The Experts below are selected from a list of 36 Experts worldwide ranked by ideXlab platform

Lewis Wheeler - One of the best experts on this subject based on the ideXlab platform.

Pau A - One of the best experts on this subject based on the ideXlab platform.

Miroslav Šilhavý - One of the best experts on this subject based on the ideXlab platform.

  • Elements of Tensor Algebra and Analysis
    The Mechanics and Thermodynamics of Continuous Media, 1997
    Co-Authors: Miroslav Šilhavý
    Abstract:

    The reader is assumed to have some familiarity with the elements of linear algebra. A convenient reference is HALMOS [1958]. A Direct Notation is used throughout, with the same conventions as in Truesdell & Noll [1965] and Gurtin [1981]. The treatment below emphasizes the differentiation of functions of tensor arguments. Thus the functional calculus for symmetric tensors is introduced and a formula is given for its derivative; as a consequence of this, e.g., the formula for the derivative of the square root is derived. Also the derivatives of the eigenvalues, singular values and eigenvectors are calculated. Some of these results can be omitted on the first reading, as they will be used only sporadically until Chap. 8 on isotropic functions and Chap. 18 on the convexity properties thereof.

H. Murakami - One of the best experts on this subject based on the ideXlab platform.

  • On the Vector-Valued Cauchy Stress 2-Form and Its Stress Rate
    Volume 10: Mechanics of Solids and Structures Parts A and B, 2007
    Co-Authors: H. Murakami
    Abstract:

    Using exterior differential forms, basic equations of continuum mechanics are presented in Direct Notation. To this end, Elie Cartan’s vector-valued Cauchy stress 2-form is introduced. Its Lie derivative along the world line becomes the Truesdell stress rate. In the presentation, the Notation adopted by Theodore Frankel (The Geometry of Physics, Cambridge, New York, 1997) is utilized. With the use of exterior differential forms, complicated computations in tensor analyses in curvilinear coordinates are dramatically simplified. As specific examples, the following subjects are presented: (i) Piola transformations of the Cauchy stress 2-form and (ii) simple shear deformation using the Lie derivative of the Cauchy stress 2-form, i.e., the Truesdell stress rate. It is known that under monotonic shear loading, if inappropriate stress-rates are used, shear stress oscillates. With the use of geometrically correct stress-rate, the shear stress monotonically increases. Thereby, the search for an appropriate stress rate reduces to the correct definition of the stress 2-form and the computation of its Lie derivative with respect to velocity.Copyright © 2007 by ASME

K. Zare - One of the best experts on this subject based on the ideXlab platform.

  • Representation and Classification of Dynamical Systems
    From Newton to Chaos, 1995
    Co-Authors: K. Zare
    Abstract:

    It is shown that any dynamical system may be represented by a system of first order differential equations of the form \({\dot x_i} = {a_{ij}}\partial Z/\partial {x_j}, \) where on the right side the (n × n) matrix-elements, a,i and the scalar potential Z depend on the coordinates x 1, x 2 ... x n and the summation convention is applied with respect to the repeated subscript j. Using Direct Notation the equation is written as \(\dot x = A(x)\partial Z/\partial x \). Invariant properties of the system under transformations are studied and the classifications of the systems are established depending on the properties of the matrix A(x). With this representation, many of the classical as well as the modern results may be shown using only linear algebra. In particular for a constant skew-symmetric matrix A, it is shown that many of the known properties of the motion for Hamiltonian systems such as form preservation, volume preservation, and variational principle remain valid.