Boundary Conditions (§ 2.5.0)

The Three Modes of Fracture

Developed by HM Westergaard in 1939, Mode I occurs when the normal stress (σ) causes the crack surface to open. For this reason, Mode I is also called the Opening Mode.

Discovered by Sih in 1966, Mode II occurs when the crack surfaces slide over one another. For this reason, Mode II is also known as the Sliding Mode or the In-Plane Shear Mode.

Mode III occurs when the crack surfaces move parallel to the leading edge of the crack and relative to each other, causing the crack surfaces to tear apart. For this reason, Mode III is also known as the Tearing Mode or the Out-of-Plane Mode.

The last step in the formulation of the stress field problem is determining the appropriate boundary conditions. Mathematically, the boundary conditions prescribe the stresses and displacements on the boundary of the elastic continuum. There are two types of boundary conditions that we will explore in this tutorial: near field and far field boundary conditions.

Near field boundary conditions are the stresses near the crack tip. If the length of the crack is small compared to the overall dimension of the elastic continuum, then these stresses are the solution of the PDE of Equation 34.

Far field boundary conditions are the stresses far from the crack tip. These stresses are not affected by the near field stresses, and they reduce to the externally applied axial stresses.

Figure 5: Boundary Conditions - Far Field (illustrated
with red and green arrows surrounding the element) and
Near Field (blue arrows within the element)

Below, we will define the near field and the far field boundary conditions for a Mode I and Mode II fracture.

Defining Near Field Boundary Conditions

The near field boundary conditions, used in solving the differential equation, are defined from the knowledge that the faces of the crack surface are traction-free. That is, the crack face does not have any stresses on it. Therefore, using Figure 5 to illustrate the axial and shear stresses, we are able to define our near field boundary conditions.

Figure 6: Near Field Boundary Conditions

The near field boundary conditions for a Mode I fracture occurs when all of the stresses equal zero on a face of the element, where y=0 and signifies beginning of the crack at the top of the element. These boundary conditions can be denoted as follows:

Near Field Boundary Conditions (Mode I)

(35)

(36)

The near field boundary conditions for a Mode II fracture will be identical to Mode I because our axial stress also implies shear stress. Therefore:

Near Field Boundary Conditions (Mode II)

(36)

(37)

Defining Far Field Boundary Conditions

Far field boundary conditions are defined in order for us to know how far from the crack we are dealing with when solving the differential equations. No matter how far from the crack we are, we will always have stress on our element. These stresses approach infinity as we move further and further away from the crack.

Our far field boundary conditions for Mode I and Mode II fractures will differ because of the externally applied stresses on the fracture. Remember that normal stress (σ) acts perpendicular to the fracture, while shear stress (τ) acts parallel to the fracture.

Exercise 3: Using your knowledge of normal and shear stresses, as well as your knowledge of Mode I and Mode II fractures, match the appropriate stress to the appropriate mode.

Shear Stress	Mode I
Normal Stress	Mode II

As you found in the exercise above, only normal stress (σ) is applied in a Mode I fracture. Thus, all shear stresses (τ) equal 0 because they are nonexistant. The far field boundary conditions for a Mode I fracture can be defined as follows:

Far Field Boundary Conditions (Mode I)

(35)

(36)

Similarly, only one type of stress is applied in a Mode II fracture, which is shear stress (τ). All normal stresses (σ) are nonexistant and thus, equal 0.

Far Field Boundary Conditions (Mode II)

(41)

(42)

By introducing a crack to an element, we have to consider the changes in our boundaries, which are defined by our boundary conditions. Both the near field and the far field boundary conditions will be used to solve the differential equations for a fracture ahead of a crack tip in the following sections of this tutorial.