World Scientific
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at customercare@wspc.com for any enquiries.

A polyconvex extension of the logarithmic Hencky strain energy

    https://doi.org/10.1142/S0219530518500173Cited by:6 (Source: Crossref)

    Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function W:GL+(n) which is equal to the classical Hencky strain energy

    WH(F)=μdevnlogU2+κ2[tr(logU)]2=μlogU2+Λ2[tr(logU)]2
    in a neighborhood of the identity matrix 𝟙; here, GL+(n) denotes the set of n×n-matrices with positive determinant, FGL+(n) denotes the deformation gradient, U=FTF is the corresponding stretch tensor, logU is the principal matrix logarithm of U, tr is the trace operator, X is the Frobenius matrix norm and devnX is the deviatoric part of Xn×n. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions WVL in the so-called Valanis–Landel form
    WVL(F)=i=1nw(λi)
    with w:(0,), where λ1,,λn denote the singular values of F.

    Dedicated to Sir John Ball on the occasion of his 70th birthday.

    AMSC: 74B20, 74G65, 26B25
    Remember to check out the Most Cited Articles!

    Check out our Differential Equations and Mathematical Analysis books in our Mathematics 2021 catalogue
    Featuring authors such as Ronen Peretz, Antonio Martínez-Abejón & Martin Schechter