The asymptotic behavior of classical solutions to the mixed initial-boundary value problem in finite thermo-viscoelasticity

Beevers, C. E.; Silhavý, M.

doi:10.1090/qam/950605

Cited by 1 publication

(6 citation statements)

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“…(i): Hence relations (29) for ' u imply (29) 113 , (36) and (37) for u4 moreover, r is uniquely determined by 3364 2 2 (ii): Again, 3314 1 for ' u implies the same relation for u and 3314 1 implies (38). To prove that there is no sequence t k 2 and no sequence 5 k of rigid body displacements such that the sequence u3t k 43 5 k W is bounded, assume that u3t k 43 5 k W c 9 2 for some sequence t k 2 and some sequence 5 k of rigid body displacements.…”

Section: Processes In Masonry Bodies 597mentioning

confidence: 96%

“…Situation (S1) occurs again1 each centered process u tends to the equilibrium set 7 5 in the sense of (29). Moreover, on the region x 9 5 : …”

Section: Examplesmentioning

confidence: 98%

“…Furthermore, in Cases (i) and (ii) the kinetic energy tends to 02 This need not be the so in Case (iii). Relations (29) establish the approach of the process to the equilibrium set 5 in the sense of the indicated quantities. If 5 is a singleton, i.e., if the equilibrium problem has a unique solution u 0 1 then 3294 4 establishes the approach of any process to u 0 in the L 2 norm.…”

Section: Large Times Behaviormentioning

confidence: 99%

“…The behavior of processes for large times is treated by using Liapunov functionals similar to those in [29] and [30,Sct. 25.2]1 we emphasize that the total energy, while being a natural Liapunov functional, is not strong enough for our purposes1 we need a modification to be described in Section 9.…”

Section: Introductionmentioning

confidence: 99%

“…Consider the pure traction problem with equilibrated loads. Let u be a general process under the loads 3d1 s1 b42 (i) If 5 1 then the process satisfies(29) 113 and there exists a unique linear family of rigid body displacements r3t4 W c for all t 0 and some c 9 24 (37)(ii) if 5 1 and I 0 6 32 then the process satisfies 3314 1 and there exists a linear family of rigid body displacements r such that7 u3t4 3 7 r3t4 H 01 (38) PROCESSES IN MASONRY BODIES 589…”

mentioning

confidence: 99%

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Processes in Masonry Bodies and the Dynamical Significance of Collapse

Padovani

Pasquinelli

Šilhavý

2008

Mathematics and Mechanics of Solids

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We consider masonry bodies with dissipation of the rate type. Global in time existence and uniqueness of weak solutions for arbitrary loads and initial conditions is established1 this includes the loads for which the masonry structure collapses. "Safe" and "collapse" loads are distinguished by different behaviors of solutions (1 processes) at large times. Three situations can arise according to the properties of the equilibrium problem. (i) The equilibrium problem has a (typically nonunique) solution in the Sobolev space W 112 of displacements1 then each process stabilizes inasmuch as the kinetic energy tends to 0 and the L 2 distance of the displacement from the set of all equilibrium displacements tends to 02 (ii) The equilibrium problem has no solution in W 112 but the infimum of the energy functional on the space of admissible displacements is finite. Then the kinetic energy tends to 0 but the W 112 norm of the displacement tends to 22 This may correspond either to a collapse or to a situation when the process approaches an equilibrium solution in a larger function space. (iii): The infimum of the energy functional on the space of admissible displacements is 322 Then the total energy approaches 32 in any process and the W 111 norm of the displacement tends to 24 the structure collapses.

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Section: Processes In Masonry Bodies 597mentioning

confidence: 96%

“…Situation (S1) occurs again1 each centered process u tends to the equilibrium set 7 5 in the sense of (29). Moreover, on the region x 9 5 : …”

Section: Examplesmentioning

confidence: 98%

Section: Large Times Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%