ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation
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Fracture Resistance Testing of Brittle Solids
Michael Jenkins, University of Washington; Johnathan Salem, NASA-Glenn Research Center
Introduction
CATASTROPHIC FAILURE best typifies the characteristic behavior of brittle solids in the presence of cracks
or crack-like flaws under ambient conditions. Examples of engineering materials that behave as brittle solids
include glasses, ceramics, and hardened metal alloys, such as bearing or brake steels. Figure 1 shows the linear-
elastic stress-strain curves and abrupt failures of a glass and a ceramic. This behavior is contrasted with the
linear/nonlinear stress-strain curves and “graceful” failure of a ductile-like material, such as a metal, or in this
case, a continuous fiber-reinforced composite.
Fig. 1 Comparison of stress-strain curves for ceramics and glasses (as examples of brittle solids) and
fiber-reinforced composites (as examples of nonbrittle solids). Source: Ref 1
Sometimes under nonambient conditions, materials that normally fail in a ductile manner may fail in a brittle
manner (e.g., carbon steel at temperatures less than their nil-ductility) and those materials that normally fail in a
brittle manner may exhibit pseudo-plasticity and failure in a ductile manner (e.g., ceramics containing glassy
secondary phases at temperatures greater than the glass-softening temperature).
Because catastrophic failure occurs without warning and can occur in any engineering material under the
“suitably wrong” conditions, it is important to characterize the fracture behavior of materials in order to
produce engineering designs that can accommodate this phenomenon. This article reviews the fracture behavior
of brittle solids and the various methods that have been developed to characterize this behavior.
Reference cited in this section
1. G.D. Quinn, Strength and Proof Testing, Ceramics and Glasses, Vol 4, Engineering Materials
Handbook, ASM International, 1991, p 585–598
Fracture Resistance Testing of Brittle Solids
Michael Jenkins, University of Washington; Johnathan Salem, NASA-Glenn Research Center
Concepts of Fracture Mechanics as Applied to Brittle Materials
Many assumptions accompany engineering analyses. For example, in fundamental mechanics of materials it is
often assumed that materials are linear elastic, homogeneous, uniform, and isotropic from a macroscopic view.
These assumptions are more or less appropriate for polycrystalline materials without any crystallographic
ordering. Microscopically, of course, these assumptions become tenuous at best, especially for dimensional
scales on the order of grain sizes.
In the study of crack and material interactions, fundamental engineering fracture mechanics also makes several
assumptions. These include the linear elastic, homogeneous, uniform, and isotropic assumptions of material
response. In addition, it is assumed that the change in stored elastic strain energy is used entirely by the fracture
process in creating new fracture surfaces and that the crack itself exists in an infinite body and is not influenced
by any boundary conditions. Using these assumptions, it is possible to write the original Griffith criterion for
fracture in terms of an applied fracture stress (Ref 1, 2, and 3):
(Eq 1)
where σ
f
is the applied stress at fracture, γ
f
is the energy required to create a unit of fractured surface area (i.e.,
fracture surface energy), E is the elastic modulus, ν is Poisson's ratio, and c is the flaw (i.e., crack) dimension
(in the case of an internal flaw, the radius). The observed fracture strengths, S
f
, of brittle solids (i.e., the applied
stress at fracture given by Eq 1 where S
f
= σ
f
) are related to the size and distributions of the strength-limiting
flaws, c, in the material (assuming γ
f
, E, and ν are deterministic material properties). Strength distributions are
therefore related to the distributions of these f laws: intrinsic flaws (e.g., those due to processing) are those that
can be treated as microcracks (short cracks) and are on the order of the microstructure; extrinsic or induced
flaws (e.g., those due to service) are those that can be treated as macrocracks (long cracks) and are on the order
of component dimensions. Note that sometimes extrinsic flaws may be of the same dimensional order as
intrinsic flaws.
The microcrack-like flaws are randomly distributed in size, location, orientation, and shape (scattered)
throughout brittle materials, causing a range of fracture strengths in otherwise identical components or parts.
This inherent scatter leads to fracture strengths that are related to the geometric size (i.e., surface area or
volume) of the component. In other words, the larger the component is, the greater the probability of a larger
(or properly oriented, or properly shaped, etc.) flaw to occur and, hence, the lower the fracture strength is.
