Stress state analysis and optimization in the vicinity of the sensor of SMART-material

Stress state analysis and optimization in the vicinity of the sensor of SMART-material

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Procedia Integrity 500(2017) StructuralStructural Integrity Procedia (2016)99–106 000–000

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2nd International Conference on Structural Integrity, ICSI 2017, 4-7 September 2017, Funchal, Madeira, Portugal

Stress state analysis optimization in the vicinity thedesensor of XV Portuguese Conference and on Fracture, PCF 2016, 10-12 February 2016,of Paço Arcos, Portugal SMART-material Thermo-mechanical modeling of a high pressure turbine blade of an aa a V.P. Matveenkoaa, I.N. Shardakov N.A. Kosheleva * *, A.Yu. Fedorova airplane gasbb, turbine engine

a b c 614990, 29 prospekt, Perm National National Research Research Polytechnic Polytechnic University, University, 29 Komsomolsky Komsomolsky prospekt, Perm, Perm, 614990, Russia Russia Institute Institute of of Continuous Continuous Media Media Mechanics Mechanics of of the the Ural Ural Branch Branch of of Russian Russian Academy Academy of of Science, Science, 11 Akademika Akademika Koroleva Koroleva str., str., 614013, 614013, Perm, Perm, Russia Russia a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Abstract Portugal c CeFEMA, Department Engineering, Instituto Superior Técnico,and Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Composite materialsofin inMechanical practical human human activity are well well positioned, their range of applications is constantly expanding. Composite materials practical activity are positioned, Portugal and their range of applications is constantly expanding. b b

aa Perm

P. Brandão , V. Infante , A.M. Deus *

Composite Composite materials materials based based on on polymer polymer are are widely widely used used among among the the different different kinds kinds of of composite composite materials. materials. The The nomenclature nomenclature of of these are different different from from each each other. other. For For these these reasons, reasons, the the complex complex structure structure of of the these materials materials is is very very wide wide and and their their properties properties are the material and material and number number of of other other factors, factors, the the traditional traditional methods methods of of assessing assessing the the strength strength and and reliability reliability of of construstions construstions from from Abstract polymeric composite materials (PCM) should be complemented. In particular, by using new and effective monitoring polymeric composite materials (PCM) should be complemented. In particular, by using new and effective monitoring systems. systems. One One of these methods diagnostics is creation SMART-materials with sensors. As the ofDuring these modern modern methods of of diagnostics is engine creationcomponents SMART-materials with embedded embedded sensors.demanding As aa sensors sensors the piezoelectric piezoelectric their operation, modern aircraft are subjected to increasingly operating conditions, sensors or fibers are They can on surface or in polymer composite material. sensors or optical optical fibers are used. used. They(HPT) can be be located located Such on the theconditions surface of of the the material materialparts or embedded embedded in the the polymer composite material. especially theofhigh pressure turbine to undergo different types ofmodelling time-dependent New methods diagnostics products from blades. these materials materials increasecause their these competitiveness. Based on numerical and New methods of diagnostics products from these increase their competitiveness. Based on numerical modelling and degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict experimental researches, researches, the the aim aim of of the the work work is is to to search search different different ways ways of of solving solving the the problem, problem, that that allow allow to to increase increase SMARTSMARTexperimental the creep behaviour of as HPT blades. Flight data records (FDR)using for athespecific aircraft, provided by a commercial aviation material reliability as well well proper registration of the the strain field, field, fiber-optic sensors and piezoelectric piezoelectric elements. The material reliability as as proper registration of strain using the fiber-optic sensors and elements. The company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D problems of of the the adhesive adhesive joint joint geometry geometry optimization optimization that that appear appear while while mounting mounting sensitive sensitive elements elements on on the the PCM PCM surface surface model and problems and needed for the fiber FEMoutputs analysis, a HPT blade in scrap scanned, and of itsadhesive chemicaljoint composition material propertieswere were designing optical were considered this was paper. The results results influence and on sensors sensors indications designing optical fiber outputs were considered in this paper. The of adhesive joint influence on indications were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified shown. Besides, Besides, the the stress-strain stress-strain state state analysis analysis that that includes includes the the singular solutions solutions for for the the points, points, where where infinite infinite stress stress may may occur, occur,3D shown. rectangular block shape, in order tothis better establish the model, singular and then with the real 3D mesh obtained from the blade scrap. The was carried out in the framework of study. was carried out in the framework of this overall expected behaviour in terms ofstudy. displacement was observed, in particular at the trailing edge of the blade. Therefore such a © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier model can be useful in the by goal of predicting turbine blade life, given a set of FDR data. © 2017 The Authors. Published Elsevier B.V. B.V. Peer-review under responsibility of the Scientific Committee of ICSI Peer-review under responsibility of the Scientific Committee ICSI 2017. 2017. Peer-review under responsibility of the Scientific Committee of ICSIof 2017 © 2016 The Authors. Published by Elsevier B.V.

