Keywords

1 Introduction

The stress–strain state (SSS) of composite structures is characterized by the concentration of stresses at the junctions of elements with different design of the boundary: special lines, points, such as the reentrant angle, etc. The complex SSS occurs in the stress concentration area, which is due to the shape of the boundary or “geometric factor” and the finite discontinuity of the given forced strains, mechanical properties, going out to an irregular point of the area boundary. The complexity of the SSS analysis in the corner cutout zone of the area boundary is due to the singularity of the solution of the homogeneous boundary value problem of elasticity.

The solution of elliptic equations for areas with non-smooth boundaries is considered in papers of Williams, Kondratiev, Fufaev, Ufliand, Kalandiia, Cherepanov [1], Bodzhi, Aksentian, Aleksandrova, Chobanian. In papers [1,2,3,4,5,6,7], the stress nature in the corner zone of a plane wedge for different boundary conditions on its sides is determined. In the neighborhood of irregular boundary points, the solution of the elliptic boundary value problem is given in the form of asymptotic series and an infinitely differentiable function. The components of these series contain solutions of homogeneous boundary value problems for model areas: a cone or a wedge. These solutions depend on local characteristics: the values of the solid and plane angle and the type of boundary conditions [7,8,9,10,11,12,13,14]. The values of the solution expansion coefficients in the neighborhood of the singular point are unknown and depend on the problem as a whole.

The application of the well-known experimental photoelastic method [9, 15,16,17,18,19,20] makes it possible to obtain a solution of the elasticity problem on the model with a corner cutout of the boundary. The most developed and effective is the method for solving plane problems of deformable body mechanics [9, 20].

Experimentally, the stress state in the neighborhood of an irregular point of the boundary area is obtained by photoelastic method using the free temperature deformation defrosting property [12, 15,16,17,18,19,20] on flat composite polymer models with different cutout corners of the boundary. Forced deformations are created in one of the areas of the model, the forced deformations gap along the area contact boundary extends to the top of the corner cutout on the boundary of the model.

This paper is aimed at analyzing SS in the zone of the corner cutout of the boundary of the region and identifying the self-balanced SS corresponding to the solution of the homogeneous boundary value problem in the area with an irregular boundary point.

2 Materials and Methods

2.1 Experimental Solution of the Boundary Corner Cutout Problem on a Flat Model by Photoelastic Method

An experimental solution of a plane problem for an area with a cutout at the boundary, the geometry of which causes the occurrence of stress concentration, is considered.

The plane area consists of parts \(\Omega _{1}\) and \(\Omega _{2}\), with elasticity module E, Poisson’s material ratio \(\nu\) and linear expansion coefficient \(\alpha\). Temperature deformations \(\alpha T\delta _{{ij}}\) are created in one of the areas—\(\Omega _{2}\), the area \(\Omega _{1}\) is not loaded. Along the area contact line \(\Gamma = \Omega _{1} \cap \Omega _{2}\), there is a deformation jump \(\Delta \varepsilon _{{ij}} = \alpha T\delta _{{ij}}\), which goes to the body boundary \(S\) to the point \(O\left( {0,0} \right)\)—the top of the corner cutout of the area boundary. For different angles of the corner cutout solution of the boundary at its top, there is a feature of the stress state, which is experimentally realized on the model made of optically sensitive material as a stress raiser (see Fig. 1).

Fig. 1
figure 1

Patterns of strips in models \(t = 3\) mm thick for different cases of the model end solution: (1) \(2\alpha = 180^{\circ }\); (2) \(\alpha + \beta = 180^{\circ }\), \(\alpha = 105^{\circ }\), \(\beta = 75^{\circ }\); (3) \(2\alpha = 270^{\circ }\); (4) \(2\alpha = 330^{\circ }\); (5) \(2\alpha = 360^{\circ }\)—narrow cutout—gap; (6) \(2\alpha = 90^{\circ }\)

