The J-Integral equation was first introduced by Jim Rice in 1968 [1]
and is probably the most fundamental quantity in nonlinear fracture mechanics.
It is an integral equation that gives the amount of energy released
per unit area of crack surface increase. This amount is referred to as \(J\),
but is also, perhaps confusingly, referred to as the J-Integral as well.
It is the analog of Griffith's energy release rate [2],
\(\mathcal{G}\), from LEFM, though \(J\) is not limited to linear elastic
analyses as \(\mathcal{G}\) is.
Derivation
Derivation of the J-Integral equation begins with the 1st Law of Thermodynamics in rate form.
\(W_{external} \; \) is work done by external forces
\(Q_{heat} \quad \; \; \, \) is heat input to, or generated by, the material
\(K_{velocity} \; \; \) is kinetic energy
\(W_{strain} \; \; \; \) is mechanical strain energy
\(U_{thermal} \; \; \) is internal energy related to temperature
\(D_{crack} \quad \; \) is dissipated mechanical energy due to the breaking of atomic bonds
(as the crack grows)
The subscripts help clarify what each term represents.
And the dot over each term represents the time derivative, i.e.,
the rate at which each energy term changes with time.
Energy Transfer
The terms on the left side of the equation represent energy input into an object
in the forms of mechanical work and heat.
The terms on the right side represent the several forms of energy into which the
external energy can go within an object. Three of these four
- kinetic energy, strain energy, and thermal energy - are reversible.
They can be stored in an object and subsequently used to do work on
other objects.
For example, if external work and heat are transferred into an object's kinetic energy,
then the object speeds up and its newly acquired 'speed-energy' can be used to
do work on other external objects. If it is a billiard ball, it may strike
another, causing that one to move.
The exception to this tale is energy that goes into crack propagation,
\(D_{crack}\). That energy is not reversible. It is not stored within
an object and cannot be used to subsequently do work on others.
Consider a situation involving only quasistatic mechanical loading such that
many of the terms are negligible. These include kinetic energy, \(K_{velocity}\),
internal energy, \(U_{thermal}\), and heat input, \(Q_{heat}\). This leaves
\[
\dot W_{external} = \dot W_{strain} + \dot D_{crack}
\]
The equation shows that as work energy, \(W_{external}\), is input into a system
by external forces, it is partitioned into internal strain energy, \(W_{strain}\),
and the breaking of atomic bonds, \(D_{crack}\), i.e., crack growth. The J-Integral
is directly related to this energy release associated with bond breaking and crack growth.
Fracture-Related Loading Scenarios
It may be helpful to consider several different loading scenarios. First,
if there is no crack propagation, \(\dot D_{crack} = 0\), and all the work
of external forces, \(W_{external}\), goes into stored internal strain energy, \(W_{strain}\).
On the other hand, when an object is being torn apart, \(\dot W_{strain}\) will likely
be negligible with the vast majority of \(W_{external}\) going directly to \(D_{crack}\).
It is also possible that an object tears apart while under static loads that do not
themselves move. In such circumstances, \(\dot W_{external} = 0\) because
\(W_{external} = \int {\bf F} \cdot d{\bf x}\) and
\(\dot W_{external} = \int {\bf F} \cdot d{\bf v}\) where \({\bf x}\) is position
and \({\bf v}\) is velocity. Both are zero if the force doesn't move.
In this case, energy is transferred directly from \(W_{strain}\) to \(D_{crack}\)
as the crack grows.
Other Scenarios
Though not directly related to fracture mechanics, it is nevertheless irresistible to note that
\[
\dot W_{external} + \dot Q_{heat} = \dot K_{velocity} + \dot W_{strain} + \dot U_{thermal} + \dot D_{crack}
\]
also reduces to other forms that will all appear familiar.
For starters, rigid-body dynamics problems involve only two terms: the work of external
forces, and kinetic energy. All others are considered negligible, leaving
\[
\dot W_{external} = \dot K_{velocity}
\]
Expressing these energy terms explicitly as functions of forces, velocities, etc, gives
\[
\int {\bf F} \cdot d{\bf v} = {d \over dt} ( {1 \over 2} m V^2 )
\]
where \({\bf v}\) is the velocity vector and \(V\) is its magnitude, i.e., speed.
Integration over a time interval gives the familiar result
\[
\int {\bf F} \cdot d{\bf x} = {1 \over 2} m V^2_2 - {1 \over 2} m V^2_1
\]
which simply states "the work done by external forces equals the change in kinetic
energy of a rigid body."
