3806:
changes slightly to accommodate the sudden increase in stress from that which the material previously experienced. At a sufficiently high load (overload), the crack grows out of the plastic zone that contained it and leaves behind the pocket of the original plastic deformation. Now, assuming that the overload stress is not sufficiently high as to completely fracture the specimen, the crack will undergo further plastic deformation around the new crack tip, enlarging the zone of residual plastic stresses. This process further toughens and prolongs the life of the material because the new plastic zone is larger than what it would be under the usual stress conditions. This allows the material to undergo more cycles of loading. This idea can be illustrated further by the
3858:
3533:
applicable direction (in most cases, this is the y-direction of a regular
Cartesian coordinate system), the crack length, and the geometry of the specimen. To estimate how this plastic deformation zone extended from the crack tip, Irwin equated the yield strength of the material to the far-field stresses of the y-direction along the crack (x direction) and solved for the effective radius. From this relationship, and assuming that the crack is loaded to the critical stress intensity factor, Irwin developed the following expression for the idealized radius of the zone of plastic deformation at the crack tip:
3940:. This curve acknowledges the fact that the resistance to fracture increases with growing crack size in elastic-plastic materials. The R-curve is a plot of the total energy dissipation rate as a function of the crack size and can be used to examine the processes of slow stable crack growth and unstable fracture. However, the R-curve was not widely used in applications until the early 1970s. The main reasons appear to be that the R-curve depends on the geometry of the specimen and the crack driving force may be difficult to calculate.
4579:
863:
31:
3819:
1572:
3907:
high toughness could not be characterized with the linear elastic fracture mechanics model. He noted that, before the fracture happened, the walls of the crack were leaving and that the crack tip, after fracture, ranged from acute to rounded off due to plastic deformation. In addition, the rounding of the crack tip was more pronounced in steels with superior toughness.
3973:. This analysis is limited to situations where plastic deformation at the crack tip does not extend to the furthest edge of the loaded part. It also demands that the assumed non-linear elastic behavior of the material is a reasonable approximation in shape and magnitude to the real material's load response. The elastic-plastic failure parameter is designated J
4149:
2561:
3456:
3805:
The same process as described above for a single event loading also applies and to cyclic loading. If a crack is present in a specimen that undergoes cyclic loading, the specimen will plastically deform at the crack tip and delay the crack growth. In the event of an overload or excursion, this model
3118:
Irwin was the first to observe that if the size of the plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress (the plastic zone) at the crack tip. In other words, a purely elastic solution
3910:
There are a number of alternative definitions of CTOD. In the two most common definitions, CTOD is the displacement at the original crack tip and the 90 degree intercept. The latter definition was suggested by Rice and is commonly used to infer CTOD in finite element models of such. Note that these
3906:
Historically, the first parameter for the determination of fracture toughness in the elasto-plastic region was the crack tip opening displacement (CTOD) or "opening at the apex of the crack" indicated. This parameter was determined by Wells during the studies of structural steels, which due to the
3841:
at the tip of a crack. One basic assumption in Irwin's linear elastic fracture mechanics is small scale yielding, the condition that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels
3801:
is large, which results in a larger plastic radius. This implies that the material can plastically deform, and, therefore, is tough. This estimate of the size of the plastic zone beyond the crack tip can then be used to more accurately analyze how a material will behave in the presence of a crack.
3495:
Next, Irwin adopted the additional assumption that the size and shape of the energy dissipation zone remains approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material. This new
820:
The processes of material manufacture, processing, machining, and forming may introduce flaws in a finished mechanical component. Arising from the manufacturing process, interior and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture
3611:
Models of ideal materials have shown that this zone of plasticity is centered at the crack tip. This equation gives the approximate ideal radius of the plastic zone deformation beyond the crack tip, which is useful to many structural scientists because it gives a good estimate of how the material
2042:
Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid. This asymptotic expression for the stress field in
1581:
Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural
913:
A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that
Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material
4553:
When a significant region around a crack tip has undergone plastic deformation, other approaches can be used to determine the possibility of further crack extension and the direction of crack growth and branching. A simple technique that is easily incorporated into numerical calculations is the
3914:
Most laboratory measurements of CTOD have been made on edge-cracked specimens loaded in three-point bending. Early experiments used a flat paddle-shaped gage that was inserted into the crack; as the crack opened, the paddle gage rotated, and an electronic signal was sent to an x-y plotter. This
3532:
Nevertheless, there must be some sort of mechanism or property of the material that prevents such a crack from propagating spontaneously. The assumption is, the plastic deformation at the crack tip effectively blunts the crack tip. This deformation depends primarily on the applied stress in the
679:
Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be used to describe the state around a crack. Fracture mechanics is used to characterise the loads on a crack, typically using a single parameter to describe the complete loading state at the crack tip. A number of
917:
To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental glass specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length
3524:
In theory the stress at the crack tip where the radius is nearly zero, would tend to infinity. This would be considered a stress singularity, which is not possible in real-world applications. For this reason, in numerical studies in the field of fracture mechanics, it is often appropriate to
1068:(surface energy − elastic energy) as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases as the crack length increases, i.e. by causing fracture. Using this procedure, Griffith found that
3529:, with a geometry dependent region of stress concentration replacing the crack-tip singularity. In actuality, the stress concentration at the tip of a crack within real materials has been found to have a finite value but larger than the nominal stress applied to the specimen.
3915:
method was inaccurate, however, because it was difficult to reach the crack tip with the paddle gage. Today, the displacement V at the crack mouth is measured, and the CTOD is inferred by assuming the specimen halves are rigid and rotate about a hinge point (the crack tip).
3962:) and G. P. Cherepanov independently developed a new toughness measure to describe the case where there is sufficient crack-tip deformation that the part no longer obeys the linear-elastic approximation. Rice's analysis, which assumes non-linear elastic (or monotonic
914:
failure before
Griffith, could not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.
4976:. As it is widely used in construction, fracture analysis and modes of reinforcement are an important part of the study of concrete, and different concretes are characterized in part by their fracture properties. Common fractures include the
3990:
1558:
For materials highly deformed before crack propagation, the linear elastic fracture mechanics formulation is no longer applicable and an adapted model is necessary to describe the stress and displacement field close to crack tip, such as on
1675:. Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy is needed for crack growth in ductile materials as compared to brittle materials.
3102:
2430:
2180:
3290:
3872:
Most engineering materials show some nonlinear elastic and inelastic behavior under operating conditions that involve large loads. In such materials the assumptions of linear elastic fracture mechanics may not hold, that is,
4983:
Bažant (1983) proposed a crack band model for materials like concrete whose homogeneous nature changes randomly over a certain range. He also observed that in plain concrete, the size effect has a strong influence on the
1864:
1685:
the dissipated energy which includes plastic dissipation and the surface energy (and any other dissipative forces that may be at work). The dissipated energy provides the thermodynamic resistance to fracture.
1662:
increases, the plastic zone increases in size until the crack grows and the elastically strained material behind the crack tip unloads. The plastic loading and unloading cycle near the crack tip leads to the
5214:(MD) simulations, AFM can provide insights into the fundamental mechanisms of crack formation and growth, the role of atomic bonds, and the influence of material defects and impurities on fracture behavior.
3606:
4699:
1503:
1969:
1916:
1129:
1582:
materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic.