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The elastic problem under the action of temperature deformations is solved by the photoelastic method using the property of “defrosting” of free temperature deformations [12, 16, 20]. A model of homogeneous epoxy material with given dimensions l = 180 mm, h = 25(23) mm is glued from two elements. The \(\Omega _{2}\) area element is cut from a plate obtained from the longitudinal shear of a hollow cylinder frozen under the action of a longitudinal uniformly distributed compression load. The height of the cylinder before “freezing” is H = 309 mm, the outer diameter is D = 180 mm, the inner diameter is 150 mm. The ends of the cylinder are cut off after freezing and only the middle part of the cylinder, which is in a flat deformed state, is considered. The compression of the cylinder after applying the load and freezing is \(\Delta H = 9\) mm, relative elongation along the longitudinal axis \(OZ\) of the cylinder is \(\varepsilon _{z} = \frac{{\Delta H}}{H} = \frac{9}{{300}} \approx 0.03\). Compression stress is \(\sigma _{z} = - \frac{P}{F} = - E\varepsilon _{z} = - 200 \cdot 0.03 = - 6\) kg/cm2, \(\sigma _{z} = \frac{{\sigma ^{{1.0}} m}}{t}\, \Rightarrow \,m = \frac{{\sigma _{z} \cdot t}}{{\sigma ^{{1.0}} }}\), where \(t\) is a shear thickness; \(\sigma ^{{1.0}}\) is a price of the model strip material; \(\sigma ^{{1.0}} = 0.341\). According to these data, in the shear \(t = 0.3\) cm thick, the number of strips is \(m = 5.3\).

The area \(\Omega _{1}\) is cut from free unloaded material. The areas \(\Omega _{1}\) and \(\Omega _{2}\) are glued together, then the model is “defrosted”. The model generates SSS due to the action in the \(\Omega _{2}\) area of temperature deformations. \(\varepsilon _{0} = - \varepsilon _{z} = - \alpha T\delta _{{ij}}\).

The pattern of the strips corresponding to the stress–strain state from the action of temperature deformations for different shapes of the model end boundary is shown in Fig. 1. Model thickness (1)…(6) is the same and equals to \(t = 3\) mm. In case (6), stress concentration at the top of point \(O\left( {0,0} \right)\) is not observed and stresses at the very top \(O\left( {0,0} \right)\) are zero.

2.2 Analysis of the Stress State Obtained Experimentally on the Photoelastic Method Model

The solution obtained experimentally on the composite model in the area of the corner cutout of the boundary under the action of discontinuous forced deformations is analyzed. An irregular point of the area boundary is considered as a top of the cutout of the end of the flat model lying on the contact boundary of the model parts.

We consider the neighborhood of an irregular point of the boundary adjacent to the singular point from which the irregular point is removed. Experimentally, the singular area on the model is defined as the neighborhood of the top of the corner cutout of the boundary, in which the interference fringes are blurred and the isochrome picture is not readable at any magnification of the neighborhood fragment.

The stress state in the neighborhood of the irregular point of the model boundary can be represented as Areas I, II, III according to Fig. 2a, where, due to the oblique symmetry of the stress diagrams, one of the areas \(\Omega _{2}\) is given.

Fig. 2
figure 2

SS scheme in the neighborhood of an irregular point of the boundary of a plane area. – – – —pure shear line

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Area I is an area adjacent to the symmetry axis of the model \(\Gamma{:}\,y = 0\), where the main compressive stresses \(\sigma _{2}\) considerably exceed the modulus of the main tensile stresses \(\sigma _{1}\). As the point moves away from the top of the cutout in area I, the \(\sigma _{1}\) stresses nature may change from tensile to compressive.

The main vector on the radial sites at the points of the area I (see Fig. 1a) has a normal component \(\sigma _{i} = \sigma _{r}\) considerably superior in modulo to the tangential one \(\sigma _{\theta }\), which agrees with the ratio of the values of the principal stresses: \(\left| {\sigma _{2} } \right| \gg \sigma _{1} ;\;\sigma _{1} \to 0\) at \(\left( {x,y} \right) \to \left( {x,0} \right);\,\,\sigma _{1} |\Gamma = 0;\,\,\Gamma{:}\,y = 0\).

