1. INTRODUCTION

If one looks into the history of the development of Fracture Mechanics, the very first breakthrough occurred as early as in 1898! This was when the first problem leading to stress concentration was solved by Kirsch. Kirsch solved the problem of an infinite plate with a circular hole for a tension strip.

Figure: Plate with circular hole (Kirsch: 1898)

Following the solution of a plate with circular hole by Kirsch, it took 15 years by Inglis (1913) to repeat the same problem with elliptical hole. For a tension strip with an elliptical hole, Inglis was able to determine that the near field stress was related to the far field stress by a factor (1+2a / b).

Figure: Elliptical hole in a flat plate (Inglis: 1913)

The problems solved by Kirsch and Inglis have been very well covered in an undergraduate course on Strength of Materials and / or Theory of Plates and Shells. It should be noted that for a plate with an elliptical hole, the moment b -> 0 (b approaches 0 means that the ellipse becomes a crack), the multiplication factor (1+2a / b) becomes infinity, which indicates;

i. the stresses become very high,

ii. the methodology of stress concentration to ascribe a crack is no longer valid.

Thus, one needs more improved mathematical approaches to handle the problem of crack because, in reality, stress can never approach infinity as plasticity will set in before that. The interpretation of Inglis meant that, once there is a crack, then, even for a small load, there will be infinite stress. This of course is not true because structures can have cracks embedded and might still not fracture.

The problem was solved by Griffith. Griffith was able to come out with a theory that energy is needed for crack growth. He formulated that an existing crack will grow provided that the total energy of the system is lowered by its growth. Through Griffith’s theory, one was able to appreciate that an existing crack will grow provided that the total energy of the system is lowered by its growth and eventually when the crack reaches its critical length, fracture occurs [unlike, what was depicted in the solution of Inglis].

Griffith’s work was confined to brittle materials and it was only in 1948, that Irwin extended the work to ductile materials.

In this article, I have attempted to elaborate in considerable detail the above contributions by Kirsch, Inglis, Griffith which form the most significant contributions towards the subject of Fracture Mechanics. This article is organized as follows;

· Section 2 of this article presents a review of Kirsch’s solution starting with a basic case of tension strip with a circular hole followed by more complex cases of biaxial tension and shear using superposition techniques. The solution for stress concentrations in finite width plates is also covered in this section.

· Section 3 of this article presents a review of Inglis’ solution for a tension strip with elliptical hole. Important to note here is the limiting case of Inglis’ solution that predicted infinite stress and formed the motivation for Griffith’s theory concerning energy requirement for crack growth.

· Summarized details of Griffith’s theory have been discussed in section 4 of this article. Concepts involving: energy release, surface energy, critical crack length, critical stress have been explained and discussed here.

1 2. KIRSCH SOLUTION: PLATE WITH A CIRCULAR HOLE

1. 2.1 KIRSCH’S SOLUTION FOR A TENSION STRIP UNDER UNIAXIAL LOADING

Kirsch solution for the case of a plate with circular hole subjected to uniaxial tension is given below. The hole has a radius ‘a’ and the radial co-ordinate is ‘r’ which is meaningless if r<a; θ = 0 aligns with the remote loading condition.

Figure: Infinite plate under uniaxial tension

The state of stress around the hole is given by (derivation not covered);

Stress concentration factor, state of stress around the hole

Stress field at the hole can be computed as;

At r = a, a / r =1, hence;

The radial stress σ_rr and the shear stress ζ_rθ = 0; because it is a free surface;

It is the hoop stress which needs attention;

At θ = 0, σ_θθ = -σ_∞, i.e. the hoop stress is negative.

The hoop stress here is acting in the direction perpendicular to the applied load. Because the stretching is uniaxial in the far-field, the strain and the stress is negative in the perpendicular / transverse direction (due to Poisson’s effect)

Important to note, is the hoop stress at θ = +90 and θ = -90. Here;

σ_θθ = 3σ_∞ (see figure below) and the factor of stress ratio occurs. The ratio is called the stress concentration factor.

It should be noted that the stress components at the hole are independent of the size of the hole itself. This is due to the fact that the plate is infinitely large so that the hole size is inconsequential relative to the size of the plate.

Figure: Plot of the hoop stress at θ = +90 and θ = -90

Stress concentration: Fluid flow analogy

In the figure below, the curved lines represent qualitatively the stress distribution around the hole analogous to fluid flow. The spacing between the lines is minimum around the sides of the hole reflecting “stress concentrations”.

Figure: Stress concentrations - fluid flow analogy

1.1 2.2 SOLUTION FOR PLATE WITH CIRCULAR HOLE UNDER BIAXIAL LOADING

The solution for stresses around the hole for a plate under biaxial loading can be obtained by superimposing the solution for uniaxial loading as shown below;

For the stress state σ₂, the solution is;

For the stress state σ₁, the solution is;

For the biaxial stress state, the resulting solution is the superposition of the solutions of stress state σ₁ and σ₂;

1.1 2.3 SOLUTION FOR PLATE WITH CIRCULAR HOLE UNDER SHEAR LOADING

Since, simple shear is a combination of simple tension along one of the diagonals (of the rectangular plate considered) and simple compression along the other diagonal, the state of stress corresponding to simple shear can be obtained by superimposing the effects of uniaxial tension and uniaxial compression at 45° as shown below;

(Note: The hole is not shown in the above pictures. The idea being to demonstrate the principle herein)

Figure: Simple shear = superposition of uniaxial tension and uniaxial compression at 45°

3. INGLIS SOLUTION: PLATE WITH AN ELLIPTICAL HOLE

Elliptical coordinate system:

In geometry, the elliptical coordinate system is a 2 dimensional orthogonal co-ordinate system in which the coordinate lines are “confocal”. Confocal means lines having the same foci (‘foci’ are special points with reference to which a variety of curves are constructed).

