PSI - Issue 50

Andrey Yu. Fedorov et al. / Procedia Structural Integrity 50 (2023) 83–90 A.Yu. Fedorov et al. / Structural Integrity Procedia 00 (2023) 000–000

84

2

(opening angles between the tangents to the singular point), the mechanical characteristics of the material in the vicin ity of the singular point (elastic moduli, Poisson’s ratios). These factors determined the direction of experimental and numerical studies related to a search for geometrical and mechanical characteristics which would provide a decrease in the stress concentration level. A good example of such studies is the work of Chobanyan (1987), which o ff ers the theoretical and experimental support for the view that combinations of the values of angles and mechanical properties corresponding to the solutions without stress singularity provide a lower stress state in the zone near the wedge vertex. Similar studies are reported in works of Wu (2004); Xu et al. (2004a,b); Wang et al. (2006); Baladi et al. (2011). The results of these studies are summarized in works Fedorov et al. (2018); Borzenkov et al. (1996), where it is shown that the optimal stress state in the vicinity of singular points can be obtained for the geometric parameters and mechani cal characteristics that determine the boundary between the solutions with and without stress singularity. Earlier this result (feature of optimal solutions) was used as the basis for developing the technique of finding the values of elastic constants of the interlayer material and its geometry at the edges of the interface, which provide the minimum level of stress concentration (Fedorov et al. (2021)). One of the ways of reducing the stress level in the vicinity of V-shaped notches is the notch apex rounding (or the use of a U-shaped notch). Another way of reducing the stress level in the vicinity of V-shaped notches is to fill its cavity with a certain material (”healing”). The optimal solutions can be useful for finding the elastic constants of the filling material which provides the greatest decrease in the level of stress concentration near the apex of the V-shaped notch, the cavity of which is filled with this material. In this case, it is necessary to construct and study singular solutions near the vertex of the corresponding closed bimaterial wedge. General solutions and transcendental equations that allow determining the eigenvalues for a composite wedge are given in the works of Bogy (1968); Dempsey et al. (1981); Paggi et al. (2008). Examples of studies, presenting the results of numerical simulation and their analysis for di ff erent variants of closed bimaterial wedges, are the works of Sinclair (2004); Bogy et al. (1971); Chen et al. (1993); Fedorov et al. (2020). In the work of Fedorov et al. (2020) a comparative analysis of numerical results on stress singularity exponents is carried out for a composite closed wedge and a homogeneous wedge with stress-free edges at di ff erent ratios of the opening angles and elastic moduli of the homogeneous parts of the composite wedge, and at di ff erent values of Poisson’s ratios. One of the main results of Fedorov et al. (2020) is the statement that the Poisson’s ratio at its value close to 0.5 has abnormal e ff ect on the character of the stress singularity, namely, leads to the disappearance of stress singularity when the stress state is symmetric relative to the bisector of the opening angle. This finding made it possible to eliminate the stress singularity at the apex of the V-shaped notch by filling its cavity with the material whose Poisson’s ratio is close to 0.5. It is demonstrated that the singular character of stresses and, hence, the concentration of stresses can be eliminated at su ffi ciently large opening angles and when the sti ff ness of the weakly compressible filling material is several orders of magnitude less than the sti ff ness of the basic material. The purpose of present work is to experimentally substantiate and evaluate the e ff ectiveness of reducing the stress concentration near a V-shaped notch when its cavity is filled with a certain material with a Poisson’s ratio close to 0.5. One of the directions, which made it possible to obtain the first results on the behavior of stresses in the vicinity of singular points, is related to the construction of eigensolutions for special regions that satisfy homogeneous equi librium equations and one of the variants of homogeneous boundary conditions (Williams (1952)). For 2D problems, these are plane wedges. The technique for constructing eigensolutions and transcendental equations for plane wedges are presented in Williams (1952); Bogy (1968); Dempsey et al. (1981); Paggi et al. (2008). For a closed bimaterial wedge with perfect contact of the constituent parts, the transcendental equation is written as (Bogy et al. (1971); Dempsey et al. (1981)) (1 + β ) 2 sin 2 p γ 1 − p 2 ( β − α )sin 2 γ 1 (1 − β ) 2 sin 2 p γ 2 − ( β − α ) 2 p 2 sin 2 γ 2 + + 1 − α 2 sin 2 p ( π − γ 1 ) 2 p 2 ( β − α ) 2 sin 2 γ 1 + 2(1 − β ) 2 sin p γ 1 sin p γ 2 − 1 − α 2 sin 2 p ( π − γ 1 ) = 0 . (1) Here p = 1 − λ , γ 1 and γ 2 are the opening angles of the wedge parts, α = Γ ( κ 1 + 1) − ( κ 2 + 1) Γ ( κ 1 + 1) + ( κ 2 + 1) and β = Γ ( κ 1 − 1) − ( κ 2 − 1) Γ ( κ 1 + 1) + ( κ 2 + 1) are the combined parameters of the elastic constants of materials (Dunders parameters) (Dundurs (1969)), κ i = 3 − 4 ν i in the plane-strain state and κ i = (3 − ν i ) / (1 + ν i ) in the plane stress state, Γ = G 2 / G 1 , G i = E i / 2(1 + ν i ), ( ν i , G i is the Poisson’s ratios and shear moduli), i = 1 , 2. 2. Framework for the experimental study