Another familiar form arises in the field of Thermodynamics. In this case, the kinetic energy term,
\(\dot K_{velocity}\), the strain energy term, \(\dot W_{strain}\), and the crack growth term,
\(\dot D_{crack}\), are all considered negligible, leaving
\[
\dot W_{external} + \dot Q_{heat} = \dot U_{thermal}
\]
In Thermo, this is usually integrated over a time interval and written, simply,
in the following familiar form.
\[
W + Q = \Delta U
\]
I desperately wish that I had fully understood this simple equation when I was a very-lost
Thermodynamics student many years ago.
Back to the J-Integral derivation. Recall that we had established the following relationship.
\[
\dot W_{external} = \dot W_{strain} + \dot D_{crack}
\]
The next step is to apply the following chain rule identity to the time derivatives
\[
{d() \over dt} = {d() \over da} {da \over dt} = {d() \over da} \dot a
\]
where \(a\) is crack length. This leads to
\[
{d W_{external} \over da \;\;\qquad} \, \dot a = {d W_{strain} \over da\;\;\quad} \, \dot a + {d D_{crack} \over da\;\;\quad} \, \dot a
\]
and \(\dot a\) can be factored out to give
\[
{d W_{external} \over da \;\;\qquad} = {d W_{strain} \over da\;\;\quad} + {d D_{crack} \over da\;\;\quad}
\]
The equation now expresses the conservation of energy per unit of crack growth. Dividing through
by \(B\), the material thickness, gives the definition of \(J\)
\[
J \equiv {1 \over B} {d D_{crack} \over da\;\;\quad} = {1 \over B} {d W_{external} \over da\;\;\qquad} - {1 \over B} {d W_{strain} \over da\;\;\quad}
\]
where \( (B \, da) \) is the increment of area created by the crack growth, \(da\).
Relationship of \(J\) to Potential Energy
The above equation clarifies that \(J\) is a measure of the energy lost per unit increase in crack
surface area. Its dimensions are \( Energy / Area \). In metric, this could be
\(Joules / m^2\), or \( N \cdot m / m^2 \), which reduces to the rather confusing \(N / m\).
The quantity \( (B \, da) \) can be written as new crack area, \(d A_{crack}\)
\[
B \, da = d A_{crack}
\]
which leads to the following expressions for the definition of \(J\).
\[
J \, \equiv \, {1 \over B} {dD_{crack} \over da\;\;\quad} \, \equiv \, {dD_{crack} \over dA_{crack}}
\]
Simple algebra can be applied to obtain an alternative, though completely equivalent, expression for \(J\).
This alternative expression often appears in fracture mechanics literature, so it will be covered here.
In the process, we will also obtain an expression for Potential Energy and an expression for
\(J\) in terms of it.
Recall that \(W_{external} = W_{strain} + D_{crack}\), where the time derivative (overhead dot) has been
dropped for convenience.
Solving for \(D_{crack}\) gives \(D_{crack} = W_{external} - W_{strain}\). So \(J\) can also be
expressed as
\[
J = {dW_{external} \over dA_{crack}} - {dW_{strain} \over dA_{crack}}
\]
Coincidentally, the definition of Potential Energy, \(\Pi\), is
\[
\Pi \equiv W_{strain} - W_{external}
\]
which is just the negative of the expression for \(D_{crack}\) above. Substituting this into the definition
of \(J\) gives
\[
J \, \equiv \, {dD_{crack} \over dA_{crack}} = - {d \, \Pi\;\;\;\;\; \over dA_{crack}}
\]
It is this relationship that sometimes leads to \(J\) being introduced as the rate of
change of an object's potential energy relative to its crack growth. I think this is very
confusing, even misleading, because it is not the fundamental definition of \(J\).
It is only a relationship arrived at via algebra applied to energy conservation expressions.
If you are confused by the definition of Potential Energy as \( \Pi \equiv W_{strain} - W_{external}\)
because you thought it was \(m g h\), then you are not alone.
I do not know how the term Potential Energy came to represent \(W_{strain} - W_{external}\).
Nevertheless, it is what it is.
In practice, it is useful to express \(W_{external}\) and \(W_{strain}\) in terms of directly
measurable quantities. For example
\[
W_{external} = \int_S {\bf T} \cdot {\bf u} \, dS
\]
where \({\bf T}\) is the traction vector, \({\bf u}\) is the displacement vector, and
\(S\) is the exterior surface area. A traction vector is simply a force vector whose
components have been divided by cross-sectional area.
See
www.continuummechanics.org/tractionvector.html for more on traction vectors
if you are not familiar with them.
The strain energy, \(W_{strain}\), is expressed in terms of strain energy density, \(w\), and
volume, \(V\).
\[
W_{strain} = \int_V w \, dV
\]
This is conceptually equivalent to saying "strain energy is the result of
multiplying strain-energy-per-unit-volume by volume."
