Fracture Mechanics and Crack Growth

Chapter 1

Introduction

The rupture of a mechanical specimen can be interpreted primarily as an interruption in the continuity of the specimen (this is in fact a particular definition of failure). In this case, the application of continuum mechanics faces a singularity due to the presence of cracks in the specimen.

Fracture mechanics is simply the application of continuum mechanics and the behavior laws of a material to a body whose boundary conditions are introduced in the presence of crack geometry.

Rupture can occur after a large deformation, usually after a plastic instability resulting from the presence of two opposite effects: one reducing the section; and the other consolidation of the material by hardening. It can, however, occur without significant prior deformation under generalized stresses that is often the case in the elastic domain. We are then in the presence of a brittle fracture.

The analysis of stresses and strains near the crack tip is a basis for understanding the behavior of cracks. Although a plastic or damaged zone is present at the tip of the crack, the linear elastic analysis provides us with an accurate enough mapping of reality for materials such as steel. In the case of ductile materials or extreme loads, however, we need to take into account the elastic-plastic behavior laws.

Fracture mechanics assumes the existence of an initial crack in the structure being studied. This introduces geometric discontinuity singularity to the stress and strain fields and deformations at the crack tip.

The phase that explains the behavior of the structure of intact state where the structure contains a macroscopic crack is called the initiation phase of the crack. Priming of a crack is usually in the vicinity of defects in the design of the structure (e.g. geometric discontinuities) due to poor execution or welding, etc. These defects create local high stresses that promote the initiation of cracks without generalized stresses that exceed the yield strength of the material.

When the cracks are initiated, their propagation can be sudden or gradual. This may result in brittle fracture or crack growth by fatigue. When the propagation of these cracks is accompanied by plastic deformations it is the plastic fracture mechanics, if not a mechanical linear elastic fracture, that will be responsible.

Table 1.1 shows the different types of failure mentioned. Indeed, each type of rupture is a set of assumptions, definitions and analysis.

Table 1.1. Types of failure according to the behavior laws

We will mainly study the two types of failure — I and III — in the context of this book.

The basic problem in linear fracture mechanics can be seen as the analysis of a stress field in plane linear elastic cracked media. This is for theoretical reasons (since the elastic plane is the means by which we find analytical solutions), and for technical and practical reasons (there are structures that are cracked, in which generalized constraints are below the elastic limit).

The definition of a failure criterion (or security) is a specific preoccupation, and is of major importance. This definition comes from a collection of reflections the engineer has on the basis of disparate elements, such as the behavior of the structure, industry, socioeconomics, etc. The safety criterion is given in Figure 1.1 and determines how it can be structured.

Figure 1.1. The safety criterion

Three essential elements exist for any judgments on the safety behavior of a structure:

– global and local geometry;

– the boundary conditions (by forces and displacements); and

– the safety criterion (or failure).

It is the comparison between the solution obtained from the first two elements and the safety criterion that is essential. In the triangle created by the geometry, the failure criterion and boundary conditions (see Figure 1.2), the intervention of one or more peaks can resize the structure.

Figure 1.2. The interaction of geometry, boundary conditions and the failure criterion

To study failure, it is essential to analyze the stress, strain and displacement fields in the cracked structure, especially near the tips of existing cracks. This is encompassed in the study of fracture mechanics. This theoretical study should interpret the phenomenological aspects of the rupture, yet these aspects cannot be addressed without experimentally observing the fracture surface, the rate of crack growth, etc. A presentation of the results of continuum mechanics and the behavior laws, however, appears to be necessary to determine the mechanical fields (displacements, strains and stresses) near the tip of a crack (or a singularity). A review of experimental observations, in light of the calculated mechanical fields gives us a better understanding of failure criteria under quasi-static loading and fatigue. Practical applications for the propagation of cracks in welded joints are detailed in Chapter 7 to illustrate the analytical process.

PART I

Stress Field Analysis Close

to the Crack Tip

Chapter 2

Review of Continuum Mechanics and the

Behavior Laws

Suppose a given structure, with known geometry and constituent materials, is subjected to boundary conditions in force (load). When there are sufficient¹ boundary conditions in the displacements (tie), a displacement field is generated in the structure that determines for each point P(x, y, z) belonging to the structure, the position P’ (x′, y′, z′) after loading where u, v, and w are the displacement of P to P’ based on x, y, z, or:

[2.1]

The displacement field generates a stress field in the structure. The stress from a point O of the structure is defined in terms of force acting on an infinitesimal area plane through O.

