# Failure Mechanics—Part I: The Coordination Between Elasticity

Download Failure Mechanics—Part I: The Coordination Between Elasticity

## Preview text

Richard M. Christensen

Professor Research Emeritus Aeronautics and Astronautics Department,

Stanford University, Stanford, CA 94305 e-mail: [email protected]

Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials

Failure mechanics is comprised of the failure theory for homogeneous and isotropic materials along with all of its implications and applications. The present failure theory is found to have an intimate connection with elasticity behavior even though plasticity may also transpire. This becomes apparent and useful when the classical theory of elasticity is renormalized to give a simpler and more transparent (but still exact) formalism. The connection or coordination between elasticity and failure then explicitly occurs through the use of the renormalized Poisson’s ratio to characterize the ductility of failure. With this uniﬁcation of failure theory and elasticity theory, failure mechanics can be extended to explain other anomalous aspects of mechanical behavior and prepare it for applications. [DOI: 10.1115/1.4027753]

Status of Failure

This work establishes an important relationship between elasticity theory and failure theory for three-dimensional homogeneous and isotropic materials. At ﬁrst exposure, one could reasonably question how there could possibly be any such connection between the elasticity formulation and that of failure. For well over a hundred years mainstream thinking has held that elastic behavior at increasing levels of stress yields into plastic behavior and that then progresses until some critical plasticity limit condition inevitably requires ﬁnal and complete rupture. Unless the failure was totally brittle, failure has nearly universally been perceived as the ultimate extreme and conclusion of plastic deformation. It is so reported in virtually all tables of properties for speciﬁc materials.

A newly developed failure theory has challenged that longstanding, conventional view. This new theory, Christensen [1], is based upon two physical postulates. The ﬁrst is that failure represents the termination of elastic behavior, not plastic behavior. This is obviously true for brittle materials, but for elastic–plastic behavior, failure is still taken to most fundamentally signify the cessation of the previous elastic behavior. The possible occurrence of plasticity simply represents a more complex transition from the elastic state to the failure state than occurs with brittle materials.

The common elastic–perfectly plastic idealization convincingly illustrates the futility of trying to use plasticity to directly and solely deﬁne failure in terms of an actual rupture. Other contrary, examples also are apparent, but have not been sufﬁciently well appreciated.

This new approach for failure theory in no way detracts from the meaning, the value, and the usefulness of plasticity theory. There are large classes of problems involving highly nonlinear and permanent deformation requiring plasticity treatments. It simply asserts that the general deﬁnition of three-dimensional failure criteria must be viewed by means other than through the lens of ultimate plastic deformation.

The second and equally important key to the new theory of failure is the hypothesis that the strengths ratio T/C covers the full

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 16, 2014; ﬁnal manuscript received May 23, 2014; accepted manuscript posted May 29, 2014; published online June 5, 2014. Editor: Yonggang Huang.

range of all possible behaviors for homogeneous and isotropic materials. Symbols T and C are the failure stresses in uniaxial tension and compression. Thus,

T

0

1

C

provides the failure descriptor for all failure types and classes. There is an important corollary and consequence to this ratio of

the uniaxial strengths as representing the full and complete spectrum of possible failure behaviors. It follows from this hypothesis that the general theory must be of a two property form, not the three or four or more parameters empirical forms that are often thought and argued to be necessary.

These two physical hypotheses are sufﬁcient to guide the way to the mathematical development of the general, complete, and self-contained theory of materials failure [1], for full density, homogeneous, and isotropic materials.

After stating the controlling forms for the new failure formalism and for linear elasticity, their explicit relationship will be closely examined in the next two sections. It will be shown that there is a type of coordination or coupling between the two formalisms. This coordination concerns the differentiation between ductile and brittle types of failures. The elasticity part of the combination reveals new and independent information on this important subject. The enabling development that makes this possible is the renormalization of the classical theory of elasticity, presented here for the ﬁrst time.

Finally, the Failure Mechanics Relationships section considers the uniﬁcation of elasticity theory and failure theory and treats the overall discipline as that of failure mechanics. This failure mechanics is for homogeneous and isotropic materials and it is completely separate from and complementary to the ﬁeld of fracture mechanics. The overall coverage and signiﬁcance of all of the historical mechanics related disciplines, including these two, are given a ﬁnal survey and assessment.

Failure Theory and Renormalized Elasticity Theory

The constitutive relations for linear, isotropic elasticity theory will be recalled for use here in one of its many but equivalent forms. The most obvious and natural form is that of the expression of the elastic energy partitioned into dilatational and distortional

Journal of Applied Mechanics

Copyright VC 2014 by ASME

AUGUST 2014, Vol. 81 / 081001-1

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

terms and calibrated by the corresponding properties. This can be done with either stresses or strains. The account here will begin with stresses and then later return to strains. This form of elastic energy is given by

1 rii2 1 U ¼ 2k 3 þ 4l sijsij (1)

where k is the bulk modulus and l is the shear modulus. The deviatoric stresses and strains are deﬁned by

sij ¼ rij À dij rkk

3

(2)

eij ¼ eij À dij ekk

3

The failure theory given by Christensen [1] has the basic crite-

rion developed by the method of polynomial invariants as

1 À T r^ii þ 3 s^ijs^ij T (3)

C

2

C

where the stress is nondimensionalized by the compressive strength

r^ij ¼ rij

(4)

C

and where T and C are the failure stresses with the brittle to ductile limits and range speciﬁed by

T

0

1

(5)

C

The failure criterion (3) is and must be supplemented by a competitive fracture criterion given by

T

T1

r^1

if

(6)

C

C 2

where r1 is the maximum principal stress. See Ref. [1] for the full background, motivation, the derivation, and for the interpretation of this new theory of failure.

The elastic energy (1) and the failure criterion (3) are from completely different derivations with each receiving an independent polynomial invariants type of expansion in the invariants of the stress tensor.

All the attention here will be upon the polynomial invariants failure criterion (3). Nothing will be done here that involves or would be superseded by the fracture criterion (6).

It will be helpful to express the elastic energy (1) in the alternative and less obvious, hybrid form using Young’s modulus, E, and Poisson’s ratio, , but still retaining the dilatational and distortional source terms. This converts (1) to

! U ¼ 21E ð1 À32Þ r2ii þ ð1 þ Þsijsij (7)

where it is clear that must have the limits

À 1 1 (8) 2

Henceforth, E will be referred to as the elastic modulus. Compare the elastic energy (7) and the failure form (3). They

are of somewhat similar forms but with some distinct differences, especially in the quadratic dependence on rii in Eq. (7) but the linear dependence on it in Eq. (3). These two forms (3) and (7) appear slightly more similar when Eq. (7) is rewritten as

!

U ¼ ð12þEÞ 13 11Àþ2 r2ii þ sijsij (9)

Using the limits for in Eq. (8) and the limits for T/C in Eq. (5)

then the range of the coefﬁcient of r^ii in Eq. (3) is from 0 to 1 while the range of the coefﬁcient of r2ii in Eq. (9) is from 0 to 1. In a sense the elasticity theory based on Eq. (7) is formalized such

that the governing property, , is normalized at the value of ¼ À1. It could be helpful if the elasticity theory were renormalized with a different nondimensional variable. Renormalization is

often used in physics for special utility or for new insights. The renormalization will be centered on ¼ 0. All properties will be taken to be unchanged at and only at ¼ 0.

As the ﬁrst step in this direction, recognize that it is only the positive values of that are physically realistic, 0 1=2. Negative values for are never found for homogeneous and isotropic

materials. There is a very good reason for this behavior and it

exists at the nanoscale. This restriction will be detailed later. But for now, it is henceforth taken that only positive values of carry

and convey physical reality.

It would furthermore be advantageous if the elasticity theory could be renormalized such that with 0 1=2 renormalizes to a variable that ranges from 0 to 1 the same as with T/C in Eq. (5).

So the renormalization of elasticity theory is the main objective here. To this end, compare the coefﬁcients of r2ii and r^ii in Eqs. (9) and (3). To bring them into closer alignment take

1 À 2 ¼ 1 À Ã

(10)

1þ

where * is a tentative new nondimensional property. Solving Eq. (10) for * gives

Ã ¼ 3

(11)

1þ

Thus * is found directly and solely from . Over the range of in Eq. (8), it follows from Eq. (11) that *

has the range

À1 Ã 1

(12)

but over the physically relevant and physically acceptable range of as

1

0

(13)

2

then from Eq. (11) the corresponding range on * is

0 Ã 1

(14)

The range of * in Eq. (14) is of the required form needed to move forward. The new property * in place of is the renormalized Poisson’s ratio.

