MCEN 2024, Spring 2008
The week of Apr 07
HW 9–Preview
The Quiz questions based upon HW9 will open on Thursday, Apr. 11
and close on Wednesday, Apr 17 at 1:30 PM.
Topic: FRACTURE
References to A&J:
Chapters 13, 14, 15 and 16
Overview:
Fracture differs from plastic yielding in that it is defined by the propagation of a crack.
Therefore, a pre–existing microcrack, or a "flaw", is the precursor to fracture. If a material is free
from flaws then the intrinsic strength of the bonds among the atoms determines the stress required
to produce fracture: this is called the ideal fracture strength, and, conceptually, is equivalent to the
ideal shear strength for plastic yielding that was discussed previously.
The essential length scale in fracture is the dimension of the flaw, which can range from just a
few micrometers to macroscopic flaws that can be seen with a naked eye.
In this session we discuss two issues: (i) what material parameter should be used in
engineering design to safeguard against fracture failure, and (ii) how do we design the
microstructure of materials in order to enhance their resistance to fracture.
The engineering design parameter is called the fracture toughness, is designated by KIC, having
he units of MPa m1/2. Note that MPa is related to the stress required for fracture, and, as we shalll
see, the length scale of meter is related to the dimension, or as often called, the size of the flaw.
Sometimes KC or simply K are also used as the nomenclature for the fracture toughness.
The materials science of the fracture process is studied by considering the work of fracture,
written as GIC or GC, and this has units of J m-2. Notice that the units have work–done per unit area.
The physical significance of these units is that the work of fracture is expressed as the work done to
propagate the crack such that its surface area is increase by a unit area.
Explanation of the Figures and the Tables
(These are now appended at the end of the HW)I
(i) The Tables for the Elastic Constants are included again, since the elastic constants are
critically important in the study of the work of fracture, GC.
(ii) Tables for the fracture toughness, KC, and for the work of fracture, GC of various materials,
taken from A&J are attached.
(iii) Often, we are concerned with fracture in beams of various cross sections. Here equation for
the tensile stress, and displacements in beams of various cross sections are important. These
equations are given at the end of this HW.
Comparison Chart for Fracture Toughness
Comparison Chart for the Work of Fracture (called as the Toughness - not fracture toughness in the book)
The equation for the maximum stress on
the right in the table for three–point
bending loading, is useful for relating
the applied load to fracture. Equations
for other geometries of the beam, and
loading, are given at the end of this HW.
1. The mechanical loading of a component for the purpose of analyzing fracture is defined by the
stress–intensity factor, KI, which depends on the applied stress as well as on the flaw size as
illustrated by the following sketch and equation:
Fracture criterion:
A common way to measure the fracture strength of glass is by the three–point bending method. The
sample, in the shape of a glass slide is supported on two roller pins. The force on a pin place above
the slide to produce fracture in the slide is measured. If this force is F, the spacing between the
roller supports at the bottom is L, the width of the glass side is w, and its thickness is b, then the
fracture strength is given by:
The fracture stress of the glass is 50 MPa, and
.
(i) Calculate the force required to break the glass slide. (What will be the force for fracture if the
glass slide is twice as thick?)
(ii) Assuming that the fracture toughness of glass is
, calculate the flaw size in the
glass slide.
(ii) Calculate the range in the flaw size in the glass slide if the fracture stress varies from 10 MPa to
100 MPa.
2. The equation for the maximum tensile stress (also called the hoop stress in such “pressure vessel”
problems) in the wall of a pressurized cylinder with an inner radius of R and a wall thickness of t
(assuming that R>>t) is given by:
• Show that the above equation has the correct units.
• The beer bottle has a radius of 5 cm. The fracture strength of the glass can vary from 10 MPa to
100 MPa. What should be the wall thickness for “safe” design if the maximum pressure within the
bottle is expected to be 3 bar (note that 1 kbar = 14,500 psi = 100 MPa), and allowing for an overall
safety factor of 2.
3. The design of a pressure vessel (the maximum pressure that is safe) can be limited by either the
yield strength or the fracture toughness of the material. Please explain the following design curves
for a pressure vessel made from aluminum:
(i) Explain the shape of the curves (why one is flat and the other curves downwards).
(ii) Derive the transition from the yield limited design to fracture limited design.
3.
(above is problem #4)
5. The following table gives the data obtained by Griffith in 1921 for the tensile strength of glass
fibers as a function of their diameter. These data led to his theory of fracture which became the
basis for modern fracture mechanics. The data are in inches (the diameter is in unites of “mils”) and
psi (recall that 14,500 psi, that is 1 kbar is equal to 100 MPa). Please convert these numbers into
micrometers and MPa.
Make a plot of the data with the fiber diameter along the x–axis and the tensile strength along the
y–axis. Estimate a range for the flaw size in the glass fibers in Griffith's experiments.
4. We wish to design the glass overhang for canyon viewing, is shown on the following page. The
length and the width of the overhang are functional design parameters that can be estimated from
the scale of the picture. The question is what should be the thickness of the glass platform,
assuming that the fracture toughness of glass is 1 MPa m1/2. How would you specify the size of the
largest flaw in the design process? How would you design the glass platform taking into account
both the physical weight of the glass and the force of the maximum loading from people standing
on it?
5. The work of fracture, defined as the mechanical work done to propagate the crack so that its
surface area increases by a unit area, is given by:
KC2
(Eqn. A)
E
(Please note that there is often a discrepancy by a factor of 2 in the equations that you will find in
the literature: this usually stems from considering fracture to be represented by two surfaces that are
created when a crack propagates, or just one surface).
GC =
a) Show that GC has units of J m-2.
b) The work of fracture for polymers is much higher than it is for a glass (like silica) – why? – yet
the fracture toughness of glass and polymers are often the same.
6. The Table for GC and KC given in the book and at the end of this problems set gives the values for
different kinds of iron–based alloys, including:
Steel
Rotor
PressVess
HighStren
MildSteel
CastIron
Make a plot of GC versus KC2 and show that the relationship given in Eq (A) is reasonably
correct.
7. Why, in Problem 6, does the elastic constant of iron remain essentially unchanged while the
fracture toughness and the work of fracture vary widely.
8. Explain why there is a trade–off between the yield stress and the fracture toughness in the
metallurgical design of steels (if the yield stress goes up, the fracture toughness goes down).
9. Pick the following classes of materials from the Table for fracture toughness and work of fracture
in the Table given the book (and appended here):
Ice
Soda Glass
Silicon Nitride
Polycarbonate
Cast Iron
Common Woods
CFRPs
Titanium Alloys
Aluminum Alloys
Rotor Steels
KC2
to check whether or not Eq. (A) is valid for a wide variety of
E
material. Since the data covers several orders of magnitude.. it would be better to plot the data on a
log-log scale: then see if the general slope for the data in this plot is equal to one.
Make a plot of GC versus
10. Please take note of the derivation for Eq. (A), which I shall do in class.