PSI - Issue 50

3

Dmitry Parshin et al. / Procedia Structural Integrity 50 (2023) 320–326 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

322

2. Aim of the study The presented work is devoted to applying a model of the above mentioned kind to mathematical describing the processes of quasi-static deformation of thin-walled arched structures being layerwise constructed of materials with rheological properties of creeping (strain heredity) and aging. The construction is realized on a horizontal foundation with the synchronous action of gravity forces on the structure. The aim of the study is to find ways to possibly control the stress state of the structure under construction due to:  a regulated change in construction pace during the process of adding new structural elements to the structure;  using in the construction structural elements pre-stretched with a regulated stress. It will be conducted a numeral research of the associated non-classical problems of continuum mechanics on the basis of an appropri ate author’s mathematical model . The focus will be on the study of possibilities to influence the distribution of contact stresses on the bottom of an arched structure being constructed. This is dictated by the practical needs in ensuring the stability of the large-sized structures on foundations and in optimizing the wear contact conditions of their interaction with the foundations (Kazakov and Manzhirov, 2008; Kazakov and Parshin, 2019; Kazakov and Kurdina, 2020; Kazakov and Sahakyan, 2020). In this study, we will consider as an example a very axially extended (in the direction of axial coordinate z ) plane strained arched structure bounded by the cylindrical surfaces ( ) a t   and b   where a t b   0 ( ) , over the angular range of [0, ]    (the construction is thus symmetrical with respect to the plane 2    ). The change in the radius a towards its decrease over time occurs due to continuous consecutive adding new layers of material to the current internal cylindrical surface of the structure during its construction. It is assumed that by the begin of constructing there exists an already made stress-free arched workpiece with the internal surface of radius a b   0  and, at the initial moment 0 t t  , this workpiece is immediately installed on a perfectly slick and rigid flat foundation, starting later, at some moment 0 start t t t   , to be consecutively built up with layers of new material. Throughout the entire constructing process of such a kind, gravity forces act on the structure parallel to the plane of its symmetry, and the structure bottom surfaces 0   and    completely remain in contact with the foundation, without being able to peel off from it, due to the organization of some appropriate retaining ties. In order to avoid excessive complications in the corresponding mechanical problems, the material of all the structure, including its workpiece part, is considered homogeneous and isotropic. 3. Material and methods Modeling is carried out within the framework of the mathematical theory of growing bodies developed by the scientific school of Professor A.V. Manzhirov (see, e.g., Manzhirov, 1995, 2017, 2018; Manzhirov and Parshin, 2015; Parshin, 2017; Manzhirov and Mikhin, 2018; Kazakov and Parshin, 2022). It was pointed out above that every element, arbitrarily small, newly added to the structure causes an additional change in the structure stress-strain state. Consequently, the latter changes incrementally. Therefore, instead of the traditional quantities themselves that characterize the strain state, we will use their speed analogues . So, instead of the displacement vector u , we will use the velocity vector d dt v u  for the continuum points displacements, and instead of the tensor of small strain E , we will use the tensor of strain rates d dt D E  . We will consider the material rheological behavior which is experimentally and theoretically described by Arutyunyan (1966). The corresponding constitutive model assumes viscoelastic deformation of material elements under the load and their aging regardless of the load. According to this model we have the following relationship for the speed quantities:

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    .

 t

 II D r

G t t 2 ( ) ( , ) T r

t 2 ( , )

G 2 ( ) 1 

2 ( , ) 

  

  

dt d

C t



D r

T r

t ( , )

( , ) 

(1)

d











m







( ) 0 r

