# Inverse Problem of Elasticity for a Plate Weakened by Hole and Cracks

Authors: Vagif M. Mirsalimov
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

An isotropic elastic plate weakened by a hole and arbitrary system of cracks is considered. Using the principle of equal strength and minimization of stress intensity factors a theoretical analysis is made of the equistrong hole shape. A criterion and a method for solving the inverse problem of fracture prevention for the plate with a hole under the action of given system of external loads are proposed. A closed system of algebraic equations allowing minimization of the stress state and stress intensity factors depending on geometrical and mechanical characteristics of the plate is obtained. The obtained optimal hole shape provides an increase in the load-bearing capacity of the plate.


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## Body

## 1. Introduction

Stress concentration is one main factor determining strength and durability of constructions. Stress concentration decrease allows obtaining more reliable, lighter, more convenient exploitation and also more economical constructions. Therefore, creation of such machines and structures with the least stress concentration is one of the important problems of engineering. For many years, a great number of responsible scientists and engineers focused on stress concentration research in different structural elements and finding ways to reduce it. A large number of responsible components of modern constructions and machines are characterized by the availability of various holes, grooves, etc. Such typical structural elements are widely used in very different fields of mechanical engineering: in cosmic, rocket, and nuclear engineering, aircrafts industry, shipbuilding, automotive industry, and so on. If, moreover, we consider that machine parts must have high strength with minimum weight, then the modern tendency to clarify the calculation of the optimal design of these structural elements for strength becomes clear. The use of more accurate stress concentration methods allows creating more convenient operation and more cost effective (equal strength) constructions as material saving, cost reduction, weight reduction, and reliability increase are achieved in this way.Plates find numerous applications for mechanical engineering and constructions. Reliability in operation, economy, and also other mechanical properties of constructions largely depends on the shape of the plates included in these constructions. Therefore, the problem of finding, in some sense, the optimal shape of a plate has an undoubted application value.The problems of finding optimal shapes of elastic plates, in general case, are reduced to solving variational problems with unknown boundaries. In some cases, in particular, in the case of finding the shape of an elastic plate with minimal stress concentration, we arrive at the elasticity theory problem with an unknown boundary (inverse problem of elasticity theory).The elasticity and elasticity-plasticity theory problems with unknown boundaries with different statements were studied in the works [1–25]. A review of the works on finding equistrong holes in defect-free constructions was given by G.P. Cherepanov [26].At the design stage of constructions, it is necessary to take into account the cases when individual elements (plates) have cracks [27]. Let us consider an elastic isotropic plate with a hole and arbitrarily placed rectilinear cracks. To increase the load-bearing capacity of the plate it is necessary to choose the optimal hole shape. The service life of the plate is determined by the distribution of stresses on the contour of the hole. Therefore, optimal design of a plate with a hole and cracks is important.The problem of optimal design theory lies in determining the characteristics of the plate in such a way that, under the action of the given loads, the plate was the best of all the other plates of the class studied. When designing the plates, the strength may be reasonably controlled by design and technological methods, especially by the geometry of the hole shape. However, until now there are no solutions of the problems of the mechanics of materials on the construction of such a geometry of the hole surface so that the stress field created by it would prevent the fracture of the plate with a hole.Thus, minimization of maximum circumferential stressσθ on the hole contour is an urgent problem and is of great practical and theoretical interest. Achievement of this goal will help to increase the service ability of plates. Changing the shape of the hole, one can reduce the stress state level in the plate.The goal of the study is to develop a mathematical model for a plate with a hole and cracks enabling the calculation of the optimal shape of the hole for given loading regimes.