Correspondingly take the renormalized elastic modulus as

EÃ ¼ E

(15)

1þ

Then the renormalized constitutive relation for linear elasticity

from Eqs. (9), (11), and (15) is given by

1 ð1 À ÃÞ

!

U ¼ 2EÃ

3 r2ii þ sijsij

(16)

where * has the general range of Eq. (12) but the physically restricted and realistic range of Eq. (14). Renormalized modulus E* is always positive.

081001-2 / Vol. 81, AUGUST 2014

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

From this point onward the range of * will always be taken as Eq. (14) even though it applies over the full range of Eq. (12). Next the consequent elastic forms that follow from Eq. (16) will be derived.

The classical two Lame0 constants and the bulk modulus are given by

k ¼ E ð1 þ Þð1 À 2Þ

l ¼ E (17) 2ð1 þ Þ

k¼ E 3ð1 À 2Þ

These properties in terms of E* and * are then given by

ÃEÃ k ¼ 3ð1 À ÃÞ l ¼ EÃ (18)

2 EÃ

k ¼ 3ð1 À ÃÞ

The variational result eij ¼ @U (19) @rij

gives the consequent strain–stress relations from Eq. (16) as

1

Ã

!

eij ¼ EÃ rij À 3 dijrkk

(20)

The other two complementary forms are found to be the energy

form in terms of strains as

EÃ

1

!

U ¼ 2 3ð1 À ÃÞ e2kk þ eijeij

(21)

and the stress–strain relations as

Ã

!

rij ¼ EÃ eij þ 3ð1 À ÃÞ dijekk

(22)

Taking mean normal stresses and strains as

rM ¼ rii

3 e

(23)

eM ¼ ii

3

then gives the slightly different forms of the four elastic constitutive relations as being

U ¼ 2E1 Ã Â3ð1 À ÃÞr2M þ sijsijÃ (24)

EÃ 3

!

U ¼ 2 ð1 À ÃÞ e2M þ eijeij

(25)

eij ¼ E1Ã Ârij À ÃdijrMÃ (26)

Ã

!

rij ¼ EÃ eij þ ð1 À ÃÞ dijeM

(27)

where E* and * are deﬁned by Eqs. (15) and (11), and subscript M is not an index. E* and *could be determined other than from

the experiments implied in Eqs. (11) and (15). This will be outlined in The Coupling Between Elasticity and Failure section.

These two groups of four relations (16) and (20)–(22) or (24)–(27) specify the renormalized forms of linear elasticity. Any one of the four relations serves to deﬁne the other three. These four simple forms remain as the same classical theory of elasticity, there are no approximations. Furthermore, these tensor results are probably the most compact, and most concise statements of the linear elastic constitutive relations. The advantage of expressing energy and strains in terms of stresses, (24) and (26), is apparent.

The four constitutive forms also reveal that the two properties E and , with subsequent E* and *, are highly preferable and more basic than the pairing of l and k. The properties l and k in these four forms lack the same symmetry and simplicity and in some cases are ambiguous because of unlimited behaviors in limiting cases.

There likely is no advantage in using this new formalism for two-dimensional or one-dimensional conditions, nor for numerical computations. It is purely for the three-dimensional formulation that is needed for basic theory purposes, especially in relation to failure conditions. It is somewhat surprising that this renormalized formulation of elasticity theory was not recognized much earlier in the past 100–150 years of intensive study.

From relation (11), some corresponding values of and * are given by

*

0

0

1/8

1/3

1/5

1/2

1/3

3/4

1/2

1

Inverting Eqs. (11) and (15) gives and E expressed in terms of * and E* as

Ã

¼ 3 À Ã

(28)

3EÃ

E ¼ 3 À Ã

(29)

Some further relationships are as follows:

EÃ ¼ E at ¼ Ã ¼ 0

EÃ ¼ 2 E at ¼ 1 ; Ã ¼ 1

(30)

3

2

Although E* and E are about the same size there is a signiﬁcant difference and the proper property is E* not E for the threedimensional calibration of elastic constitutive equations.

E* is the universal and dimensional property expressing a materials general capability to resist deformation while bearing load. Property * is the universal and nondimensional new property with what will be found to be a deep and insightful interpretation of what must be happening at the atomic scale that controls the balance of the energy sources involving the dilatational versus the distortional resistance.

Both the elastic energy form (16) and the failure form (3) involve a single dimensional property and a single nondimensional property. It follows that the limits of the nondimensional property in Eq. (16) associates with either a ductile or a brittle failure condition in uniaxial tension as given by

Ã ¼ 0 Totally Brittle

Ã ¼ 1 Perfectly Ductile

(31)

The basis for relations (31) is as follows. At * ¼ 0, the constitutive relation (26) shows that the stress and strain tensors are co-

Journal of Applied Mechanics

AUGUST 2014, Vol. 81 / 081001-3

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

axial, there is no Poisson’s ratio effect. This means that there is no coupling between the stress components. Then the related failure behavior corresponds to the maximum normal stress criterion of Lame0 and Rankine, which has always been interpreted as that of brittle fracture. Therefore this limit of * ¼ 0 is taken as that of the totally brittle state.

At the other limit of * ¼ 1, relation (24) shows that there is no dilatational contribution to the energy and Eq. (25) shows that there is no dilatation, eM ¼ 0. The deformation is purely distortional. This is the state of incompressibility which usually corresponds to the behavior of elastomers and approached by extremely ductile metals. This is the state of perfect ductility.

Relations (31) give the ﬁrst indication of a possible coupling behavior between the two very different theories of mechanical behavior, elastic energy, and failure. The ductile versus brittle behavior as a function of T/C is fully explored in Ref. [1]. That the Poisson’s ratio through * also associates with brittle versus ductile behavior comes as an unexpected result from this derivation. However, it is not completely unexpected from a practical, observational point of view. Beryllium has a Poisson’s ratio of virtually 0 and it is very brittle. Gold has a very large Poisson’s ratio and it is very ductile. It is this possible relationship between * and ductility that will be pursued further.

As a prelude to the further developments, it can now be rationalized that there could conceivably be some relationship between elasticity theory properties and failure theory properties, especially since failure is perceived as the termination of elastic behavior.

The Coupling Between Elasticity and Failure

As derived in Ref. [1], the state of uniaxial tension has a speciﬁc measure of the degree of ductility D through

D ¼ T Uniaxial Tension (32) C

This extraordinarily simple ductility measure goes from 0 to 1 meaning from the state of total brittleness to the state of perfect ductility. Similarly more complicated forms for the ductility measures can be derived for any stress state and are done so in Ref. [1]. But the interests here are solely with uniaxial tension since it is by far the most important special case, and it is universally used to calibrate comparative ductility levels for different materials. The ductile brittle transition for uniaxial tension in Eq. (32) is at T/C ¼ 1/2. This also is explained and explored at length in Ref. [1] and will later be derived here by a different method.

In view of the ductility limits shown in Eq. (31), it follows that the renormalized elastic property * may have some use as a continuous measure of the degree of ductility in uniaxial tension. To this end, take D* as the possible measure of ductility in terms of *. At this point it is only known from Eq. (31) that D* as a function of * has the limiting behaviors

DÃðÃÞ ¼ 1 at Ã ¼ 1

DÃðÃÞ ¼ 0 at Ã ¼ 0

(33)

ð1 À3 ÃÞ r2ii þ sijsij ¼ 2EÃU (35)

and

1 À T r^ii þ 3 s^ijs^ij T (36)

C

2

C

where the limits on * are given by Eq. (14) and those on T/C by Eq. (5).

Note that all the forms and properties in the two relations (35) and (36) are either quadratic or linear in form. In accordance with these characteristics, the most likely values for exponent n in Eq. (34) are either n ¼ 1 or n ¼ 2, linear or quadratic.

From Eq. (36), it is seen that T/C in the ﬁrst term is linear and this may therefore favor taking n ¼ 1 in the ductility measure (34). On the other hand, it is also seen that T/C is associated with the linear term r^ii in Eq. (36) while * is associated with the quadratic term r2ii in Eq. (35). This may favor taking n as being quadratic, n ¼ 2.

More speciﬁc reasoning to ascertain the proper value for n is as follows. The ductile brittle transition in terms of T/C is at T/C ¼ 1/2. This was developed by a general and elaborate methodology in Ref. [1]. A much simpler approach will be sufﬁcient for the restricted case of uniaxial tension and will now be given.

