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

Mirsalimov, Vagif M.

doi:https://doi.org/10.1155/2019/4931489

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2019 | Article ID 4931489 | https://doi.org/10.1155/2019/4931489

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

Vagif M. Mirsalimov^1,2

Academic Editor: Francesco Tornabene

Received25 Oct 2018

Revised30 Jan 2019

Accepted04 Mar 2019

Published25 Mar 2019

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.

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 stress 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 function 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 (Figure 1). At infinity the plate is subjected to homogeneous stresses , , and . The faces of rectilinear cracks are free from external loads. The origins of local system of coordinates whose axes coincide with the crack lines and form angles with the axis are located at the centre of the k-th crack.

The boundary conditions in the problem under consideration have the formHere , , are stress tensor components in the plate; and are natural coordinates; .

We need to determine the shape of the hole (function ) in the plate so that the stress field created by it prevents the crack growth.

To find the function 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 function ) 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 , 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 formHere is the circumferential stress; is the optimal value of normal tangential stress on the contour of the hole; , 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 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 length (Figure 2).

We will look for the unknown in advance contour L of the hole in the plate in the class of contours close to circular one. The unknown contour L is represented in the formwherein the function is to be determined in the process of solving the optimization problem. Here is a small parameter; is the greatest height of irregularity of the profile of the contour of the hole from the circumference .

Without loss of generality of the stated optimization problem, we assume that the sought-for function may be represented in the form of Fourier series

The desired functions (stresses, displacements, and stress intensity factors) are sought in the form of expansions in a small parameter of :wherein, for simplification, the terms containing 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 : .

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 of , we obtain the values of stress tensor components . Using the known formulas [28] for stress components and the boundary conditions of the problem on the contour at the crack faces will take the following form:

for zero approximation

for the first approximationHere

We can represent the stress tensor components and displacement vectors in the plane problem of elasticity theory [28] by means of Kolosov-Muskhelishvili complex potentialswhere is the shear module of the plate material; is the Muskhelishvili constant dependent on the Poisson ratio ; , are a polar system of coordinates.

By means of formula (11), the boundary conditions of the problem in zero approximation are written as follows:where is the affix of a rectilinear crack in zero approximation; and 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 potentials and are determined by the formulaswhere ; is the affix of the points of the crack contour; ; is the coordinates of the crack centre; is the sought-for function characterizing the opening of crack facesHere and 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 :where ; ; ; , , , and are dimensionless variables referred to ; , are determined by the known formulas [29].

To the singular integral equation (15) it is necessary to add an additional equalityproviding 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 algebraic equations with respect to approximate values of the sought-for function at the nodal pointswhere ; .

For stress intensity factors in zero approximation we havewhere are the stress intensity factors in the vicinity of the crack tip for and for , 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 and 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

The solution of boundary value problem (19) analogous to zero approximation is sought in the formwhere the complex potentials and are determined by the formula similar to (13)

The coefficients and are found by the formulas

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 :where ; , , , and are dimensionless variables referred to .

To the integral equation (23) it is necessary to add the additional conditionproviding 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 linear algebraic equations with respect to approximate values of the sought-for function at nodal pointswhere ; .

For stress intensity factors in the first approximation we have

For the given function 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) and it is necessary to find the circumferential stress on the hole’s contour. The stresses on the hole’s contour are determined by means of formula (9).

Using the solution obtained, we find the on the surface contour to within first order quantities with respect to the small parameter

Now 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

The optimization problem is to find such values of unknown parameters , (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 , , and will be the ones under which the sum of squares of deviations will be least, i.e.,

The function and the stress intensity factors at the crack tips depend on the sought-for coefficients , . The stated problem of optimal design is reduced to the conditional extremum problem of the function when the coefficients , are connected with additional complex conditions

It is required to find the minimal value of the function , its arguments are not independent, and they are subjected to four real additional conditions (30) .

To solve the problem, on the conditional extremum problem we use the Lagrange undefined multiplier method. Introduce four undetermined multipliers and consider the auxiliary function

For the function we write necessary extremum conditions

The obtained (2m + 2) equations, together with four additional equations (30), compose the system of (2m + 2 + 4) equations with (2m + 6) unknowns . 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 and the values . We divide the segment of change of variable into equal parts .

As the function and the stress intensity factors at the crack tips are linear with respect to unknown parameters , 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 be rectilinear cracks of length (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 approximationin the first approximationBoundary conditions (33) of the problem, in zero approximation by means of formula (11), are written in the formwhere is the affix of the points of the -th () crack; and 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 potentials and have the formwhere ; ; are the sought-for functions characterizing the opening of cracks faces

Requiring that function (36) satisfies the boundary condition (35) on the crack faces, a system of singular integral equations with respect to the functions is obtained:where ; , , , and are dimensionless quantities referred to ; are determined by the known formulas [29].

To the system of singular integral equations (38) it is necessary to add the additional equalitiesthat 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 of N₁×M algebraic equations with respect to approximate values of the sought-for functions at the nodal points where ; .

Passing to complex conjugate quantities in system (40) one more system of N₁×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:

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.

For , the functions and are found by formulas (10).

The boundary conditions of problem (34) for the first approximation are written as follows:

The solution of the boundary value problem (34) is looked for similar to zero approximation in the form:where the complex potentials and are determined by the relations similar to (36), in which should be replaced by , and the analytic functions and are sought in the form of power series (21). The coefficients and 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 complex singular integral equations with respect to the unknown functions where .

To the system of singular integral equations (44) it is necessary to add the additional conditionsensuring 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 of N₁×M linear algebraic equations with respect to the approximate values of the unknown functions at the nodal points

Passing in system (46) to complex conjugate quantities, we get one more system of N₁×M linear algebraic equations. Under the given function 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:

To construct the missing equations it is necessary to find circumferential stress . By means of the found solution the for to within first order quantities with respect to the small parameter ε in the form similar to (27).

To construct the missing equations it is required to minimize the function U under the constraintsHere

If a part of cracks N₂ 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 parameters and 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 factors , 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 crack ; ; . The results of calculations of coefficients and of expansion of the function of the hole’s shape for a plate with the material of Poisson ratio ν = 0.33 and Young’s modulus MPa are given in Table 1 (the coefficients are given in mm).

5.2. The Case of Several Cracks

The results of calculations of the function 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 = 10°, = 0.075, ; = 30°, = 0.025, ; = 45°, = 0.05, and . The plate material and the external load were the same.

The optimal solution, i.e., the found design parameters and of the design function 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 stress is maximal compared with maximal value for any other hole contours. Therefore, the plate with hole having found contour 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.

Data Availability

No data were used to support this study.

Disclosure

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2019 Vagif M. Mirsalimov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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