Therefore, fracture strength is not a deterministic property in brittle materials unless the flaws are extremely
uniform and consistent. Factors of safety in the conventional sense cannot be used. Strength values vary
significantly with size and shape of the component (or test specimen) and with processing conditions. There
may even be batch-to-batch differences as a consequence of material inconsistencies. These factors, coupled
with the “inherent” brittleness of the materials, mean that either extremely conservative stress/strength-based
deterministic design philosophies or probabilistic reliability methods must be used for components fabricated
from brittle materials.
The concepts of engineering fracture mechanics can be applied when a flaw has a measurable crack size. In this
case, the stress field at the crack tip is described in terms of stress intensity factor, which can be written as (Ref
4, 5):
K = Yσ
(Eq 2)
where K is the stress intensity factor, σ is an applied stress, Y is a geometry correction factor, and a is the
macrocrack dimension. Three “modes” of fracture are related to the “mode” of loading (Fig. 2). Mode I, the
“opening mode,” is considered to be the limiting case for the tendency to fracture because a tensile normal
stress “opens” the crack with the resulting stresses in the material “carried” at the crack tip (as opposed to
partially distribute through interaction of the crack faces as in modes II and III).
Fig. 2 Modes of fracture. Mode I (opening), mode II (sliding), and mode III (tearing). Source: Ref 4
The critical mode I condition for brittle fracture in a component with a crack-like flaw is reached at a
combination of the crack/component geometry correction factor, Y; a sufficiently high tensile, normal stress, σ;
and a sufficiently long, sharp crack, a. If the fracture resistance of the material is not a function of crack length
then catastrophic fracture will occur in the component when the stress intensity factor is equal to the critical
stress intensity factor at fracture in the component:
Brittle fracture if K
I
= K
Ic
(Eq 3)
where K
I
is the mode I stress intensity factor and K
Ic
is the fracture toughness
*
of the material (i.e., resistance to
fracture), which can be related to fundamental fracture behavior of the material through the Griffith approach
(Ref 1, 4):
(Eq 4)
From a practical view, materials scientists often prefer to use the Griffith approach and γ
f
(Eq 1) to describe the
fracture characteristics of materials and their relation to the fundamental aspects of the material. However,
designers and engineers prefer the fracture-mechanics approach and K
Ic
(Eq 2 and 3) because fracture
characteristics of the component can be related to the applied stress and the crack size.
Note that Eq 3 is a necessary, but not sufficient, condition for fracture. From the Griffith fracture criterion,
fracture does not occur at the extreme value of the energy balance as a function of crack length, U = f(c), but
rather when the derivative of this energy with respect to the crack length, dU/dc, is equal to zero (Ref 3).
If the resistance of the material to fracture is denoted as R, then, according to Ref 3, the conditions for unstable
(brittle catastrophic) fracture can be written as:
(Eq 5a)
Stable (noncatastrophic) fracture is represented as:
(Eq 5b)
where dK/dc and dR/dc are the derivatives of the stress intensity factor, K, and the fracture resistance of the
material, R, with respect to the crack length, c, respectively. Equations 5a(a) and 5b(b) can be illustrated as a
fracture resistance or R-curve, as shown in Fig. 3. Note that the R term can be further described in parts such
that (Ref 3):
R = R
o
+ R(c)
(Eq 6)
where R
o
can be considered the intrinsic fracture resistance, and R(c) is the R-curve component. Note that at
some finite length of crack extension, R becomes constant and R(c) is no longer a function of c. This “steady-
state” value of fracture resistance [R
∞
≠ f(c)] corresponds to fully developed “toughening mechanisms” (Ref 1,
3). R-curve behavior in brittle materials typically develops because of microstructural effects, as shown in Fig.
4. R-curve effects often confuse and frustrate attempts to experimentally measure R
o
(or K
Ic
) because the effects
of crack growth history add to experimental scatter if R (or K
Ic
) is measured and reported outside the context of
the crack extension.
Fig. 3 Schematic representation of R-curve behavior
Fig. 4 Microstructural features responsible for fracture resistance as a function of crack length (R-curve
effects). Source: Ref 1
Part of the debate surrounding the development of standardized test methods (Ref 7, 8, 9, and 10) for
determining the fracture resistance of brittle ceramics has been whether the test method should measure the
intrinsic fracture resistance, R
o
, or the fully developed fracture resistance, R
∞
. To address this question, the
operational aspects of the test method must take into account intrinsic versus extrinsic flaws (or cracks), the
degree of crack extension prior to measurement of the fracture resistance, the resulting linearity (or
nonlinearity) of the loading curve, features on the fracture surfaces, and other “clues” indicating brittle or
nonbrittle fracture.
