Strain Energy To Failure Definition Essay

LECTURE 35

THEORIES OF ELASTIC FAILURE

              While dealing with the design of structures or machine elements or any component of a particular machine the physical properties or chief characteristics of the constituent materials are usually found from the results of laboratory experiments in which the components are subject to the simple stress conditions. The most usual test is a simple tensile test in which the value of stress at yield or fracture is easily determined.

              However, a machine part is generally subjected simultaneously to several different types of stresses whose actions are combined therefore, it is necessary to have some basis for determining the allowable working stresses so that failure may not occur. Thus, the function of the theories of elastic failure is to predict from the behavior of materials in a simple tensile test when elastic failure will occur under any conditions of applied stress.

A number of theories have been proposed for the brittle and ductile materials.

Strain Energy: The concept of strain energy is of fundamental importance in applied mechanics. The application of the load produces strain in the bar. The effect of these strains is to increase the energy level of the bar itself. Hence a new quantity called strain energy is defined as the energy absorbed by the bar during the loading process. This strain energy is defined as the work done by load provided no energy is added or subtracted in the form of heat. Some times strain energy is referred to as internal work to distinguish it from external work ‘W'. Consider a simple bar which is subjected to tensile force F, having a small element of dimensions dx, dy and dz.

The strain energy U is the area covered under the triangle

A three dimension state of stress respresented by s1, s2 and s3 may be throught of consisting of two distinct state of stresses i.e Distortional state of stress

Deviatoric state of stress and dilational state of stress

Hydrostatic state of stresses.

Thus, The energy which is stored within a material when the material is deformed is termed as a strain energy. The total strain energy Ur

UT = Ud+UH

Ud is the strain energy due to the Deviatoric state of stress and UH is the strain energy due to the Hydrostatic state of stress. Futher, it may be noted that the hydrostatic state of stress results in change of volume whereas the deviatoric state of stress results in change of shape.

Different Theories of Failure : These are five different theories of failures which are generally used

(a)   Maximum Principal stress theory ( due to Rankine )

(b)   Maximum shear stress theory ( Guest - Tresca )

(c)   Maximum Principal strain ( Saint - venant ) Theory

(d)  Total strain energy per unit volume ( Haigh ) Theory

(e)  Shear strain energy per unit volume Theory ( Von – Mises & Hencky )

In all these theories we shall assume.

sYp = stress at the yield point in the simple tensile test.

s1, s2, s3- the three principal stresses in the three dimensional complex state of stress systems in order of magnitude.

(a) Maximum Principal stress theory :

This theory assume that when the maximum principal stress in a complex stress system reaches the elastic limit stress in a simple tension, failure will occur.

Therefore the criterion for failure would be

s1 = syp

For a two dimensional complex stress system s1 is expressed as

                                         

Where sx, sy and txy are the stresses in the any given complex stress system.

(b) Maximum shear stress theory:

This theory states that teh failure can be assumed to occur when the maximum shear stress in the complex stress system is equal to the value of maximum shear stress in simple tension.

The criterion for the failure may be established as given below :

For a simple tension case

(c) Maximum Principal strain theory :

This Theory assumes that failure occurs when the maximum strain for a complex state of stress system becomes equals to the strain at yield point in the tensile test for the three dimensional complex state of stress system.

For a 3 - dimensional state of stress system the total strain energy Ut per unit volume in equal to the total work done by the system and given by the equation

                  

(d) Total strain energy per unit volume theory :

The theory assumes that the failure occurs when the total strain energy for a complex state of stress system is equal to that at the yield point a tensile test.

Therefore, the failure criterion becomes

It may be noted that this theory gives fair by good results for ductile materials.

(e) Maximum shear strain energy per unit volume theory :

This theory states that the failure occurs when the maximum shear strain energy component for the complex state of stress system is equal to that at the yield point in the tensile test.

Hence the criterion for the failure becomes

As we know that a general state of stress can be broken into two components i.e,

(i)   Hydrostatic state of stress ( the strain energy associated with the hydrostatic state of stress is known as the volumetric strain energy )

(ii)  Distortional or Deviatoric state of stress ( The strain energy due to this is known as the shear strain energy )

As we know that the strain energy due to distortion is given as

This is the distortion strain energy for a complex state of stress, this is to be equaled to the maximum distortion energy in the simple tension test. In order to get we may assume that one of the principal stress say ( s1 ) reaches the yield point ( syp ) of the material. Thus, putting in above equation s2 = s3 = 0 we get distortion energy for the simple test i.e

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Griffith's Energy Release Rate

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Introduction

Alan Arnold Griffith's energy-based analysis of cracks in 1920 is considered to be the birth of the field of fracture mechanics [1]. A copy of his paper can be found here. He was motivated by Inglis's linear elastic solution for stresses around an elliptical hole [2], which predicted that the stress level approached infinity as the ellipse flattened to form a crack.