Elliptical coordinates are the generalisation of 2D polar coordinates suited to problems involving ellipses, just as polar co-ordinates are suited to problems involving circles. Elliptical coordinates are defined through the following equations;

The equations relate (x,y) to (c,β) with α being a fixed parameter based on the aspect ratio of the ellipse.

From the above,

which is an ellipse with major axis equal to c cos h(α) and minor axis equal to c sin h(α).

At very small values of α, the ellipse approaches a crack as it flattens out and at larger values of α, the elliptical coordinates become the polar co-ordinates.

The parameter α can be determined from the ratio of the maximum and minimum axis of the ellipse. An ellipse of height 2b and width 2a has α determined by;

State of stress around the hole:

Considering a plate with an elliptical hole subjected to uniaxial tension with the far field stress given by σ_∞ as indicated in the figure below;

Figure: Plate with an elliptical hole under uniaxial tension

The derivation of stresses at the tip of the ellipse is not covered in this article. The maximum stress at the tip of the ellipse is related to the size and shape of the ellipse and is given by;

Thus, Inglis’ solution reduces to ;

4. GRIFFITH’S THEORY

Griffith’s dilemma:

The immediate consequence of Inglis’ solution was that even for a small load, the crack would grow and break the component into pieces as the stress level approached infinity. This was (then) labelled as “Griffith’s dilemma” because a practical observation was that a solid might contain a crack and still remain intact. This could be possible only if some other mechanisms operate which help the solid to sustain solid forms – this was the key to Griffith’s analysis.

Concept of surface energy:

Analogous to the surface tension in a liquid, all solid surfaces are associated with surface energies or free energies. These energies are developed because atoms close to a surface behave differently from atoms in the interior of a solid.

The interior atoms are attracted or repelled by the neighbouring atoms more or less uniformly from all directions. On the contrary, an atom on the (free) surface has no neighbouring atoms on one side of the surface, thus, resulting in a different equilibrium. In fact, the atoms at the free surface have to “re-adjust” themselves to form the equilibrium and this develops strain in the material close to the free surface. Such surface deformation requires energy and this is termed as “surface energy”.

Griffith’s explanation:

Griffith realised that a crack in a body will not grow unless energy was “released” to “overcome” the energy need to form two new surfaces: one below and one above the crack plane.

The surface energy of a material depends upon its material properties, the magnitude being small to the order of 1J/m².

The table below lists the surface energies of some common solids:

Table: Surface energies of some common materials

Griffith’s analysis:

Let us consider a plate with no prior crack. It is pulled and then maintained in tension between 2 rigid supports as shown in the figure.

Now, with a knife a crack is cut at the corner of the plate with the crack plane normal to the tensile stress. A critical stage is reached when the crack starts growing on its own without further need of a knife. The length of the crack at which it starts growing on its “own” is called “critical crack length”.

Figure: (a) an unstretched plate (b) a stretched plate (c) introducing a crack at the centre of the plate using a knife

Prediction of critical crack length:

[Critical crack length is the length beyond which the crack grows on its own]

To understand Griffith’s analysis, let us carry out an approximate analysis. The plate is chosen to have its dimensions much larger than the longest crack that is to be considered. Then, the stress at points far away from the crack is assumed to remain constant.

When the crack has grown into the solid to a depth “a”, a region of the material adjacent to the free surface is unloaded and the strain energy is released. For the sake of convenience, a major area of the plate from where the strain energy is released may be taken a triangle on each side of the crack (note: the assumption of triangular shape is assumed just to keep the algebra simple).

The height of the triangle is λ(2a) where λ is a constant. Then, the total release of energy is determined by multiplying the area of the triangles by the plate thickness “B” and the strain energy density σ² / 2E. Thus, the released energy is given by;

The value of λ for thin plates can be shown to be = π / 2.

Thus,

Energy is required to create 2 new surfaces.

If γ is the surface energy per unit area of the surface, the surface energy required to create 2 (one above and one below the crack plane) new surfaces is;

Energy release (E_R) vs. Energy required (E_s) for creating two new surfaces:

The above relations for the energy released (E_R) and the energy required to create two new surfaces (E_S) are plotted below. These relationships of E_R and E_Sare plotted against an increasing crack length ‘a’ as shown in the figure below.

Figure: Variation of energy released and the required surface energy Es plotted against crack length

It can be seen that E_R increases parabolically with crack length ‘a’ whereas E_S increases linearly. For an increase in crack length ∆a, the increase in E_R is smaller than the corresponding increase in E_S. Thus, the energy releases is not sufficient to create two new surfaces. The crack would not grow on its own and would remain dormant.

If the length of the crack is increased further (in this case by the operator using the “knife”, a stage is reached when the increase in E_R is equal to the energy required to create two new surfaces. The crack length corresponding to this stage is known as the “critical crack length” because beyond this stage the crack continues to grow on its “own” without the operator having to use the knife. Beyond the critical crack length, the system loses it energy on its own, thus allowing the crack to grow. Beyond the critical crack length, the crack growth is spontaneous and catastrophic.

Mathematical expression for critical crack length:

Thus, based on the above discussion, for the crack length to be critical, the rate of increase of ER should be greater than or equal to the energy required to create two new surfaces (ES).

That is;