## 2. Problem Statement

Let an unbounded elastic isotropic plate under the conditions of plane stress state be weakened by rectilinear cracks arbitrarily placed near the hole. Increasing the load-bearing capacity of the plate is achieved by choosing the optimal shape of the hole. The function of the geometry of the hole shape is not known in advance and is to be determined. There are no restrictions on the mutual location and relative size of the hole and cracks. It is assumed that the rectilinear cracks do not intersect each other.The coefficients of expansion of the functionρ(θ) of the hole shape geometry are taken as control parameters (design parameters). As a mathematical model of the problem of reducing the stress state of an elastic plate, we accept a differential equation of plane theory of elasticity.Combine the origin of the system of coordinates with geometrical centre of the hole (Figure1). At infinity the plate is subjected to homogeneous stresses σx∞, σy∞, and τxy∞. The faces of rectilinear cracks are free from external loads. The origins of local system of coordinates xkOkyk whose axes xk coincide with the crack lines and form angles αk with the axis x are located at the centre of thek-th crack.Figure 1
Design scheme of the inverse problem.The boundary conditions in the problem under consideration have the form(1)σn-iτnt=0onr=ρθ,(2)σyk=0,τxkyk=0onthecrackfaces.Here σn, σt, τnt are stress tensor components in the plate; n and t are natural coordinates; i=-1.We need to determine the shape of the hole (functionρ(θ)) in the plate so that the stress field created by it prevents the crack growth.To find the functionρ(θ) of the shape of the hole in the plate, it is necessary to supplement the problem statement with a criterion for choosing the hole shape. Obviously, the lower the stress intensity in the plate, the higher its operation life. According to Irwin-Orowan theory of quasi-brittle fracture, a parameter characterizing the stressed state in the vicinity of the crack is the stress intensity factor. Thus, the value of stress intensity factor in the vicinity of the crack vertices may be regarded responsible for fracture of plate’s material.As a criterion for determining the optimal hole shape (the functionρ(θ)) we accept the condition of equal strength on the contour of the hole with additional conditions that the stress intensity factor equals zero at the cracks tips. Thus, it is required to determine of the hole shape ρ(θ), so that the stress field created in the loading process of the plate would ensure the fulfillment of the condition of equal strength on the contour of the hole and prevent the cracks growth.The adopted additional optimization conditions is written in the form(3)σθ=σ∗onr=ρθ,(4)KIlk=0,KIIlk=0,KI-lk=0,KII-lk=0.Here σθ is the circumferential stress; σ∗ is the optimal value of normal tangential stress on the contour of the hole; KI, KII are the stress intensity factors for modes I and II, respectively.These additional conditions allow determination of the sought-for function of the shape of the hole.Earlier in the papers of N.V. Banichuk [9, 12] it was strictly proven that the accepted equal strength condition (3) along the hole’s contour provides minimum distribution of circumferential normal stress σθ compared with maximum stress on any other contours of holes. In other words, an equistrong contour has no stress concentrations.


