PSI - Issue 59

Hryhorii Habrusiev et al. / Procedia Structural Integrity 59 (2024) 494–501 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

495

2

mathematical model aimed at studying the influence of the initial stress on the propagation of waves in a hollow infinite multilayered composite cylinder. A geometrically linearized theory for incompressible materials was derived by Jesenko and Schmidt (2021) from the nonlinear elasticity theory in the regime of small displacement. The stress strain state of a prestrained thick plate in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter was studied by Habrusiev et al (2022) using a linearized formulation of elasticity theory. It should be noted that the interaction of complex shape indentors with prestressed bodies is still not sufficiently studied. The article demonstrates the method of determining the axisymmetric stress-strain state of a prestressed body using as an example a plate during its contact interaction with a rigid indenter. Also the influence of the indenter shape, the plate thickness and its initial deformations on the distribution of contact stresses and vertical displacements of the plate surface was investigated in the paper.

Fig. 1. Schematic diagram of contact interaction

2. Problem statement Consider the problem of insertion with a constant force P of an indenter into a prestrained plate, which lies without friction on rigid base. The plate is modeled by an elastic layer. We introduce a cylindrical coordinate system Or z  in such a way that the coordinate plane Or  should coincide with the upper boundary plane of layer, and the Oz axis with the line of action of the force P (Fig. 1). The boundary conditions of the problem are   ,0 0, 0 rz r r     ; (1)   ,0 0, zz r a r    ; (2)     ,0 , 0 z u r f r r a    ; (3)   , 0, 0 rz r h r      ;   , 0, 0 z u r h r     . (4) Function   f r corresponds to the shape of surface bounding the indenter. We assume that residual stresses arisen in the layer are homogeneous. Then we may write expressions for the components of stress tensor and displacement vector (Habrusiev et al, 2022) on the upper boundary plane of layer ( 0 z  )

0  

0  

  ,0

    0 A As J r d  1 2

  ,0

    B B s J r d  2 0 1 1

,

,

zz r

c

rz r

c





3 

 





3 

 

33

31

 m B B s J r d        ,    2 1 2 1 0

  ,0

(5)

z u r

0