Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals

“…This is exactly in the spirit of the Murat-Tartar theory of compensated compactness ( [Tar79,Mur78,Mur81], see also [Tar90]) that develops conditions for weak semicontinuity of bilinear expressions if one has control on certain differential expressions of the sequences. In the context of variational problems this is closely related to the notions of quasiconvexity [Mor52] and more generally of A-quasiconvexity [Dac82,FM99,DF02] which define essentially necessary and sufficient conditions for weak lower semicontinuity. The sufficiency statement, however, requires growth conditions which are undesirable in nonlinear elasticity (since they force the energy to remain finite even at infinite elastic compression).…”

Section: Resultsmentioning

confidence: 99%

Lower semicontinuity and existence of minimizers in incremental finite‐strain elastoplasticity

Mielke

Müller

2006

Z Angew Math Mech

View full text Add to dashboard Cite

We study incremental problems in geometrically nonlinear elastoplasticity. Using the multiplicative decomposition Dϕ = F el F pl we consider general energy functionals of the formwhich occur as the sum of the stored energy and the dissipation in one time step.Here G(F pl ) is the dislocation tensor which takes the form 1 detImposing the usual constraint det F pl ≡ 1 and suitable growth and polyconvexity conditions on U we show that the minimum of I is attained in the natural Sobolev spaces. Moreover, we are able to treat multiple time steps by controlling the stored and dissipated energies. We also address the relation of the incremental problem to the time-continuous energetic formulation of elastoplasticity.

show abstract

“…As pointed out already by L. Tartar, the weak lower semicontinuity of V(t, ·) follows, in fact, from a weaker assumption on W , the so-called A-quasiconvexity (cf. [6,11,12]) for A being an operator whose kernel consists of all symmetric rotation-free fields.…”

Section: ω1∪ω2mentioning

confidence: 99%

Quasistatic delamination problem

Roubíček

Scardia

Zanini

2009

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

We study delamination of two elastic bodies glued together by an adhesive that can undergo a unidirectional inelastic rate-independent process. The quasistatic delamination process is thus activated by time-dependent external loadings, realized through body forces and displacements prescribed on parts of the boundary. The novelty of this work consists of considering the glue as infinitesimally thin and ideally rigid in the sense that a crack in the glue cannot be seen before, speaking "microscopically", all macromolecular links of the adhesive are fully debonded. The concept of energetic solution is applied and existence of such solutions is proved by showing Γ -convergence of a suitable approximation that, in addition, allows for a direct computer implementation, unlike the original problem.

show abstract

Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals

Cited by 254 publications

References 0 publications

Existence Criteria in Some Extremum Problems Involving Eigenvalues of Elliptic Operators

Existence Criteria in Some Extremum Problems Involving Eigenvalues of Elliptic Operators

Lower semicontinuity and existence of minimizers in incremental finite‐strain elastoplasticity

Quasistatic delamination problem

Contact Info

Product

Resources

About