Keywords: Smart materials; fiber singularity; composite Keywords: Smartunder materials; fiber optic optic sensors; sensors; singularity;Committee composite materials materials Peer-review responsibility of the Scientific of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

* * Corresponding Corresponding author. author. Tel.: Tel.: +7(342)2378308 +7(342)2378308 E-mail E-mail address: address: [email protected] [email protected] 2452-3216 2452-3216 © © 2017 2017 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. Peer-review under responsibility the * Corresponding Tel.: +351of Peer-review underauthor. responsibility of218419991. the Scientific Scientific Committee Committee of of ICSI ICSI 2017. 2017. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216  2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ICSI 2017 10.1016/j.prostr.2017.07.074

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1. Introduction Materials on polymer basis are widely used among composite materials. The variety of these materials and their wide application in products that require high reliability leads to the need of new approaches. Along with traditional methods new approaches will be used for evaluating the strength and reliability of structures made of polymer composite materials. One of such approaches connected with the use of effective monitoring systems. Discussing the topic of composite materials monitoring, it is reasonable to use the term smart materials. According to the classification of smart materials that was given in [1], smart materials should include one or several of the following structures: sensing elements, actuator systems, real-time information processing systems. To implement the concept of smart structures composite materials are match to each other in the best way because during its manufacturing the suitable elements could be embedded. Primarily, systems containing sensitive elements (sensors) that allow to monitor the composite material by various parameters have been spread among the smart materials. At the present time, the most promising design of smart materials that contains sensors is associated with the use of optical fibers and piezoelectric materials. Sensors based on the fiber optics can measure different physical and mechanical quantities, capable to withstand the strains which are comparable to the strains of the composite. They are light and affordable in manufacturing, immune to electrical interference and under severe conditions have advantages, including sensitivity, compared to other sensors. Also fiber optic sensors can be easily integrated with other equipment for remote monitoring and allow observing composite structure during all stages of its manufacturing: design, testing, operation. One of the main tasks of using fiber optic sensors in composite materials is related to the strain measurement. Problems that encountered in the implementation of this feature are related to the strain measurement of the fiber Bragg gratings and described in the details in the review [2]. The questions of the influence of embedded fiber on the properties of polymer composite material, the issues of input-output optical fiber from the composite material, the tasks of choosing a model that linked the strain of the optical fiber at location of a fiber Bragg grating with measurement of its resonant wavelength and the problem of calibration are observed. There are a number of other problems that need to be investigated, for example, the influence of temperature, time, and complex stress. One of the problems that appeared when sensors are embedded into a composite material is related to the optimization of the stress state nearby the sensors. In this paper, the problems of optimizing the stress state in an adhesive joint when the sensor is mounted to the material surface, optimizing the stress state in the optical fiber exit zone from the composite material, and the effect of the adhesive joint on the fiber optic sensors readings are studied. 2. Optimization of stress state in adhesive lap joints A lap joint is the most common type of adhesive joints when mounting the sensors on the material surface. The design diagrams of such adhesive joints contain singular points. In problems of the elasticity theory, there are singular solutions, which are due to the presence of infinite stress values at individual points (lines) of the domain, known as singular points (lines). Among these points are points on the body surface where the surface smoothness condition is violated, the type of boundary conditions changes, and different materials contact, or internal points where, for example, the condition of smoothness of the contact surface of different materials is violated [3, 4]. Infinite stress values appear in an idealized model of a real object considered in the framework of the linear elasticity theory. Singular solutions usually indicate that the simulated object has clearly expressed stress concentration areas. There are various ways to reduce the stresses in the adhesive lap joints. One of them is to change the shape of the external surface of the adhesive layer at the ends of the contact area. Fig. 1 shows two variants of adhesive lap joints: with and with no spew fillet (lN and lW are the lap area lengths for joints without and with a spew fillet, respectively). The design models of these joints have singular points A and B on the body surface, at which different materials are connected, and a point C, at which the smoothness of the contact surface of the two materials is violated. In the lap joint with a direct end of the adhesive layer obtained by removing the excessive adhesive from the edges of the contact area, a nonuniform stress distribution on the contact surface and stress concentrations at the ends of the contact area are observed. Changes in the rectangular shape of the end of the adhesive layer by forming an external excessive adhesive (spew fillet) help distribute the load on a larger area and provide a more uniform stress distribution. It was shown in [5–12] that the use of spew fillets at the ends of the contact area of adhesive joints decreases the stress