For uniaxial stress, r, the failure criterion (36) reduces to

Tr

r2 T

1ÀC C þ C C

"

"

(37)

Dilatational Distortional

For uniaxial tensile failure at r ¼ T and for equality in Eq. (37), it becomes

1 À T þ T ¼ 1 (38)

C

C

The separate sources from dilatation and distortion are so labeled in Eq. (37). At the two limits, Eq. (38) becomes

At T ¼ 1; 0 þ 1 ¼ 1

C

"

(39)

Distortional

and

At T ¼ 0; 1 þ 0 ¼ 1

C

"

(40)

Dilatational

Let the possible ductility measure take the general power law form

DÃðÃÞ ¼ ðÃÞn; n ! 1

(34)

The form (34) satisﬁes the required limits in Eq. (33). The exponent n in Eq. (34) must be taken to satisfy some as yet unspeciﬁed but logical condition. This condition must involve compatibility with the observed behaviors of the entire spectrum of isotropic materials types.

At this point, the elastic energy and the polynomial invariants failure criterion from Eqs. (16) and (3) will be rewritten here for direct comparison with each other. These are

At the ﬁrst limit (39), the distortion source dominates and the behavior is ductile. Of course, plasticity behavior will ensue near the T/C ¼ 1 end of the scale since the distortional source dominates. At the second limit (40), the dilatation source dominates and the failure behavior is brittle.

At the value of T/C ¼ 1/2, Eq. (38) becomes

At T ¼ 1 ; 1 þ 1 ¼ 1 (41) C 2 22

and the dilatational source and the distortional source are in balance and this is taken as the ductile brittle transition. Thus the value T/C ¼ 1/2 denotes the ductile brittle transition in uniaxial

081001-4 / Vol. 81, AUGUST 2014

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

tension for this failure criterion. This is in accordance with the much more general method from Ref. [1].

Just as T/C ¼ 1/2 gives the ductile brittle transition, the corresponding ductile brittle transition in terms of * will be taken as midway between the limits for D* of 0 and 1, namely at D* ¼ 1/2. The two candidate measures for D* then give the respective ductile brittle transition as occurring at

n ¼ 1; DÃ ¼ Ã ¼ 1

(42)

2

or at

n ¼ 2; DÃ ¼ ðÃÞ2¼ 1 ; Ã ¼ p1ﬃﬃ

(43)

2

2

From relation (28), it further follows that the conceivable ductile brittle transition occurs at

n ¼ 1; ¼ 1 (44) 5

or at

n ¼ 2; ¼ pﬃ1ﬃ ¼ 0:3084

(45)

3 2À1

The n ¼ 1 prediction of the ductile brittle transition as occurring at ¼ 1/5 is completely unrealistic. Materials such as carbon (diamond) and silicon are near this value of Poisson’s ratio and they

are distinctly brittle in failure behavior. The case for the ductility measure at n ¼ 1 in Eq. (34) is com-

pletely unacceptable and it is discarded. This leaves the n ¼ 2 case of Eq. (34) as

DÃ ¼ ðÃÞ2

(46)

and as being the most likely measure of ductility in terms of the renormalized Poisson’s ratio *. This measure of ductility in uniaxial tension is still provisional at this juncture, pending veriﬁcation with explicit materials comparisons. The comparisons will be conducted shortly.

The signiﬁcant values for D*, *, and from Eqs. (46) and (28) are shown in Table 1. The ductile brittle transition at D* ¼ 1/2 has already been discussed. Also shown are the values at D* ¼ 1/4. For lack of a better term this is called a special case because it does appear to have some physical signiﬁcance. Very few materials have Poisson’s ratios of less than 1/5. The coefﬁcient of the dilatational term in Eq. (24) is half way between its maximum and minimum values of 1 and 0 at this value of Poisson’s ratio.

It is possible to determine E* and * other than from experiments directly on E and . From Eq. (18), it follows that:

EÃ ¼ 2l

(47)

and

Ã ¼ 1 À 2l

(48)

3k

In most cases it is probably far easier to use Eqs. (11) and (15) directly. However, relation (47) does incisively show that E* is proportional to the distortional source of resistance while in Eq. (48), * involves the interplay between the two independent physical effects, distortional and dilatational resistances to deformation. Relations (47) and (48) should be compared with the complex classical forms relating these four properties.

The comparison of ductility levels predicted by Eq. (46) with actual materials is given next as a critical test of the result (46).

Table 1 Ductility D*, Eq. (46)

Ductility D* ¼ (*)2

*

Totally brittle

0

0

0

Special case

1/4

1/2

1/5

D/B transition

1/2

0.71

0.31

3/4

0.87

0.41

Perfectly ductile

1

1

1/2

Table 2 shows a list of the solids forming common elements from the periodic table. The Poisson’s ratios are from the sources referenced in Ref. [1]. The prediction of ductility levels in the last column of Table 2 is in accordance with general understanding of the spread of brittle to ductile behaviors for the different materials types. The ductile brittle transition of Eqs. (43) and (45) occurs at the elements of nickel and cobalt which also is in reasonable accord with practice. This ductile brittle transition is not a transition in the thermodynamic sense of the term. But it is a meaningful transition as the division between predominately brittle versus predominately ductile groupings.

Even though Poisson’s ratio is difﬁcult to determine with high accuracy, the overall results of Table 2 establish the validity of using renormalized Poisson’s ratio in Eq. (46) as a characterization of the ductility in uniaxial tension. It should be noted that the results in Table 2 are identical to the results found from a nanoscale analysis in Ref. [1].

The nanoscale analysis just mentioned was based upon differentiating the effects of bond stretching versus bond bending. The ductile end of the scale has no bond bending resistance compared with the bond stretching resistance. The brittle end of the scale maximizes the bond bending resistance relative to the bond stretching resistance. The present analysis has the ductile end of the scale as having no dilatational energy in Eq. (35) at * ¼ 1. The brittle end of the scale has the dilatational energy in Eq. (35) as maximized at * ¼ 0. These macroscopic scale explanations and understandings of behavior are completely compatible with the nanoscale explanations and understandings.

Table 2 Ductility of the solids forming elements

Element

Ductility D* ¼ (*)2

Absolute limit, perfect ductility

1/2

1

Gold

0.44

0.84

Lead

0.44

0.84

Niobium

0.40

0.73

Palladium

0.39

0.71

Platinum

0.38

0.68

Silver

0.37

0.66

Vanadium

0.37

0.66

Tin

0.36

0.63

Aluminum

0.35

0.60

Copper

0.34

0.58

Tantalum

0.34

0.58

Titanium

0.32

0.53

Nickel

0.31

0.50

Cobalt

0.31

0.50

Magnesium

0.29

0.45

Iron

0.29

0.45

Tungsten

0.28

0.43

Zinc

0.25

0.36

Manganese

0.23

0.31

Uranium

0.23

0.31

Silicon

0.22

0.29

Plutonium

0.21

0.27

Chromium

0.21

0.27

Carbon (diamond)

0.20

0.25

Limit for most elements/materials

1/5

1/4

Beryllium

0.032

0.01

Absolute limit, total brittleness

0

0

Journal of Applied Mechanics

AUGUST 2014, Vol. 81 / 081001-5

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

The nanoscale analysis in Ref. [1] also served the important purpose of proving that the physically unrealistic range of bond bending relative to bond stretching resistance disallows negative values of Poisson’s ratio. Finally, it must be emphasized that the relation between Poisson’s ratio and ductility has notable exceptions and its use here is mainly for studying fundamental effects rather than for general utility. In such exceptions, other physical effects are tailored to intercede, as with steel versus iron.

When interests are in a comprehensive treatment of ductile brittle matters, the account in Ref. [1] should be followed where the failure number, Fn, is derived and deﬁned to specify the ductility level for any state of stress. The purpose here was entirely different. It was to establish the foundation of the new failure theory by carefully examining the connection between elasticity theory and failure theory. The ductility characterization for uniaxial tension was shown to be the strong and supporting linkage between the two theories. From a failure point of view, it would be appropriate to say that this new development provided not just the coupling between the two theories but the integration into and uniﬁcation of failure theory with elasticity theory.

Failure Mechanics Relationships

The term failure mechanics is here taken to cover and embrace failure theory for homogeneous and isotropic materials with all of its associations and all its implications and applications. It is the latest in a long line of mechanics related and formulated disciplines.

It all began with the monumental and almost unthinkably difﬁcult creation of classical mechanics by Isaac Newton. Then over about the following two centuries the disciplines of solid mechanics and ﬂuid mechanics came into being and useful forms gradually evolved through the dedicated and inspired efforts of such great scientists as Cauchy, Bernoulli(s), Euler, Navier, Lame0, Poisson, Stokes, and many others.

The dominant mechanics achievements of the 20th century were the two disciplines of quantum mechanics and fracture mechanics. It is pointless to try to directly compare them because one is at the atomic scale and the other at the macroscopic scale. Both became vitally important.