Area III is the area adjacent to the model cutout boundary, where the tensile principal stresses \(\sigma _{1}\) considerably exceed the modulus of compressive stresses \(\sigma _{2}{:}\,\,\sigma _{1} \gg \left| {\sigma _{2} } \right|\); \(\sigma _{2} \to 0\) at \(\left( {x,y} \right) \to S\); \(S\)—model boundary; \(S{:}\,y = ax\), \(a = tg\alpha\), \(2\alpha\)—angle of the model end opening; \(\sigma _{2} |S = 0\).

Area II is a transitional area in which sharp angles, isochrome and significant gradients of the isocline parameter forming loops are observed. The largest isocline parameter is observed for the inner, nested in the remaining isocline loops, and varies from 45°…50° up to 70° when increasing the opening of the model end from \(2\alpha = 180^{\circ }\) to \(2\alpha = 260^{\circ }\).

There is a change in the direction of the main sites rotation in area II. If the main sites rotate clockwise in area III, then, passing area II, they change the direction of rotation to the opposite, so that the rotation of the main sites at the points of area I is observed in a clockwise direction as before.

In the transition area II (see Fig. 2a), a pure shear site is observed. The principal stresses at these points are equal in modulo and opposite in sign: \(\sigma _{1} = - \sigma _{2}\). The sites tilted at an angle of 45° to the main ones are in pure shear conditions, i.e. \(\tau _{{\max }} = \frac{{\sigma _{1} - \sigma _{2} }}{2} = \sigma _{1}\), and the normal stresses across these sites are zero: \(\sigma _{i} = \frac{{\left( {\sigma _{1} + \sigma _{2} } \right)}}{2} = 0\). According to the experimental data, the pure shear sites coincide with the radial ones. The radial stress as the normal stress over the pure shear site is zero: \(\sigma _{r} = 0\), and shear stresses \(\tau _{{\max }} = \frac{{m\sigma ^{{1.0}} }}{{2t}} \ne 0\). Therefore, the isochrome orders m at the points of pure shear sites are not equal to zero. According to the experimental data, pure shear sites are observed at the tops of sharp angles of isochromes and their vicinities.

The stresses are continuous in the area \(\Omega _{2} {:}\,0 \le \theta \le \alpha\), so there is a subarea in transition area II in which pure shear and a stress state close to it are observed. For this “pure shear” subarea, there are sites where the normal stresses coincide or are close to the radial stresses and are equal or close to zero: \(\sigma _{i} \cong \sigma _{r} \cong 0\) (see Fig. 2).

Main stresses \(\sigma _{2}\) continuously varying from the highest value along the contact line of the area Γ: y = 0 in area I, going to area II “pure shear” subarea, then quickly decrease to zero values in area III. The sign may also change for main stresses \(\sigma _{2}\) when approaching the boundary of the area S. Similar, tensile stresses \(\sigma _{1}\), changing continuously from the highest values at the model cutout boundary in area III, passing to the “pure shear” subarea II, then rapidly decreasing to zero values in area I. For the main stresses \(\sigma _{1}\) the sign may also change at the transition to the area contact line Γ = Ω1 ∩ Ω2, Γ: y = 0.

Radial stresses in area I: \(\sigma _{r} \cong \sigma _{2}\) at \(\left( {x,y} \right) \to \left( {x,0} \right)\), and in area III: \(\sigma _{r} \cong \sigma _{1}\) at \(\left( {x,y} \right) \to \left( {x,\alpha x} \right)\), \(S{:}\,y = ax\), \(a = tg\alpha\). In the transition area II on pure shear sites and close to them \(\sigma _{r} = 0\) or \(\sigma _{r} \approx 0\). Such a distribution of radial stresses in the neighborhood of the top of the model end cutout corresponds to the theoretical radial stress distribution of the form:

$$ \sigma _{r} = cf\left( r \right)f\left( \theta \right) = \frac{{\alpha ET}}{{r^{{1 - \lambda }} }}\left( {c_{1} \cos \theta + c_{2} \sin \theta } \right),\,\sigma _{\theta } = \tau _{{r\theta }} = 0 $$
(1)

At the pure shear sites, at \(\theta = \theta _{0} ,\,r > r_{0}\), \(r_{0}\)—sufficiently small, \(\sigma _{r} = 0\), \(\sigma _{r} = \frac{{\alpha ET}}{{r^{{1 - \lambda _{0} }} }}\left( {c_{1} \cos \theta _{0} + c_{2} \sin \theta _{0} } \right) = 0\, \Rightarrow \,c_{1} = - c_{2} \left( {\frac{{\sin \theta _{0} }}{{\cos \theta _{0} }}} \right)\).