1011:
An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a linear
1322:
4812:
2958:
1006:
4444:
2420:
2366:
4536:
1247:
99:
4748:
2882:
will be different for geometries other than the center-cracked infinite plate, as discussed in the article on the stress intensity factor. Consequently, it is necessary to introduce a
1637:
3256:
3484:
is the stress intensity factor in mode I. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode I,
1206:
4362:
2026:
1732:
4144:{\displaystyle J=\int _{\Gamma }(w\,dy-T_{i}{\frac {\partial u_{i}}{\partial x}}\,ds)\quad {\text{with}}\quad w=\int _{0}^{\varepsilon _{ij}}\sigma _{ij}\,d\varepsilon _{ij}}
2700:
4332:
2213:
1432:
5206:
when subjected to fracture. It integrates concepts from fracture mechanics with atomistic simulations to understand how cracks initiate, propagate, and interact with the
4611:
3799:
3745:
3718:
3664:
963:
4928:
4895:
5192:
5069:
802:
5172:
5112:
5088:
5009:
4477:
4300:
4175:
2257:
1755:
1345:
1172:
4641:
2757:
2556:{\displaystyle K_{c}={\begin{cases}{\sqrt {EG_{c}}}&{\text{for plane stress}}\\\\{\sqrt {\cfrac {EG_{c}}{1-\nu ^{2}}}}&{\text{for plane strain}}\end{cases}}}
2337:
2287:
5132:
5029:
4955:
4862:
4255:
4226:
3772:
3691:
3637:
2880:
2727:
2637:
2590:
2068:
1782:
1553:
3032:
2657:
3451:{\displaystyle G=G_{I}={\begin{cases}{\cfrac {K_{I}^{2}}{E}}&{\text{plane stress}}\\{\cfrac {(1-\nu ^{2})K_{I}^{2}}{E}}&{\text{plane strain}}\end{cases}}}
2850:
6135:
4280:
3024:
2824:
829:
analysis the safe operation of a structure. Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new.
5152:
5049:
4832:
4197:
3001:
2981:
2904:
2801:
2777:
2610:
2307:
2237:
2076:
1997:
1526:
1385:
1365:
1270:
1152:
1044:
936:
884:
717:
842:
How long does it take for a crack to grow from a certain initial size, for example the minimum detectable crack size, to the maximum permissible crack size?
4957:
for engineering alloys is 100 mm and for ceramics is 0.001 mm. If we assume that manufacturing processes can give rise to flaws in the order of
3720:
is low, one knows that the material is more ductile. The ratio of these two parameters is important to the radius of the plastic zone. For instance, if
1057:
Fix the boundary so that the applied load does no work and then introduce a crack into the specimen. The crack relaxes the stress and hence reduces the
821:
mechanics is the analysis of flaws to discover those that are safe (that is, do not grow) and those that are liable to propagate as cracks and so cause
6369:
3666:, are of importance because they illustrate many things about the material and its properties, as well as about the plastic zone size. For example, if
5662:
5632:
4563:
5941:
Bažant, Z. P., and Pang, S.-D. (2006) “Mechanics based statistics of failure risk of quasibrittle structures and size effect on safety factors.”
1046:
in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was:
3837:
But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo significant
648:
5389:
3612:
behaves when subjected to stress. In the above equation, the parameters of the stress intensity factor and indicator of material toughness,
2780:
5732:
5582:
1793:
6128:
3888:
the local conditions for initial crack growth which include the nucleation, growth, and coalescence of voids (decohesion) at a crack tip.
4961:, then, it can be seen that ceramics are more likely to fail by fracture, whereas engineering alloys would fail by plastic deformation.
773:. Crack growth occurs when the parameters typically exceed certain critical values. Corrosion may cause a crack to slowly grow when the
5439:
5790:
777:
stress intensity threshold is exceeded. Similarly, small flaws may result in crack growth when subjected to cyclic loading. Known as
4562:
and
Dugdale in the early 1960s. The relationship between the Dugdale-Barenblatt models and Griffith's theory was first discussed by
3539:
4646:
769:
The characterising parameter describes the state of the crack tip which can then be related to experimental conditions to ensure
672:
to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to
845:
What is the service life of a structure when a certain pre-existing flaw size (e.g. a manufacturing defect) is assumed to exist?
6121:
1447:
6045:
5352:
3981:
using the equation below. Also note that the J integral approach reduces to the
Griffith theory for linear-elastic behavior.
1921:
1872:
1682:
the stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for fracture.
5893:
5202:
Atomistic
Fracture Mechanics (AFM) is a relatively new field that studies the behavior and properties of materials at the
3880:
the size and shape of the plastic zone may change as the applied load is increased and also as the crack length increases.
1075:
3516:, found in the plane strain condition, which is accepted as the defining property in linear elastic fracture mechanics.
2910:, is given by empirically determined series and accounts for the type and geometry of the crack or notch. We thus have:
6425:
5324:
1278:
909:
The theoretical stress needed for breaking atomic bonds of glass is approximately 10,000 MPa (1,500,000 psi).
5274:, a formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures
4753:
1651:(NRL) during World War II realized that plasticity must play a significant role in the fracture of ductile materials.
6073:
5784:
2916:
971:
641:
4388:
2381:
2342:
804:
experienced by the crack due to the applied loading. Fast fracture will occur when the stress intensity exceeds the
4486:
6396:
2375:
replaced surface weakness energy. Both of these terms are simply related to the energy terms that
Griffith used:
1252:
For the simple case of a thin rectangular plate with a crack perpendicular to the load, the energy release rate,
1211:
781:, it was found that for long cracks, the rate of growth is largely governed by the range of the stress intensity
684:
at the tip of the crack is small relative to the crack length the stress state at the crack tip is the result of
55:
4704:
6809:
6497:
3901:
3884:
Therefore, a more general theory of crack growth is needed for elastic-plastic materials that can account for:
1648:
763:
614:
1602:
1022:
The growth of a crack, the extension of the surfaces on either side of the crack, requires an increase in the
6242:
5295:
3924:
3134:
315:
152:
6863:
6444:
6352:
5654:
3865:
3807:
634:
355:
241:
750:– Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front).
6853:
6565:
6276:
6210:
5624:
3842:
though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures.
898:– to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts:
310:
219:
102:
6014:
5757:
5692:"A path independent integral and the approximate analysis of strain concentration by notches and cracks"
1177:
839:
What crack size can be tolerated under service loading, i.e. what is the maximum permissible crack size?
4337:
3124:
3113:
1560:
1249:
gives excellent agreement of
Griffith's predicted fracture stress with experimental results for glass.
848:
During the period available for crack detection how often should the structure be inspected for cracks?
34:
The loads at a crack tip can be reduced to a combination of three independent stress intensity factors.
6081:
2002:
1695:
1061:
near the crack faces. On the other hand, the crack increases the total surface energy of the specimen.
6293:
5719:
5289:
2665:
1065:
774:
226:
24:
3318:
2452:
6724:
6719:
6602:
6570:
6485:
6225:
6215:
3857:
3128:
may then be calculated as the change in elastic strain energy per unit area of crack growth, i.e.,
2289:
are functions that depend on the crack geometry and loading conditions. Irwin called the quantity
521:
516:
305:
298:
131:
4307:
2188:
1398:
668:
concerned with the study of the propagation of cracks in materials. It uses methods of analytical
6784:
6465:
6381:
6271:
5369:
5259:
4985:
4589:
3777:
3723:
3696:
3642:
2037:
941:
723:
found any state could be reduced to a combination of three independent stress intensity factors:
697:
584:
579:
248:
4900:
4867:
4566:
in 1967. The equivalence of the two approaches in the context of brittle fracture was shown by
3877:
the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size
6858:
6364:
6237:
6031:
5714:
5691:
5551:
2906:, in order to characterize the geometry. This correction factor, also often referred to as the
1784:
is the plastic dissipation (and dissipation from other sources) per unit area of crack growth.