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concentration both in the adhesive layer and in the adherends. In this case, only one shape of spew fillets is usually considered. An exception is Ref. [11], where various shapes of spew fillets are subjected to a comparative analysis. Note that the conclusions in [11] about the impact of the spew fillet shape on the stress level in adhesive joints have information on the decisive role of the angle of attachment of the free surface of the spew fillet to the adherend. At the same time, there is no conclusion about the optimality of the determined spew fillet shape and the effectiveness of the proposed variants for adhesive joints made of other adherends and adhesives.

a B A

1 2

lN

b

В

В A

А 1

1 2

С

В С

lW

А 1

Fig. 1. Variants of adhesive lap joints of materials: (a) without and (b) with a spew: 1 − material; 2 − adhesive.

In the numerical analysis of the stress-strain state of the considered adhesive joints, it is difficult to estimate the accuracy and convergence of solutions in the vicinity of singular points. If the finite element method is used, one of the ways to solve this problem is to refine the finite element mesh. In this case, we can achieve the required accuracy of stress calculations outside a certain vicinity of the singular point. The size of this vicinity depends on the degree of clustering of the finite element mesh. An effective estimation of the accuracy of results obtained by the numerical solution of the problem on the basis of the principle of virtual displacements is the accuracy of satisfaction of natural boundary conditions on the load-free external surface of the samples and the contact surface of two different materials. In the present study, we used sampling that ensures the fulfillment of natural boundary conditions outside three or four elements adjacent to a singular point with an error less than 1%. Various shapes of spew fillets were analyzed in [6–9, 11, 12]. Most of the authors considered the triangular geometry of spew fillets with the angle of coupling of the external surface of the adhesive with the adherend surface usually equal to 45°. Lang and Mallick [11] studied eight spew geometries, among which the most significant decrease in the stress peaks is provided by a shape formed of a circle arc with a center and a radius at which the coupling angles are gА = gB = 0° (see Fig. 2). It should be noted that this shape is sufficiently producible. In this work, the shape of the free surface of the spew fillet is also chosen as a circle arc with a center, which makes it possible to obtain different angles gА and gB of coupling of the external surface of the adhesive with the surface of the adherends at the points A and B. When estimating the stress states obtained by the finite element method, we are interested in data on the singular nature of the stresses at singular points. These data may be obtained by analyzing eigensolutions for composite wedges formed by tangents at singular points to the surfaces of the material, adhesive, and contact surface (see Fig. 2). In a polar coordinate system with the origin at the wedge apex, the eigensolutions have the form [3]:

ur  r ,  r k rk    ,

u  r ,  r k k    ,

    where r is the distance from the singular point, k are the eigenvalues, r    and     are the eigenforms, and ur  r,  , u  r,  are the displacement components. The procedure of constructing eigensolutions for a composite wedge [13] includes finding the eigenvalues of transcendental equations: k

k

4  1    sin 2 pg m  p 2      sin 2 g m   1    sin 2 pg a  p 2     sin 2 g a  

 1     sin 2 pg m  p 2 sin 2 g m   1     sin 2 pg a  p 2 sin 2 g a   2 1   2   sin pg m sin p  a cos p  g a  g m   p 2 sin g m sin  a cos  g a  g m    0. 2

2

(1)

4102

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Here p = 1 – ; gm, ga are the angels of the composite wedges;  

Г (km  1)  (ka  1) Г (km  1)  (ka  1) ,  — Г (km  1)  (ka  1) Г (km  1)  (ka  1)

combined parameters of elastic material constants; ki  3  4 i , i = {m, a}; m, a are Poisson’s rations of materials and adhesive; Γ = Ga /Gm ; Gm , Ga are the shear modulus. В

t

g g a

gm

B

R C

ga gA

A

gm

Fig. 2. Shapes of the free surface of the spew fillet.