The theoretical radial distribution of stresses in the neighborhood of the cutout of the model of the form (1) is understood as a self-balanced “own” stress state in the neighborhood of an irregular border point obtained as a solution of a plane homogeneous boundary value problem.

2.3 Example of Stress State Analysis Obtained on a Flat Model with a Straight End

The stress state is analyzed in the neighborhood of the irregular point of the boundary where the gap of forced deformations exits, using the straight end model as an example. According to the theoretical model [1,2,3,4, 12, 13], the stress state in the neighborhood of an irregular point of the area boundary can be represented as:

$$ \sigma _{{ij}} = \sigma _{{ij}}^{o} + \sigma _{{ij}}^{l} $$
(2)

where \(\sigma _{{ij}}\)—stresses in the vicinity of an irregular point of the area boundary; \(\sigma _{{ij}}^{o}\)—eigensolution of the homogeneous boundary value problem in the vicinity of an irregular boundary point; \(\sigma _{{ij}}^{l}\)—stress caused by the action of the specified loads or the total stress field.

Internal stresses \(\sigma _{{ij}}^{o}\) in the area of the end of the model with a straight end have a radial view [1,2,3,4, 12, 13]:

$$ \sigma _{r} = \frac{{\alpha ET}}{r}\left( {c_{1} \cos \theta _{0} + c_{2} \sin \theta _{0} } \right),\quad \sigma _{\theta } = \tau _{{r\theta }} = 0 $$
(3)

With a continuous distribution of stresses in the area \(\Omega _{2}\) (similarly in the area \(\Omega _{1}\)) of the form (3) in the cross-section \(r = r_{0}\) there will be a point at which \(\sigma _{r} = 0\). Consider one area of the straight-end model, for example, \(\Omega _{2}\): E, \(\nu ,\;\alpha T\). Pattern of the model strips for the area \(\Omega _{2}\) is given in Fig. 2b. According to the experimental data in area II of the model, the point with the net shear area is observed in the cross section at an angle \(\theta _{0} \approx 60^{\circ }\). Figure 2 shows the “pure shear” line, which has a slope angle \(\theta _{0} \approx 57.5^{\circ }\) in the vicinity of point \(O\left( {0,0} \right)\). According to the experimental findings

$$ \sigma _{r} = 0\,\,{\text{at}}\,\,\theta _{0} \approx 57.5^{\circ } $$
(4)

Taking into account the theoretical distribution (1), the radial stresses will be recorded:

$$ \sigma _{r} = \frac{{c_{0} }}{r}\left( {c_{1} \cos \theta + c_{2} \sin \theta } \right) = 0, $$
(5)

or

$$ c_{1} = - c_{2} \frac{{\sin \theta _{0} }}{{\cos \theta _{0} }} = - c_{2} \frac{{\sin 57.5^{\circ } }}{{\cos 57.5^{\circ } }} = - 1.57c_{2} $$
(6)

Stress (3) taking into account (6) will write over:

$$ \sigma _{r} = \frac{c}{r}\left( { - 1.57\cos \theta + \sin \theta } \right) $$
(7)

where \(c = c_{0} \;c_{2}\), c is an unknown multiplier.

