Fracture of Nonmetals and Composites

1051 Pages·1972·21.936 MB·English

by  H. Liebowitz (Eds.)

FRACTURE
An Advanced Treatise
EDITED BY
H. LIEBOWITZ
I : Microscopic and Macroscopic Fundamentals
II: Mathematical Fundamentals
III: Engineering Fundamentals and Environmental Effects
IV: Engineering Fracture Design
V: Fracture Design of Structures
VI : Fracture of Metals
VII: Fracture of Nonm étais and Composites
FRACTURE
An Advanced Treatise
EDITED BY
H. LIEBOWITZ
SCHOOL· OF ENGINEERING AND APPLIED SCIENCE
THE GEORGE WASHINGTON UNIVERSITY
WASHINGTON, D.C.
VOLUME VII
Fracture of Nonmetals and Composites
1972
®
ACADEMIC PRESS New York San Francisco London
A Subsidiary ofH arc ou rt Brace Jovanovich, Publishers
LIST OF CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors' contributions begin.
A. ASSUR (879), U.S. Army Cold Regions Research and Engineering
Laboratory, Hanover, New Hampshire
J. P. BERRY (37), Rubber and Plastics Research Association of Great
Britain, Shropshire, England
R. L. COBLE (243), Department of Metallurgy and Materials Science,
Massachusetts Institute of Technology, Cambridge, Massachusetts.
H. T. CORTEN (675), Department of Theoretical and Applied Mechanics,
University of Illinois, Urbana, Illinois
NORRIS F. Dow (611), G. E. Space Sciences Laboratory, King of Prussia,
Pennsylvania
F. R. EIRICH (351), Polytechnic Institute of Brooklyn, Brooklyn, New York
A. N. GENT (315), Institute of Polymer Science, The University of Akron,
Akron, Ohio
J. GURLAND (841), Division of Engineering, Brown University, Providence,
Rhode Island
GEORGE HERRMANN* (771), Department of Civil Engineering, The Techno-
logical Institute, Northwestern University, Evanston, Illinois
HAROLD LIEBOWITZ (771), School of Engineering and Applied Science,
The George Washington University, Washington, D.C.
LEONARD OBERT (93), U.S. Bureau of Mines, Denver, Colorado
N. M. PARIKH (243, 841), IIT Research Institute, Chicago, Illinois
C. J. PHILLIPS (1), Department of Ceramics, Rutgers—The State Univer-
sity, New Brunswick, New Jersey
B. WALTER RoSENf (611), G. E. Space Sciences Laboratory, King of
Prussia, Pennsylvania
THOR L. SMITH (351), I.B.M. Research Laboratory, San Jose, California
R. J. STOKES (157), Honeywell Corporate Research Center, Hopkins, Min-
nesota
W. F. WEEKS (879), U.S. Army Cold Regions Research and Engineering
Laboratory, Hanover, New Hampshire
* Present affiliation: Department of Applied Mechanics, Stanford University, Stanford,
California.
f Present affiliation: Materials Science Corporation, Blue Bell, Pennsylvania.
V
PREFACE
In this, the seventh and final volume of the Advanced Treatise on Frac-
ture, examination is extended to the fracture of nonmetals and composites.
Progress in the understanding of fracture and in application of that
knowledge depends largely on the successful integration of continuum
mechanics with the scientific disciplines of materials science, physics,
mathematics, and chemistry. Since few people have equal experience in
all these fields, the text of this treatise has been designed so that the reader
may acquire pertinent information by self-study. Most chapters have been
written in detail and, insofar as possible, have been made to fill a significant
gap by also providing, when appropriate, the details of complicated and
involved mathematical derivations in appendixes. Whenever possible, only
a level of college calculus on the part of the reader has been assumed.