In quantum mechanics the problem was to understand the behavior of the electrons orbiting the atomic nucleus under their opposite charges. Were the electrons best viewed as being particles or as waves? The evidence was contradictory. Then the possibility emerged that the behaviors were not deterministic but rather must be seen as probabilistic. That possibility was then enlarged to more general concepts of the uncertainty of predictability. The complexity of the behaviors was enfolded and enshrouded in the full and glorious complications of threedimensional interactions.

Over a prolonged and frenzied period of activities in the 1920s Bohr and Heisenberg and Dirac and Schrodinger and Born and others sorted through, debated, and argued all the possibilities. The community ultimately reached a conclusion, something of an amalgam of all of the above. The almost unsolvable complexity of the quantum mechanics problem was at least partially due to its three-dimensional nature. It was almost but not quite unsolvable. There ﬁnally was a workable resolution to the problem and it has carried forward and endured.

Fracture mechanics, like quantum mechanics, evolved and progressed through various stages in its development. Grifﬁth [2] originated it and conceived the crack growth problem as an energy balance between the release of elastic strain energy against the work to create new crack surface. Irwin [3] had the insight to characterize the failure inception as related to the nature of the elastic singularity at the crack tip. This offered a normalizing and comparative capability. Rice [4] effected a further breakthrough by relating crack tip conditions to the properties of an invariant contour integral enclosing the crack. Barenblatt [5] re-examined

crack tip conditions and found the nonsingular stress state that helped to explain much of the behavior.

All of these efforts in combination formed the discipline of fracture mechanics. It has become enormously successful in design applications where materials are made into load bearing structures containing a single worst ﬂaw (or ﬂaws) that could be idealized as a crack (or cracks). Although fracture mechanics can be formulated in some special three-dimensional cases, its fundamental and wide ranging applicability is of a two-dimensional nature and type. The success of fracture mechanics proved that the general subject of failure characterization no longer need be relegated to the state of just being an undisciplined agglomeration of empirical formulas.

Perhaps the completing mechanics related discipline will be that of the failure mechanics needed to characterize the threedimensional failure behavior of homogeneous and isotropic materials. The fact that it is coming into existence only now was not for lack of effort over the years and over the centuries. Its great difﬁculty of development comes from the same source as that for quantum mechanics, namely, its fully three-dimensional nature having extremely complex interactions.

It is highly unlikely that failure mechanics, as envisioned here, could have been developed before the advent of fracture mechanics. Part of this is no doubt due to the clean and clear twodimensional nature of fracture mechanics compared with the much greater complexity of general failure in three dimensions. Also, for the failure of homogeneous materials, there is no guiding certainty as to the site of failure initiation, as occurs with the preexisting, stress concentrating cracks in fracture mechanics. Fracture mechanics had to lead the way for progress on failure. So the emergence of failure mechanics at this latter time has a reasonable coordination with the order and sequence of development of the other mechanics related disciplines.

There is no need to summarize the derivation and capability of failure mechanics. That is comprehensively treated in Ref. [1] under the heading of failure theory. The special interpretation developed herein however merits particular attention. Elasticity theory and failure theory for homogeneous and isotropic materials are not absolutely independent constructs. One without the other is incomplete. Although there is no single parameter that appears in both theories and bridges between them, there nevertheless is a more subtle but no less effective coordination or coupling between them.

To explain this coupling between the two theories, it is important and even necessary to recognize that any failure theory that does not fully treat ductile versus brittle failure conditions is not complete. Even more serious, such an incomplete failure theory cannot be assessed and evaluated, its status must be uncertain, even of doubtful validity until proven otherwise. The failure theory developed in Ref. [1] has the strengths ratio T/C and the stress states as determining the ductile versus brittle behaviors. More speciﬁcally, for the most important special case of uniaxial tension, the ratio T/C itself directly delineates the degree of ductility over the range from 0 to 1.

In the case of linear elasticity theory, the work here has shown that renormalized Poisson’s ratio, *, also gives a direct measure of the degree of ductility for uniaxial tension. This occurs through Eq. (46) over the same range of 0 to 1 as that for T/C.

It is thus the ductility measures from both the failure theory and from elasticity theory that provides the bridge between the two theories. At ﬁrst it is exceedingly surprising that elasticity theory could say anything at all about the ductility in failure. But on deeper reﬂection it is not so surprising. The atomic scale characteristics of bond stretching and bond bending directly determine and differentiate not only T and C but also . Poisson’s ratio, , actually contains a great deal of information that has heretofore been ignored or vastly under-appreciated.

Poisson’s ratio, , is not at all secondary to the elastic modulus, E, it is the full and equal partner to E and of great signiﬁcance in its own right. This new and enlarging insight became fairly

081001-6 / Vol. 81, AUGUST 2014

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

apparent when the renormalized theory of elasticity was obtained with the renormalized Poisson’s ratio. Contrast this with the classical form of elasticity theory, usually using the two Lame0 constants k and l. The k property has no physical interpretation at all. E and or better yet their renormalized forms E* and * represent the most meaningful statements of elastic properties, even with direct implications for failure. In association with E* and *, the uniaxial strengths T and C turn out to give the most meaningful and useful calibrating anchors for failure theory.

It is the failure theory of Ref. [1] (and herein) in its broadest context that comprises the discipline of failure mechanics. The comparison of failure mechanics (for solids) with ﬂuid mechanics and with quantum mechanics is both illuminating and technically useful. In ﬂuid mechanics and for slow ﬂows, the Navier–Stokes equations give a linear behavior between the shear stress and the velocity gradient. But as the velocities increase, the nonlinear effect of the inertial source invalidates the previous linear relations and leads to turbulence.

For failure mechanics a similar situation evolves. Small deformation states produce linear relationships among the ﬁeld variables of elasticity theory. But the nonlinear intercession of failure theory invalidates the previous conservative, linear behaviors. Nonlinear intercessions provide the limits to the linear behaviors of both solids and ﬂuids. Elasticity theory without failure theory is open ended and incomplete. They must go together to provide closure.

The comparison between failure mechanics and quantum mechanics of course does not contain any analogies or congruencies between them. But both do share an overriding common characteristic that had much to do with their histories of development. Perhaps this would be better described as the common barriers that greatly impeded their respective developments. The commonality is simply this. Neither of these two mechanics disciplines yielded to the standard and ordinary expectations. Both solutions turned out to be strongly counterintuitive and it took major expenditures of time and effort to ultimately synthesize them, one over

a highly intensive time period and the other over an agonizingly long and drawn out time extension.

All six of these mechanics related ﬁelds: Classical mechanics, solid mechanics, ﬂuid mechanics, quantum mechanics, fracture mechanics, and failure mechanics comprise basic sources of mechanistic knowledge. The ﬁrst ﬁve of them are mature, scientiﬁc disciplines. The newest and sixth one, failure mechanics, cannot be said to be mature, it is still being assimilated. The comprehensive and complete theory was given in Ref. [1], the full, broad, and inclusive understanding of it necessarily will take longer.

The present article “Failure Mechanics, Part I” will later be followed by “Failure Mechanics, Part II: The Central and Decisive Role of Graphene in Deﬁning the Elastic and Failure Properties for all Isotropic Materials.” This coming work will go much further toward completing the foundation upon which failure mechanics is built.

Acknowledgment

The author was appreciative of support for this work from the Ofﬁce of Naval Research, Dr. Y. D. S. Rajapakse, Program Manager.

References

[1] Christensen, R. M., 2013, The Theory of Materials Failure, Oxford University Press, Oxford, UK.

[2] Grifﬁth, A. A., 1921, “The Phenomena of Rupture and Flow in Solids,” Trans. R. Soc. London A, 221(582–593), pp. 163–197.

[3] Irwin, G. R., 1957, “Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate,” ASME J. Appl. Mech., 24(3), pp. 361–364.

[4] Rice, J. R., 1968, “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks,” ASME J. Appl. Mech., 35(2), pp. 379–386.

[5] Barenblatt, G. I., 1962, “The Mathematical Theory of Equilibrium of Cracks in Brittle Fracture,” Adv. Appl. Mech., 7, pp. 55–129.