136:
5177:
5054:
3284:(opening mode) the strain energy release rate and the stress intensity factor are related by:
832:
Fracture mechanics should attempt to provide quantitative answers to the following questions:
784:
6769:
6347:
5283:
5229:
5157:
5097:
5073:
4994:
4977:
4449:
4285:
4160:
3097:{\displaystyle Y\left({\frac {a}{W}}\right)={\sqrt {\sec \left({\frac {\pi a}{W}}\right)}}\,}
2242:
1740:
1330:
1157:
559:
177:
6053:
4616:
2732:
2312:
2262:
6832:
6506:
6410:
6405:
6376:
6342:
6322:
6307:
6288:
6259:
6200:
5982:
5955:
5864:
5822:
5706:
5472:
5460:
5235:
5117:
5014:
4933:
4840:
4233:
4204:
3750:
3669:
3615:
2883:
2858:
2705:
2615:
2568:
2216:
2046:
1760:
1531:
891:
720:
685:
397:
214:
194:
182:
126:
5855:
Willis, J. R. (1967), "A comparison of the fracture criteria of
Griffith and Barenblatt",
4972:
is part of fracture mechanics that studies crack propagation and related failure modes in
3489:
3485:
3281:
2642:
2175:{\displaystyle \sigma _{ij}=\left({\cfrac {K_{I}}{\sqrt {2\pi r}}}\right)~f_{ij}(\theta )}
8:
6804:
6656:
6595:
6575:
6391:
6337:
6220:
6179:
6145:
5265:
4548:
3861:
3838:
3526:
2829:
1643:) predicted by Griffith's theory is usually unrealistically high. A group working under
1012:
825:
of the flawed structure. Despite these inherent flaws, it is possible to achieve through
599:
447:
340:
46:
6061:
6054:
Fracture
Mechanics of Electromagnetic Materials: Nonlinear Field Theory and Applications
6006:, Cambridge Solid State Science Series, Eds. Clarke, D.R., Suresh, S., Ward, I.M. (1998)
5868:
5826:
5710:
5476:
5262: – Science of predicting if, when, and how a given material will fail under loading
4262:
3845:
Linear-elastic fracture mechanics is of limited practical use for structural steels and
3006:
2983:
is a function of the crack length and width of sheet given, for a sheet of finite width
2806:
6470:
6254:
5838:
5574:
5431:
5247:
5211:
5137:
5034:
4817:
4182:
3963:
3846:
3498:
3474:
3466:
2986:
2966:
2889:
2786:
2762:
2595:
2372:
2292:
2222:
1982:
1511:
1388:
1370:
1350:
1255:
1137:
1029:
921:
869:
822:
805:
778:
770:
702:
619:
209:
204:
5776:
5570:
5385:
6523:
6460:
6439:
6434:
6359:
6327:
6069:
6041:
5876:
5842:
5834:
5780:
5435:
5348:
5320:
5277:
890:
Fracture mechanics was developed during World War I by English aeronautical engineer
236:
187:
5578:
744:
acting parallel to the plane of the crack and perpendicular to the crack front), and
6759:
6668:
6661:
6651:
6627:
6492:
6010:
5961:
5872:
5830:
5772:
5724:
5566:
5480:
5423:
5411:
5381:
4559:
3959:
3830:
3823:
1976:
1918:. For ductile materials such as steel, the plastic dissipation term dominates and
1051:
826:
809:
574:
549:
462:
437:
432:
387:
6774:
6709:
6686:
6622:
6607:
6590:
6540:
6528:
6415:
6386:
6303:
6281:
6264:
6230:
6164:
6159:
6060:
A.N. Gent, W.V. Mars, In: James E. Mark, Burak Erman and Mike Roland, Editor(s),
6035:
5253:
3891:
a global energy balance criterion for further crack growth and unstable fracture.
3492:(tearing mode) stress intensity factors for the most general loading conditions.
1659:
669:
564:
488:
452:
402:
333:
322:
267:
169:
6113:
5250: – Initiation and propagation of cracks in a material due to cyclic loading
6779:
6646:
6560:
6518:
6332:
5238: – Damage to concrete affecting its mechanical strength and its durability
5207:
5203:
1058:
1023:
731:
569:
427:
392:
293:
199:
5965:
5427:
6847:
6827:
6794:
6691:
6555:
6420:
5687:
4567:
3955:
1016:
609:
442:
19:
This article is about predicting fracture. For the science of fractures, see
3929:
An early attempt in the direction of elastic-plastic fracture mechanics was
6799:
6789:
6764:
5522:
Analysis of stresses and strains near the end of a crack traversing a plate
5485:
5271:
4480:
2339:
is dimensionless, the stress intensity factor can be expressed in units of
1869:
For brittle materials such as glass, the surface energy term dominates and
1787:
The modified version of Griffith's energy criterion can then be written as
1392:
741:
594:
589:
554:
286:
6107:
5370:"The energy release rate for a Griffith crack in a piezoelectric material"
1654:
In ductile materials (and even in materials that appear to be brittle), a
1587:
Griffith's theory provides excellent agreement with experimental data for
6714:
4958:
3930:
1664:
1644:
604:
507:
3911:
two definitions are equivalent if the crack tip blunts in a semicircle.
6729:
6641:
6480:
6174:
5241:
4578:
4375:
to characterise fracture toughness, a relation has been used to reduce
3970:
3949:
2371:
Stress intensity replaced strain energy release rate and a term called
808:
of the material. The prediction of crack growth is at the heart of the
759:
526:
422:
5728:
5552:"An improved semi-analytical solution for stress at round-tip notches"
5232: – Failure mode of anchors in concrete submitted to tensile force
3119:
may be used to calculate the amount of energy available for fracture.
6751:
6741:
6736:
6678:
6585:
6169:
5930:
Fracture and Size Effect in Concrete and Other Quasibrittle Materials
4864:., and depends on the material properties of the structure. When the
1555:, this is the criterion for which the crack will begin to propagate.
665:
498:
493:
327:
5813:
Dugdale, D. S. (1960), "Yielding of steel sheets containing slits",
3818:
1859:{\displaystyle \sigma _{f}{\sqrt {a}}={\sqrt {\cfrac {E~G}{\pi }}}.}
862:
30:
6701:
6612:
6580:
6205:
5758:"The mathematical theory of equilibrium cracks in brittle fracture"
4973:
3693:
is high, then it can be deduced that the material is tough, and if
2781:
different ways of loading a material to enable a crack to propagate
1972:
673:
477:
382:
362:
348:
20:
6068:, Fourth edition, Academic Press, Boston, 2013, pp. 473–516,
5508:, International Journal of Solids and Structures, 37, pp. 171–183.
1434:. The strain energy release rate can physically be understood as:
1015:
material is infinite. To avoid that problem, Griffith developed a
836:
What is the strength of the component as a function of crack size?
6513:
6453:
6101:
6095:
5286: – Behavior of solid objects subject to stresses and strains
3966:
1655:
1592:
1588:
754:
When the size of the plastic zone at the crack tip is too large,
231:
4643:. Based on fracture mechanics, the material will fail at stress
1571:
6617:
5226:– Fracture mechanics and fatigue crack growth analysis software
5223:
3810:
of Aluminum with a center crack undergoing overloading events.
3601:{\displaystyle r_{p}={\frac {K_{C}^{2}}{2\pi \sigma _{Y}^{2}}}}
1668:
372:
6819:
6548:
6315:
5611:
Aerospace Structures- an Introduction to Fundamental Problems
1678:
Irwin's strategy was to partition the energy into two parts:
1596:
903:
276:
5268: – Externally-produced indentation in a planar material
4930:
the failure is governed by fracture mechanics. The value of
4694:{\displaystyle \sigma _{\text{fail}}=K_{Ic}/{\sqrt {\pi a}}}
4558:
method which is based on concepts proposed independently by
4177:
is an arbitrary path clockwise around the apex of the crack,
3444:
2549:
1672:
1436:
the rate at which energy is absorbed by growth of the crack
1054:
stored in a perfect specimen under a uniaxial tensile load.
