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Time-discrete integration of finite deformations
Fiala, Zdeněk
Some necessary implications for the time discrete integration of finite deformations will be discussed together with particular Runge--Kutta--Munthe-Kaas schemes, when the geometrical structure of the space of Cauchy-Green deformation tensors, implicitly contained in the principle of virtual power, is taken into account.
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Je logaritmická časová derivace jednoduše Zarembova-Jaumannova derivace?
Fiala, Zdeněk
The paper raises a question whether the logarithmic time derivative, expressed in specific coordinate system, is the Zaremba-Jaumann time derivative, and if not why. In fact, it has been already proved that the Z-J derivative represents the geometrically consistent linearization of tensor fields along deformation process in terms of the covariant derivative in the space of right Cauchy-Green deformation tensors C.
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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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Stress rate and incremental principle of virtual work in finite deformations
Fiala, Zdeněk
Solution of finite deformation problems is sought in the space of all deformation tensor fields. Representation of a deformation process here as a trajectory makes us possible to further classify symmetric second-order tensor fields either as points, vectors, or covectors, and, as a consequence, assign them the corresponding time derivatives. However, as the space of all deformation tensor fields has proved non-euclidean, the time derivative of vector, and covector fields along the trajectory should be defined by the covariant derivative. This approach enables us coherently to formulate an incremental principle of virtual work, and propose the corresponding procedure in solving finite deformation problems.
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Large deformation - large amount of unknown
Fiala, Zdeněk
Finite deformations are exposed from a viewpoint of differential geometry. From this perspective infinite dimensional Riemannian geometry of Riemannian metrics is emplyed to propose novel objective time derivative by means of covariant derivative.
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Mechanical Properties of Rheonomic Materials
Minster, Jiří
There are introduced with consideration for engineering applications, constitutive equations small and moderate deformation behaviour of both non-ageing and ageing rheonomic materials, basic methods for identification of their mechanical characteristics and possibilities of an ultimate behaviour assessment.
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