Single phase energy minimizers for materials with nonlocal spatial dependence

Authors:
Roger L. Fosdick and Darren E. Mason

Journal:
Quart. Appl. Math. **54** (1996), 161-195

MSC:
Primary 73V25; Secondary 49J20, 73C50

DOI:
https://doi.org/10.1090/qam/1373845

MathSciNet review:
MR1373845

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a one-dimensional body and assume that the total stored energy functional depends not only on the local strain field but also on the spatial average of the strain field over the body, weighted with an influence kernel. We investigate the problem of minimizing the total stored energy subject to given end displacements. The general existence theory for this problem is reviewed. Then, we narrow our study and concentrate on certain fundamental aspects of nonlocal spatial dependence by restricting our consideration to the case of a convex local energy and an exponential-type influence function for the nonlocal part. We find explicit solutions and show their characteristic properties as a function of the parameter that measures the extent of influence in the nonlocal kernel. We then study in detail the behavior that results when the total stored energy functional loses its coercivity. In this case, issues concerning the local and global stability of extremal fields are considered.

**[1]**S. S. Antman,*Monotonicity and invertibility conditions in one dimensional nonlinear elasticity*. In: Nonlinear Elasticity, Proceedings of a Symposium Conducted by the Mathematics Research Center, University of Wisconsin--Madison, ed. by R. W. Dickey, Academic Press, 1973, pp. 57-92**[2]**S. S. Antman,*Ordinary differential equations of nonlinear elasticity*II:*Existence and regularity theory for conservative boundary value problems*, Arch. Rational Mech. Anal.**61**, 353-393 (1976)**[3]**J. Carr, M. E. Gurtin, and M. Slemrod,*Structured phase transitions on a finite interval*, Arch. Rational Mech. Anal.**86**, 317-351 (1984)**[4]**R. Courant and D. Hilbert,*Methods of Mathematical Physics*, Vol. I, John Wiley and Sons, 1962**[5]**B. Dacorogna,*Direct Methods in the Calculus of Variations*, Springer-Verlag, 1989**[6]**J. E. Dunn and R. L. Fosdick,*The morphology and stability of material phases*, Arch. Rational Mech. Anal.**74**, 1-99 (1980)**[7]**I. Ekeland and R. Temam,*Convex Analysis and Variational Problems. Studies in Mathematics and its Applications*, Volume I, North-Holland Publishing Co., 1976**[8]**J. L. Ericksen,*Equilibrium of Bars*, J. Elasticity**5**, 191-201 (1975)**[9]**R. L. Fosdick and D. E. Mason,*Energy Minimizers in Nonlocal Continuum Mechanics*(Work in progress)**[10]**T. Lin and R. C. Rogers,*Computation of Phase Transitions Using Nonlocal Regularization*, ICAM Report, Virginia Polytechnic Institute, Department of Mathematics, 14 September 1993**[11]**I. Stakgold,*Boundary Value Problems of Mathematical Physics, Vol*. I, The Macmillan Company, 1967**[12]**J. D. van der Waals,*The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density*. (In Dutch), Verhandel. Konink. Akad. Weten. Amsterdam (Sect. 1),**1**, No. 8 (1893)**[13]**J. W. Walter,*On a Nonlocal Model of Fluid Phases*, Ph.D. Thesis, University of Minnesota, Dept. of Aerospace Engineering and Mechanics, 1985**[14]**R. L. Wheeden and A. Zygmund,*Measure and Integral: An Introduction to Real Analysis*, Marcel Dekker, Inc., 1977**[15]**K. Yosida,*Functional Analysis*, Springer-Verlag, 1965

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73V25,
49J20,
73C50

Retrieve articles in all journals with MSC: 73V25, 49J20, 73C50

Additional Information

DOI:
https://doi.org/10.1090/qam/1373845

Article copyright:
© Copyright 1996
American Mathematical Society