Journal of Applied Mechanics

AUGUST 2014, Vol. 81 / 081001-7

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

Professor Research Emeritus Aeronautics and Astronautics Department,

Stanford University, Stanford, CA 94305 e-mail: [email protected]

Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials

Failure mechanics is comprised of the failure theory for homogeneous and isotropic materials along with all of its implications and applications. The present failure theory is found to have an intimate connection with elasticity behavior even though plasticity may also transpire. This becomes apparent and useful when the classical theory of elasticity is renormalized to give a simpler and more transparent (but still exact) formalism. The connection or coordination between elasticity and failure then explicitly occurs through the use of the renormalized Poisson’s ratio to characterize the ductility of failure. With this uniﬁcation of failure theory and elasticity theory, failure mechanics can be extended to explain other anomalous aspects of mechanical behavior and prepare it for applications. [DOI: 10.1115/1.4027753]

Status of Failure

This work establishes an important relationship between elasticity theory and failure theory for three-dimensional homogeneous and isotropic materials. At ﬁrst exposure, one could reasonably question how there could possibly be any such connection between the elasticity formulation and that of failure. For well over a hundred years mainstream thinking has held that elastic behavior at increasing levels of stress yields into plastic behavior and that then progresses until some critical plasticity limit condition inevitably requires ﬁnal and complete rupture. Unless the failure was totally brittle, failure has nearly universally been perceived as the ultimate extreme and conclusion of plastic deformation. It is so reported in virtually all tables of properties for speciﬁc materials.

A newly developed failure theory has challenged that longstanding, conventional view. This new theory, Christensen [1], is based upon two physical postulates. The ﬁrst is that failure represents the termination of elastic behavior, not plastic behavior. This is obviously true for brittle materials, but for elastic–plastic behavior, failure is still taken to most fundamentally signify the cessation of the previous elastic behavior. The possible occurrence of plasticity simply represents a more complex transition from the elastic state to the failure state than occurs with brittle materials.

The common elastic–perfectly plastic idealization convincingly illustrates the futility of trying to use plasticity to directly and solely deﬁne failure in terms of an actual rupture. Other contrary, examples also are apparent, but have not been sufﬁciently well appreciated.

This new approach for failure theory in no way detracts from the meaning, the value, and the usefulness of plasticity theory. There are large classes of problems involving highly nonlinear and permanent deformation requiring plasticity treatments. It simply asserts that the general deﬁnition of three-dimensional failure criteria must be viewed by means other than through the lens of ultimate plastic deformation.

The second and equally important key to the new theory of failure is the hypothesis that the strengths ratio T/C covers the full

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 16, 2014; ﬁnal manuscript received May 23, 2014; accepted manuscript posted May 29, 2014; published online June 5, 2014. Editor: Yonggang Huang.

range of all possible behaviors for homogeneous and isotropic materials. Symbols T and C are the failure stresses in uniaxial tension and compression. Thus,

T

0

1

C

provides the failure descriptor for all failure types and classes. There is an important corollary and consequence to this ratio of

the uniaxial strengths as representing the full and complete spectrum of possible failure behaviors. It follows from this hypothesis that the general theory must be of a two property form, not the three or four or more parameters empirical forms that are often thought and argued to be necessary.

These two physical hypotheses are sufﬁcient to guide the way to the mathematical development of the general, complete, and self-contained theory of materials failure [1], for full density, homogeneous, and isotropic materials.

After stating the controlling forms for the new failure formalism and for linear elasticity, their explicit relationship will be closely examined in the next two sections. It will be shown that there is a type of coordination or coupling between the two formalisms. This coordination concerns the differentiation between ductile and brittle types of failures. The elasticity part of the combination reveals new and independent information on this important subject. The enabling development that makes this possible is the renormalization of the classical theory of elasticity, presented here for the ﬁrst time.

Finally, the Failure Mechanics Relationships section considers the uniﬁcation of elasticity theory and failure theory and treats the overall discipline as that of failure mechanics. This failure mechanics is for homogeneous and isotropic materials and it is completely separate from and complementary to the ﬁeld of fracture mechanics. The overall coverage and signiﬁcance of all of the historical mechanics related disciplines, including these two, are given a ﬁnal survey and assessment.

Failure Theory and Renormalized Elasticity Theory

The constitutive relations for linear, isotropic elasticity theory will be recalled for use here in one of its many but equivalent forms. The most obvious and natural form is that of the expression of the elastic energy partitioned into dilatational and distortional

Journal of Applied Mechanics

Copyright VC 2014 by ASME

AUGUST 2014, Vol. 81 / 081001-1

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

terms and calibrated by the corresponding properties. This can be done with either stresses or strains. The account here will begin with stresses and then later return to strains. This form of elastic energy is given by

1 rii2 1 U ¼ 2k 3 þ 4l sijsij (1)

where k is the bulk modulus and l is the shear modulus. The deviatoric stresses and strains are deﬁned by

sij ¼ rij À dij rkk

3

(2)

eij ¼ eij À dij ekk

3

The failure theory given by Christensen [1] has the basic crite-

rion developed by the method of polynomial invariants as

1 À T r^ii þ 3 s^ijs^ij T (3)

C

2

C

where the stress is nondimensionalized by the compressive strength

r^ij ¼ rij

(4)

C

and where T and C are the failure stresses with the brittle to ductile limits and range speciﬁed by

T

0

1

(5)

C

The failure criterion (3) is and must be supplemented by a competitive fracture criterion given by

T

T1

r^1

if

(6)

C

C 2

where r1 is the maximum principal stress. See Ref. [1] for the full background, motivation, the derivation, and for the interpretation of this new theory of failure.

The elastic energy (1) and the failure criterion (3) are from completely different derivations with each receiving an independent polynomial invariants type of expansion in the invariants of the stress tensor.

All the attention here will be upon the polynomial invariants failure criterion (3). Nothing will be done here that involves or would be superseded by the fracture criterion (6).

It will be helpful to express the elastic energy (1) in the alternative and less obvious, hybrid form using Young’s modulus, E, and Poisson’s ratio, , but still retaining the dilatational and distortional source terms. This converts (1) to

! U ¼ 21E ð1 À32Þ r2ii þ ð1 þ Þsijsij (7)

where it is clear that must have the limits

À 1 1 (8) 2

Henceforth, E will be referred to as the elastic modulus. Compare the elastic energy (7) and the failure form (3). They

are of somewhat similar forms but with some distinct differences, especially in the quadratic dependence on rii in Eq. (7) but the linear dependence on it in Eq. (3). These two forms (3) and (7) appear slightly more similar when Eq. (7) is rewritten as

!

U ¼ ð12þEÞ 13 11Àþ2 r2ii þ sijsij (9)

Using the limits for in Eq. (8) and the limits for T/C in Eq. (5)

then the range of the coefﬁcient of r^ii in Eq. (3) is from 0 to 1 while the range of the coefﬁcient of r2ii in Eq. (9) is from 0 to 1. In a sense the elasticity theory based on Eq. (7) is formalized such

that the governing property, , is normalized at the value of ¼ À1. It could be helpful if the elasticity theory were renormalized with a different nondimensional variable. Renormalization is

often used in physics for special utility or for new insights. The renormalization will be centered on ¼ 0. All properties will be taken to be unchanged at and only at ¼ 0.

As the ﬁrst step in this direction, recognize that it is only the positive values of that are physically realistic, 0 1=2. Negative values for are never found for homogeneous and isotropic

materials. There is a very good reason for this behavior and it

exists at the nanoscale. This restriction will be detailed later. But for now, it is henceforth taken that only positive values of carry

and convey physical reality.

It would furthermore be advantageous if the elasticity theory could be renormalized such that with 0 1=2 renormalizes to a variable that ranges from 0 to 1 the same as with T/C in Eq. (5).

So the renormalization of elasticity theory is the main objective here. To this end, compare the coefﬁcients of r2ii and r^ii in Eqs. (9) and (3). To bring them into closer alignment take

1 À 2 ¼ 1 À Ã

(10)

1þ

where * is a tentative new nondimensional property. Solving Eq. (10) for * gives

Ã ¼ 3

(11)

1þ

Thus * is found directly and solely from . Over the range of in Eq. (8), it follows from Eq. (11) that *

has the range

À1 Ã 1

(12)

but over the physically relevant and physically acceptable range of as

1

0

(13)

2

then from Eq. (11) the corresponding range on * is

0 Ã 1

(14)

The range of * in Eq. (14) is of the required form needed to move forward. The new property * in place of is the renormalized Poisson’s ratio.

Correspondingly take the renormalized elastic modulus as

EÃ ¼ E

(15)

1þ

Then the renormalized constitutive relation for linear elasticity

from Eqs. (9), (11), and (15) is given by

1 ð1 À ÃÞ

!

U ¼ 2EÃ

3 r2ii þ sijsij

(16)

where * has the general range of Eq. (12) but the physically restricted and realistic range of Eq. (14). Renormalized modulus E* is always positive.