It follows from Eq. (1) that the eigenvalues depend on the angles gm and ga and on the mechanical characteristics νm, νa, and Γ = Ga /Gm . In this case, the values of k with the real part 0 < Re k < 1 correspond to the singular solutions and determine the nature of the stress changes in the vicinity of the singular points. As an example, consider the following parameters of adhesive joint [11]. The elasticity moduli and Poisson’s ratios of the plate material and adhesive have the following values: Em = 1.567· 1010 Pa, νm = 0.46, Ea = 3.81 · 109 Pa, a = 0.48, plate thickness t = 2.54 mm, thickness of adhesive layer t = 0.762 mm. The adhesive joint is under the influence of the stretching force P = 445 N. Calculations were carried out for different angles ga = gB and ga = gA. In the examples considered that gm at the point A is equal to π, and at the point B is equal to π/2. The numerical analysis results show that for angles gm, ga, where singular solutions are occurred (values satisfying the condition Re k < 1), the stress concentration takes place nearby the singular points. In this case, the stresses in the vicinity of a singular point, where singular solutions take place, are determined by the degree of mesh refinement. While the dimensions of the finite elements are decreasing, the stresses at the singular point asymptotically tend to infinity, which in the numerical solution indicates as the existence of a singular solution. In the absence of singular solutions (all values of k satisfy the condition Re k > 1), the stresses at the singular point tend asymptotically to zero as the size of the finite elements decreases. At a minimum value of Re k, which is equal to one, the stress at the singular point has a finite value. It should be noted that: the lower the value of Re k < 1, the brighter the picture of stress concentration nearby the singular points. It should be noted that in the works [6–8, 11] for determination of the best version of the spew fillet geometry, the comparison was done using the stresses at the point A or B obtained by the finite element method. In the presence of a stress singularity, such analysis is not completely correct, since in this case the stress values that were obtained at singular points, are determined by the degree of mesh refinement. It was shown in [14, 15] that, in the vicinity of singular points, optimal geometries in terms of reduction of stress concentrations, have a common property: the parameters of optimal geometries in the vicinity of singular points (angles gm, ga) and mechanical characteristics of the material (νm, νa, Γ = Ga /Gm ) define the boundary between the solutions with and without the singularity. We use this property of optimal geometries to choose an optimal variant of the circle arc connecting the points A and B. In this case, the consideration of a limited class of surfaces in for choosing the optimal geometry is not critical and can be justified by the technology of adhesive joint manufacturing. The analysis of the eigenvalues in Eq. (1) for gm = 180° shows that eigenvalues with 0 < Re λ1 < 1 are obtained for all gА > 0. According to the property of optimal geometries in the vicinity of singular points, the angle of coupling of the free surface of the adhesive layer with the adherend surface at the point A should be zero for the optimal choice of the circle arc. At the singular point B, for the considered materials, all the eigenvalues have real parts greater than unity at gВ < 63°, and eigenvalues with 0 < Re λ1 < 1corresponding to singular solutions appear at gВ > 63°. Consequently, the optimal angle at the point B is gВ ≤ 63°. There are no singular solutions at the point C only at gm = ga = 180°, but this variant does not make sense in



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this task. Zhao et al. [12] proposed rounding of the internal right angle at the point C. The rounding radius in this work equals the plate thickness. Fig. 3 shows the stress intensity distribution σi on the adhesion surface for the optimal variant of the spew geometry with gА = 0° and gВ = 63° in the case of rounding at the point C and the distribution σi for the variant of the external surface of the spew fillet with the coupling angles gА = gВ = 9.3°, which, according to [11], provides the lowest stress concentration. In contrast to [11], this paper has an additional rounding of the right angle at the point C. The analysis of the results shows that the stresses in the vicinity of singular points for the obtained optimal geometry are much smaller than for the spew fillet geometries discussed in [6–9, 11, 12]. i , MPa

8 6 4

1 2

2 0

0

4

8

12

16

20

24 x, mm

Fig. 3. Stress intensity distribution σi over the adhesive joint surface: curve 1 refers to the variant of the spew geometry [11] (gА = gВ = 9.3°) and curve 2 refers to the optimal variant of the spew geometry (gА = 0°, gВ = 63°).