Inglis's result had prompted much discussion concerning the fact that it could not be "correct" because no material can support an infinite stress without yielding and failing. Therefore, everything should immediately fail under even the smallest load if a crack were present. (But this obviously doesn't happen.) So Griffith looked to propose an energy-based failure criterion that effectively side-stepped Inglis's infinite-stress prediction, while nevertheless making direct use of his linear elastic solution.

Griffith compared the work required to break atomic bonds to the strain energy released as a crack grows. This page begins by covering these topics separately, then combines them to form Griffith's energy release rate criterion.


Energy Principles

In order to understand Griffith's work, one must understand the basic principles of work and energy, and especially strain energy density. Work, \(W\), is the mechanical form of energy and is given by

\[ W = \int {\bf F} \cdot {\bf dx} \]
For mechanics calculations, it is often convenient to calculate work (or energy) in terms of stress and strain rather than force and displacement. To do so, multiply and divide the above equation through by the volume, \(V\), but express \(V\) in the denominator as \(A L\), the area times length, and group them as follows

\[ W \; \; = \; \; \int {\bf F} \cdot {\bf dx} \left( V \over A L \right) \; \; = \; \; \int \left ( { {\bf F} \over A} \right ) \cdot \left ( { d{\bf x} \over L} \right) \; V \]
But \(({\bf F} / A)\) is stress, \(\boldsymbol{\sigma}\), and \(d{\bf x} / L\) is strain, \(d \boldsymbol{\epsilon}\). This gives

\[ U = \int \boldsymbol{\sigma} : d{\boldsymbol{\epsilon}} \; V \]
In this case, \(U\) has been used instead of \(W\) to indicate that strain energy is the specific form of work (and energy) being considered. The integral \(\int \boldsymbol{\sigma} : d{\boldsymbol{\epsilon}}\) is strain energy density. It is strain energy per unit volume and is represented by \(U'\) as

\[ U' = \int \boldsymbol{\sigma} : d{\boldsymbol{\epsilon}} \qquad \text{and} \qquad U = U' * V \]
For a linear elastic material in uniaxial tension, Hooke's Law reduces to \(\sigma = E \epsilon\). Inserting this into the integral for strain energy density gives

\[ U' = \int \boldsymbol{\sigma} : d{\boldsymbol{\epsilon}} = \int E \, \epsilon \, d\epsilon = {1 \over 2} E \, \epsilon^2 \]
And then the \(\sigma = E \epsilon\) relation can be substituted to form several different equivalent expressions

\[ U' \; = \; {1 \over 2} E \, \epsilon^2 \; = \; {1 \over 2} \sigma \epsilon \; = \; {\sigma^2 \over 2 E} \]
We will return to the last expression shortly because it appears in the derivation of Griffith's energy criterion.


Atomic Bonds and Surface Energy

Solids have an equilibrium spacing distance between pairs of atoms making up a material. Too close, and the atoms repel each other. Too far, and the atoms attract each other. And at an intermediate distance, there is no net force between them. They are in equilibrium.

Imagine an atom being pulled away from its neighbor. The x-axis below represents the displacement of an atom from its equilibrium position. The graph shows that the force required to move the atom initially increases as the atoms are separated. (The more the atoms separate, the more they attract each other.) But as the distance increases, the atoms eventually become separated so far that they no longer attract each other. By this point, the force will have leveled out, then begun to decrease, before eventually returning to zero.

The key principle here is that the area under the force-displacement curve represents the energy of an atomic bond.




The above discussion presented the concept of atomic bond energy. In order to break a bond, an amount of work equal to the bond energy must be performed on the system. The next figure extends this to the case of a crack growing in a solid.

The figure shows a crack that has grown to length, \(a\), and in the process, has broken several atomic bonds along the way, each requiring a certain amount of work to overcome the atomic bond energy. The total energy is expressed as

\[ E_{bond} = 2 \, \gamma_s a \, B \]
where \(\gamma_s\) represents the energy required to break atomic bonds per unit surface area created by the crack. (It does not stand for shear strain here.) The surface area is \(a * B\) where \(a\) is the crack length and \(B\) is the part thickness. Curiously, it is convention to measure \(\gamma_s\) relative to free surface area (the \(s\) stands for surface), and since there are two free surfaces created by a crack, a top and a bottom, the 2 is necessary in the equation.