Strain Energy Density
Strain energy density is, as the name implies, the amount of energy per unit volume
used to deform an object. It is the differential element analog of total strain energy
in an object. While energy is \(W = \int {\bf F} \cdot d{\bf x}\),
strain energy density, \(w\), is
\[
w = \int \boldsymbol{\sigma} : d\boldsymbol{\epsilon}
\]
If the material is linear elastic, then the integral can be easily evaluated and gives
At this point, one might ask, "The integrals are over the surface and volume of what?!"
This is indeed an excellent question. The answer is "Any volume enclosing the crack tip."
It is not yet clear why the word, any, is included, but it is correct nevertheless.
The next page on path independence will show why this is indeed the case.
The above equation could be used to compute \(J\). But it is inconvenient because calculation
of the derivatives requires evaluation of the displacement and strain energy density fields
at two different crack lengths.
It was at this point that Jim Rice [1] imposed some clever physical insight onto the problem
in order to simplify the calculation of \(J\). (Yes, \(J\) stands for Jim, at least so I'm told.)
He asserted that the change in stresses, strains, displacements, etc., due to the increment in
crack length, \(da\), would be the same as the differences in the values between two different
locations separated by the distance \(da\) while the crack length is held constant. In effect
\[
{d() \over da} = - {\partial () \over \partial x}
\]
where the \(x\) coordinate is measured along the direction of the crack. The \(J\)-Integral
expression then becomes
\[
J = {1 \over B} \int_V {\partial w \over \partial x} \, dV -
{1 \over B} \int_S {\bf T} \cdot {\partial {\bf u} \over \partial x} \, dS
\]
This offers the great advantage of requiring only one analysis at a single crack length
instead of two analyses at two crack lengths.
The next step is to apply the divergence theorem to the first term to obtain another useful form.
\[
J = {1 \over B} \int_S w \; n_x \, dS -
{1 \over B} \int_S {\bf T} \cdot {\partial {\bf u} \over \partial x} \, dS
\]
In this case, \(n_x\) is the value of the \(x\) component of the unit normal to the surface.
Divergence Theorem
Recall that the Divergence Theorem states
\[
\int_V \nabla w \, dV = \int_S w \, {\bf n} \, dS
\]
Expanding the notation out gives
\[
\int_V \left( {\partial w \over \partial x} \, , {\partial w \over \partial y} \, ,
{\partial w \over \partial z} \, \right) dV
= \int_S \left( w \, n_x \, , w \, n_y \, , w \, n_z \, \right) dS
\]
which is in fact vector notation for three separate equations, one for each
of the x, y, and z components. Since only \(\partial w / \partial x\) is present in
the volume integral of the J-Integral equation, application of the
Divergence Theorem includes only the \(w \, n_x\) term, so
A final simplification can be imposed, and is indeed made in virtually every case, by
recognizing that cracks propagate through objects that are "plate like" in nature.
They inherently have a thickness, which is ironically the objects'
thinnest dimension. In such cases, the thickness direction is perpendicular
to \(x,\) the crack propagation dimension, therefore \(n_x = 0\) on the faces
of the plate, and the J-Integral's first term is zero as well.
\[
{1 \over B} \int_S w \; n_x \, dS = 0 \qquad \text{on front and back faces}
\]
Also, the faces are negligibly loaded, if at all, i.e., \({\bf T} = 0.\)
Therefore, the second integral is zero here as well.
\[
{1 \over B} \int_S {\bf T} \cdot {\partial {\bf u} \over \partial x} \, dS = 0
\qquad \text{on front and back faces}
\]
This means the only contributions to the J-Integral occur within the plate.
And in it, the surface increment, \(dS\), can be expressed as \(B \, d\Gamma\).
Inserting this relationship into the J-Integral equation gives
\[
J = {1 \over B} \oint w \; n_x \, B \, d\Gamma -
{1 \over B} \oint {\bf T} \cdot {\partial {\bf u} \over \partial x} \, B \, d\Gamma
\]
and the thickness, \(B,\) cancels out, giving the final, common form of the
J-Integral equation.
\[
J = \oint w \; n_x \, d\Gamma -
\oint {\bf T} \cdot {\partial {\bf u} \over \partial x} \, d\Gamma
\]
The expressions are path integrals.
This is signified by the \(\oint\) integral symbol.
One last step that is sometimes taken is to note that \(n_x \, d\Gamma = dy.\)
(Yes, it is indeed \(dy\) and not \(dx.)\) This gives
Rice, J.R., "A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks." Journal of Applied Mechanics, Vol. 35, pp. 379-386, 1968.
Griffith, A.A., "The Phenomena of Rupture and Flow in Solids," Philosophical Transactions, Series A, Vol. 221, pp. 163-198, 1920.