3. Optimization of stress state in the output zone of a fiber optic sensor embedded into composite material One of the problems of using fiber optic sensors embedded into the polymer composite material is to ensure their connection to the measuring equipment. This goal can be achieved through physical contact or without it. In both cases, there is an input-output of optical fiber or dismountable elements on the surface of the composite material. Due to the properties heterogeneity of the optical fiber on the surface, special points are appeared in the schemes for stress state calculating, where singular solutions can exist in the framework of linear theory of elasticity, reflecting in the actual product the presence of pronounced stress concentration zones. According to this and developing the approaches outlined in the previous chapter, it is possible to propose options for optimizing the stress state in the fiber-optic inputoutput zone of the composite material. For the numerical simulation of the stressed state, consider the subregion in the input-output zone with the following dimensions 2l by d, where l = 1.5 mm, d = 3 mm (highlighted in color in Fig. 4a). Two variants of a fiberoptic sensor in the form of a quartz fiber and a quartz fiber with a polymer coating are considered. Here, the quartz fiber has a diameter of 0.124 mm and the mechanical characteristics Eq = 71.4 · 109 Pa, q = 0.17, thickness of polymer coating 0.012 mm and the mechanical characteristics Epc = 5 · 109 Pa, pc = 0.35. Mechanical characteristics of the material: Em = 26.3 · 109 Pa, m = 0.18. Fig. 5 shows the stress intensity distribution over the surface of a quartz fiber without and with polymer coating under loading. The results demonstrate the presence of the high stress concentration zone in the vicinity of a singular point on the surface where various materials (composite material–quartz, composite material–polymer coating) come into the contact. The presence of such stress concentration zone can lead to the destruction of the optical fiber. One of the options for eliminating the established stress concentration is related to the design of the geometry in the fiber input-output zone in accordance with the properties of the optimal geometries nearby the singular points. In the present problem, such geometry addition can be formed from epoxy binder at the stage of the polymer composite formation. In this case for exclusion of singular solutions the coupling angles to a fiber and material surfaces must be zero. This geometry will correspond to the geometry of the additional region, which has a quarter-circle shape in the cross section (it is indicated by dashed lines in Fig. 4a). The radius of this circle is equal to 0.82 mm. For this variant, the stress intensity distribution over the surface of the quartz fiber is shown in Fig. 5. These results demonstrate the effectiveness of stress concentration elimination. Also, it should be noted that, due to the small diameter of the optical fiber, the technological implementation of the proposed solution is quite complex.

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Comparing the results of stress distribution along the fiber boundary with the polymer coating and without it, it can be stated that the local stress concentration, which is caused by singular points (point A) on the outer surface where the polymer coating and composite material have a contact, decreases along the thickness of the polymer coating. Based on this result, an additional element in the input-output zone can be introduced in the form of coating with a length of 2l. In this example it is a fluoroplastic coating with 0.2 mm thickness and the following mechanical characteristics: Efc = 0.545 · 109 Pa, νfc = 0.466. Fig. 5c shows the stress intensity distribution over the surface of the quartz fiber guarded by the fluoroplastic coating without and with a modified geometry nearby the singular point (see Fig. 4b). The results show that the fluoroplastic coating with the corresponding thickness eliminates the local stress concentration in the optical fiber. In this case, the change in the geometry in the vicinity of the singular point A practically does not affect on the stress state picture in the optical fiber. P

P

a

z

l

O

b

l r

A

A l

l d

d

Fig. 4. Options for the input-output of the optical fiber from the material.

a

 i2 P 1 .5

1 2

0 .5

-0 .0 5

0

0 .0 5

z0 .1l

1 .5

1

1

1

0 .5

2

0 .5

0 -0 .1

c

 2i P

1 .5

1

0 -0 .1

b

 2i P

-0 .0 5

0

0 .0 5

z0 .1l

0 -0 .1

1 2 -0 .0 5

0

0 .0 5

0z.1l

Fig. 5. The distribution of stress intensity over the surface of a quartz fiber: a) without a polymer coating, b) with a polymer coating, c) with a polymer and ftoroplastic coatings; 1 — without rounding; 2 — with rounding.