The bottom line is that \(E_{bond} = 2 \, \gamma_s a \, B\) gives the amount of work (energy) that must be input into an object to overcome the atomic bond energy of the atoms forming a crack of length, \(a\).



Conventions

It is common in fracture mechanics to use \(B\) for thickness, perhaps because \(t\) and \(T\) are usually assigned to time and temperature, respectively. And \(a\) is used for crack length, probably a carry-over from using \(a\) and \(b\) for the major and minor axes of an ellipse.

Strain Energy Release

The concept of Strain Energy Release with crack growth is simple. Consider a bar pulled in tension by a stress, \(\sigma\). Its strain energy will be

\[ U = {\sigma^2 \over 2 E} \, V \]
according to the Energy Principles discussed above. Now assume that a crack propagates all the way across the bar, causing it to snap in two halves. Both halves will now be unloaded following the break, so they can have no strain energy. The original strain energy, \(U\), has been released as a result of the propagation of the crack across the part.

Griffith was the first to compute the strain energy release associated with crack growth. He did so for the case of an infinite plate in uniaxial tension. Why this scenario? Easy, because the infinite plate in uniaxial tension was the very case that Inglis had just solved seven years earlier in 1913. Griffith used Ingils' limiting case of an ellipse flattened to form a crack and integrated the stress and strain fields to obtain the strain energy as a function of crack length, \(a\). He obtained the following result (for one-half of the infinite plate).

\[ U = {\sigma^2 \over 2 E} \, V - {\sigma^2 \over 2 E} \, B \, \pi \, a^2 \]
The result is surprisingly simple. It is a combination of a baseline value, \({\sigma^2 \over 2 E} \, V\), corresponding to zero crack length, and a second term that subtracts strain energy away as the crack length increases according to \(a^2\). It is interesting that the quadratic dependence on crack length means that the change in strain energy with crack length at short lengths is very small, but then becomes very sensitive to crack length at longer lengths.

One might wonder about the utility of this result given that the plate is infinite in size and therefore, the strain energy is infinite because \(V\) is infinite. Nevertheless, this equation will prove to be useful because we will eventually differentiate it, thereby eliminating the infinite volume term because it is constant.

Quick Geometric Interpretation

There is an easy way to remember Griffith's result, and it is shown in the figure below. It is to think of an unloaded portion of material above and below the crack that is triangular in nature, having width \(a\) and height \(\pi a\). This gives an area of \(\pi a^2 / 2\), and since there are two triangles, one above and one below, the area of both triangles is \(\pi a^2\). Finally since the thickness is \(B\), the volume is \(B \pi a^2\). This matches the volume term in Griffith's equation.

So the easy way to think of this is there is a volume of material near the crack equal to \(B \pi a^2\) that subtracts out the strain energy density, \(\sigma^2 / 2 E\). Keep in mind that this is just an easy way to remember the result. It is not meant to imply that the triangular region is in fact a hard boundary separating nonzero strain energy regions from zero strain energy regions.


Griffith's Failure Criterion

Recall that the energy input required to break the atomic bonds and grow the crack is given by

\[ E_{bond} = 2 \, \gamma_s a \, B \]
whereas as the crack grows, the stored mechanical strain energy decreases according to

\[ U = {\sigma^2 \over 2 E} \, V - {\sigma^2 \over 2 E} \, B \, \pi \, a^2 \]
The total energy in the system is simply the sum of the two expressions. This is reflected by the red curve in the figure below.

\[ E_\text{total} = 2 \, \gamma_s a \, B + {\sigma^2 \over 2 E} \, V - {\sigma^2 \over 2 E} \, B \, \pi \, a^2 \]


The key insight is to recognize that for short crack lengths (left of the max value on the graph), the total energy of the object increases with increasing crack length. Therefore, additional energy must be input into the material in order to cause the crack to grow. It is stable.

However, at longer crack lengths (right of the max curve value), an increase in crack length leads to a decrease in total energy. This means the crack can grow without any additional external input. This is an unstable situation that can lead to catastrophic failure as a crack suddenly propagates completely through a part. In technical lingo, it is said that the crack can grow spontaneously.