The orientation of this area can be described by the unit normal vector. The force is also a vector. It is appropriate to describe the stress in the form of the components of these two vectors in a coordinate system that has been defined previously. Each area and force vector has three components in three dimensions, so that it is expected to describe the stress in nine terms. The nine components (terms) of the stress are plotted on a three-dimensional element (dx, dy, dz) in Cartesian coordinates, see Figure 2.1.

Figure 2.1. Representation of the components of stress in Cartesian coordinates

Each component of the stress is defined by two indices. The first indicates the side on which the stress is applied (1 for x, 2 for y and 3 for z). The second index indicates the direction of the force-generating component of the stress.

We can take the example: , which is a normal stress perpendicular to side A1 (perpendicular to x) in the direction of x. The stress state at the point O (the stress tensor) is called σij . σij represents nine components where i and j independently take the values 1, 2, and 3.

When the stress components are in equilibrium, some components must have the same values and σ 12 = σ 21, σ 13 = σ 31 σ 23 = σ 32 (σij = σji ), to avoid all rotation action on the three-dimensional element.

The components of stress in another axial system are shown in Figure 2.2. This uses cylindrical coordinates (z, r and θ), where components σrr, σθθ, σzz are the normal stresses and σrθ, σrz, σθz, σθr, σzθ and σzr are the shear stresses. During equilibrium, σθz = σzθ, σθr = σrθ and σzr = σrz , we use the notation σij , which is similar to the Cartesian axial system, to describe the stress state in the vicinity of the origin of coordinates.

Figure 2.2. Representation of the components of stresses usingcylindrical coordinates

The displacement field also generates a strain field, where the strain is defined as the relative displacement of points belonging to a structure to each other. The strain is closely related to stress by a behavior law and is written in the form of the εij tensor, which consists of nine components:

2.1. Kinematic equations

Suppose that point P belonging to a deformed body, with coordinates P (x, y, z ), is associated with point Q at a distance, ds, from P. Its coordinates are: Q (x + dx, y + dy, z + dz ). Thus, we have: . Then a load is applied on this body. Segment PQ will move to P′Q′ with the following coordinates and .

Thus: (see Figure 2.3).

Figure 2.3. A solid deformation

[2.2]

By definition:

[2.3]

Replacing [2.3] in [2.2], we have:

[2.4]

Considering that the environment (deformable body) that belongs to PQ is continuous, we can write:

[2.5]

Replacing [2.5] in [2.4]:

[2.6]

with:

[2.7]

we notice that the left-hand side of equation [2.6] may be written as:

Naming lengthening of segment PQ, we have:

Equation [2.6] can then be written as:

[2.8]

Neglecting (∆ ℓ )², equation [2.8] gives a physical significance to each term in the strain tensors εij of equation [2.7].

Noting that: , from equations [2.7] it is easy to show that εij = εji , therefore reducing the number of strain components to six.

Equation [2.7] is written with the index notation in the form of:

[2.9]

where k takes the values of 1, 2 and 3 for every pair of values given to i and j . We consider that u 1 ≡ u, u 2 ≡ v, u 3 ≡ w and x 1 ≡x, x 2 ≡ y , x3 ≡ z.

For example:

Equations [2.7] and [2.9] are called kinematic equations.

In the case of small strains (small displacements), by neglecting the second-order derivatives equation [2.9] can be written:

[2.10]

or:

[2.11]

From equations [2.10] or [2.11], in the case of small displacements we can write the displacement field as:

[2.12]

or using index notation:

[2.13]

or for the (i) given, (j) takes the values 1, 2 and 3.

Considering a two-dimensional application, suppose that point P(x,y ) moves after loading to P’(x’,y’ ), where:

with:

Figure 2.4 shows an element as a plane (dxdy) described in a Cartesian coordinate system (element PACB). After loading, this element becomes (P’A’C’B’), with a left warping and a shift in translation.

Figure 2.4. Deformations of a volume element in a two-dimensional medium with Cartesian coordinates

The displacement of point A to A’ occurs via a translation ν , and an increase in ν due to a shift of P’ to A’ over x, or , etc. The rotation of segment P’B’ relative to P’A’is therefore equal to .

Similar to stress fields, strain fields can also be written in symmetrical tensor form, where εij= εji , or:

In the case of polar coordinates, where:

with x = r cos θ and y = r sin θ , Figure 2.5 shows the strain components where ABCD, the volume element (plane) in the polar coordinates becomes A’B’C’D’ after deformation. Note from Figure 2.5 that:

[2.14]

where ur and uθ are the displacement of point A (r, θ ) (which becomes A’ (r’,θ ) after the strain) based on the axes :

The strains εrr and εθθ are the relative addition of sides AB and AD:

[2.15]

Figure 2.5. Deformations of a volume element in a two dimensional medium shown bypolar coordinates

These relations can be deduced from the formulas of the transformation of the strain tensor from Cartesian coordinates to polar coordinates:

[2.16]

ε11, ε22 and ε12 are given by equation [2.11], where u and v are linked to ur and uθ by the following relations:

[2.17]

Thus equations [2.14] and [2.15] can be obtained.