## 3. The Case of a Single Crack

Let an elastic isotropic plate with a hole have one rectilinear crack of length2l1 (Figure 2).Figure 2
Design scheme of the inverse problem for the case of a single crack.We will look for the unknown in advance contourL of the hole in the plate in the class of contours close to circular one. The unknown contourL is represented in the form(5)ρθ=R+εHθ,wherein the function H(θ) is to be determined in the process of solving the optimization problem. Here ε=R0/R is a small parameter; R0 is the greatest height of irregularity of the profile of the contour L of the hole from the circumference r=R.Without loss of generality of the stated optimization problem, we assume that the sought-for functionH(θ) may be represented in the form of Fourier series(6)Hθ=∑k=0∞ak∗cos⁡kθ+bk∗sin⁡kθ.The desired functions (stresses, displacements, and stress intensity factors) are sought in the form of expansions in a small parameter ofε:(7)σn=σn0+εσn1+⋯;τnt=τnt0+ετnt1+⋯;σt=σt0+εσt1+⋯;u=u0+εu1+⋯;v=v0+εv1+⋯;KI=KI0+εKI1+⋯;KII=KII0+εKII1+⋯,wherein, for simplification, the terms containing ε higher than the first degree are ignored.The ability to truncate the series in (7) to the first order requires a condition on the parameter ε: ε≤1/3.Each of approximations satisfies the system of differential equations of the plane problem of elasticity theory.Expanding in series the expressions for stresses in the vicinity ofr=ρ(θ), we obtain the values of stress tensor components r=R. Using the known formulas [28] for stress components σn and τnt the boundary conditions of the problem on the contour r=R at the crack faces will take the following form:for zero approximation(8)σr0=0,τrθ0=0,σy10=0,τx1y10=0onthecrackfaces;for the first approximation(9)σr1=N,τrθ1=T,σy11=0,τx1y11=0onthecrackfaces.Here(10)N=2τrθ0RdHθdθ-Hθ∂σr0∂rforr=R,T=σθ0-σr01RdHθdθ-Hθ∂τrθ0∂r.We can represent the stress tensor componentsσr,σθ,τrθ and displacement vectors u,v in the plane problem of elasticity theory [28] by means of Kolosov-Muskhelishvili complex potentials(11)σx+σy=σr+σθ=2Φz+Φz¯,σy-σx+iτxy=σθ-σr+2iτrθe-2iθ=2z-Φ′z+Ψz,2μu+iv=κφz-zΦz¯-ψz¯,where μ is the shear module of the plate material; κ=(3-ν)/(1+ν) is the Muskhelishvili constant dependent on the Poisson ratio ν; r, θ are a polar system of coordinates.By means of formula (11), the boundary conditions of the problem in zero approximation are written as follows:(12)Φ0z+Φ0z¯-e2iθz-Φ0′z+Ψ0z=0forz=Reiθ,Φ0x1+Φ0x1¯+x1Φ0′x1¯+Ψ0x1¯=0onthecrackfaces,where x1 is the affix of a rectilinear crack in zero approximation; Φ0(z) and Ψ0(z) are Kolosov-Muskhelishvili complex potentials in zero approximation.The solution of boundary value problem (12) for a plane with circular hole and arbitrarily placed crack is known [29].In this case, the complex potentialsΦ0(z) and Ψ0(z) are determined by the formulas(13)Φ0z=12π∫-l1l1g10tdtt-z1+12π∫-l1l1-1z-T-1-zT1eiα1g10t+1-T1T-1T-11-zT-12e-iα1g10t¯dt+∑k=0∞ak0z-k,Ψ0z=12πe-2iα1∫-l1l1g10t¯t-z1-T-1eiα1t-z12g10tdt+12πz∫-l1l11T1z-T-1z1-T-1z+T-121-T-1z2eiα1g10t+1zT-1+21-e-iα1z2T-12-z+T1z1-zT-12-2T-1ze-iα1-T11-T-1z3e-iα1g10t¯dt+∑k=0∞bk0z-k,where T1=teiα1+z10; t is the affix of the points of the crack contour; z1=e-iα1z-z10; z10=x10+iy10 is the coordinates of the crack centre; g1(x1) is the sought-for function characterizing the opening of crack faces(14)g10x1=2μi1+κ∂∂x1u10+-u10-+iv10+-v10-,a10=14σx∞+σy∞,b00=12σy∞-σx∞+iτxy∞,a10=0,b10=0,ak0=0k≥3,bk0=0k≥5,a20=b-00R2,b20=2a00R2,b30=0,b40=3R2a20.Here