We need to differentiate the total energy curve with respect to crack length, \(a\), and then set the derivative to zero, in order to find the length at which the unstable crack growth (and part failure) can occur.

\[ {dE_\text{total} \over da} \, = \, 2 \, \gamma_s \, B \, - \, {\sigma^2 \over E} \, B \, \pi \, a \, = \, 0 \]
Cancelling out \(B\) from the terms and solving for \(\sigma\) gives

\[ \sigma_f = \sqrt{2 \, \gamma_s E \over \pi \, a} \]
An \(_f\) subscript has been added to \(\sigma\) to signify failure. The equation shows that the failure stress increases with the square root of bond energy, \(\gamma_s\), but decreases with the square root of crack length, \(a\). Finally, the quantity \(2 \gamma_s\) is combined into the Griffith Critical Energy Release Rate, \(G_c\).

\[ \sigma_f = \sqrt{G_c E \over \pi \, a} \]

Example of a Crack in Glass

It is well known that glass panes are brittle, and are especially susceptible to shattering when cracks are present. Griffith's criterion can be used to estimate the critical failure stress for a given crack length. Let's calculate the critical stress of a typical glass pane when a 1" (25.4 mm) crack is present.

The modulus of glass is \(E = 70,000\;\text{MPa}\) and the critical energy release rate is about \(G_c = 7\;\text{J/m}^2\).

\[ \sigma_f \; = \; \sqrt{G_c E \over \pi \, a} \; = \; \sqrt{(7\;\text{J/m}^2) (7\text{E}10\;\text{N/m}^2) \over (\pi) (0.025\;\text{m})} \; = \; 2.5\text{E}6\;\text{N/m}^2 \; = \; 2.5\;\text{MPa} \]
This is a very low failure stress. It can be easily exceeded when bending loads are imposed on a pane of glass. For comparison, the yield strength of aircraft-grade aluminum (e.g., Al 7075-T73) is approximately \(400\;\text{MPa}\).

Metal Plasticity and Griffith's Criterion

It is important to recognize that the equation \(E_{bond} = 2 \gamma_s B a\) neglects any energy associated with metal plasticity. It applies only to brittle materials, such as glass, because it only accounts for energy associated with atomic bond breaking. Therefore, Griffith's energy-based failure criterion only applies to brittle materials as well. In fact, it was glass that Griffith used in tests to experimentally confirm his failure criterion.

It was soon discovered that his criterion greatly underestimated the critical failure strength of many metals. It turns out that in metallic materials, there is a great deal more energy dissipation associated with plastic deformation near the crack tip than with atomic bond breaking. This was addressed by Irwin [3] and Orowan [4], who modified Griffith's initial development by adding \(\gamma_p\), the energy due to plastic deformation at the crack tip per unit surface area created by crack propagation.

\[ E_{bond} = 2 (\gamma_s + \gamma_p) B a \]
Therefore, Griffith's critical energy release rate becomes

\[ G_c = 2 (\gamma_s + \gamma_p) \]
Nevertheless, the equation relating crack length, modulus, critical stress, and critical release rate remains unchanged. It's just that \(G_c\) now includes plastic energy dissipation.

\[ \sigma_f \; = \; \sqrt{G_c E \over \pi \, a} \]

Stress Intensity Factors

Solving Griffith's equation for \(G_c\) gives

\[ G_c \; = \; { \sigma_f^2 \, \pi \, a \over E} \]
which in itself is nothing special. But in 1957, Irwin introduced the critical Stress Intensity Factor, \(K_c\), defined as [5]

\[ K_c = \sigma_f \sqrt{\pi \, a} \]
and this means that Griffith's equation can also be written as

\[ G_c = { K_c^2 \over E} \]
This has served as a "teaser" to the complete discussion of stress intensity factors, which comes later. It turns out that over time, the use of stress intensity factors became much more popular than energy release rates for linear elastic problems.

References

  1. Griffith, A.A., "The Phenomena of Rupture and Flow in Solids," Philosophical Transactions, Series A, Vol. 221, pp. 163-198, 1920.
  2. Inglis, C.E., "Stresses in Plates Due to the Presence of Cracks and Sharp Corners," Transactions of the Institute of Naval Architects, Vol. 55, pp. 219-241, 1913.
  3. Irwin, G.R., "Fracture Dynamics," Fracturing of Metals, American Society for Metals, Cleveland, OH, pp. 147-166, 1948.
  4. Orowan, E., "Fracture and Strength of Solids," Reports on Progress in Physics, Vol. XII, p. 185, 1948.
  5. Irwin, G.R., "Analysis of Stresses and Strains near the End of a Crack Traversing a Plate," Journal of Applied Mechanics, Vol. 24, pp. 361-364, 1957.


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