Bureau of Standards  B. Mathematics and Mathematical Physics, Vol. 67B, No. July 3September 1963
 ↑ AL Cauchy: Sur l'equilibre et le mouvement intérieur des corps considérés comme des masses continues , Oeuvres 9, 243269 (1829)
 ^ C. Jaumann: System of physical and chemical differential laws , session reports of the Academy of Sciences in Vienna (IIa) 120 , 385530 (1911), Chapter IX.
 ^ S. Zaremba: Sur une forme perfectionnée de la théorie de la relaxation , Bulletin International de l'Academie des Sciences de Cracovie, 1903 , 534–614.
 ^ F. Fromm: Material laws of the isotropic continuum , IngenieurArchiv 4 , pp. 432466 (1933), equation 53a
 ^ C. Truesdell: Hypoelasticity , Journal of Rational Mech. Anal. 4 (1955), pp. 83183 and 10191020
 ↑ after James G. Oldroyd (1921  1982)
 ↑ 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])

↑ The Fréchet derivative of a scalar function with respect to a tensor
is the tensor for which  if it exists  applies:
 ↑ W. Noll: On the continuity of the solid and fluid states , Journal of Rational Mechanics and Analysis 4 , 381 (1955)
 ^ B. Bernstein: Hypoelasticity and elasticity , Archive for Rational Mechanics and Analysis 6 (1960), pp. 89104 and B. Bernstein: Relation between hypoelasticity and elasticity , Transaction of the Society of Rheology 4 (1960) , Pp. 2328.