v10+-v10- and (u10+-u10-) are normal and tangential components of the opening of crack faces in zero approximation.Satisfying the boundary condition (12) on the crack faces by functions (13), we arrive at a singular integral equation with respect to the function g1(x1):(15)∫-l1l1R11t,x1g10t+S11t,x1g10t¯dt=πF0x1x1≤l1,where F0(x1)=-Φ10(x1)+Φ10(x1)¯+x1Φ10′(x1)¯+Ψ10(x1)¯; Φ10(x1)=∑k=0∞ak0x1-k; Ψ10(x1)=∑k=0∞bk0x1-k; x1, t, z10, and l1 are dimensionless variables referred to R; R11, S11 are determined by the known formulas [29].To the singular integral equation (15) it is necessary to add an additional equality(16)∫-l1l1g10tdt=0providing uniqueness of displacements when tracing the contours of the inner crack.Under additional conditions (16), the singular integral equation (15) is reduced [29–31] to the system of M algebraic equations with respect to approximate values g10tm(m=1,2,…,M) of the sought-for function at the nodal points(17)1M∑m=1Ml1g10tmR11l1tm,l1xr+g10tm¯S11l1tm,l1xr=F0xr,∑m=1Mg10tm=0,where tm=cos⁡2m-1/2Mπ(m=1,2,…,M); xr=cos⁡πr/M(r=1,2,…,M-1).For stress intensity factors in zero approximation we have(18)KI0l1-iKII0l1=πl1∑m=1M-1mg10tmcot⁡2m-14Mπ,KI0-l1-iKII0-l1=πl1∑m=1M-1m+Mg10tmtan⁡2m-14Mπ,where KI0l1,KII0l1 are the stress intensity factors in the vicinity of the crack tip for x1=l1 and KI0-l1,KII0-l1 for x1=-l1, respectively.The stress components in the plate in zero approximation are found from formulas (11) and (13). Knowing the stressed state in zero approximation, we formally find the functions N and T by formulas (10).After finding the solutions in zero approximation, we arrive at the solution of the problem in the first approximation. The boundary conditions for the first approximation are written in the form(19)Φ1z+Φ1z¯-e2iθz-Φ1′z+Ψ1z=N-iTforz=Reiθ,Φ1x1+Φ1x1¯+x1Φ1′x1¯+Ψ1x1¯=0onthecrackfaces.The solution of boundary value problem (19) analogous to zero approximation is sought in the form(20)Φ1z=Φ01z+Φ11z;Ψ1z=Ψ01z+Ψ11z,where the complex potentials Φ01z and Ψ01z are determined by the formula similar to (13)(21)Φ01z=12π∫-l1l1g11tdtt-z1+12π∫-l1l1-1z-T̄1-zT1eiα1g11t+1-T1T̄1T̄11-zT̄12e-iα1g11t¯dt,Ψ01z=12πe-2iα1∫-l1l1g11t¯t-z1-T̄1eiα1t-z12g11tdt+12πz∫-l1l11T1z-T̄1z1-T̄1z+T̄121-T̄1z2eiα1g11t+1zT̄1+21-e-iα1z2T̄12-z+T1z1-zT̄12-2T̄1ze-iα1-T11-T̄1z3e-iα1g11t¯dt;Φ11z=∑k=0∞akz-k,Ψ11z=∑k=0∞bkz-k.The coefficientsak and bk are found by the formulas(22)a0=0,b0=0,a1=A-1R1+κ,b1=-κA1R1+κ,an=A-nRnn=2,b2=-A0R2,bn=n-1R2an-2-RnA-n+2n≥3,N-iT=∑k=-∞∞Akeikθ.Satisfying the boundary conditions on the crack faces by functions (20)-(21), after some transformations we get a singular integral equation with respect to the unknown function g11x1:(23)∫-l1l1R11t,xg11t+S11t,xg11t¯dt=πF1xx≤l1,where F1(x)=-Φ11(x)+Φ11(x)¯+xΦ11′(x)¯+Ψ11(x)¯; x, t, z10, and l1 are dimensionless variables referred to R.To the integral equation (23) it is necessary to add the additional condition(24)∫-l1l1g11tdt=0providing uniqueness of displacements when tracing the crack’s contour in the first approximation.Under condition (24), the singular integral equation (23) similar to the zero approximation is reduced [29–31] to the system of M linear algebraic equations with respect to approximate values g11tm(m=1,2,…,M) of the sought-for function at nodal points(25)1M∑m=1Ml1g11tmR11l1tm,l1xr+g11tm¯S11l1tm,l1xr=F1xr,∑m=1Mg11tm=0,where tm=cos⁡2m-1/2Mπ(m=1,2,…,M); xr=cos⁡πr/M(r=1,2,…,M-1).For stress intensity factors in the first approximation we