081001-2 / Vol. 81, AUGUST 2014

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

From this point onward the range of * will always be taken as Eq. (14) even though it applies over the full range of Eq. (12). Next the consequent elastic forms that follow from Eq. (16) will be derived.

The classical two Lame0 constants and the bulk modulus are given by

k ¼ E ð1 þ Þð1 À 2Þ

l ¼ E (17) 2ð1 þ Þ

k¼ E 3ð1 À 2Þ

These properties in terms of E* and * are then given by

ÃEÃ k ¼ 3ð1 À ÃÞ l ¼ EÃ (18)

2 EÃ

k ¼ 3ð1 À ÃÞ

The variational result eij ¼ @U (19) @rij

gives the consequent strain–stress relations from Eq. (16) as

1

Ã

!

eij ¼ EÃ rij À 3 dijrkk

(20)

The other two complementary forms are found to be the energy

form in terms of strains as

EÃ

1

!

U ¼ 2 3ð1 À ÃÞ e2kk þ eijeij

(21)

and the stress–strain relations as

Ã

!

rij ¼ EÃ eij þ 3ð1 À ÃÞ dijekk

(22)

Taking mean normal stresses and strains as

rM ¼ rii

3 e

(23)

eM ¼ ii

3

then gives the slightly different forms of the four elastic constitutive relations as being

U ¼ 2E1 Ã Â3ð1 À ÃÞr2M þ sijsijÃ (24)

EÃ 3

!

U ¼ 2 ð1 À ÃÞ e2M þ eijeij

(25)

eij ¼ E1Ã Ârij À ÃdijrMÃ (26)

Ã

!

rij ¼ EÃ eij þ ð1 À ÃÞ dijeM

(27)

where E* and * are deﬁned by Eqs. (15) and (11), and subscript M is not an index. E* and *could be determined other than from

the experiments implied in Eqs. (11) and (15). This will be outlined in The Coupling Between Elasticity and Failure section.

These two groups of four relations (16) and (20)–(22) or (24)–(27) specify the renormalized forms of linear elasticity. Any one of the four relations serves to deﬁne the other three. These four simple forms remain as the same classical theory of elasticity, there are no approximations. Furthermore, these tensor results are probably the most compact, and most concise statements of the linear elastic constitutive relations. The advantage of expressing energy and strains in terms of stresses, (24) and (26), is apparent.

The four constitutive forms also reveal that the two properties E and , with subsequent E* and *, are highly preferable and more basic than the pairing of l and k. The properties l and k in these four forms lack the same symmetry and simplicity and in some cases are ambiguous because of unlimited behaviors in limiting cases.

There likely is no advantage in using this new formalism for two-dimensional or one-dimensional conditions, nor for numerical computations. It is purely for the three-dimensional formulation that is needed for basic theory purposes, especially in relation to failure conditions. It is somewhat surprising that this renormalized formulation of elasticity theory was not recognized much earlier in the past 100–150 years of intensive study.

From relation (11), some corresponding values of and * are given by

*

0

0

1/8

1/3

1/5

1/2

1/3

3/4

1/2

1

Inverting Eqs. (11) and (15) gives and E expressed in terms of * and E* as

Ã

¼ 3 À Ã

(28)

3EÃ

E ¼ 3 À Ã

(29)

Some further relationships are as follows:

EÃ ¼ E at ¼ Ã ¼ 0

EÃ ¼ 2 E at ¼ 1 ; Ã ¼ 1

(30)

3

2

Although E* and E are about the same size there is a signiﬁcant difference and the proper property is E* not E for the threedimensional calibration of elastic constitutive equations.

E* is the universal and dimensional property expressing a materials general capability to resist deformation while bearing load. Property * is the universal and nondimensional new property with what will be found to be a deep and insightful interpretation of what must be happening at the atomic scale that controls the balance of the energy sources involving the dilatational versus the distortional resistance.

Both the elastic energy form (16) and the failure form (3) involve a single dimensional property and a single nondimensional property. It follows that the limits of the nondimensional property in Eq. (16) associates with either a ductile or a brittle failure condition in uniaxial tension as given by

Ã ¼ 0 Totally Brittle

Ã ¼ 1 Perfectly Ductile

(31)

The basis for relations (31) is as follows. At * ¼ 0, the constitutive relation (26) shows that the stress and strain tensors are co-

Journal of Applied Mechanics

AUGUST 2014, Vol. 81 / 081001-3

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

axial, there is no Poisson’s ratio effect. This means that there is no coupling between the stress components. Then the related failure behavior corresponds to the maximum normal stress criterion of Lame0 and Rankine, which has always been interpreted as that of brittle fracture. Therefore this limit of * ¼ 0 is taken as that of the totally brittle state.

At the other limit of * ¼ 1, relation (24) shows that there is no dilatational contribution to the energy and Eq. (25) shows that there is no dilatation, eM ¼ 0. The deformation is purely distortional. This is the state of incompressibility which usually corresponds to the behavior of elastomers and approached by extremely ductile metals. This is the state of perfect ductility.

Relations (31) give the ﬁrst indication of a possible coupling behavior between the two very different theories of mechanical behavior, elastic energy, and failure. The ductile versus brittle behavior as a function of T/C is fully explored in Ref. [1]. That the Poisson’s ratio through * also associates with brittle versus ductile behavior comes as an unexpected result from this derivation. However, it is not completely unexpected from a practical, observational point of view. Beryllium has a Poisson’s ratio of virtually 0 and it is very brittle. Gold has a very large Poisson’s ratio and it is very ductile. It is this possible relationship between * and ductility that will be pursued further.

As a prelude to the further developments, it can now be rationalized that there could conceivably be some relationship between elasticity theory properties and failure theory properties, especially since failure is perceived as the termination of elastic behavior.

The Coupling Between Elasticity and Failure

As derived in Ref. [1], the state of uniaxial tension has a speciﬁc measure of the degree of ductility D through

D ¼ T Uniaxial Tension (32) C

This extraordinarily simple ductility measure goes from 0 to 1 meaning from the state of total brittleness to the state of perfect ductility. Similarly more complicated forms for the ductility measures can be derived for any stress state and are done so in Ref. [1]. But the interests here are solely with uniaxial tension since it is by far the most important special case, and it is universally used to calibrate comparative ductility levels for different materials. The ductile brittle transition for uniaxial tension in Eq. (32) is at T/C ¼ 1/2. This also is explained and explored at length in Ref. [1] and will later be derived here by a different method.

In view of the ductility limits shown in Eq. (31), it follows that the renormalized elastic property * may have some use as a continuous measure of the degree of ductility in uniaxial tension. To this end, take D* as the possible measure of ductility in terms of *. At this point it is only known from Eq. (31) that D* as a function of * has the limiting behaviors

DÃðÃÞ ¼ 1 at Ã ¼ 1

DÃðÃÞ ¼ 0 at Ã ¼ 0

(33)

ð1 À3 ÃÞ r2ii þ sijsij ¼ 2EÃU (35)

and

1 À T r^ii þ 3 s^ijs^ij T (36)

C

2

C

where the limits on * are given by Eq. (14) and those on T/C by Eq. (5).

Note that all the forms and properties in the two relations (35) and (36) are either quadratic or linear in form. In accordance with these characteristics, the most likely values for exponent n in Eq. (34) are either n ¼ 1 or n ¼ 2, linear or quadratic.

From Eq. (36), it is seen that T/C in the ﬁrst term is linear and this may therefore favor taking n ¼ 1 in the ductility measure (34). On the other hand, it is also seen that T/C is associated with the linear term r^ii in Eq. (36) while * is associated with the quadratic term r2ii in Eq. (35). This may favor taking n as being quadratic, n ¼ 2.

More speciﬁc reasoning to ascertain the proper value for n is as follows. The ductile brittle transition in terms of T/C is at T/C ¼ 1/2. This was developed by a general and elaborate methodology in Ref. [1]. A much simpler approach will be sufﬁcient for the restricted case of uniaxial tension and will now be given.

For uniaxial stress, r, the failure criterion (36) reduces to

Tr

r2 T

1ÀC C þ C C

"

"

(37)

Dilatational Distortional

For uniaxial tensile failure at r ¼ T and for equality in Eq. (37), it becomes

1 À T þ T ¼ 1 (38)

C

C

The separate sources from dilatation and distortion are so labeled in Eq. (37). At the two limits, Eq. (38) becomes

At T ¼ 1; 0 þ 1 ¼ 1

C

"

(39)

Distortional

and

At T ¼ 0; 1 þ 0 ¼ 1

C

"

(40)

Dilatational

Let the possible ductility measure take the general power law form

DÃðÃÞ ¼ ðÃÞn; n ! 1

(34)

The form (34) satisﬁes the required limits in Eq. (33). The exponent n in Eq. (34) must be taken to satisfy some as yet unspeciﬁed but logical condition. This condition must involve compatibility with the observed behaviors of the entire spectrum of isotropic materials types.

