General stability result of swelling porous elastic soils with a viscoelastic damping
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
General stability result of swelling porous elastic soils with a viscoelastic damping Tijani A. Apalara Abstract. We consider a swelling porous-elastic system with a single memory term as the only damping source. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous elastic soils. Unexpectedly, using the multiplier method, we establish a general decay result irrespective of the wave speeds of the system. Mathematics Subject Classifications. 93D20, 35B40. Keywords. Swelling porous problem, Memory term, General decay.
1. Introduction Towards the end of the nineteenth century, Eringen [1] advanced a continuum theory for a mixture of elastic solids, viscous fluid, and gas. Furthermore, he highlighted the balance laws for each component of the mixture and obtained the field equations for a heat-conducting mixture. We refer the reader to [2] for detailed historical development/review related to the general theory of the mixtures. In this work, we focus on the asymptotic behavior of swelling (also called expansive) soils that have been characterized (see [3]) under porous media theory. It is to be recognized that swelling soils contain clay minerals that attract and absorb water, which may lead to increase pressure (see [4]). In architectural and civil engineering, swelling soils are considered problematic/harmful. If the pressure of the soil is higher compared to the main structure, it could result in heaving [5]. The more the initial dry density of a soil, the more its potential to swell due to capillary action that accompanies absorption of underground water or shrinkage due to dryness as a result of changes in weather condition. Swelling soils cause serious engineering problems such as uneven foundation settlement especially when it exceeds 10% in most of expansive clay. Uneven foundation settlement could lead to minor or major cracking in building structure or cause an undulation in road pavements [6]. The evidence available has shown that expansive soils spread across the continents from Asia (India), Australia, North America (US), South America, Africa (South Africa), and Middle East [7]. Estimate indicates that about 20–25% of land area in the USA is covered with such problematic soils with the accompanied economic loss of 5.5 to 7 billions USD in 2003 [8]. Hence, it is crucial to study the ways to annihilate or at least minimize such damages. Reader is referred to [9] for other details concerning swelling soil. As established by Ie¸san [10] and simplified by Quintanilla [11], the basic field equations for the linear theory of swelling porous elastic soils is mathematically given by ρz ztt = P1x − G1 + H1 ρu utt = P2x + G2 + H2 ,
(1.1)
where the constituents z and u represent the displacement of the fluid and the elastic solid material, respectively. The duo positive constant coefficients ρz and ρu are the densities of each constituent. The functions (P1 , G1 , H1 ) represent the partial tension, internal body
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