965:) was nearly constant, which is expressed by the equation:
6634:
5465:
Philosophical Transactions of the Royal Society of London
5342:
3984:
The mathematical definition of J-integral is as follows:
2259:
is the angle with respect to the plane of the crack, and
2043:
mode I loading is related to the stress intensity factor
1575:
The plastic zone around a crack tip in a ductile material
1498:{\displaystyle G_{c}={\frac {\pi \sigma _{f}^{2}a}{E}}\,}
719:. Although the load on a crack can be arbitrary, in 1957
412:
6004:
An Introduction to the Mechanical Properties of Ceramics
5981:
Buckley, C.P. "Material Failure", Lecture Notes (2005),
4897:, the failure is governed by plastic yielding, and when
3969:) deformation ahead of the crack tip, is designated the
1174:
is the surface energy density of the material. Assuming
6027:, Cambridge Solid State Science Series, 2nd Edn. (1993)
5916:
Lecture Notes in Fracture Mechanics by Victor E. Saouma
5463:(1921), "The phenomena of rupture and flow in solids",
3421:
3375:
3348:
3324:
3222:
3207:
3167:
3152:
2512:
2490:
2122:
2103:
1964:{\displaystyle G\approx G_{p}=1000\,\,{\text{J/m}}^{2}}
1911:{\displaystyle G\approx 2\gamma =2\,\,{\text{J/m}}^{2}}
1839:
1821:
1658:
zone develops at the tip of the crack. As the applied
1107:
1089:
3424:
3378:
3351:
3327:
3225:
3210:
3170:
3155:
2515:
2493:
2125:
2106:
1842:
1824:
1110:
1092:
5180:
5160:
5140:
5120:
5100:
5076:
5057:
5037:
5017:
4997:
4936:
4903:
4870:
4843:
4820:
4756:
4707:
4649:
4619:
4592:
4489:
4452:
4391:
4340:
4310:
4288:
4265:
4236:
4207:
4185:
4163:
3993:
3852:
3780:
3753:
3726:
3699:
3672:
3645:
3618:
3542:
3293:
3277:
are constant while evaluating the above expressions.
3137:
3035:
3009:
2989:
2969:
2919:
2892:
2861:
2832:
2809:
2789:
2765:
2759:
and is considered a material property. The subscript
2735:
2708:
2668:
2645:
2618:
2598:
2571:
2433:
2384:
2345:
2315:
2295:
2265:
2245:
2225:
2191:
2079:
2049:
2005:
1985:
1924:
1875:
1796:
1763:
1743:
1698:
1605:
1534:
1514:
1450:
1401:
1373:
1353:
1333:
1281:
1258:
1214:
1180:
1160:
1140:
1078:
1032:
974:
944:
924:
872:
787:
705:
58:
5604:
5602:
5347:. Edward Arnold and Delftse Uitgevers Maatschappij.
4701:. Based on plasticity, the material will yield when
3509:. Today, it is the critical stress intensity factor
2702:. For the special case of plane strain deformation,
1124:{\displaystyle C={\sqrt {\cfrac {2E\gamma }{\pi }}}}
852:
5292: – Growth of cracks in a corrosive environment
5256: – Fracture or discontinuity in displaced rock
1019:approach to explain the relation that he observed.
680:different parameters have been developed. When the
5186:
5166:
5146:
5126:
5106:
5082:
5063:
5043:
5023:
5003:
4949:
4922:
4889:
4856:
4826:
4806:
4742:
4693:
4635:
4605:
4530:
4471:
4438:
4356:
4326:
4294:
4274:
4249:
4220:
4191:
4169:
4143:
3793:
3766:
3739:
3712:
3685:
3658:
3631:
3600:
3450:
3250:
3096:
3018:
2995:
2975:
2952:
2898:
2874:
2844:
2818:
2795:
2771:
2751:
2721:
2694:
2651:
2631:
2604:
2584:
2555:
2414:
2360:
2331:
2301:
2281:
2251:
2231:
2207:
2174:
2062:
2020:
1991:
1963:
1910:
1858:
1776:
1749:
1726:
1631:
1547:
1520:
1497:
1426:
1379:
1359:
1339:
1316:
1264:
1241:
1200:
1166:
1146:
1123:
1038:
1000:
957:
930:
878:
796:
711:
93:
6143:
6037:Fatigue and Fracture Mechanics of High-Risk Parts
5755:
5599:
5317:Fracture Mechanics: Fundamentals and Applications
4980:that form around anchors under tensile strength.
2309:the stress intensity factor. Since the quantity
1317:{\displaystyle G={\frac {\pi \sigma ^{2}a}{E}}\,}
6845:
4807:{\displaystyle a=K_{Ic}^{2}/\pi \sigma _{Y}^{2}}
1395:should be divided by the plate stiffness factor
1026:. Griffith found an expression for the constant
6110:by Piet Schreurs, TU Eindhoven, The Netherlands
6102:Notes on Fracture of Thin Films and Multilayers
5197:
4257:are the components of the displacement vectors,
3003:containing a through-thickness crack of length
2953:{\displaystyle K_{I}=Y\sigma {\sqrt {\pi a}}\,}
1001:{\displaystyle \sigma _{f}{\sqrt {a}}\approx C}
5992:, Cambridge Solid State Science Series, (1979)
5857:Journal of the Mechanics and Physics of Solids
5815:Journal of the Mechanics and Physics of Solids
5812:
5537:. Reports on Progress in Physics XII, 185–232.
4439:{\displaystyle K_{Ic}={\sqrt {E^{*}J_{Ic}}}\,}
4228:are the components of the vectors of traction,
2415:{\displaystyle K_{I}=\sigma {\sqrt {\pi a}}\,}
2361:{\displaystyle {\text{MPa}}{\sqrt {\text{m}}}}
6129:
5854:
5314:
4964:
4531:{\displaystyle E^{*}={\frac {E}{1-\nu ^{2}}}}
642:
6104:by Prof. John Hutchinson, Harvard University
6098:by Prof. John Hutchinson, Harvard University
5894:"Fracture Mechanics for Structural Concrete"
5244: – Sudden movement of the Earth's crust
3122:The energy release rate for crack growth or
1979:temperature, we have intermediate values of
5524:, Journal of Applied Mechanics 24, 361–364.
5500:
5498:
5496:
4368:Since engineers became accustomed to using
1242:{\displaystyle \gamma =1\ {\text{J/m}}^{2}}
1154:is the Young's modulus of the material and
886:is in the middle an infinity large material
94:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
16:Study of propagation of cracks in materials
6136:
6122:
5924:
5922:
4743:{\displaystyle \sigma _{fail}=\sigma _{Y}}
4582:Failure stress as a function of crack size
2031:
649:
635:
5718:
5608:
5516:
5514:
5484:
5453:
5412:"Analysis of a crack at a weak interface"
5367:
4435:
4124:
4063:
4016:
3519:
3093:
2949:
2411:
1948:
1947:
1895:
1894:
1494:
1313:
906:is around 100 MPa (15,000 psi).
688:forces within the material and is termed
5613:. West Lafayette, IN: Purdue University.
5493:
5459:
5338:
5336:
4577:
4282:is an incremental length along the path
3856:
3817:
3265:is the elastic energy of the system and
3107:
1632:{\displaystyle \sigma _{f}{\sqrt {a}}=C}
1570:
861:
758:can be used with parameters such as the
29:
5957:Atomistic Modeling of Materials Failure
5919:
5888:
5886:
5682:
5680:
5545:
5543:
5308:
5298: – Field of structural engineering
5210:of materials. By using techniques like
4573:
3251:{\displaystyle G:=\left_{P}=-\left_{u}}
1566:
857:
6846:
6009:
5910:
5848:
5511:
4542:
6117:
5928:Bažant, Z.P., and Planas, J. (1998).