Determine what force is statically equivalent to the radial stresses (7), acting over a small radius cross section \(r = r_{0}\) of the area \(\Omega _{2}\) of the model end [4, 12, 16, 17].

$$ P_{V} = \sum X = \mathop \int \limits_{0}^{{\frac{\pi }{2}}} \sigma _{r} \cos \theta \text{d}F\;\; = \;\mathop \int \limits_{0}^{{\frac{\pi }{2}}} \sigma _{r} \cos \theta rt\text{d}\theta $$
(8)

where t is a model thickness. Considering (7), force \(P_{V}\) equals:

$$ P_{V} = \mathop \int \limits_{0}^{{\frac{\pi }{2}}} \frac{c}{r}\left[ { - 1.57\cos \theta + \sin \theta } \right]\;\cos \theta r\;t\text{d}\theta \;\; = \; - \;\frac{{ct}}{r}\left( {1.57\pi + 1} \right) $$
(9)

Let’s find the horizontal component of the acting forces:

$$ P_{H} = \sum Y = \mathop \int \limits_{0}^{{\frac{\pi }{2}}} \sigma _{r} \sin \theta \;\text{d}F\;\; = \;\mathop \int \limits_{0}^{{\frac{\pi }{2}}} \sigma _{r} \sin \theta rt\text{d}\theta $$
(10)

Considering (7), force \(P_{H}\) equals zero:

$$ P_{H} = \mathop \int \limits_{0}^{{\frac{\pi }{2}}} \frac{c}{r}\left( { - 1.57\cos \theta + \sin \theta } \right)\sin \theta r t\text{d}\theta = 0 $$
(11)

Then force action angle \(\gamma\), statically equivalent to radial stresses of the form (7) in the area \(\Omega _{2}\): \(tg\gamma = \frac{{P_{H} }}{{P_{V} }} = 0 \to \gamma = 0^{0}\).

Finally, the radial stresses of the form (7):

$$ \sigma _{r} = \frac{c}{r}\left( { - 1.57\cos \theta + \sin \theta } \right) $$
(12)

in the small radius cross section \(r = r_{0}\) in the area \(\Omega _{2}\) are statically equivalent to the action of a vertical force directed along the axis \(OX\):

$$ P_{V} = \sum X = - \frac{{ct}}{4}\left( {1.57\pi + 1} \right) $$
(13)

and the horizontal thrust is zero: \(P_{H} = \sum Y = 0\).

The resulting scheme of force action \(P_{V} ,\,P_{H}\) in the small radius cross section \(r = r_{0}\) of the model end coincides with the corresponding forces for the wedge with an opening angle \(\theta = \frac{\pi }{2}\) under the action of a concentrated force at the top of the wedge [4, 12, 16, 17].

3 Results

The experimentally obtained position of the pure shear line in the irregular point area of the model boundary shows full coincidence with the position of the neutral axis of the radial stress state for the wedge in the area of the top \(O\left( {0,0} \right)\) under concentrated force action.

The coincidence of the slope angle of the pure shear line in the area of the model end and the neutral axis of the model wedge during the action of the radial stress state, the coincidence of the isocline parameter in the model isochrome tops and the calculated slope angle of the main sites confirm experimentally the existence of the self-balanced radial stress state in the neighborhood of the model end top, corresponding to the internal stress state \(\sigma _{{ij}}^{o}\) in the general presentation of the SS: \(\sigma _{{ij}} = \sigma _{{ij}}^{o} + \sigma _{{ij}}^{l}\) of the model end.

4 Discussion

Experimentally, the singular area on the model is defined as the neighborhood of the top of the corner cutout of the boundary, in which the interference fringes are blurred and the isochrome pattern is not readable at any magnification of the neighborhood fragment. A contradiction arises with a continuous change of stresses in the punctured vicinity of an irregular boundary point, “sharp angles” of isochromes occur. The above analysis is applicable to the neighborhood of an irregular point of the boundary adjacent to the singular point from which the irregular point is removed. The validity of the analysis data in the selected area of the irregular boundary point is confirmed by the coincidence of the experimental and theoretical stress values.

5 Conclusions

The analysis of SS in the area of the irregular point of the boundary, in which the gap of forced deformations exits, shows the existence of a self-balanced SS \(\sigma _{{ij}}^{o}\), corresponding to the solution of the homogeneous boundary value elasticity problem. It explains the growth of strips orders observed from the inside of the stress concentration area, and not at the very top of the cutout of the area. The inequation to zero of orders of the strips in area II of pure shear in the neighborhood of the top of the model end shows the existence of a self-balanced SS caused by the action of the given loads or the total stress field.