have(26)KI1l1-iKII1l1=πl1∑m=1M-1mg11tmcot⁡2m-14Mπ,KI1-l1-iKII1-l1=πl1∑m=1M-1m+Mg11tmtan⁡2m-14Mπ.For the given functionH(θ) of the shape of the hole, the obtained linear algebraic systems are closed and allow studying the stress-strain state of the plate with a hole and finding the stress intensity factors in the vicinity of the crack tips.To construct the missing equations that allow determination of the sought-for coefficients (design parameters)ak∗ and bk∗ it is necessary to find the circumferential stress σθ on the hole’s contour. The stresses σθ on the hole’s contour are determined by means of formula (9).Using the solution obtained, we find theσθ on the surface contour L(r=ρ(θ)) to within first order quantities with respect to the small parameter ε(27)σθ=σθ0θ∣r=R+εHθ∂σθ0∂r+σθ1θ∣r=RNow we pass to the problem of optimal design.To construct missing equations we require additional conditions (3) to be fulfilled. We verify minimization of stresses on the contour of the hole using the equal strength condition and the principle of least squares. We achieve stress reduction using the criterion(28)∑i=1nσθθi-σ∗2⟶min.The optimization problem is to find such valuesσθ(θi) of unknown parameters ak∗, bk∗ (control parameters) that will ensure the values of the function of normal tangential stress according to condition (2) subject to additional conditions (3) in the best way. The principle of least squares affirms that the best probable values of the sought-for parameters ak∗, bk∗, and σ∗ will be the ones under which the sum of squares of deviations εi=σθθi-σ∗(i=1,2,…,n) will be least, i.e.,(29)U=∑i=1nσθθi-σ∗2⟶min.The functionU and the stress intensity factors at the crack tips depend on the sought-for coefficients ak∗, bk∗. The stated problem of optimal design is reduced to the conditional extremum problem of the function Uak∗,bk∗,σ∗,ε when the coefficients ak∗, bk∗ are connected with additional complex conditions(30)∑m=1M-1mg10tm+εg11tmcot2m-14Mπ=0,∑m=1M-1m+Mg10tm+εg11tmtan⁡2m-14Mπ=0.It is required to find the minimal value of the functionUak∗,bk∗,σ∗,ε, its 2k+2 arguments are not independent, and they are subjected to four real additional conditions (30) (2k+2>4).To solve the problem, on the conditional extremum problem we use the Lagrange undefined multiplier method. Introduce four undetermined multipliersλ1,λ2,λ3,λ4 and consider the auxiliary function(31)U0=U+λ1KIl1+λ2KI-l1+λ3KIIl1+λ4KII-l1.For the functionU0ak∗,bk∗,σ∗,λ1,λ2,λ3,λ4,ε we write 2k+2 necessary extremum conditions(32)∂U0∂ak∗=0,∂U0∂bk∗=0,k=1,2,…,m,∂U0∂σ∗=0,∂U0∂ε=0.The obtained (2m + 2) equations, together with four additional equations (30), compose the system of (2m + 2 + 4) equations with (2m + 6) unknowns ak∗,bk∗,σ∗,λ1,λ2,λ3,λ4,ε. Adding these (2m + 6) equations to earlier received algebraic systems (17), (22), (25), and (32) we have a closed algebraic systems for finding all unknowns including the coefficients ak∗,bk∗(k=1,2,…,m) and the values σ∗,ε. We divide the segment [0,2π] of change of variable θ into n equal parts Δθ=2π/n.As the functionσθθi,ak∗,bk∗ and the stress intensity factors at the crack tips are linear with respect to unknown parameters ak∗,bk∗, then the construction and the solution of system (32) are greatly simplified. We can write the system of (32) in the form of a normal system.If the crack with one end arrives at the hole surface, then equality (16) and (24) in each approximation is replaced by an additional condition expressing the finiteness of stresses at the edge crack on the hole’s contour.