5749:
5665:from the original on 21 November 2016
5635:from the original on 21 November 2016
5409:
5333:
5280: – Sudden transient acceleration
4586:Let a material have a yield strength
3496:material property was given the name
3269:is the crack length. Either the load
696:) and can be characterised using the
6076:, 10.1016/B978-0-12-394584-6.00010-8
6066:The Science and Technology of Rubber
5883:
5686:
5677:
5540:
5343:H.L. Ewalds; R.J.H. Wanhill (1984).
4613:and a fracture toughness in mode I
3977:and is conventionally converted to K
3747:is small, then the squared ratio of
2239:is the distance from the crack tip,
6062:Chapter 10 – Strength of Elastomers
5806:
5659:Handbook for Damage Tolerant Design
5629:Handbook for Damage Tolerant Design
5549:
3936:, Crack growth resistance curve or
902:The stress needed to fracture bulk
13:
6096:Nonlinear Fracture Mechanics Notes
5975:
5609:Weisshaar, Terry (July 28, 2011).
4364:are the stress and strain tensors.
4289:
4164:
4054:
4039:
4005:
3853:Elastic–plastic fracture mechanics
3227:
3212:
3172:
3157:
1201:{\displaystyle E=62\ {\text{GPa}}}
866:A Griffith crack (flaw) of length
788:
756:elastic-plastic fracture mechanics
734:normal to the plane of the crack),
14:
6875:
6089:
5571:10.1016/j.engfracmech.2015.10.004
5416:International Journal of Fracture
5368:McMeeking, Robert M. (May 2004).
4357:{\displaystyle \varepsilon _{ij}}
3525:represent cracks as round tipped
1639:still holds, the surface energy (
853:Linear elastic fracture mechanics
690:linear elastic fracture mechanics
6057:, Imperial College Press, (2012)
5932:. CRC Press, Boca Raton, Florida
4986:critical stress intensity factor
4199:is the density of strain energy,
3934:crack extension resistance curve
2021:{\displaystyle {\text{J/m}}^{2}}
1727:{\displaystyle G=2\gamma +G_{p}}
5990:Mechanical Behavior of Ceramics
5948:
5935:
5796:from the original on 2023-04-17
5738:from the original on 2008-08-28
5647:
5617:
5588:from the original on 2018-07-13
5535:Fracture and strength of solids
5442:from the original on 2023-04-17
5392:from the original on 2022-02-13
4079:
4073:
2884:dimensionless correction factor
2783:. It refers to so-called "mode
2695:{\displaystyle K_{I}\geq K_{c}}
5559:Engineering Fracture Mechanics
5527:
5403:
5374:Engineering Fracture Mechanics
5361:
5174:= maximum aggregate size, and
4750:. These curves intersect when
4070:
4010:
3902:Crack tip opening displacement
3813:
3399:
3380:
2169:
2163:
1649:U.S. Naval Research Laboratory
1591:materials such as glass. For
1421:
1402:
1367:is half the crack length, and
938:) and the stress at fracture (
812:mechanical design discipline.
764:crack tip opening displacement
1:
5777:10.1016/s0065-2156(08)70121-2
5765:Advances in Applied Mechanics
5625:"Crack Tip Plastic Zone Size"
5550:Liu, M.; et al. (2015).
5386:10.1016/S0013-7944(03)00135-8
5302:
5296:Structural fracture mechanics
3943:
3925:Crack growth resistance curve
2803:" loading as opposed to mode
815:
5945:103 (25), pp. 9434–9439
5877:10.1016/0022-5096(67)90029-4
5835:10.1016/0022-5096(60)90013-2
5699:Journal of Applied Mechanics
5198:Atomistic Fracture Mechanics
4988:, and proposed the relation
4327:{\displaystyle \sigma _{ij}}
3866:American Airlines Flight 587
2639:the fracture toughness, and
2208:{\displaystyle \sigma _{ij}}
1441:However, we also have that:
1427:{\displaystyle (1-\nu ^{2})}
7:
5943:Proc. Nat'l Acad. Sci., USA
5217:
5114:= stress intensity factor,
4606:{\displaystyle \sigma _{Y}}
3794:{\displaystyle \sigma _{Y}}
3740:{\displaystyle \sigma _{Y}}
3713:{\displaystyle \sigma _{Y}}
3659:{\displaystyle \sigma _{Y}}
958:{\displaystyle \sigma _{f}}
10:
6880:
6025:Fracture of Brittle Solids
6016:The brittle fracture story
5756:Barenblatt, G. I. (1962),
4970:Concrete fracture analysis
4965:Concrete fracture analysis
4923:{\displaystyle a>a_{t}}
4890:{\displaystyle a<a_{t}}
4546:
3947:
3922:
3918:
3899:
3868:, leading to a fatal crash
3849:testing can be expensive.
3125:strain energy release rate
3114:Strain energy release rate
3111:
2035:
1757:is the surface energy and
1689:Then the total energy is:
1561:fracture of soft materials
1064:Compute the change in the
18:
6818:
6750:
6700:
6677:
6539:
6302:
6188:
6152:
6034:, and Glassco, J. (1997)
5966:10.1007/978-0-387-76426-9
5290:Stress corrosion cracking
5194:= an empirical constant.
25:Fracture (disambiguation)
6486:Compact tension specimen
6206:Conservation of momentum
5999:, Open University (2004)
5187:{\displaystyle \lambda }
5064:{\displaystyle \lambda }
3639:, and the yield stress,
3280:Irwin showed that for a
1599:, although the relation
1391:, which for the case of
797:{\displaystyle \Delta K}
153:Clausius–Duhem (entropy)
103:Fick's laws of diffusion
6566:Navier–Stokes equations
6466:Material failure theory
6454:Material failure theory
6085:, SpringerLink, (2012).
6011:Tipper, Constance Fligg
5428:10.1023/A:1011041409243
5410:Lenci, Stefano (2001).
5260:Material failure theory
5167:{\displaystyle \delta }
5107:{\displaystyle \sigma }
5083:{\displaystyle \delta }
5004:{\displaystyle \sigma }
4472:{\displaystyle E^{*}=E}
4295:{\displaystyle \Gamma }
4170:{\displaystyle \Gamma }
3895:
3864:, which separated from
2252:{\displaystyle \theta }
2038:Stress intensity factor
2032:Stress intensity factor
1750:{\displaystyle \gamma }
1347:is the applied stress,
1340:{\displaystyle \sigma }
1167:{\displaystyle \gamma }
698:stress intensity factor
311:Navier–Stokes equations
249:Material failure theory
6238:Conservation of energy
6051:Chen, X., Mai, Y.-W.,
6040:, Chapman & Hall.
5486:10.1098/rsta.1921.0006
5315:T.L. Anderson (1995).
5188:
5168:
5148:
5128:
5108:
5092:
5084:
5065:
5045:
5025:
5005:
4951:
4924:
4891:
4858:
4828:
4808:
4744:
4695:
4637:
4636:{\displaystyle K_{Ic}}
4607:
4583:
4532:
4473:
4440:
4358:
4328:
4296:
4276:
4251:
4222:
4193:
4171:
4145:
3869:
3834:
3833:while in harbor, 1943.