Typically, brittle materials exhibit low values of fracture toughness (K
Ic
= R
o
) (<1 MPa for many glasses,
<10 MPa for many monolithic ceramics, and <20 MPa for cast irons and hardened steel alloys).
These low fracture toughnesses combined with moderate strengths mean that from a Griffith approach, intrinsic
flaw sizes may only be on the order of 1 to 50 μm, which often lead to low, broad strength distributions. From a
fracture-mechanics approach, lower fracture toughness combined with detectable macrocrack sizes severely
limit the design stresses allowable in components comprised of brittle materials.
Thus, the real utility of measuring the fracture resistance of brittle materials may not be in its direct application
to crack- or flaw-based design (e.g., when does K
I
= K
Ic
?) through either Griffith or fracture-mechanics
approaches. Instead, measures of fracture resistance (e.g., K
Ic
) may find their greatest utility when combined
with other material properties in relative comparison indices, such as those that describe wear or brittleness.
One possible proposed abrasive wear model for ceramics is (Ref 11, 12):
(Eq 7)
where is the volume-loss rate of material, P is the applied load, K
c
is the fracture resistance, and H is the
hardness. Although the exponents in Eq 7 are specific to the application and material, it is apparent from the
model that to minimize abrasive wear at high loads, P, the material must possess high K
c
, high H, and low E.
A possible proposed erosive wear model for ceramics has the form (Ref 12, 13):
(Eq 8)
where ε
v
is the volume of material removed, ν is the impact velocity of the particle with diameter d, and ρ is the
density of the particle. Again, note that for high velocities of impacting particles, the erosive wear of the
material can be minimized by a high K
c
, low E, and low ρ.
Finally, the susceptibility of a material to cracking rather than deformation under the action of a concentrated
stress can be defined in terms of its brittleness (Ref 14, 15):
(Eq 9)
where B is the brittleness. In this case, in order to decrease the tendency toward brittle behavior (i.e., cracking
rather than plastic deformation), B needs to be decreased by decreasing H and E and increasing K
Ic
.
These examples of relative comparison indices show that quantifying the fracture resistance of brittle solids has
direct and important applications other than those conventionally thought of from a fracture-mechanics
viewpoint. In addition, as indicated in Eq 7, 8, and 9, depending on the application, the exponent on the
fracture-resistance term can have a significant influence on the resulting value of the index being calculated.
Thus, accurate and precise measures of the fracture resistance of brittle materials are of paramount importance
for making sensible predictions of material behavior using relative comparison indices.
Footnote
*
ASTM Committee E08 denotes the symbol K
Ic
to represent the “plane strain fracture toughness” of a material.
Plane strain fracture toughness is defined as: “the crack extension resistance under conditions of crack-
tip
plane strain… (and)… the value of stress intensity factor designated as K
Ic
as measured using the operational
procedure (and satisfying all validity requirements) specified in …(ASTM E399)… which provides for the
measurement of crack-
extension resistance at the start of crack extension and provides operational definitions
of crack-tip sharpness, start of crack extension and crack-tip plane strain” (Ref 6).
References cited in this section
1. G.D. Quinn, Strength and Proof Testing, Ceramics and Glasses, Vol 4, Engineering Materials
Handbook, ASM International, 1991, p 585–598
2. A.A. Griffith, The Phenomena of Rupture and Flow in Solids, Philos. Trans. R. Soc. (London) A, Vol
221, 1920, p 163–198
3. B. Lawn, Fracture of Brittle Solids, Cambridge University Press, 1993
4. K.E. Amin, Toughness, Hardness, and Wear, Ceramics and Glasses, Vol 4, Engineered Materials
Handbook, S.J. Schneider, J., Ed., ASM International, 1991, p 599–609
5. G.R. Irwin, Fracture I, Handbuch der Physik, Vol 6, Springer-Verlag, Berlin, Germany, 1958, p 558–
590
6. “Standard Test Method for Plane Strain Fracture Toughness of Metallic Materials,” E 399-90, Annual