## 4. Case of Arbitrary Number of Cracks

Let near the hole there beN1 rectilinear cracks of length 2lk(k=1,2,…,N1) (Figure 1). The solution of the problem of minimization of the stress state of the plate due to the choice of the shape of the hole, in this case, is similar to the solution of the problem for the case of a single crack.Using the perturbations method, the boundary conditions of the problem accept the form: in zero approximation(33)σr0-iτrθ0=0onthecontouroftheholer=R,σyk0-iτxkyk0=0k=1,2,…,N1onthecracksfaces;in the first approximation(34)σr1-iτrθ1=N-iTonthecontouroftheholer=R,σyk1-iτxkyk1=0k=1,2,…,N1onthecracksfaces.Boundary conditions (33) of the problem, in zero approximation by means of formula (11), are written in the form(35)Φ0z+Φ0z¯-e2iθz-Φ0′z+Ψ0z=0forz=Reiθ,Φ0xk+Φ0xk¯+xkΦ0′xk¯+Ψ0xk¯=0onthecracksfaces,where xk is the affix of the points of the k-th (k=1,2,…,N1) crack; Φ0(z) and Ψ0(z) are Kolosov-Muskhelishvili complex potentials in zero approximation.The solution of boundary value problem (35) for a plane with a circular hole and arbitrarily placed rectilinear cracks is known [29].In this case, the complex potentialsΦ0(z) and Ψ0(z) have the form(36)Φ0z=12π∑k=1N1∫-lklkgk0tdtt-zk+∫-lklk-1z-T-k1-zT-keiαkgk0t+gk0t¯e-iαk1-TkT-kT-k1-zT-k2dt+Φ10z,Ψ0z=12π∑k=1N1e-2iαk∫-lklkgk0t¯t-zk-T-keiαkt-zk2gk0tdt+12πz∑k=1N1∫-lklkgk0teiαk1zTk-2z2-T-kz1-zT-k+T-k21-zT-k2+e-iαkgk0t¯1-TkT-kzT-k1-zT-k2-11-zTk-21-TkT-k1-zT-k3dt+Ψ10z,where Tk=teiαk+zk0; zk=e-iαkz-zk0; gk(xk)(k=1,2,…,N1) are the sought-for functions characterizing the opening of cracks faces(37)gk0xk=2μi1+κ∂∂xkuk0+-uk0-+ivk0+-vk0-.Requiring that function (36) satisfies the boundary condition (35) on the crack faces, a system of N1 singular integral equations with respect to the functions gk0xk is obtained:(38)∑k=1N1∫-lklkRnkt,xngk0t+Snkt,xngk0t¯dt=πFn0xnxn≤lnn=1,2,…,N1,where Fn0xn=-Φ10xn+Φ10xn¯+xnΦ10′xn¯+Ψ10xn¯; xn, t, zn0, and ln are dimensionless quantities referred to R; Rnk,Snk are determined by the known formulas [29].To the system of singular integral equations (38) it is necessary to add the additional equalities(39)∫-lklkgk0tdt=0k=1,2,…,N1that ensure uniqueness of displacements when tracing the contours of inner cracks.Under additional equalities (39), the system of complex singular integral equations is reduced [29–31] to the finite system ofN1×M algebraic equations with respect to approximate values of the sought-for functions gk∗xk at the nodal points(40)1M∑k=1N1∑m=1Mlkgk0tmRnklktm,lnxr+gk0tm¯Snklktm,lnxr=Fn0xr,∑m=1Mgk0tm=0, where tm=cos⁡2m-1/2Mπ(m=1,2,…,M); xr=cos⁡πr/M(r=1,2,…,M-1).Passing to complex conjugate quantities in system (40) one more system ofN1×M linear algebraic equations is obtained. By means of the solution of algebraic systems one can study the stress-strain state of a plate with a hole and cracks and determine the stress intensity factors in the vicinity of all crack tips in zero approximation.For stress intensity factors in zero approximation the following relations are obtained:(41)KI0lk-iKII0lk=πlk∑m=1M-1mgk0tmcot⁡2m-14Mπ,KI0-lk-iKII0-lk=πlk∑m=1M-1m+Mgk0tmtan⁡2m-14Mπk=1,2,…,N1.The stress components on the plane in zero approximation are found from relations (11) and (36). Then, knowing the stress state in zero approximation the solution of the problem in the first approximation is constructed.Forr=R, the functions N and T are found by formulas (10).The boundary conditions of problem (34) for the first approximation are written as follows:(42)Φ1z+Φ1z¯-e2iθz-Φ1′z+Ψ1z=N-iTforz=Reiθ,Φ1xk+Φ1xk¯+xkΦ1′xk¯+Ψ1xk¯=0onthecracksfacesk=1,2,…,N1.The solution of the boundary value problem (34) is looked for similar to zero approximation in the form:(43)Φ1z=Φ01z+Φ11z,Ψ1z=Ψ01z+Ψ11z,where the complex potentials Φ01z and Ψ01z are determined by the relations similar to (36), in which gk0t should be replaced by gk1t, and the analytic functions Φ11z and Ψ11z are sought in the form of power series (21). The coefficients ak and bk are found from formulas of type (22).Satisfying the boundary conditions on the crack faces by analytic functions of type (36), after some transformations we get the system of N1 complex singular integral equations with respect to the unknown functions gk1xk(44)∑k=1N1∫-lklkPnkt,xgk1t+Snkt,xgk1t¯dt=πFn1xx≤lnn=1,2,…,N1,where Fn1(x)=-Φ11(x)+Φ11(x)¯+xΦ11′(x)¯+Ψ11(x)¯.To the system of singular integral equations (44) it is necessary to add the additional conditions(45)∫-lklkgk1tdt=0k=1,2,…,N1ensuring the uniqueness of displacements when tracing the cracks contours in the first approximation.Under conditions (45), by means of algebraization procedure [29–31] the system of complex singular integral equations (44) is reduced to the finite system ofN1×M linear algebraic equations with respect to the approximate values of the unknown functions gk1xk at the nodal points(46)1M∑k=1N1∑m=1Mlkgk1tmRnklktm,lnxr+gk1tm¯Snklktm,lnxr=Fn1xr,∑m=1Mgk1tm=0r=1,2,…,M-1;n=1,2,…,N1.Passing in system (46) to complex conjugate quantities, we get one more system ofN1×M linear algebraic equations.Under the given function H(θ) of the hole’s shape, the obtained linear algebraic equations will be closed. In this case, by solving the algebraic system of equations, one can study the stress-strain state of a plate with a hole and cracks and determine the stress intensity factor in the vicinity of all crack tips.For stress intensity factor in the first approximation the following formulas are obtained:(47)KI1lk-iKII1lk=πlk∑m=1M-1mgk1tmcot⁡2m-14Mπ,KI1-lk-iKII1-lk=πlk∑m=1M-1m+Mgk1tmtan⁡2m-14Mπk=1,2,…,N1.To construct the missing equations it is necessary to find circumferential stressσθ. By means of the found solution the σθ for r=ρ(θ) to within first order quantities with respect to the small parameterε in the form similar to (27).To construct the missing equations it is required to minimize the functionU under the constraints(48)KIlk=0,KIIlk=0,KI-lk=0,KII-lk=0k=1,2,…,N1.Here(49)KIlk=KI0lk+εKI1lk,KIIlk=KII0lk+εKII1lk,KI-lk=KI0-lk+εKI1-lk,KII-lk=KII0-lk+εKII1-lk.If a part of cracksN2 with one end arrives, the hole’s surface, then equalities (39) and (45) in each approximation are replaced by additional conditions expressing finiteness of stresses at the cracks end on the hole’s contour.Changing the values of the parametersαk and zk0(k=1,2,…,N1) one can study different cases of placement of cracks in the material of the plate near the hole and their influence on optimal shape of the hole in the plate.In the case of another contour, the stress concentration will take place along the contour and furthermore, that is more dangerous, the stress intensity factorsKIlk, KIIlk will not be zero. This may lead to crack growth and total failure of the construction. At the same time, the obtained results provide absence of stress concentration along the hole contours and equality to zero of stress intensity factors in all crack tips. Therefore, there is no doubt that the obtained results are optimal. This optimal shape of the hole prevents failure of the construction.


