History and Overview
1.5 A BRIEF REVIEW OF DIMENSIONAL ANALYSIS
At first glance, a section on dimensional analysis may seem out of place in the introductory chapter of a book on fracture mechanics. However, dimensional analysis is an important tool for developing mathematical models of physical phenomena, and it can help us understand existing models. Many difficult concepts in fracture mechanics become relatively transparent when one considers the relevant dimensions of the problem. For example, dimensional analysis gives us a clue as to when a particular model, such as linear elastic fracture mechanics, is no longer valid.
Let us review the fundamental theorem of dimensional analysis and then look at a few simple applications to fracture mechanics.
1.5.1 THE BUCKINGHAM Π-THEOREM
The first step in building a mathematical model of a physical phenomenon is to identify all of the parameters that may influence the phenomenon. Assume that a problem, or at least an idealized version of it, can be described by the following set of scalar quantities: {u, W1, W2,…, Wn}. The dimensions of all quantities in this set is denoted by {[u], [W1], [W2],…, [Wn]}. Now suppose that we wish to express the first variable u as a function of the remaining parameters:
(1.6) Thus, the process of modeling the problem is reduced to finding a mathematical relationship that represents f as best as possible. We might accomplish this by performing a set of experiments in which we measure u while varying each Wi independently. The number of experiments can be greatly reduced, and the modeling processes simplified, through dimensional analysis. The first step is to identify all of the fundamental dimensional units (fdu’s) in the problem: {L1, L2,…, Lm}.
For example, a typical mechanics problem may have {L1 = length, L2 = mass, L3 = time}. We can express the dimensions of each quantity in our problem as the product of the powers of the fdu’s;
i.e., for any quantity X, we have
(1.7) The quantity X is dimensionless if [X] = 1.
In the set of Ws, we can identify m primary quantities that contain all of the fdu’s in the problem.
The remaining variables are secondary quantities, and their dimensions can be expressed in terms of the primary quantities:
(1.8) Thus, we can define a set of new quantities πi that are dimensionless:
(1.9) Similarly, the dimensions of u can be expressed in terms of the dimensions of the primary quantities:
(1.10) u f W W= ( , , , )1 2Wn
[ ]X =L L1a1 2a2,,Lmam
[Wm j+ ] [= W1]am j+( )1,, [Wm]am j m+( ) (j=1 2, ,,n m−
πi a m j
ma
W
W m j W m j m
= + + +
1 ( )1,, ( )
[ ] [u = W1] ,a1, [Wm]am
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and we can form the following dimensionless quantity:
(1.11) According to the Buckingham Π-theorem, π depends only on the other dimensionless groups:
(1.12) This new function F is independent of the system of measurement units. Note that the number of quantities in F has been reduced from the old function by m, the number of fdu’s. Thus dimensional analysis has reduced the degrees of freedom in our model, and we need vary only n – m quantities in our experiments or computer simulations.
The Buckingham Π-theorem gives guidance on how to scale a problem to different sizes or to other systems of measurement units. Each dimensionless group (πi) must be scaled in order to obtain equivalent conditions at two different scales. Suppose, for example, that we wish to perform wind tunnel tests on a model of a new airplane design. Dimensional analysis tells us that we should reduce all length dimensions in the same proportion; thus we would build a ‘‘scale” model of the airplane.
The length dimensions of the plane are not the only important quantities in the problem, however. In order to model the aerodynamic behavior accurately, we would need to scale the wind velocity and the viscosity of the air in accordance with the reduced size of the airplane model. Modifying the viscosity of the air is not practical in most cases. In real wind tunnel tests, the size of the model is usually close enough to full scale that the errors introduced by not scaling viscosity are minor.
1.5.2 DIMENSIONAL ANALYSISIN FRACTURE MECHANICS
Dimensional analysis proves to be a very useful tool in fracture mechanics. Later chapters describe how dimensional arguments play a key role in developing mathematical descriptions for important phenomena. For now, let us explore a few simple examples.
Consider a series of cracked plates under a remote tensile stress , as illustrated in Figure 1.13.
Assume that each is a two-dimensional problem; that is, the thickness dimension does not enter into the problem. The first case, Figure 1.13(a), is an edge crack of length a in an elastic, semi-infinite plate. In this case infinite means that the plate width is much larger than the crack size. Suppose that we wish to know how one of the stress components σij varies with position. We will adopt a polar coordinate system with the origin at the crack tip, as illustrated in Figure 1.9. A generalized functional relationship can be written as
(1.13) where
ν = Poisson’s ratio
σkl= other stress components
εkl = all nonzero components of the strain tensor
We can eliminate σkl and εkl from f1 by noting that for a linear elastic problem, strain is uniquely defined by stress through Hooke’s law and the stress components at a point increase in proportion to one another. Let σ∞ and a be the primary quantities. Invoking the Buckingham Π-theorem gives
20 Fracture Mechanics: Fundamentals and Applications
(a) (b)
(c)
FIGURE 1.13 Edge-cracked plates subject to a remote tensile stress: (a) edge crack in a wide elastic plate, (b) edge crack in a finite width elastic plate, and (c) edge crack with a plastic zone at the crack tip.
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When the plate width is finite (Figure 1.13(b)), an additional dimension is required to describe the problem:
(1.15)
Thus, one might expect Equation (1.14) to give erroneous results when the crack extends across a significant fraction of the plate width. Consider a large plate and a small plate made of the same material (same E and ν), with the same a/W ratio, loaded to the same remote stress. The local stress at an angle θ from the crack plane in each plate would depend only on the r/a ratio, as long as both plates remained elastic.
When a plastic zone forms ahead of the crack tip (Figure 1.13(c)), the problem is complicated further. If we assume that the material does not strain harden, the yield strength is sufficient to define the flow properties. The stress field is given by
(1.16) The first two functions, F1 and F2, correspond to LEFM, while F3 is an elastic-plastic relationship.
Thus, dimensional analysis tells us that LEFM is valid only when ry << a and <<σYS. In Chapter 2, the same conclusion is reached through a somewhat more complicated argument.
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σ
22 Fracture Mechanics: Fundamentals and Applications
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