Numerical examples showing the engineering applications have been in-
cluded; also, photographs and drawings have been greatly utilized. When-
ever possible and appropriate, reference has been made to both the
theoretical and experimental results and also to the interrelationship between
the microscopic and macroscopic viewpoints. Of particular importance are
the sections near the end of each chapter identifying the technical problems
and the specific research areas where efforts are required to fill present and
anticipated gaps in our understanding of the subject.*
Throughout, an attempt has been made to integrate the atomistic and
continuum approaches as much as possible, particularly by inviting many
outstanding people in the fields of structures and materials to contribute.
In this way, it is hoped that an effective interdisciplinary approach has been
achieved.
Seven major areas are covered in this treatise. They are: (1) microscopic
and macroscopic fundamentals; (2) mathematical fundamentals; (3) engi-
neering fundamentals of fracture and environmental effects ; (4) engineering
fracture design ; (5) fracture design of structures ; (6) fracture of metals ; and
(7) fracture of nonmetals and composites (this volume).
In the first chapter (Phillips), the fracture behavior of glass, as it is com-
monly observed, is reviewed, and the overall complexity of the phenomena
* Imposed editorial requirements precluded the possibility of indicating the references
after 1966.
vii
viii PREFACE
is emphasized. The use of maximum tension as the failure criterion is
reviewed and justified. Methods of estimating ultimate or theoretical
strength are outlined, and some of the difficulties with these simple models
are explained. The high theoretical strengths are compared with the usually
very much lower observed strengths, and the concept of stress concentra-
tion around flaws is introduced to explain the discrepancy. The Inglis and
Griffith criteria are developed and compared with each other and with
other methods of calculating ultimate strength. The importance of surface
condition is stressed, and the information which can often be derived from
fracture surfaces is reviewed. The statistical theories based on the flaw
concept are discussed, and their strengths and shortcomings are outlined.
It is pointed out that, to fully explain static fatigue in glass, it is necessary
to assume some kind of stress corrosion, probably stress dependent, which
can strongly affect stress concentration at the crack tips. To explain the
effects of elevated temperature, it is also necessary to invoke a surface-
weakening mechanism based on contamination from dust, water, devitri-
fication, or all three. Recent ion-exchange experiments are discussed, and
the reality of several types of manmade microcracks is emphasized. The
four observable ranges of glass strength are reviewed and correlated, so far
as possible, with the concepts of notch sensitivity, stress corrosion, and
surface contamination. It is shown that some observations cannot presently
be explained in this way. New fracture criteria are discussed, and several
areas for future research are outlined.
Berry's chapter is concerned with the fracture of polymeric glasses.
Glassy polymers may fail in a brittle or a ductile manner, depending on the
experimental conditions of temperature and time scale. Although these
extremes of behavior and the transition between them can be considered in
terms of the Ludwik hypothesis, it is necessary to determine the mechanism
of the failure process to elucidate the factors that govern the behavior
displayed under any particular set of conditions. Application of the Griffith
flaw theory to brittle fracture indicates that the fracture surface energy and
the inherent flaw size are significant material parameters. The influence of
changing experimental conditions and materials on the values of these
parameters indicates that they are interdependent. A major contribution to
the first arises from the energy required for the formation of a layer of
modified structure at the fracture plane, while the second is related to the
crazes that develop in these materials when they are stressed. Detailed
examination of the structure of crazes reveals that they are planar regions
formed by a hydrostatic tension and consist of oriented material containing
about 50% by volume of interconnecting voids. The fracture surface layer
is believed to possess a similar structure, and the process of brittle fracture
in these materials involves the formation and rupture of craze material.
PREFACE IX
Time-dependent effects are also important, and a large amount of ex-
perimental data can be systematized by a phenomenological theory which
is of the same form as those obtained from molecular considerations. Un-
fortunately, these theories have tended to ignore structure effects, just
as the structure theories have largely ignored time effects, since they are
not readily accommodated within the Griffith approach. Consequently,
there is not yet a completely satisfactory comprehensive theory of polymer
fracture, and it is suggested that the elucidation of the structural factors is a
necessary step in the formulation of such a theory.