In the case of cylindrical coordinates, where we have:

the kinematic equations are written as follows:

[2.18]

These equations are equivalent to equations [2.11] in Cartesian coordinates, and therefore only analyze small strain cases.

2.2. Equilibrium equations in a volume element

If we consider a volume element (dx dy dz) belonging to a deformable body, there are six facets of this element on which there are nine pairs of stress components. There are nine stress components on three facets formed by three planes — xoy, xoz and yoz — and nine other stress components on the three facets opposite. Figure 2.6 shows three pairs of components in the direction of x.

Figure 2.6. Equilibrium in a volume element

When loading is applied to the structure, it is assumed that it is in equilibrium. In other words, any volume element belonging to this structure (body deformation) is in equilibrium. If we write the balance of forces from Figure 2.6 along x, we obtain:

Considering dx dy dz = dV = volume of the element ≠ 0, we obtain:

Applying the six equilibrium equations, we obtain:

[2.19]

Equations [2.19] are known as the Cartesian equilibrium equations in a volume element. In this case, we have ignored the volume forces acting on the element, and we are in a quasi-static state.

Equations [2.19] are written in index form as follows:

[2.20]

and in vectorial form as follows:

[2.21]

When volume forces are considered, we have:

[2.22]

where:

In the planar case, considering volume forces equations [2.19] are written as follows:

[2.23]

Figure 2.7 shows the stress components on a planar volume element (dr rdθ) of uniform thickness in the polar coordinate case.

By projecting along the normal , by considering equilibrium we obtain:

(dθ) being infinitesimal, we have: cos (dθ) ≅ 1, sin (dθ) ≅ dθ.

Figure 2.7. Equilibrium of a two-dimensional volume element in polar coordinates

Neglecting the infinitesimal third-order terms, we obtain:

[2.24]

In the context of cylindrical coordinates, neglecting volume forces the equilibrium equations of the volume elementare as follows:

[2.25]

2.3. Behavior laws

A behavior law is a relationship between the components of stress and components of strain. This relationship depends on the variables intrinsic to the material. In fact, it was experimentally observed in the tensile test specimens of a simple one-dimensional ε11 that the strain varies with the stress, σ11.

The shape of the (σ11 ~ε11) curve obtained is closely related to the quality of the material in the specimen. Hooke observed that with a simple loading generating a small value of σ11, the strain ε11 is linearly related to σ11 with:

where E is Young’s modulus (which is intrinsic to the material). Hooke also noted that when the load generating σ11 is removed, ε11 becomes zero, resulting in a behavior that is termed reversible. Note that when you exceed a threshold σ11, known as σy, the relation between σ11 and ε11 becomes nonlinear, and when the load is removed a permanent strain εp remains, which is called the plastic residual strain (see Figure 2.8).

Figure 2.8. Schematic of a one-dimensional behavior law

When reloading, the elastic range is exceeded with a higher value of σ11 placed on the loading monotonic curve. In other words, the value of σy varies during cyclic loading in a phenomenon known as hardening. During hardening, a material has an increase in yield strength and its plastic range is restricted.

Figure 2.9. Different types of one-dimensional behavior

Several types of behavior may occur at point M of the behavior law, σ11 ~ε11 (see Figure 2.9):

– unloading: where we are back in the elastic region;

– continued monotonic loading, with the continuity of the work hardening phenomenon;

– maintenance of the stress level and where the evolution of ε11 is observed as a function of time. This is the creep phenomenon that is often observed in the thermomechanical beyond 400°C in the case of steel; and

– maintenance of the strain level where the evolution of σ11 is observed as a function of time. This refers to the relaxation phenomenon:

[2.26]

Figure 2.10. Modeling of a one-dimensional behavior law

2.3.1. Modeling the linear elastic constitutive law

The linear elastic behavior law linearly connects the stress field to the strain field. It is written as follows for a one-dimensional element:

[2.27]

Factor E is the elasticity modulus (Young’s modulus) and factor (1/E ) is known as the elastic compliance modulus. The value of E is in the region of 20,000 MPa/mm² for most steels:

= initial section of the element

Figure 2.11. Schematic of Poisson’s ratio

Based on this one-dimensional case, the Poisson ratio, ν, is defined as being the ratio between lateral contraction, δt, and longitudinal extension δl :

[2.28] (see Figure 2.11):

The value of ν is equal to 0.28 to 0.33 for most metals.