4. Evaluation of the influence of various structural factors of the fiber-optic sensor on the substrate on the values of the calibration factor To register axial strain on the surface of deformable bodies, fiber optical sensors (FOS) with Bragg gratings on substrates in various designs are used. The common thing that unites them is that they represent the structure that consists of heterogeneous interacting elements. At least there are three elements in them: an optical fiber with a Bragg grating, a substrate and an adhesive that connects the fiber to the substrate. All components of the sensor and their interaction determine how the strain values on the surface of the deformed body are correlated with the strain values of the optical fiber which are recorded with the help of the Bragg grating. Also, the method of the sensor mounting to the surface, which is under investigation, is an important factor affecting to this correspondence. The correspondence between the strain values on the surface of the deformable body ( ) and the values of the optical fiber strain (d) are determined by the calibration factor:

V.P. Matveenko et al. / et Structural Integrity Procedia 00 (2017) 000–000 N.A. Kosheleva al. / Procedia Structural Integrity 5 (2017) 99–106



K   d ,

1057

(2)

Mathematical modeling of the strain interaction between the sensor elements and the sensor element itself with investigated surface makes it possible to evaluate effectively the dependence of the calibration factor on all the above factors. Modeling was carried out within the framework of elastic deformation of all sensor elements. For numerical implementation, finite element software (ANSYS) was used. Fig. 6a shows the general structural scheme of the FOS, where the substrate dimensions are 60×10×0.2 mm, the diameter of the quartz fiber is 125 μm, the thickness of the inner acrylic coating is 125 μm, the thickness of the outer polymer coating is 125 μm, the diameter and thickness of the adhesive point is 3 mm and 2.5 mm, the diameter of the technological holes is 1 mm. Materials of sensor design elements have the following mechanical characteristics: substrate Es = 2.1 · 1011 Pa, s = 0.3; the optical fiber Eo = 71.4 · 109 Pa, o = 0.17; the inner coating Eic = 1.56 · 109 Pa, ic = 0.13; the outer coating Eoc = 3.1·109 Pa, oc = 0.15; epoxy adhesive Eea = 3 · 109 Pa, ea = 0.32. 3

1

а 4

b 4

3

2

Fig. 6. а) Structural scheme of FOS (1 − substrate, 2 − optical fiber, 3 − adhesive point, 4 − technological holes); b) ¼ sensor symmetric part.

4% 42%

25% 29%

a b c d

Fig. 7. The degree of influence of various design factors on the values of calibration factors Kε: a − optical fiber; b − mounting; c − adhesive; d − technological hole.

The analysis of numerical experiments results was done with a mathematical model that described the threedimensional spatial distribution of the strain of all FOS elements on a substrate. The sensitivity of the calibration coefficient to the type of adhesive joint of the fiber and the presence of technological holes in the metal substrate was established, as well as to the method of mounting the substrate to the study surface. In particular for determination the most rational way of adhering the FOS metal substrate to the investigated surface, the strain response of the fiber sensor to a homogeneous uniaxial strain (ε) was established with two fixing methods. The first method is to adhering only on the side faces of the substrate. The second method consists of adhering the entire contour of the substrate. It was assumed that from the two methods that was considered above, the one for which the calibration coefficient is closest to unity was chosen as the rational one. For the first method, the coefficient is Kε = 1.23, for the second Kε = 1.007. The choice of the mounting the FOS substrate to the investigated surface by adhering the substrate throughout the contour was indicated by this ratio of the values of the coefficients. From the analysis of the totality of the numerical experiments results, the influence degree of various design factors on the values of the calibration coefficients Kε was established. As such factors were considered: the physical and mechanical properties of optical fiber; the presence of a technological hole on a metal substrate; physical and mechanical properties of the adhesive; conditions for adhesive joint the metal substrate. The influence degree of these factors on the calibration factor as a percentage is shown in Fig. 7.