Obert's chapter considers the mechanics of the fracture process in rock,
with emphasis on the engineering viewpoint. The development of a con-
sistent fracture mechanics for rock is complicated by several factors, such as
the extreme constitutive variability of rock. These factors are discussed in
some detail. Next, empirical procedures for determining limiting states of
stress are reviewed. In these procedures, the state of stress in the specimen
is either homogeneous or inhomogeneous. However, the limiting states of
stress, as determined in homogeneous and inhomogeneous tests, are often
in disagreement, and possible reasons for differences are discussed. The
Coulomb-Navier, Mohr, and Griffith theories of fracture are reviewed, and
the merits and defects of each considered. Processes by which initial
and branch fractures extend and ultimately produce terminal failure are
examined. The chapter is concluded with a summary in which the state of
the art is evaluated. Areas in which information regarding the fracture
mechanism in rock is deficient or inconsistent are indicated, and topics for
future research are suggested.
Stokes reviews the fracture behavior of simple, single-phase ceramics.
Ceramics may be subdivided into completely brittle, semibrittle, and duc-
tile categories. A further distinction is made between low-temperature and
high-temperature behavior.
Completely brittle ceramics undergo no plastic deformation. Their frac-
ture behavior is governed by the introduction and propagation of flaws. In
polycrystalline material, internal stresses and intergranular flaws contribute
a weakening effect.
Semibrittle ceramics can undergo plastic deformation, but on restricted
slip systems. This leads to accommodation problems which play a role in all
stages of fracture. Crack initiation can occur by the introduction of surface
flaws or, more fundamentally, from accommodation problems caused by
the interaction of slip bands with structural discontinuities such as other
slip bands, kink boundaries, or grain boundaries. Crack extension to critical
dimensions and the low fracture surface energy associated with crack
propagation in semibrittle ceramics is also due to restricted slip.
Ductile ceramics undergo unrestricted slip which permits complete ac-
X PREFACE
commodation between plastically deforming grains and structural dis-
continuities. Deformation then continues to a ductile fracture.
At high temperatures, completely brittle ceramics generally become semi-
brittle. In the polycrystalline form, accommodation problems at grain
boundaries lead to intergranular sliding and intergranular rupture.
The fifth chapter, by Coble and Parikh, constitutes a review of empirical
information about, and our level of understanding of, fracture in poly-
crystalline ceramics. The review is limited to the materials on which wide
ranges of experimental variables have been investigated and from which the
factors governing fracture can be assessed (principally A1 0 and MgO).
2 3
The high strengths observed in single crystal whiskers and fire-polished
macroscopic crystals is taken as confirmation of the calculated theoretical
strengths. Therefore, the low strengths of polycrystalline ceramics require
the assumed presence of preexisting flaws. Two general courses of behavior
are delineated : that the flaws give rise to plastic deformation, or propagate
directly when the stress level satisfies the modified Griffith-Orowan criter-
ion. That is, the strength is greater than (Εγ/d)1^2, in which E is the elastic
modulus, y is the surface energy, and d is the grain size. In rock salt
structure materials at low temperatures, the flaws initially present first give
rise to plastic deformation by slip on the primary slip planes. The distri-
bution of slip governs the hardening and the ease of crack propagation
simultaneously. The ultimate fracture criterion becomes complex, but the
main point is that, in these cases, deformation precedes fracture. The fact
that the observed fracture stresses exceed the Griffith-Orowan criterion
can be accounted for by the fact that the effective surface energy accompany-
ing crack propagation is higher than the true surface energy because of
deformation at the crack tips.