3795:
3768:
3741:
3714:
3687:
3660:
3633:
3602:
3520:Crack tip plastic zone
3452:
3252:
3098:
3020:
2997:
2977:
2954:
2908:geometric shape factor
2900:
2876:
2846:
2820:
2797:
2779:arises because of the
2773:
2753:
2752:{\displaystyle K_{Ic}}
2723:
2696:
2653:
2633:
2606:
2586:
2557:
2416:
2362:
2333:
2332:{\displaystyle f_{ij}}
2303:
2283:
2282:{\displaystyle f_{ij}}
2253:
2233:
2209:
2176:
2064:
2022:
1993:
1965:
1912:
1860:
1778:
1751:
1728:
1633:
1585:
1576:
1549:
1522:
1499:
1428:
1381:
1361:
1341:
1318:
1266:
1243:
1202:
1168:
1148:
1125:
1040:
1002:
959:
932:
887:
880:
798:
713:
95:
35:
23:. For other uses, see
6603:Archimedes' principle
6571:Bernoulli's principle
5284:Strength of materials
5230:Concrete cone failure
5189:
5169:
5149:
5129:
5127:{\displaystyle \tau }
5109:
5085:
5066:
5046:
5026:
5024:{\displaystyle \tau }
5006:
4990:
4978:cone-shaped fractures
4952:
4950:{\displaystyle a_{t}}
4925:
4892:
4859:
4857:{\displaystyle a_{t}}
4829:
4809:
4745:
4696:
4638:
4608:
4581:
4533:
4474:
4441:
4359:
4329:
4297:
4277:
4252:
4250:{\displaystyle u_{i}}
4223:
4221:{\displaystyle T_{i}}
4194:
4172:
4146:
3860:
3821:
3796:
3769:
3767:{\displaystyle K_{C}}
3742:
3715:
3688:
3686:{\displaystyle K_{c}}
3661:
3634:
3632:{\displaystyle K_{C}}
3603:
3453:
3253:
3108:Strain energy release
3099:
3021:
2998:
2978:
2955:
2901:
2877:
2875:{\displaystyle K_{I}}
2847:
2821:
2798:
2774:
2754:
2724:
2722:{\displaystyle K_{c}}
2697:
2662:Fracture occurs when
2654:
2634:
2632:{\displaystyle K_{c}}
2607:
2587:
2585:{\displaystyle K_{I}}
2558:
2417:
2363:
2334:
2304:
2284:
2254:
2234:
2210:
2177:
2065:
2063:{\displaystyle K_{I}}
2023:
1994:
1966:
1913:
1861:
1779:
1777:{\displaystyle G_{p}}
1752:
1729:
1634:
1578:
1574:
1550:
1548:{\displaystyle G_{c}}
1523:
1500:
1429:
1382:
1362:
1342:
1319:
1267:
1244:
1203:
1169:
1149:
1126:
1041:
1003:
960:
933:
881:
865:
799:
714:
306:Bernoulli's principle
299:Archimedes' principle
96:
33:
6833:William Prager Medal
6411:Rock mass plasticity
6308:Structural mechanics
6201:Conservation of mass
6189:Laws and Definitions
5983:University of Oxford
5471:(582–593): 163–198,
5236:Concrete degradation
5178:
5158:
5154:= size of specimen,
5138:
5134:= tensile strength,
5118:
5098:
5074:
5055:
5035:
5015:
4995:
4934:
4901:
4868:
4841:
4836:transition flaw size
4818:
4754:
4705:
4647:
4617:
4590:
4574:Transition flaw size
4487:
4450:
4389:
4338:
4308:
4286:
4263:
4234:
4205:
4183:
4161:
3991:
3778:
3751:
3724:
3697:
3670:
3643:
3616:
3540:
3488:(sliding mode), and
3291:
3273:or the displacement
3135:
3033:
3007:
2987:
2967:
2917:
2890:
2859:
2830:
2807:
2787:
2763:
2733:
2706:
2666:
2659:is Poisson's ratio.
2652:{\displaystyle \nu }
2643:
2616:
2596:
2569:
2431:
2382:
2343:
2313:
2293:
2263:
2243:
2223:
2189:
2077:
2047:
2003:
1983:
1922:
1873:
1794:
1761:
1741:
1696:
1603:
1567:Irwin's modification
1532:
1512:
1448:
1399:
1371:
1351:
1331:
1279:
1256:
1212:
1178:
1158:
1138:
1076:
1030:
972:
942:
922:
870:
858:Griffith's criterion
785:
703:
398:Cohesion (chemistry)
220:Infinitesimal strain
56:
6864:Structural analysis
6576:Poiseuille equation
6353:Membrane elasticity
6338:Transverse isotropy
6180:Rigid body dynamics
6146:continuum mechanics
5869:1967JMPSo..15..151W
5827:1960JMPSo...8..100D
5711:1968JAM....35..379R
5477:1921RSPTA.221..163G
5266:Notch (engineering)
4803:
4780:
4556:cohesive zone model
4549:Cohesive zone model
4543:Cohesive zone model
4110:
3862:Vertical stabilizer
3839:plastic deformation
3594:
3572:
3423:
3416:
3377:
3350:
3343:
3326:
3224:
3209:
3169:
3154:
2855:The expression for
2845:{\displaystyle III}
2514:
2492:
2124:
2105:
1999:between 2 and 1000
1841:
1823:
1484:
1109:
1091:
316:Poiseuille equation
47:Continuum mechanics
41:Part of a series on
6854:Fracture mechanics
6725:Electrorheological
6720:Magnetorheological
6476:Fracture mechanics
6243:Entropy inequality
6108:Fracture Mechanics
6082:Fracture Mechanics
5533:Orowan, E., 1949.
5506:Fracture Mechanics
5504:E. Erdogan (2000)
5380:(7–8): 1149–1163.
5345:Fracture Mechanics
5212:Molecular Dynamics
5184:
5164:
5144:
5124:
5104:
5080:
5061:
5041:
5021:
5001:
4947:
4920:
4887:
4854:
4824:
4804:
4789:
4763:
4740:
4691:
4633:
4603:
4584:
4528:
4469:
4436:
4354:
4324:
4292:
4275:{\displaystyle ds}
4272:
4247:
4218:
4189:
4167:
4141:
4086:
3964:deformation theory
3870:
3847:Fracture toughness
3835:
3791:
3764:
3737:
3710:
3683:
3656:
3629:
3598:
3580:
3558:
3499:fracture toughness
3448:
3443:
3430:
3418:
3402:
3357:
3345:
3329:
3248:
3234:
3219:
3179:
3164:
3094:
3019:{\displaystyle 2a}
3016:
2993:
2973:
2950:
2896:
2872:
2842:
2819:{\displaystyle II}
2816:
2793:
2769:
2749:
2719:
2692:
2649:
2629:
2612:stress intensity,
2602:
2582:
2553:
2548:
2534:
2509:
2412:
2373:fracture toughness
2358:
2329:
2299:
2279:
2249:
2229:
2205:
2172:
2139:
2119:
2060:
2018:
1989:
1961:
1908:
1856:
1848:
1836:
1774:
1747:
1724:
1629:
1595:materials such as
1577:
1545:
1518:
1495:
1470:
1424:
1377:
1357:
1337:
1314:
1262:
1239:
1198:
1164:
1144:
1121:
1116:
1104:
1036:
998:
955:
928:
888:
876:
806:fracture toughness
794:
740:– Sliding mode (a
730:– Opening mode (a
709:
662:Fracture mechanics
522:Magnetorheological
517:Electrorheological
254:Fracture mechanics
91:
36:
6841:
6840:
6524:Bending of plates
6498:Johnson-Holmquist
6461:Drucker stability
6435:Contact mechanics
6382:Cauchy elasticity
6360:Equation of state
6046:978-0-412-12991-9
5729:10.1115/1.3601206
5354:978-0-7131-3515-2
5278:Shock (mechanics)
5147:{\displaystyle d}
5044:{\displaystyle d}
4827:{\displaystyle a}
4689:
4657:
4538:for plane strain.