## 5. Numerical Examples

### 5.1. The Case of a Single Crack

The numerical calculation is carried out by Gauss method choosing the principal element. The calculation was carried out for the case when the plate is weakened by a crackα1=45°; l1/R=0.02; z10=1.35Reiπ/8. The results of calculations of coefficients ak∗ and bk∗ of expansion of the function H(θ) of the hole’s shape for a plate with the material of Poisson ratioν = 0.33 and Young’s modulus E=2.04·105 MPa are given in Table 1 (the coefficients are given in mm).Table 1
The values of Fourier coefficients for equistrong shape of the hole in the plate for the case of a single crack.
a 0 ∗ a 1 ∗ a 2 ∗ a 3 ∗ a 4 ∗ a 5 ∗ a 6 ∗ a 7 ∗ a 8 ∗ a 9 ∗ 0.2019 0.1923 -0.1721 0.1105 -0.0907 0.0816 0.0654 0.0569 0.0376 0.0251  b 1 ∗ b 2 ∗ b 3 ∗ b 4 ∗ b 5 ∗ b 6 ∗ b 7 ∗ b 8 ∗ b 9 ∗ 0.1872 -0.1523 0.0932 0.0765 0.0639 -0.0531 0.0358 0.0257 0.0191


### 5.2. The Case of Several Cracks

The results of calculations of the functionH(θ) of the hole’s shape in the plate (the coefficients are given in mm) are given in Table 2. The numerical analysis was carried out for the case when the plate has three rectilinear cracks α1 = 10°, l1/R = 0.075, z10=1.25Reiπ/10; α2 = 30°, l2/R = 0.025, z20=1.30Reiπ/12; α3 = 45°, l3/R = 0.05, and z30=1.40Reiπ/6. The plate material and the external load were the same.Table 2
The values of Fourier coefficients for equistrong shape of the hole in the plate for the case of three cracks.
a 0 ∗ a 1 ∗ a 2 ∗ a 3 ∗ a 4 ∗ a 5 ∗ a 6 ∗ a 7 ∗ a 8 ∗ a 9 ∗ 0.2911 0.2379 -0.2023 0.1865 -0.1541 0.1197 0.0889 0.0623 0.0490 0.0304  b 1 ∗ b 2 ∗ b 3 ∗ b 4 ∗ b 5 ∗ b 6 ∗ b 7 ∗ b 8 ∗ b 9 ∗ 0.2158 -0.1902 0.1605 0.1322 0.0941 -0.0703 0.0578 0.0364 0.0253The optimal solution, i.e., the found design parametersak∗ and bk∗ of the design function H(θ) of the hole’s shape in the plate, contributes to an increase in the load carrying capacity of the plate.Comparison of the function ensuring equal strength on the hole contour with other function contours shows that the stressσθ=σ∗ is maximal compared with maximal value σθ for any other hole contours. Therefore, the plate with hole having found contour q(θ) has the property of the greatest strength.