For aluminum oxide at low temperatures, the occurrence of plastic
deformation with fracture is variously reported. Twinning and plastic de-
formation by dislocation movement have been associated with fracture·;
whether they precede crack initiation or occur during crack propagation is
not known with certainty. The higher strengths observed in fracture of
polycrystalline alumina—higher than the strengths predicted by the Griffith
criterion—have been rationalized by three different models : (1) propagation
through a grain with subsequent crack blunting at the grain boundary prior
to intersection of the adjacent grain; (2) that high effective surface energies
are due to plastic deformation accompanying propagation; or (3) that the
initial flaws from which propagation begins have larger radii of curvature
than assumed in the Griffith-Orowan criterion. Further studies are needed
to determine whether cracking proceeds catastrophically from an initial
flaw, or whether cracks form following plastic deformation. Multiple crack-
ing prior to fracture has been observed for a number of rocks subjected to
PREFACE XI
compressive loading and, in a few instances, for natural ceramics in bending
and tensile loading. These observations imply that cracks are initiated and
subsequently stopped in the polycrystalline matrix prior to complete frac-
ture for a reason not yet clearly identified.
High-strength, small grain size materials are found to be susceptible to
strengthening by careful surface preparation ; this suggests that the presence
of surface flaws and, perhaps, grain boundary grooves are most important in
governing fracture in routinely handled specimens. Porous specimens and
those of larger grain sizes and of those containing other impurities are
found not to be susceptible to strengthening by various surface treatments.
For these, it is assumed that the pores present provide sites for fracture
initiation and cause sufficient strength reductions such that surface treat-
ments are to no avail.
At low temperatures, the influence of temperature on strength is dis-
cussed with respect to the independent changes with temperature of the
elastic modulus, surface energy, the possible influence of plastic deforma-
tion, and internal stresses due to thermal expansion anisotropy. The de-
crease in transcrystalline fracture with increasing temperature is not now
interprétable for temperatures below that at which grain boundary sliding
begins because we have, in effect, no basic knowledge about the boundary
structure and properties to permit prediction of the changes in strength
at the grain boundaries to compare with those in the crystals. The effect of
internal stress due to thermal expansion anisotropy is considered, and ap-
pears only to be important as the cause of spontaneous cracking in various
materials of large grain size.
The effect of porosity on strength is considered in relationship to the
change in elastic modulus with porosity. It is found that, for some materials,
an assumed constant stress-concentration factor, considered with the
average stress (which can be evaluated from the elastic modulus) gives the
strength as a function of porosity. The change in stress-concentration
factor with a change in pore shape leads to strength changes with porosity
greater than the elastic modulus changes with porosity. For cases when the
strength changes less rapidly with porosity than does the elastic modulus,
a change in fracture mechanism is assumed.
At high temperatures, pore formation at boundaries, grain boundary
sliding, and intergranular fracture have been observed for both MgO and
A1 0 , though different deformation modes (general slip and diffusion-
2 3
controlled creep, respectively) were operative. For either case, there is not
enough data to establish, or to test, a fracture criterion.
Gent, in Chapter 6, considers the fracture of elastomers. Elastomers are
never perfectly elastic; part of the energy spent in deforming them is
dissipated in overcoming the viscous resistance to motion of the molecular
Xll PREFACE
chains and in breaking structures associated with dispersed solid particles
(fillers) or crystalline regions. These energy losses have recently been shown
to govern the resistance of elastomers to various types of fracture: tensile
rupture, tearing, surface cracking by ozone, cracking and fracture under
repeated deformations (fatigue), and abrasive wear. These findings make it
possible to bring the diverse behavior of different elastomers over a wide
range of temperature and fracture speed into a common pattern; They also
point to the importance of the mechanics of fracture in a particular case.
Both the manner in which elastic energy is transformed into molecular
rupture and the efficiency of the transformation depend on mechanical
features of the fracture process which differ from one type of fracture to
another. Attention is also drawn to modes of failure which might properly
be termed "elastic instabilities,,' as they can be predicted quantitatively
from the elastic properties alone.
