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Vliv volby konstitutivního vztahu na modelování šíření vln v předepjatém prostředí
Kruisová, Alena ; Plešek, Jiří ; Červ, Jan
The most common constitutive relation in acoustoelasticity is the second-order constitutive relation expressed in terms of the Green-Lagrange strain tensor. This material model is characterized by the high sensitivity of material parameters to the small measurement errors. This rather bad property leads to the proposal of another material model with the second-order constitutive relation expressed in terms of the logarithmic strain tensor. For this material model the acoustic wave velocities were derived for three different types of homogeneous pre-stress.
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Geometrie konečných deformací, linearizace a inkrementální deformace při počátečním stavu napětí/přetvoření
Fiala, Zdeněk
As usual in continuum mechanics, deformation and stress tensors at a point are considered to form vector (i.e. Euclidean) spaces. However, we can also regard the space of deformation tensors (i.e. positive definite matrices) as a Riemannian manifold of constant negative curvature and prove that the stress tensors then form covectors. From this standpoint we can simply geometrically interpret logarithmic strain and consecutively generalize it for states with initial deformation, naturally and unambiguously introduce objective time derivative (in particular the stress rate) and thus linearize in a geometrically consistent way, and exactly formulate incremental approach.
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Exponenciála matice a geometrický význam pole logaritmického tenzoru přetvoření
Fiala, Zdeněk
On the space of all symmetric positive definite matrices (the space of deformation tensor fields) one can introduce a Riemannian geometry, so that the matrix exponential represents ageodesic (i.e. a generalised straight line, the shortest connecting line of two points) emanating from an initial point - the identity matrix, in a direction given by a vector - the prescribed matrix. Based on this approach, we prove that the logarithmic strain can be interpreted as a vector, determined by a geodesic connecting an undeformed and a deformed states.
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Pitfalls in nonlinear elasticity
Höschl, Cyril
Assumed linear dependence of 2nd Piola-Kirchhoff stress tensor on Green-Lagrange strain tensor leads to a paradox even in simplest case of one-dimonsional compression. Such an extension of Hooke´s law to nonlinear elasticity is therefore dubious. The behavior of rubber and the theory of pure shear based on linear dependence of energetically conjugate Cauchy stress on logarithmic strain is shortly discussed, with the aim to avoid possible misunderstanding.
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