Existence theory for the equations of inelastic material behavior of metals : transformation of interior variables and energy estimates

• Autorzy: Alber Hans-Dietmer , Chełmiński Krzysztof

Identyfikatory

DOI

10.14708/ma.v25i39.1844

• Języki publikacji: EN

Abstrakty

EN

The system of equations, which we study, consists of linear partial differential equations and of nonlinear ordinary differential equations for internal variables. The existence theory for such systems was studied first by the french mathematicians G. Duvaut and J.L. Lions [1]. Next we can find in the literature a work of C. Johnson [2] on a quasi-static problem for a special model. Then in the nineties we can find more works in the domain. This work consists of two parts. In the first part we will classify constitutive equations and therefore we define constitutive equations of monotone type. Moreover by transformation of internal variables we will enlarge the class of constitutive equations, for which we can prove a. global in time existence theorem for large initial data. But there exist models, which are not of monotone type and which we can not transform to monotone type. Therefore we must study such models with other methods. This is the second part, of the work. We write about the energy method for the model of Bodner-Partom.

Słowa kluczowe

EN

time-dependent problems; equations of mathematical physics

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Matematyka Stosowana : matematyka dla społeczeństwa
• Rocznik: 1996
• Tom: Nr 39
• Strony: 3--15
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Alber Hans-Dietmer

• Technische Hochschule Darmstadt, Germany

autor

Chełmiński Krzysztof

Bibliografia

• [1] ’72 G. Duvaut, J. L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag.
• [2] ’78 C. Johnson, On Plasticity with Hardening, Jour. of Math. Analysis and App. 62 p. 325-336, 1978.
• [3] ’90 P. Le Tallec, Numerical analysis of viscoelastic problems. Masson and Springer-Verlag.
• [4] ’95 A. Nonri, M. Rascle, A global existence and uniquness theorem for a model in dynamic elasto-plasticity with isotropic strain-hardening. - SIAM Jour. of Math. Analysis 26, 1995.
• [5] ’89 H. D. Alber, On a system of equations from the theory of nonlinear viscoplasticity. - Preprint Nr. 1265 FB Mathematik TH Darmstadt 1989.
• [6] ’95 Alber, Mathematische Theorie des inelastischen Materialverhaltens von Metallen. - Mitt. Ges. Ang. Math. Mech. p. 9-38, 1995.
• [7] ’95 Alber, Global existence and boundness of large solutions to nonlinear equations of viscoelasticity with hardening. Commun. Math. Phys. 166, p. 565-601, 1995.
• [8] ’96 Alber, The boundary-value problem from the theory of inelastic material behavior of metals - Transformation of interior variables. - preprint in preparation.
• [9] ’93 K. Chełmiński, Global in time existence of solutions to the constitutive model of Bodner-Partom with isotropic hardening. - shall appear in Demonstratio Mathematica 1995
• [10] ’95 K. Chełmiński, On large solutions for the quasistatic problem in nonlinear viscoelasticity with the constitutive equations of Bodner-Partom. - shall appear in Math. Meth. in the App. S. 1995.
• [11] ’95 K. Chełmiński, Energy estimates and global in time residts for a problem from nonlinear viscoelasticity. - shall appear in Bull. of the Polish Acad. of Sciences: Math.
• [12] ’96 K. Chełmiński, Stress L ∞ -estimates and the uniqueness problem for the model of Bodner-Partom - preprint in preparation.