In the seventh chapter, Eirich and Smith state that elastomers are dilute
networks of otherwise fluid chain molecules which are strong only when an
optimal internal friction at suitable temperature ranges or rates of extension
leads to chain alignment and partial crystallinity. They become strong also
when the induction of microtriaxial strains due to filler particles reduce
nonuniform stress distribution and invoke volume dilatation without des-
troying self-reinforcement and energy dissipation around flaws. When the
viscosity becomes too high, the strength increases further, but the elas-
tomeric usefulness ends because the rubber turns first leathery and then
hard and glassy. When self-reinforcement, load distribution, and energy
dissipation diminish on account of low internal viscosity, the rubbers
become cheesy and weak.
Practically all elastomers today are made of molecular chains with carbon
backbones. The exceptions are silicones and polyurethanes and the less
important epoxies, but, even in these cases, the characteristics of the effective
chains are not basically altered. It is, therefore, not too surprising that,
when the essential factors of chain stiffness and network extensibility are
taken into account by normalizing to corresponding temperatures or strain
rates and to equal crosslink densities, the mechanical responses of most
rubbers become superimposable, and their maximum strength comparable.
Important progress has been made in the stress analysis and engineering
design of viscoelastic materials. This includes failure criteria of viscoelastic
bodies, the treatment of structured continua, and thermal stresses. Failure
surfaces in coordinates of stress, strain, and temperature-reduced time are
available which afford reasonable predictions of short-, medium-, and
long-time mechanical responses. The failure envelope, as the intersection
of such surfaces with the positive stress-strain plane, has been found to be
particularly useful. It is proposed that the failure envelope can be under-
PREFACE Xlll
stood to be the composite locus of three individual failure criteria, namely,
that of the glassy body, that of the energy-dissipating, fully stretchable,
tough rubber, and the low-energy dissipating, practically ideally elastic, but
nonuniform, rubbery network. The weakness of the latter is proposed to be
due to the absence of viscous stress transmission and various self-reinforcing
mechanisms that can act to reduce stress concentrations in the network.
Next, Rosen and Dow undertake a review of analyses of the failure
mechanics of fibrous composites. The mechanical and geometrical charac-
teristics of fibers which lead to the unique-failure modes of these composites
are described. The influence of the matrix properties on the failure levels is
emphasized.
A description is presented of analyses of the strength of a uniaxial fibrous
composite in simple tension and compression in the fiber direction, and in
shear, both in the fiber plane and in the transverse plane. The use of these
strength theories in the definition of a laminate failure criterion is then
described.
Experimental methods for the measurement of composite material
strength are discussed and new methods are proposed.
Results demonstrate that, while composites exhibit many new modes of
failure, they include materials with a far greater strength and stiffness
potential than commonly available homogeneous materials. The use of
studies of the internal mechanics of fibrous composites to define guidelines
for the development of improved materials is emphasized.
In a chapter on fracture mechanics of composites, Corten introduces the
concepts of linear elastic fracture mechanics and illustrates their application
to the interpretation of fracture behavior of composite materials. Linear
elastic fracture mechanics is based on a description of the linear elastic stress
field around the tip of a crack. The equations for stresses close to a crack tip
in a homogeneous isotropic plane plate are developed. These equations lead
directly to the definition of the stress-intensity factor K a single parameter
y
characterization of the crack tip stress field. The level of K corresponding
to crack extension and fracture is a measure of the fracture toughness of a
material.
For fibrous composites, crack tip stress field equations and the stress-
intensity factors for a linear elastic special orthotropic homogeneous mate-
rial are introduced. Small-scale plastic behavior at the crack tip and its
influence on the crack tip stress field and stress-intensity factor K are
discussed. Crack extension force ô is defined and related to the stress-
intensity factor K. Crack tip stress fields for two-material members with
cracks at and near the interface are presented.
Fracture analysis for two particulate composites systems, a WC rein-
forced cobalt alloy and a W particle reinforced glass composite are re-