4526:
4433:
4192:{\displaystyle w}
4077:
4061:
3954:In the mid-1960s
3596:
3439:
3432:
3422:
3376:
3366:
3359:
3349:
3325:
3236:
3223:
3208:
3181:
3168:
3153:
3091:
3085:
3051:
2996:{\displaystyle W}
2976:{\displaystyle Y}
2947:
2899:{\displaystyle Y}
2796:{\displaystyle I}
2772:{\displaystyle I}
2605:{\displaystyle I}
2544:
2537:
2536:
2513:
2491:
2477:
2470:
2409:
2356:
2355:
2349:
2302:{\displaystyle K}
2232:{\displaystyle r}
2149:
2141:
2137:
2123:
2104:
2010:
1992:{\displaystyle G}
1953:
1900:
1851:
1850:
1840:
1831:
1822:
1812:
1621:
1521:{\displaystyle G}
1492:
1380:{\displaystyle E}
1360:{\displaystyle a}
1311:
1265:{\displaystyle G}
1231:
1226:
1196:
1192:
1147:{\displaystyle E}
1119:
1118:
1108:
1090:
1039:{\displaystyle C}
990:
931:{\displaystyle a}
879:{\displaystyle a}
712:{\displaystyle K}
659:
658:
534:
533:
468:
467:
237:Contact mechanics
160:
159:
89:
6871:
6662:Combined gas law
6657:Gay-Lussac's law
6628:Capillary action
6493:Damage mechanics
6138:
6131:
6124:
6115:
6114:
6020:
6019:. Cambridge U.P.
5995:Demaid, Adrian,
5970:
5969:
5952:
5946:
5939:
5933:
5926:
5917:
5914:
5908:
5907:
5905:
5903:
5898:
5890:
5881:
5879:
5852:
5846:
5845:
5810:
5804:
5803:
5802:
5801:
5795:
5762:
5753:
5747:
5745:
5744:
5743:
5737:
5722:
5720:10.1.1.1023.7604
5696:
5684:
5675:
5674:
5672:
5670:
5661:. LexTech, Inc.
5651:
5645:
5644:
5642:
5640:
5631:. LexTech, Inc.
5621:
5615:
5614:
5606:
5597:
5596:
5594:
5593:
5587:
5556:
5547:
5538:
5531:
5525:
5520:Irwin G (1957),
5518:
5509:
5502:
5491:
5489:
5488:
5457:
5451:
5450:
5448:
5447:
5407:
5401:
5400:
5398:
5397:
5365:
5359:
5358:
5340:
5331:
5330:
5312:
5193:
5191:
5190:
5185:
5173:
5171:
5170:
5165:
5153:
5151:
5150:
5145:
5133:
5131:
5130:
5125:
5113:
5111:
5110:
5105:
5089:
5087:
5086:
5081:
5070:
5068:
5067:
5062:
5050:
5048:
5047:
5042:
5030:
5028:
5027:
5022:
5010:
5008:
5007:
5002:
4956:
4954:
4953:
4948:
4946:
4945:
4929:
4927:
4926:
4921:
4919:
4918:
4896:
4894:
4893:
4888:
4886:
4885:
4863:
4861:
4860:
4855:
4853:
4852:
4833:
4831:
4830:
4825:
4814:. This value of
4813:
4811:
4810:
4805:
4802:
4797:
4785:
4779:
4774:
4749:
4747:
4746:
4741:
4739:
4738:
4726:
4725:
4700:
4698:
4697:
4692:
4690:
4682:
4680:
4675:
4674:
4659:
4658:
4655:
4642:
4640:
4639:
4634:
4632:
4631:
4612:
4610:
4609:
4604:
4602:
4601:
4537:
4535:
4534:
4529:
4527:
4525:
4524:
4523:
4504:
4499:
4498:
4478:
4476:
4475:
4470:
4462:
4461:
4445:
4443:
4442:
4437:
4434:
4432:
4431:
4419:
4418:
4409:
4404:
4403:
4363:
4361:
4360:
4355:
4353:
4352:
4333:
4331:
4330:
4325:
4323:
4322:
4301:
4299:
4298:
4293:
4281:
4279:
4278:
4273:
4256:
4254:
4253:
4248:
4246:
4245:
4227:
4225:
4224:
4219:
4217:
4216:
4198:
4196:
4195:
4190:
4176:
4174:
4173:
4168:
4150:
4148:
4147:
4142:
4140:
4139:
4123:
4122:
4109:
4108:
4107:
4094:
4078:
4075:
4062:
4060:
4052:
4051:
4050:
4037:
4035:
4034:
4009:
4008:
3960:Brown University
3831:brittle fracture
3800:
3798:
3797:
3792:
3790:
3789:
3773:
3771:
3770:
3765:
3763:
3762:
3746:
3744:
3743:
3738:
3736:
3735:
3719:
3717:
3716:
3711:
3709:
3708:
3692:
3690:
3689:
3684:
3682:
3681:
3665:
3663:
3662:
3657:
3655:
3654:
3638:
3636:
3635:
3630:
3628:
3627:
3607:
3605:
3604:
3599:
3597:
3595:
3593:
3588:
3571:
3566:
3557:
3552:
3551:
3457:
3455:
3454:
3449:
3447:
3446:
3440:
3437:
3433:
3431:
3429:
3419:
3417:
3415:
3410:
3398:
3397:
3373:
3367:
3364:
3360:
3358:
3356:
3346:
3344:
3342:
3337:
3322:
3309:
3308:
3257:
3255:
3254:
3249:
3247:
3246:
3241:
3237:
3235:
3233:
3220:
3218:
3205:
3192:
3191:
3186:
3182:
3180:
3178:
3165:
3163:
3150:
3103:
3101:
3100:
3095:
3092:
3090:
3086:
3081:
3073:
3061:
3056:
3052:
3044:
3025:
3023:
3022:
3017:
3002:
3000:
2999:
2994:
2982:
2980:
2979:
2974:
2959:
2957:
2956:
2951:
2948:
2940:
2929:
2928:
2905:
2903:
2902:
2897:
2881:
2879:
2878:
2873:
2871:
2870:
2851:
2849:
2848:
2843:
2825:
2823:
2822:
2817:
2802:
2800:
2799:
2794:
2778:
2776:
2775:
2770:
2758:
2756:
2755:
2750:
2748:
2747:
2728:
2726:
2725:
2720:
2718:
2717:
2701:
2699:
2698:
2693:
2691:
2690:
2678:
2677:
2658:
2656:
2655:
2650:
2638:
2636:
2635:
2630:
2628:
2627:
2611:
2609:
2608:
2603:
2591:
2589:
2588:
2583:
2581:
2580:
2562:
2560:
2559:
2554:
2552:
2551:
2545:
2543:for plane strain
2542:
2538:
2535:
2533:
2532:
2531:
2510:
2508:
2507:
2506:
2488:
2487:
2482:
2478:
2476:for plane stress
2475:
2471:
2469:
2468:
2456:
2443:
2442:
2421:
2419:
2418:
2413:
2410:
2402:
2394:
2393:
2367:
2365:
2364:
2359:
2357:
2353:
2352:
2350:
2347:
2338:
2336:
2335:
2330:
2328:
2327:
2308:
2306:
2305:
2300:
2288:
2286:
2285:
2280:
2278:
2277:
2258:
2256:
2255:
2250:
2238:
2236:
2235:
2230:
2214:
2212:
2211:
2206:
2204:
2203:
2181:
2179:
2178:
2173:
2162:
2161:
2147:
2146:
2142:
2140:
2138:
2127:
2120:
2118:
2117:
2116:
2101:
2092:
2091:
2069:
2067:
2066:
2061:
2059:
2058:
2027:
2025:
2024:
2019:
2017:
2016:
2011:
2008:
1998:
1996:
1995:
1990:
1977:glass transition
1970:
1968:
1967:
1962:
1960:
1959:
1954:
1951:
1940:
1939:
1917:
1915:
1914:
1909:
1907:
1906:
1901:
1898:
1865:
1863:
1862:
1857:
1852:
1849:
1847:
1837:
1835:
1829:
1819:
1818:
1813:
1808:
1806:
1805:
1783:
1781:
1780:
1775:
1773:
1772:
1756:
1754:
1753:
1748:
1733:
1731:
1730:
1725:
1723:
1722:
1638:
1636:
1635:
1630:
1622:
1617:
1615:
1614:
1554:
1552:
1551:
1546:
1544:
1543:
1527:
1525:
1524:
1519:
1504:
1502:
1501:
1496:
1493:
1488:
1483:
1478:
1465:
1460:
1459:
1433:
1431:
1430:
1425:
1420:
1419:
1386:
1384:
1383:
1378:
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1323:
1321:
1320:
1315:
1312:
1307:
1303:
1302:
1289:
1271:
1269:
1268:
1263:
1248:
1246:
1245:
1240:
1238:
1237:
1232:
1229:
1224:
1207:
1205:
1204:
1199:
1197:
1194:
1190:
1173:
1171:
1170:
1165:
1153:
1151:
1150:
1145:
1130:
1128:
1127:
1122:
1120:
1117:
1115:
1105:
1103:
1087:
1086:
1052:potential energy
1045:
1043:
1042:
1037:
1007:
1005:
1004:
999:
991:
986:
984:
983:
964:
962:
961:
956:
954:
953:
937:
935:
934:
929:
894:– thus the term
885:
883:
882:
877:
827:damage tolerance
810:damage tolerance
803:
801:
800:
795:
775:stress corrosion
718:
716:
715:
710:
664:is the field of
651:
644:
637:
483:
482:
448:Gay-Lussac's law
438:Combined gas law
388:Capillary action
273:
272:
116:
115:
100:
98:
97:
92:
90:
88:
80:
72:
38:
37:
6879:
6878:
6874:
6873:
6872:
6870:
6869:
6868:
6844:
6843:
6842:
6837:
6814:
6746:
6710:Viscoelasticity
6696:
6687:Acoustic theory
6673:
6623:Surface tension
6541:Fluid mechanics
6535:
6529:Sandwich theory
6421:Yield criterion
6416:Viscoplasticity
6387:Viscoelasticity
6348:hyperelasticity
6298:
6282:Antiplane shear
6265:Stress measures
6184:
6165:Fluid mechanics
6160:Solid mechanics
6148:
6142:
6092:
6079:Zehnder, Alan.