A problem with modeling the elastic behavior law occurs in the real three-dimensional case where we have six independent stress components and six independent strain components, and a linear relationship between εij and σij . In this case, 36 constants are essential:

[2.29]

After considering the linearity assumption, two other assumptions can be made. The medium is homogeneous and isotropic; implying that the axes of the stress and strain components are similar in the three axes, x, y and z, considered (homogeneity).

Equations [2.29], considering the two previous assumptions, become:

[2.30]

where μ and λ are known as the Lamé coefficients:

These equations can then be written as follows:

[2.31]

where:

– δij =1, when i = j;

– δij = 0 when i≠ j; and

– δij is known as Kroneker coefficient.

By inverting equation [2.31], we obtain:

[2.32]

with:

σkk = σ 11 + σ 22 + σ 33

[2.33]

or vice versa:

NOTE 2.1.– Incompressibility: when the medium is considered incompressible (dilatation = 0), the diagonal of the strain tensor ε 11+ε 22 + ε 33 = 0. By using equation [2.32], we obtain , where representing the maximum value of ν obtained in incompressible media.

2.3.2. Definitions

Principal stresses and strains

In three-dimensions, the stress is applied to the faces of an orthonormal system, with an arbitrary orientation passing through point O (see Figure 2.1). This stress is expressed with six components, σij . It is possible to determine three specific orientations X, Y and Z perpendicular to six faces of a three-dimensional element on which no shear stress is present. The three faces perpendicular to X, Y and Z are known as the principal faces. The three orthonormal vectors and are known as the principal directions. Finally, the three normal stresses σI , σII and σIII on the principal faces are known as principal stresses. Conventionally, σI > σII > σIII . σI is the highest stress component in the structure. The values σI, σII and σIII are obtained by diagonalizing the stress tensor in order to obtain a stress tensor having only values σI , σII and σIII on the diagonal (not shear). Thus σI , σII and σIII willbe the eigenvalues of the tensor {σij} and and will be the eigenvectors.

2.3.2.1. Calculation of eigenvalues

[2.34]

leading to:

[2.35]

with:

[2.36]

the quantities I 1, I 2 and I 3 stay unchanged irrespective of the choice of axes. Thus, these quantities are known as the invariants of stress tensors [σij ]. Equation [2.35] gives the three eigenvalues λ 1 = σI , λ 2 = σII and λ 3 = σIII.

2.3.2.2. Calculation of eigenvectors

If we consider ℓ, m and n to be the direction cosines of each eigenvector, for each value of λ (that is σI , σII and σIII ) we have the following equations:

[2.37]

with:

[2.38]

The maximum shear stresses act on facets that are equal to 45° with the principal facets. The value of the maximum shear stress is given by:

[2.39]

Analogically with the stresses, it is possible to determine the axial system defining the faces in which there is no shear strain. For isotropic solids, we can show that the principal axes X, Y and Z of the principal stresses are identical to the principal strains.

2.3.2.3. Two-dimensional applications

In a two-dimensional stress medium σi3 = 0, we consider a volume element in the axis system (xy). By rotation with θ, we suppose that we reach the principal faces AB and AC in the X, Y axis system. Figure 2.12 shows the principal stress components σI and σII applied on these faces and σ 11 and σ 12 applied on the BC face of the volume element before rotation. Writing the equilibrium of forces on ABC, we have the two following equations:

and:

Figure 2.12. Representation of the principal stresses in a plane medium

By dividing the two equations by (dy), and knowing that:

we obtain:

[2.40]

where:

[2.41]

By analogy:

From equations [2.41], we determine σI , σII and θ as functions of σ 11, σ 22 and σ 12:

[2.42]

Equations [2.42] can be represented in a graphical form that is commonly known as Mohr’s circle (see Figure 2.13).

Point M represents the stress state (σ 11, σ 22 and σ 12). M M’’ and M’ M’’ are the facets where the principal stresses σI and σII act with the values on the normal stress axes. Equations [2.41] and [2.42] can also be obtained from equations [2.35] and [2.37] in a two-dimension medium.

Figure 2.13. Representation of a stress state in Mohr’s circle

2.3.2.4. Equations of compatibility

One of the principles of continuum mechanics is that the strains must be continuous; this is known as the compatibility condition. The equation of the compatibility condition can be more clearly illustrated in a two-dimensional medium. In the case of small deformations, the equations of kinematics are expressed, from equation [2.10], by differentiating ε 11 twice with respect to y , ε 22, twice with respect to x , and ε 12 with respect to x and y :

[2.43]

This equation is known as the compatibility equation for a two-dimensional medium.