## 5.1. The Case of a Single Crack

The numerical calculation is carried out by Gauss method choosing the principal element. The calculation was carried out for the case when the plate is weakened by a crackα1=45°; l1/R=0.02; z10=1.35Reiπ/8. The results of calculations of coefficients ak∗ and bk∗ of expansion of the function H(θ) of the hole’s shape for a plate with the material of Poisson ratioν = 0.33 and Young’s modulus E=2.04·105 MPa are given in Table 1 (the coefficients are given in mm).Table 1
The values of Fourier coefficients for equistrong shape of the hole in the plate for the case of a single crack.
a 0 ∗ a 1 ∗ a 2 ∗ a 3 ∗ a 4 ∗ a 5 ∗ a 6 ∗ a 7 ∗ a 8 ∗ a 9 ∗ 0.2019 0.1923 -0.1721 0.1105 -0.0907 0.0816 0.0654 0.0569 0.0376 0.0251  b 1 ∗ b 2 ∗ b 3 ∗ b 4 ∗ b 5 ∗ b 6 ∗ b 7 ∗ b 8 ∗ b 9 ∗ 0.1872 -0.1523 0.0932 0.0765 0.0639 -0.0531 0.0358 0.0257 0.0191


## 5.2. The Case of Several Cracks

The results of calculations of the functionH(θ) of the hole’s shape in the plate (the coefficients are given in mm) are given in Table 2. The numerical analysis was carried out for the case when the plate has three rectilinear cracks α1 = 10°, l1/R = 0.075, z10=1.25Reiπ/10; α2 = 30°, l2/R = 0.025, z20=1.30Reiπ/12; α3 = 45°, l3/R = 0.05, and z30=1.40Reiπ/6. The plate material and the external load were the same.Table 2
The values of Fourier coefficients for equistrong shape of the hole in the plate for the case of three cracks.
a 0 ∗ a 1 ∗ a 2 ∗ a 3 ∗ a 4 ∗ a 5 ∗ a 6 ∗ a 7 ∗ a 8 ∗ a 9 ∗ 0.2911 0.2379 -0.2023 0.1865 -0.1541 0.1197 0.0889 0.0623 0.0490 0.0304  b 1 ∗ b 2 ∗ b 3 ∗ b 4 ∗ b 5 ∗ b 6 ∗ b 7 ∗ b 8 ∗ b 9 ∗ 0.2158 -0.1902 0.1605 0.1322 0.0941 -0.0703 0.0578 0.0364 0.0253The optimal solution, i.e., the found design parametersak∗ and bk∗ of the design function H(θ) of the hole’s shape in the plate, contributes to an increase in the load carrying capacity of the plate.Comparison of the function ensuring equal strength on the hole contour with other function contours shows that the stressσθ=σ∗ is maximal compared with maximal value σθ for any other hole contours. Therefore, the plate with hole having found contour q(θ) has the property of the greatest strength.


## 6. Conclusions

A criterion and a method for solving the problem on prevention fracture of a plate with a hole and rectilinear cracks under the action of the given system of external loads are suggested. A closed system of algebraic equations that allows the solution of the problem of optimal design of a plate weakened by a hole and cracks depending on mechanical and geometrical characteristics of the plate is obtained.The main resolving equations obtained in the article allow under the given shape of the hole, by means of determination of stress intensity factors, predicting the crack growth in the plate, to set up admissible level of deficiency and maximal value of working loads ensuring sufficient reliability resource. The solution of optimal design problem on definition of the shape of the hole at the design stage enables choosing geometric parameters of the plate, ensuring increase in load-bearing ability. The found hole shape ensures a reduction in stress concentration on the hole contour and minimization of stress intensity factors. The results of the present theoretical study open new possibilities for optimal design of a plate with a hole and cracks.



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