Book of ASTM Standards, Vol 03.01, ASTM, 1998
7. STP 678, Fracture Mechanics Applied to Brittle Materials, S.W. Freiman, Ed., ASTM, 1979
8. T. Fujii and T. Nose, Evaluation of Fracture Toughness of Ceramic Materials, IJIS Int., Vol 29 (No. 9),
1989, p 717–725
9. G.D. Quinn, M.G. Jenkins, J.A. Salem, and I. Bar-On, Standardization of Fracture Toughness Testing of
Ceramics in the United States, Kor J. Ceram., Vol 4 (No. 4), 1998, p 311–322
10. M. Sakai and R.C. Bradt, Fracture Toughness Testing of Brittle Materials, Int. Mater. Rev., Vol 38 (No.
2), 1993, p 53–58
11. J. Larsen-Basse, Abrasive Wear of Ceramics, Friction and Wear of Ceramics, S. Jahanmir, Ed., Marcel
Dekker, Inc., New York, 1994, p 99–118
12. A. Evans and D. Marshall, Wear Mechanisms in Ceramics, Fundamentals of Friction and Wear of
Materials, D. Rigney, Ed., American Society for Metals, 1981, p 439–452
13. J. Ritter, Erosion Damage in Structural Ceramics, Material Sci. Eng. Vol 71, 1985, p 195–201
14. J.B. Quinn and G.D. Quinn, Hardness and Brittleness of Ceramics, Ceram. Eng. Sci. Proc., Vol 17 (No.
3), 1996, p 59–68
15. J.B. Quinn and G.D. Quinn, Indentation Brittleness of Ceramics: A Fresh Approach, J. Mater. Sci., Vol
32, 1997, p 4331–4346
Fracture Resistance Testing of Brittle Solids
Michael Jenkins, University of Washington; Johnathan Salem, NASA-Glenn Research Center
Fracture Toughness and R-Curve Testing at Ambient Temperature
In 1970, the fracture-mechanics and metals communities developed the premiere fracture—toughness testing
standard for determining plane strain fracture toughness of metals: ASTM E 399, “Standard Test Method for
Plane Strain Fracture Toughness of Metallic Materials” (Ref 6). ASTM Committee E-8 on “Fatigue and
Fracture” has defined the term fracture toughness and the symbol, K
Ic
, based on the operational methods of
ASTM E 399. This test method includes several necessary aspects for an acceptable fracture test method:
• All test specimens must have atomistically sharp precracks at the onset of fracture
• Precracks must be well characterized and measurable at the conclusion of the fracture measurement
• Crack geometry must have a valid stress intensity factor
• Plane strain conditions must exist at the crack tip for the measurement of plane strain fracture
toughness.
ASTM E 399 was developed and is appropriate for measuring plane strain fracture toughness in ductile metals
but does not necessarily address the unique aspects of brittle solids (e.g., cyclic fatigue precracking is not easily
accomplished in brittle materials). In addition, test specimens are rather large, as are the notches, cracks, and
precracks, presumably for ease of implementation as well as to minimize the influence of dimensional
variations on the repeatability and reproducibility of the fracture-resistance measurement. In recent years,
ASTM committees have developed numerous standard test methods for metals; recently, a “unified” standard
test method for fracture testing was introduced (ASTM E 1820, “Standard Test Method for Measurement of
Fracture Toughness”) (Ref 16). Although a variety of precracking, crack length, fracture parameter (e.g.,
fracture toughness, R-curve, etc.), gripping, and test-specimen preparation techniques are detailed in these
standards, many aspects may not be directly applicable to brittle materials.
Since the introduction of ASTM E 399, standards-writing bodies worldwide have labored to develop similar
robust and well-accepted test methods for brittle solids, particularly advanced ceramics (Ref 7, 8, 9, and 10).
The following sections, where applicable, discuss general aspects of these test methods for extrinsic flaws
(macrocracks) and intrinsic flaws (microcracks) in test specimens. The reader is referred to the standards
themselves for specific details when such standards exist and to appropriate references when a useful technique
is described.
Standard Test Methods for Fracture Toughness at Ambient Temperature
ASTM Standard for Fracture Toughness of Ceramics. After nearly eight years of focused effort, which included
downselecting from five to three operational procedures and the introduction of an interim limited-life (two
years) provisional standard (Ref 17), ASTM C 1421 “Standard Test Methods for the Determination of Fracture
Toughness of Advanced Ceramics at Ambient Temperature” (Ref 18) was approved as a full consensus
standard in early 1999. A technical and historical overview of the evolution of these test methods has been
published (Ref 9). Only a summary of the test methods is discussed here.
Generally, the test methods involve the flexural testing of extrinsically flawed bend bars, as shown in Fig. 5.
Regardless of test method, a minimum of four valid tests is required to complete a test series.