If we consider the kinematic equations expressed for a two-dimensional medium, and the polar coordinates given in equation [2.18], the compatibility equation is written as:

[2.44]

In the context of small strains, equations [2.43] and [2.44] are viable irrespective of the behavior law.

The continuity of the medium in question is provided by the satisfaction of this equation. Equation [2.43] can be expressed using the law of material behavior. In the case of a linear elastic behavior law (equation [2.32]), we obtain:

[2.45]

In a three-dimensional environment, the condition of compatibility is written in the form of six equations developed from the equations of kinematics in the case of small strains:

[2.46]

2.3.2.5. Boundary conditions

If we consider a solid in stable equilibrium, the boundary conditions applied to the solid are of two natures:

– boundary conditions for displacements applied on the surface Su of the structure (solid) where the displacements are given and the forces (reactions) are unknown; and

– boundary conditions for forces applied on surface SF of the solid where the forces are given and the displacements are unknown.

The combination of these displacements and forces (Su and SF) represent the entire surface of the structure.

The loads applied to a structure are always on its surface. We can, however, list some forces that are applied in the volume, such as: inertial forces, thermal forces and volume forces.

Displacements applied to a structure may represent a recess (where rotations and displacements are completely blocked), a joint or movement imposed by actuators, spring, etc. In any case, we consider that any force applied to the solid three-dimensional component is:

All surface displacement under force also has three components u, v and w.

Figure 2.14. Boundary conditions under loading

Let us consider a solid, D, in a three-dimensional environment composed of infinite volume elements. On the surface of the structure, the load is applied to an oblique surface where the normal is .

By taking force and the internal forces generated by the stress components σij at equilibrium, we have:

[2.47]

where A is the surface abc, and A.n 1, A.n 2 and A.n 3 are the projections of this surface on planes yoz, xoz and xoy, respectively.

We name: the force vector spread over the abc surface boundary. We therefore obtain:

[2.48]

This equation represents the boundary conditions of force applied to the surface SF of the structure.

NOTE 2.2.– In order to apply the boundary conditions, we use the Saint-Venant assumption, thus eliminating the local effects.

2.3.2.6. The position of the problem of computing a structure

The data for the calculation of a structure are of three types:

– the geometry of the structure: this refers to the overall geometry and local geometry of this structure;

– the intrinsic mechanical characteristics of the material(s) constituting the structure; and

– the boundary conditions, see Table 2.1, which are:

– the load applied to the structure on the SF boundary (force boundary conditions), and

– the fasteners in the structure applied on the Su boundary (displacement boundary conditions).

The total surface or boundary of the structure S = Su ∪ SF.

From these data, before finding a solution, a definition of the solution must be explained. A solution to a structural analysis is defined as the knowledge of the stress tensor σij , strain tensor εij , and displacement tensor u, v, w at any point in the structure. To achieve this, we go through a black box, known as analysis. This black box comprises three systems of equations allowing the attainment of a solution from the data.

The three systems of equations are:

– the equilibrium equations in a volume element (see equations [2.19] to [2.25]);

– the kinematic equations (see equations [2.11] or [2.18]); and

– the behavior law (see equations [2.31] or [2.32] on elasticity and section 2.3.3 on elastoplasticity).

The first two systems use continuum mechanics, in which the nature of the material is not mentioned. The third system takes into account the nature of the material.

In the three-dimensional case, the three systems of equations represent 15 equations, namely:

– three equilibrium equations in a volume element,

– six kinematic equations; and

– six equations for the behavior law.

These contain 15 unknowns (three displacements u, v and w; six εij strain components; and six σij stress components) for each volume element. This is therefore a well-posed problem. Solving these equations is quite difficult, however, because we have partial differential equations. Indeed, the integration of these equations has led to integration constants and their determination is based on the boundary conditions of forces and displacements. Similarly, to ensure the continuity of displacement and strain fields, we should check the compatibility equations.

In a two-dimensional case, the three systems’ equations are reduced to eight equations with eight unknown functions of (x, y) which are: (u, v, ε 11, ε 22, ε 12, σ 11, σ 22 and σ 12). The two-dimensional problems may be solved analytically in some cases. In the case of a one-dimensional element, the three systems’ equations are reduced to three equations.

The solution of the problem is thus obvious if the boundary conditions are known.

2.3.2.7. Mechanical properties of displacement and stress fields

There are a number of mechanical properties to be considered:

– The admissible kinematic (CA) displacement. For a displacement