At this point, the elastic energy and the polynomial invariants failure criterion from Eqs. (16) and (3) will be rewritten here for direct comparison with each other. These are

At the ﬁrst limit (39), the distortion source dominates and the behavior is ductile. Of course, plasticity behavior will ensue near the T/C ¼ 1 end of the scale since the distortional source dominates. At the second limit (40), the dilatation source dominates and the failure behavior is brittle.

At the value of T/C ¼ 1/2, Eq. (38) becomes

At T ¼ 1 ; 1 þ 1 ¼ 1 (41) C 2 22

and the dilatational source and the distortional source are in balance and this is taken as the ductile brittle transition. Thus the value T/C ¼ 1/2 denotes the ductile brittle transition in uniaxial

081001-4 / Vol. 81, AUGUST 2014

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

tension for this failure criterion. This is in accordance with the much more general method from Ref. [1].

Just as T/C ¼ 1/2 gives the ductile brittle transition, the corresponding ductile brittle transition in terms of * will be taken as midway between the limits for D* of 0 and 1, namely at D* ¼ 1/2. The two candidate measures for D* then give the respective ductile brittle transition as occurring at

n ¼ 1; DÃ ¼ Ã ¼ 1

(42)

2

or at

n ¼ 2; DÃ ¼ ðÃÞ2¼ 1 ; Ã ¼ p1ﬃﬃ

(43)

2

2

From relation (28), it further follows that the conceivable ductile brittle transition occurs at

n ¼ 1; ¼ 1 (44) 5

or at

n ¼ 2; ¼ pﬃ1ﬃ ¼ 0:3084

(45)

3 2À1

The n ¼ 1 prediction of the ductile brittle transition as occurring at ¼ 1/5 is completely unrealistic. Materials such as carbon (diamond) and silicon are near this value of Poisson’s ratio and they

are distinctly brittle in failure behavior. The case for the ductility measure at n ¼ 1 in Eq. (34) is com-

pletely unacceptable and it is discarded. This leaves the n ¼ 2 case of Eq. (34) as

DÃ ¼ ðÃÞ2

(46)

and as being the most likely measure of ductility in terms of the renormalized Poisson’s ratio *. This measure of ductility in uniaxial tension is still provisional at this juncture, pending veriﬁcation with explicit materials comparisons. The comparisons will be conducted shortly.

The signiﬁcant values for D*, *, and from Eqs. (46) and (28) are shown in Table 1. The ductile brittle transition at D* ¼ 1/2 has already been discussed. Also shown are the values at D* ¼ 1/4. For lack of a better term this is called a special case because it does appear to have some physical signiﬁcance. Very few materials have Poisson’s ratios of less than 1/5. The coefﬁcient of the dilatational term in Eq. (24) is half way between its maximum and minimum values of 1 and 0 at this value of Poisson’s ratio.

It is possible to determine E* and * other than from experiments directly on E and . From Eq. (18), it follows that:

EÃ ¼ 2l

(47)

and

Ã ¼ 1 À 2l

(48)

3k

In most cases it is probably far easier to use Eqs. (11) and (15) directly. However, relation (47) does incisively show that E* is proportional to the distortional source of resistance while in Eq. (48), * involves the interplay between the two independent physical effects, distortional and dilatational resistances to deformation. Relations (47) and (48) should be compared with the complex classical forms relating these four properties.

The comparison of ductility levels predicted by Eq. (46) with actual materials is given next as a critical test of the result (46).

Table 1 Ductility D*, Eq. (46)

Ductility D* ¼ (*)2

*

Totally brittle

0

0

0

Special case

1/4

1/2

1/5

D/B transition

1/2

0.71

0.31

3/4

0.87

0.41

Perfectly ductile

1

1

1/2

Table 2 shows a list of the solids forming common elements from the periodic table. The Poisson’s ratios are from the sources referenced in Ref. [1]. The prediction of ductility levels in the last column of Table 2 is in accordance with general understanding of the spread of brittle to ductile behaviors for the different materials types. The ductile brittle transition of Eqs. (43) and (45) occurs at the elements of nickel and cobalt which also is in reasonable accord with practice. This ductile brittle transition is not a transition in the thermodynamic sense of the term. But it is a meaningful transition as the division between predominately brittle versus predominately ductile groupings.

Even though Poisson’s ratio is difﬁcult to determine with high accuracy, the overall results of Table 2 establish the validity of using renormalized Poisson’s ratio in Eq. (46) as a characterization of the ductility in uniaxial tension. It should be noted that the results in Table 2 are identical to the results found from a nanoscale analysis in Ref. [1].

The nanoscale analysis just mentioned was based upon differentiating the effects of bond stretching versus bond bending. The ductile end of the scale has no bond bending resistance compared with the bond stretching resistance. The brittle end of the scale maximizes the bond bending resistance relative to the bond stretching resistance. The present analysis has the ductile end of the scale as having no dilatational energy in Eq. (35) at * ¼ 1. The brittle end of the scale has the dilatational energy in Eq. (35) as maximized at * ¼ 0. These macroscopic scale explanations and understandings of behavior are completely compatible with the nanoscale explanations and understandings.

Table 2 Ductility of the solids forming elements

Element

Ductility D* ¼ (*)2

Absolute limit, perfect ductility

1/2

1

Gold

0.44

0.84

Lead

0.44

0.84

Niobium

0.40

0.73

Palladium

0.39

0.71

Platinum

0.38

0.68

Silver

0.37

0.66

Vanadium

0.37

0.66

Tin

0.36

0.63

Aluminum

0.35

0.60

Copper

0.34

0.58

Tantalum

0.34

0.58

Titanium

0.32

0.53

Nickel

0.31

0.50

Cobalt

0.31

0.50

Magnesium

0.29

0.45

Iron

0.29

0.45

Tungsten

0.28

0.43

Zinc

0.25

0.36

Manganese

0.23

0.31

Uranium

0.23

0.31

Silicon

0.22

0.29

Plutonium

0.21

0.27

Chromium

0.21

0.27

Carbon (diamond)

0.20

0.25

Limit for most elements/materials

1/5

1/4

Beryllium

0.032

0.01

Absolute limit, total brittleness

0

0

Journal of Applied Mechanics

AUGUST 2014, Vol. 81 / 081001-5

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

The nanoscale analysis in Ref. [1] also served the important purpose of proving that the physically unrealistic range of bond bending relative to bond stretching resistance disallows negative values of Poisson’s ratio. Finally, it must be emphasized that the relation between Poisson’s ratio and ductility has notable exceptions and its use here is mainly for studying fundamental effects rather than for general utility. In such exceptions, other physical effects are tailored to intercede, as with steel versus iron.

When interests are in a comprehensive treatment of ductile brittle matters, the account in Ref. [1] should be followed where the failure number, Fn, is derived and deﬁned to specify the ductility level for any state of stress. The purpose here was entirely different. It was to establish the foundation of the new failure theory by carefully examining the connection between elasticity theory and failure theory. The ductility characterization for uniaxial tension was shown to be the strong and supporting linkage between the two theories. From a failure point of view, it would be appropriate to say that this new development provided not just the coupling between the two theories but the integration into and uniﬁcation of failure theory with elasticity theory.

Failure Mechanics Relationships

The term failure mechanics is here taken to cover and embrace failure theory for homogeneous and isotropic materials with all of its associations and all its implications and applications. It is the latest in a long line of mechanics related and formulated disciplines.

It all began with the monumental and almost unthinkably difﬁcult creation of classical mechanics by Isaac Newton. Then over about the following two centuries the disciplines of solid mechanics and ﬂuid mechanics came into being and useful forms gradually evolved through the dedicated and inspired efforts of such great scientists as Cauchy, Bernoulli(s), Euler, Navier, Lame0, Poisson, Stokes, and many others.

The dominant mechanics achievements of the 20th century were the two disciplines of quantum mechanics and fracture mechanics. It is pointless to try to directly compare them because one is at the atomic scale and the other at the macroscopic scale. Both became vitally important.

In quantum mechanics the problem was to understand the behavior of the electrons orbiting the atomic nucleus under their opposite charges. Were the electrons best viewed as being particles or as waves? The evidence was contradictory. Then the possibility emerged that the behaviors were not deterministic but rather must be seen as probabilistic. That possibility was then enlarged to more general concepts of the uncertainty of predictability. The complexity of the behaviors was enfolded and enshrouded in the full and glorious complications of threedimensional interactions.