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5. Conclusions The problems of the adhesive joint geometry optimization that appear while mounting sensitive elements on the PCM surface and designing optical fiber outputs were considered in this paper. The results of adhesive joint influence on sensors indications were shown. Besides, the stress-strain state analysis that includes the singular solutions for the points, where infinite stress may occur, was carried out in the framework of this study. Mathematical simulation of the strain interaction between the sensor elements and the sensor element itself with investigated surface makes it possible to evaluate effectively the dependence of the calibration factor. The analysis of numerical experiments results was done with a mathematical model that described the three-dimensional spatial distribution of the strain of all FOS elements on a substrate. The sensitivity of the calibration coefficient to the type of adhesive joint of the fiber and the presence of technological holes in the metal substrate was established, as well as to the method of mounting the substrate to the investigated surface. Acknowledgements The research was performed at Perm National Research Polytechnic University with support of the Russian Science Foundation (project №15-19-00243). References [1] Committee on New Materials for Advanced Civil Aircraft, National Materials Advisory Board, Aeronautics and Space Engineering Board, Commission on Engineering and Technical Systems, National Research Council, 1996, New materials for next-generation commercial transports, National Academy Press, Washington, D.C, pp. 98. [2] Makhsidov, V.V., Fedotov, M.Yu., Shiyonok, A.M., Zuev, M.A., 2014, For an issue of embedded optical fiber in CFRP and strain measurement with fiber Bragg gratings sensors, Journal on Composite Mechanics and Design 20(4), 568–574. [3] Williams, M.L., 1952, Stress singularities resulting from various boundary conditions in angular corners of plates in extension, Journal of Applied Mechanics 19(4), 526–528. [4] Matveenko, V.P., Nakaryakova, T.O., Sevodina, N.V., Shardakov, I.N., 2008, Stress singularity at the vertex of homogeneous and composite cones for different boundary conditions, J. Appl. Math. Mech. 72(3), 331–337. [5] Adams, R.D., Peppiatt, N.A., 1974, Stress analysis of adhesive-bonded lap joints, J. Strain Anal. Engng. 9(3), 185–196. https://doi.org/ 10.1243/03093247V093185. [6] Crocombe, A.D., Adams, R.D., 1981, Influence of the spew fillet and other parameters on the stress distribution in the single lap joint, The Journal of Adhesion 13, 141–155. http://dx.doi.org/10.1080/00218468108073182. [7] Adams, R.D., Atkins, R.W., Harris, J.A., Kinloch, A.J., 1986, Stress analysis and failure properties of carbon-fibre-reinforced-plastic/steel double-lap joints, The Journal of Adhesion 20, 29–53. http://dx.doi.org/10.1080/00218468608073238. [8] Adams, R.D., Harris, J.A., 1987, The influence of local geometry on the strength of adhesive joints, International Journal of Adhesion and Adhesives 2(1), 69–80. http://dx.doi.org/10.1016/0143-7496(87)90092-3. [9] Dorn, L., Liu, W., 1993, The stress state and failure properties of adhesive-bonded plastic/metal joints, International Journal of Adhesion and Adhesives 13(1), 21–31. http://dx.doi.org/10.1016/0143-7496(93)90005-T. [10] Tsai, M.Y., Morton, J., 1995, The effect of a spew fillet on adhesive stress distributions in laminated composite single-lap joints, Composite Structures 32, 123–131. http://dx.doi.org/10.1016/0263-8223(95)00059-3. [11] Lang, T. P., Mallick, P. K., 1998, Effect of spew geometry on stresses in single lap adhesive joints, International Journal of Adhesion and Adhesives 18(1), 167–177. http://dx.doi.org/10.1016/S0143-7496(97)00056-0. [12] Zhao, X., Adams, R. D., da Silva, L.F.M., 2011, Single lap joints with rounded adherend corners: stress and strain analysis, Journal of Adhesion Science and Technology 25(8), 819–836. http://www.tandfonline.com/doi/abs/10.1163/016942410X520871. [13] Dempsey, J.P., Sinclair, G.B., 1981, On the singular behavior at the vertex of a bi-material wedge, Journal of Elasticity 11(3), 317–327. http://dx.doi.org/10.1007/BF00041942. [14] Matveyenko, V.P., Borzenkov, S.M., 1996, Semianalytical singular element and its application to stress calculation and optimization, Int. J. Numer. Meth. Eng. 39(10), 1659–1680. http://dx.doi.org/10.1002/(SICI)1097-0207(19960530)39:10<1659::AID-NME919>3.0.CO;2-W. [15] Matveenko, P.V., Fedorov, A.Yu., 2011, Optimization of the geometry of compound elastic bodies with aim to improve strength test procedures for adhesive joints, Vychisl. Mekh. Splosh. Sred. 4(4), 63–70. http://dx.doi.org/10.7242/1999-6691/2011.4.4.40.