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On the positivity of the second variation in finite elasticity. (English) Zbl 0657.73027
The problem of the minimization of the integral $$I(f)=\int_{\Omega}W(x,\nabla f(x))dx$$ is investigated, where f is a vector-valued function on a bounded open domain $$\Omega \in {\mathbb{R}}^ n$$ and satisfies $$f=d$$ on a portion of the boundary $${\mathcal D}$$, here d is a given function.
The authors deal with the following questions: (a) What are the necessary and sufficient conditions for the second variation of I to be uniformly positive? (b) What are the sufficient conditions for the second variation of I to be coercive on a certain domain?
In elasticity I(f) is the total stored energy of an elastic body when it is deformed by the deformation f. Applications of the proved results to the linear isotropic elastic body are given in detail. Some results are presented in connection with the operator $${\mathcal L}$$ given by the formula $${\mathcal L}[u]=-div C_ f[\nabla u]+\lambda u$$, where u is the displacement, $$C_ f$$ is the elasticity tensor. The behaviour of the spectrum of $${\mathcal L}$$ and the existence condition of the solution of the equation $${\mathcal L}[u]=b$$ are also analyzed.
Reviewer: I.Ecsedi

##### MSC:
 74B20 Nonlinear elasticity 74S30 Other numerical methods in solid mechanics (MSC2010) 49J27 Existence theories for problems in abstract spaces 49J45 Methods involving semicontinuity and convergence; relaxation 47A10 Spectrum, resolvent 47B38 Linear operators on function spaces (general) 49J20 Existence theories for optimal control problems involving partial differential equations 49K99 Optimality conditions
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##### References:
 [1] Agmon, S., The coerciveness problem for integro-differential forms. J. d’Analyse Math. 6 (1958), 183–223. · Zbl 0119.32302 · doi:10.1007/BF02790236 [2] Agmon, S., Remarks on self-adjoint and semi-bounded elliptic boundary value problems. Proc. International Symposium on Linear Spaces, The Israel Academy of Sciences and Humanities. Jerusalem, 1961, 1–13. [3] Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math. 15 (1962), 119–147. · Zbl 0109.32701 · doi:10.1002/cpa.3160150203 [4] Agmon, S., A. Douglis & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17 (1964), 35–92. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104 [5] Coral, M., On the necessary conditions for the minimum of a double integral. Duke Math. J. 3 (1937), 585–592. · JFM 63.1066.02 · doi:10.1215/S0012-7094-37-00346-6 [6] De Figueiredo, D. G., The coerciveness problem for forms over vector valued functions. Comm. Pure Appl. Math. 16 (1963), 63–94. · Zbl 0136.09502 · doi:10.1002/cpa.3160160109 [7] Ekeland, I., & R. Temam, Convex Analysis and Variational Problems. New York: American Elsevier, 1976. · Zbl 0322.90046 [8] Fichera, G., Existence theorems in elasticity. Handbuch der Physik VIa/2 (C. Truesdell, ed.). Berlin Heidelberg New York: Springer-Verlag, 1972. · Zbl 0269.73028 [9] Friedman, A., Partial Differential Equations. New York: Holt, 1969. · Zbl 0224.35002 [10] Graves, L. M., The Weierstrass condition for multiple integral variation problems. Duke Math. J. 5 (1939), 656–660. · Zbl 0021.41403 · doi:10.1215/S0012-7094-39-00554-5 [11] Gurtin, M. E., The linear theory of elasticity. Handbuch der Physik VIa/2 (C. Truesdell, ed.). Berlin Heidelberg New York: Springer-Verlag, 1972. · Zbl 0317.73002 [12] Hadamard, J., Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique. Paris: Hermann, 1903. · JFM 34.0793.06 [13] Hestenes, M. R., & E. J. McShane, A theorem on quadratic forms and its application in the calculus of variations. Trans. AMS 47 (1940), 501–512. · Zbl 0023.33301 · doi:10.1090/S0002-9947-1940-0002839-X [14] Hörmander, L., Linear Partial Differential Operators. Berlin Heidelberg New York: Springer-Verlag, 1969. · Zbl 0175.39201 [15] Kato, T., Perturbation Theory for Linear Operators. Berlin Heidelberg New York: Springer-Verlag, 1984. · Zbl 0531.47014 [16] Kolsky, H., Stress Waves in Solids. London: Oxford University Press, 1953. · Zbl 0052.42502 [17] Lions, J. L., & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. I. Berlin Heidelberg New York: Springer-Verlag, 1972. · Zbl 0223.35039 [18] Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Fourth Edition. New York: Dover, 1944. · Zbl 0063.03651 [19] Marsden, J. E., & T. J. R. Hughes, Mathematical Foundations of Elasticity. Englewood Cliffs: Prentice-Hall, 1983. · Zbl 0545.73031 [20] Mikhlin, S. G., The spectrum of a family of operators in the theory of elasticity. Russian Math. Surveys 28 # 3 (1973), 45–88. · Zbl 0291.35065 · doi:10.1070/RM1973v028n03ABEH001563 [21] Morrey, C. B., Multiple Integrals in the Calculus of Variations. Berlin Heidelberg New York: Springer-Verlag, 1966. · Zbl 0142.38701 [22] Peetre, J., Another approach to elliptic boundary problems. Comm. Pure. Appl. Math. 14 (1961), 711–731. · Zbl 0104.07303 · doi:10.1002/cpa.3160140404 [23] Potier-Ferry, M., On the mathematical foundations of elastic stability theory. I. Arch. Rational Mech. Anal. 78 (1982), 55–72. · Zbl 0488.73043 · doi:10.1007/BF00253224 [24] Rayleigh, Lord, On waves propagated along the plane surface of an elastic solid. London Math. Soc. Proc. 17 (1887), 4–11. · JFM 17.0962.01 · doi:10.1112/plms/s1-17.1.4 [25] Riesz, F., & B. Sz.-Nagy, Functional Analysis. New York: Frederick Ungar, 1955. [26] Schechter, M., General boundary value problems for elliptic partial differential equations. Comm. Pure Appl. Math. 12 (1959), 457–486. · Zbl 0087.30204 · doi:10.1002/cpa.3160120305 [27] Simpson, H. C., & S. J. Spector, On failure of the complementing condition and nonuniqueness in linear elastostatics. J. Elasticity 15 (1985), 229–231. · Zbl 0576.73012 · doi:10.1007/BF00041996 [28] Simpson, H. C., & S. J. Spector, On bifurcation in finite elasticity: Buckling of a rectangular rod. In preparation. · Zbl 1163.74017 [29] Spector, S. J., On uniqueness for the traction problem in finite elasticity. J. Elasticity 12 (1982), 367–383. · Zbl 0506.73043 · doi:10.1007/BF00042210 [30] Terpstra, F. J., Die Darstellung biquadratischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung. Math. Ann. 116 (1939), 166–180. · Zbl 0019.35203 · doi:10.1007/BF01597353 [31] Thompson, J. L., Some existence theorems for the traction boundary value problem of linearized elastostatics. Arch. Rational Mech. Anal. 32 (1969), 369–399. · Zbl 0175.22108 · doi:10.1007/BF00275646 [32] Truesdell, C., & W. Noll, The Non-linear Field Theories of Mechanics. Handbuch der Physik III/3 (S. Flügge, ed.). Berlin Heidelberg New York: Springer-Verlag, 1965. · Zbl 0779.73004 [33] van Hove, L., Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Proc. Koninklijke Nederlandsche Akademie Van Wetenschappen 50 # 1 (1947), 18–23. · Zbl 0029.26802 [34] van Hove, L., Sur le signe de la variation seconde des intégrales multiples à plusieurs fonctions inconnues. Acad. Royale Sci. Belgique, Brussels. Memoires 24 # 5 (1949), 1–68. · Zbl 0036.34501
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