Over a prolonged and frenzied period of activities in the 1920s Bohr and Heisenberg and Dirac and Schrodinger and Born and others sorted through, debated, and argued all the possibilities. The community ultimately reached a conclusion, something of an amalgam of all of the above. The almost unsolvable complexity of the quantum mechanics problem was at least partially due to its three-dimensional nature. It was almost but not quite unsolvable. There ﬁnally was a workable resolution to the problem and it has carried forward and endured.

Fracture mechanics, like quantum mechanics, evolved and progressed through various stages in its development. Grifﬁth [2] originated it and conceived the crack growth problem as an energy balance between the release of elastic strain energy against the work to create new crack surface. Irwin [3] had the insight to characterize the failure inception as related to the nature of the elastic singularity at the crack tip. This offered a normalizing and comparative capability. Rice [4] effected a further breakthrough by relating crack tip conditions to the properties of an invariant contour integral enclosing the crack. Barenblatt [5] re-examined

crack tip conditions and found the nonsingular stress state that helped to explain much of the behavior.

All of these efforts in combination formed the discipline of fracture mechanics. It has become enormously successful in design applications where materials are made into load bearing structures containing a single worst ﬂaw (or ﬂaws) that could be idealized as a crack (or cracks). Although fracture mechanics can be formulated in some special three-dimensional cases, its fundamental and wide ranging applicability is of a two-dimensional nature and type. The success of fracture mechanics proved that the general subject of failure characterization no longer need be relegated to the state of just being an undisciplined agglomeration of empirical formulas.

Perhaps the completing mechanics related discipline will be that of the failure mechanics needed to characterize the threedimensional failure behavior of homogeneous and isotropic materials. The fact that it is coming into existence only now was not for lack of effort over the years and over the centuries. Its great difﬁculty of development comes from the same source as that for quantum mechanics, namely, its fully three-dimensional nature having extremely complex interactions.

It is highly unlikely that failure mechanics, as envisioned here, could have been developed before the advent of fracture mechanics. Part of this is no doubt due to the clean and clear twodimensional nature of fracture mechanics compared with the much greater complexity of general failure in three dimensions. Also, for the failure of homogeneous materials, there is no guiding certainty as to the site of failure initiation, as occurs with the preexisting, stress concentrating cracks in fracture mechanics. Fracture mechanics had to lead the way for progress on failure. So the emergence of failure mechanics at this latter time has a reasonable coordination with the order and sequence of development of the other mechanics related disciplines.

There is no need to summarize the derivation and capability of failure mechanics. That is comprehensively treated in Ref. [1] under the heading of failure theory. The special interpretation developed herein however merits particular attention. Elasticity theory and failure theory for homogeneous and isotropic materials are not absolutely independent constructs. One without the other is incomplete. Although there is no single parameter that appears in both theories and bridges between them, there nevertheless is a more subtle but no less effective coordination or coupling between them.

To explain this coupling between the two theories, it is important and even necessary to recognize that any failure theory that does not fully treat ductile versus brittle failure conditions is not complete. Even more serious, such an incomplete failure theory cannot be assessed and evaluated, its status must be uncertain, even of doubtful validity until proven otherwise. The failure theory developed in Ref. [1] has the strengths ratio T/C and the stress states as determining the ductile versus brittle behaviors. More speciﬁcally, for the most important special case of uniaxial tension, the ratio T/C itself directly delineates the degree of ductility over the range from 0 to 1.

In the case of linear elasticity theory, the work here has shown that renormalized Poisson’s ratio, *, also gives a direct measure of the degree of ductility for uniaxial tension. This occurs through Eq. (46) over the same range of 0 to 1 as that for T/C.

It is thus the ductility measures from both the failure theory and from elasticity theory that provides the bridge between the two theories. At ﬁrst it is exceedingly surprising that elasticity theory could say anything at all about the ductility in failure. But on deeper reﬂection it is not so surprising. The atomic scale characteristics of bond stretching and bond bending directly determine and differentiate not only T and C but also . Poisson’s ratio, , actually contains a great deal of information that has heretofore been ignored or vastly under-appreciated.

Poisson’s ratio, , is not at all secondary to the elastic modulus, E, it is the full and equal partner to E and of great signiﬁcance in its own right. This new and enlarging insight became fairly

081001-6 / Vol. 81, AUGUST 2014

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

apparent when the renormalized theory of elasticity was obtained with the renormalized Poisson’s ratio. Contrast this with the classical form of elasticity theory, usually using the two Lame0 constants k and l. The k property has no physical interpretation at all. E and or better yet their renormalized forms E* and * represent the most meaningful statements of elastic properties, even with direct implications for failure. In association with E* and *, the uniaxial strengths T and C turn out to give the most meaningful and useful calibrating anchors for failure theory.

It is the failure theory of Ref. [1] (and herein) in its broadest context that comprises the discipline of failure mechanics. The comparison of failure mechanics (for solids) with ﬂuid mechanics and with quantum mechanics is both illuminating and technically useful. In ﬂuid mechanics and for slow ﬂows, the Navier–Stokes equations give a linear behavior between the shear stress and the velocity gradient. But as the velocities increase, the nonlinear effect of the inertial source invalidates the previous linear relations and leads to turbulence.

For failure mechanics a similar situation evolves. Small deformation states produce linear relationships among the ﬁeld variables of elasticity theory. But the nonlinear intercession of failure theory invalidates the previous conservative, linear behaviors. Nonlinear intercessions provide the limits to the linear behaviors of both solids and ﬂuids. Elasticity theory without failure theory is open ended and incomplete. They must go together to provide closure.

The comparison between failure mechanics and quantum mechanics of course does not contain any analogies or congruencies between them. But both do share an overriding common characteristic that had much to do with their histories of development. Perhaps this would be better described as the common barriers that greatly impeded their respective developments. The commonality is simply this. Neither of these two mechanics disciplines yielded to the standard and ordinary expectations. Both solutions turned out to be strongly counterintuitive and it took major expenditures of time and effort to ultimately synthesize them, one over

a highly intensive time period and the other over an agonizingly long and drawn out time extension.

All six of these mechanics related ﬁelds: Classical mechanics, solid mechanics, ﬂuid mechanics, quantum mechanics, fracture mechanics, and failure mechanics comprise basic sources of mechanistic knowledge. The ﬁrst ﬁve of them are mature, scientiﬁc disciplines. The newest and sixth one, failure mechanics, cannot be said to be mature, it is still being assimilated. The comprehensive and complete theory was given in Ref. [1], the full, broad, and inclusive understanding of it necessarily will take longer.

The present article “Failure Mechanics, Part I” will later be followed by “Failure Mechanics, Part II: The Central and Decisive Role of Graphene in Deﬁning the Elastic and Failure Properties for all Isotropic Materials.” This coming work will go much further toward completing the foundation upon which failure mechanics is built.

Acknowledgment

The author was appreciative of support for this work from the Ofﬁce of Naval Research, Dr. Y. D. S. Rajapakse, Program Manager.

References

[1] Christensen, R. M., 2013, The Theory of Materials Failure, Oxford University Press, Oxford, UK.

[2] Grifﬁth, A. A., 1921, “The Phenomena of Rupture and Flow in Solids,” Trans. R. Soc. London A, 221(582–593), pp. 163–197.

[3] Irwin, G. R., 1957, “Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate,” ASME J. Appl. Mech., 24(3), pp. 361–364.

[4] Rice, J. R., 1968, “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks,” ASME J. Appl. Mech., 35(2), pp. 379–386.

[5] Barenblatt, G. I., 1962, “The Mathematical Theory of Equilibrium of Cracks in Brittle Fracture,” Adv. Appl. Mech., 7, pp. 55–129.

Journal of Applied Mechanics

AUGUST 2014, Vol. 81 / 081001-7

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/27/2014 Terms of Use: http://asme.org/terms

## Categories

## You my also like

### Elkem Chart for Graphite Structuresin Cast Iron

302.8 KB62.4K26.2K### GUIDELINES Installation Guide for Ductile Iron Pipe

3.6 MB13.7K6.9K### Quick Refresher Guide Machine Design

533.9 KB28K6.2K### The Demand Side of the Market

362.6 KB79.4K32.6K### Chapter 5: Elasticity and Its Application Principles of

138.8 KB59.2K20.1K### Elasticity and its Application Premium PowerPoint Slides by

92.7 KB48.6K14.6K### Relation between modulus of elasticity obtained by ultrasound

594.6 KB44.3K19K### Point Elasticity Versus Arc Elasticity: On Different

344 KB34K11.6K### Elasticity and Its Application

423.6 KB20K3.6K