J integral - AbsoluteAstronomy.com

The J-integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

represents a way to calculate the strain energy release rate

Strain energy release rate

Strain energy release rate is the energy dissipated during fracture per unit of newly created fracture surface area. This quantity is central to fracture mechanics because the energy that must be supplied to a crack tip for it to grow must be balanced by the amount of energy dissipated due to the...

, or work (energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by Cherepanov and in 1968 by Jim Rice independently, who showed that an energetic contour path integral (called J) was independent of the path around a crack

Crack

Crack may refer to:* Crack cocaine, the freebase form of cocaine that can be smoked* Crack, a fracture or discontinuation in a bodyCrack may also refer to:-Music:* Crack , an album by Z-RO* Cracks , an album by Nabiha...

.

Later, experimental methods were developed, which allowed measurement of critical fracture properties using laboratory-scale specimens for materials in which sample sizes are too small and for which the assumptions of Linear Elastic Fracture Mechanics

Fracture mechanics

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.In...

(LEFM) do not hold, and to infer a critical value of fracture energy

. The quantity

defines the point at which large-scale plastic

Plasticity (physics)

In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...

yielding during propagation takes place under mode one loading.

The J-integral is equal to the strain energy release rate

Strain energy release rate

for a crack in a body subjected to monotonic loading. This is true, under quasistatic conditions, both for linear elastic

Linear elasticity

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...

materials and for materials that experience small-scale yield

Yield (engineering)

The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...

ing at the crack tip.

Two-dimensional J-integral

The two-dimensional J-integral was originally defined as (see Figure 1 for an illustration)

where

is the strain energy density,

are the coordinate directions,

is the surface traction

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

vector,

is the normal to the curve

is the Cauchy stress

Stress (physics)

tensor, and

is the displacement vector. The strain energy density is given by

The J-Integral around a crack tip is frequently expressed in a more general form (and in index notation

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

) as

where

is the component of the J-integral for crack opening in the

direction and

is a small region around the crack tip.
Using Green's theorem

Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...

we can show that this integral is zero when the boundary

is closed and encloses a region that contains no singularities

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

and is simply connected. If the faces of the crack do not have any surface tractions on them then the J-integral is also path independent.

Rice also showed that the value of the J-integral represents the energy release rate for planar crack growth.
The J-integral was developed because of the difficulties involved in computing the stress close to a crack in a nonlinear elastic

Elasticity (physics)

In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

or elastic-plastic

Plasticity (physics)

material. Rice showed that if monotonic loading was assumed (without any plastic unloading) then the J-integral could be used to compute the energy release rate of plastic materials too.

Proof that the J-integral is zero over a closed path

To show the path independence of the J-integral, we first have to show that the value of

is zero over a closed contour in a simply connected domain. Let us just consider the expression for

which is

We can write this as

From Green's theorem

Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...

(or the two-dimensional divergence theorem

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

) we have

Using this result we can express

where

is the area enclosed by the contour

. Now, if there are no body forces present, equilibrium (conservation of linear momentum) requires that

Also,

Therefore,

From the balance of angular momentum we have

. Hence,

The J-integral may then be written as

Now, for an elastic material the stress can be derived from the stored energy function

using

Then, using the chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

of differentiation,

Therefore we have

for a closed contour enclosing a simply connected region without any stress singularities.

Proof that the J-integral is path-independent

Consider the contour

. Since this contour is closed and encloses a simply connected region, the J-integral around the contour is zero, i.e.

assuming that counterclockwise integrals around the crack tip have positive sign. Now, since the crack surfaces are parallel to the

axis, the normal component

on these surfaces. Also, since the crack surfaces are traction free,

. Therefore,

Therefore,

and the J-integral is path independent.

J-integral and fracture toughness

For isotropic, perfectly brittle, linear elastic materials, the J-integral can be directly related to the fracture toughness

Fracture toughness

In materials science, fracture toughness is a property which describes the ability of a material containing a crack to resist fracture, and is one of the most important properties of any material for virtually all design applications. The fracture toughness of a material is determined from the...

if the crack extends straight ahead with respect to its original orientation.

For plane strain, under Mode I

Fracture mechanics

loading conditions, this relation is

where

is the critical strain energy release rate,

is the fracture toughness in Mode I loading,

is the Poisson's ratio, and E is the Young's modulus

Young's modulus

Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

of the material.

For Mode II

Fracture mechanics

loading, the relation between the J-integral and the mode II fracture toughness (