6030:Farahmand, B.,
5988:Davidge, R.W.,
5978:
5976:Further reading
5973:
5954:
5953:
5949:
5940:
5936:
5927:
5920:
5915:
5911:
5901:
5899:
5896:
5892:
5891:
5884:
5853:
5849:
5811:
5807:
5799:
5797:
5793:
5787:
5760:
5754:
5750:
5741:
5739:
5735:
5694:
5685:
5678:
5668:
5666:
5653:
5652:
5648:
5638:
5636:
5623:
5622:
5618:
5607:
5600:
5591:
5589:
5585:
5554:
5548:
5541:
5532:
5528:
5519:
5512:
5503:
5494:
5461:Griffith, A. A.
5458:
5454:
5445:
5443:
5408:
5404:
5395:
5393:
5366:
5362:
5355:
5341:
5334:
5327:
5313:
5309:
5305:
5254:Fault (geology)
5220:
5200:
5179:
5176:
5175:
5159:
5156:
5155:
5139:
5136:
5135:
5119:
5116:
5115:
5099:
5096:
5095:
5075:
5072:
5071:
5056:
5053:
5052:
5036:
5033:
5032:
5016:
5013:
5012:
4996:
4993:
4992:
4967:
4941:
4937:
4935:
4932:
4931:
4914:
4910:
4902:
4899:
4898:
4881:
4877:
4869:
4866:
4865:
4848:
4844:
4842:
4839:
4838:
4819:
4816:
4815:
4798:
4793:
4781:
4775:
4767:
4755:
4752:
4751:
4734:
4730:
4712:
4708:
4706:
4703:
4702:
4681:
4676:
4667:
4663:
4654:
4650:
4648:
4645:
4644:
4624:
4620:
4618:
4615:
4614:
4597:
4593:
4591:
4588:
4587:
4576:
4551:
4545:
4519:
4515:
4508:
4503:
4494:
4490:
4488:
4485:
4484:
4457:
4453:
4451:
4448:
4447:
4424:
4420:
4414:
4410:
4408:
4396:
4392:
4390:
4387:
4386:
4381:
4374:
4345:
4341:
4339:
4336:
4335:
4315:
4311:
4309:
4306:
4305:
4287:
4284:
4283:
4264:
4261:
4260:
4241:
4237:
4235:
4232:
4231:
4212:
4208:
4206:
4203:
4202:
4184:
4181:
4180:
4162:
4159:
4158:
4132:
4128:
4115:
4111:
4100:
4096:
4095:
4090:
4074:
4053:
4046:
4042:
4038:
4036:
4030:
4026:
4004:
4000:
3992:
3989:
3988:
3980:
3976:
3952:
3946:
3927:
3921:
3904:
3898:
3855:
3829:split apart by
3816:
3785:
3781:
3779:
3776:
3775:
3758:
3754:
3752:
3749:
3748:
3731:
3727:
3725:
3722:
3721:
3704:
3700:
3698:
3695:
3694:
3677:
3673:
3671:
3668:
3667:
3650:
3646:
3644:
3641:
3640:
3623:
3619:
3617:
3614:
3613:
3589:
3584:
3573:
3567:
3562:
3556:
3547:
3543:
3541:
3538:
3537:
3522:
3515:
3508:
3502:and designated
3483:
3475:Poisson's ratio
3467:Young's modulus
3442:
3441:
3436:
3434:
3425:
3420:
3411:
3406:
3393:
3389:
3379:
3374:
3372:
3369:
3368:
3363:
3361:
3352:
3347:
3338:
3333:
3328:
3323:
3321:
3314:
3313:
3304:
3300:
3292:
3289:
3288:
3242:
3226:
3221:
3211:
3206:
3204:
3200:
3199:
3187:
3171:
3166:
3156:
3151:
3149:
3145:
3144:
3136:
3133:
3132:
3116:
3110:
3074:
3072:
3068:
3060:
3043:
3039:
3034:
3031:
3030:
3008:
3005:
3004:
2988:
2985:
2984:
2968:
2965:
2964:
2939:
2924:
2920:
2918:
2915:
2914:
2891:
2888:
2887:
2866:
2862:
2860:
2857:
2856:
2831:
2828:
2827:
2808:
2805:
2804:
2788:
2785:
2784:
2764:
2761:
2760:
2740:
2736:
2734:
2731:
2730:
2713:
2709:
2707:
2704:
2703:
2686:
2682:
2673:
2669:
2667:
2664:
2663:
2644:
2641:
2640:
2623:
2619:
2617:
2614:
2613:
2597:
2594:
2593:
2576:
2572:
2570:
2567:
2566:
2547:
2546:
2541:
2539:
2527:
2523:
2516:
2511:
2502:
2498:
2494:
2489:
2486:
2483:
2480:
2479:
2474:
2472:
2464:
2460:
2455:
2448:
2447:
2438:
2434:
2432:
2429:
2428:
2401:
2389:
2385:
2383:
2380:
2379:
2351:
2346:
2344:
2341:
2340:
2320:
2316:
2314:
2311:
2310:
2294:
2291:
2290:
2270:
2266:
2264:
2261:
2260:
2244:
2241:
2240:
2224:
2221:
2220:
2217:Cauchy stresses
2196:
2192:
2190:
2187:
2186:
2154:
2150:
2126:
2121:
2112:
2108:
2107:
2102:
2100:
2096:
2084:
2080:
2078:
2075:
2074:
2054:
2050:
2048:
2045:
2044:
2040:
2034:
2012:
2007:
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2004:
2001:
2000:
1984:
1981:
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1955:
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1931:
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1464:
1455:
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1411:
1400:
1397:
1396:
1389:Young's modulus
1372:
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704:
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670:solid mechanics
655:
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544:
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489:Viscoelasticity
480:
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469:
457:
407:
403:Surface tension
367:
270:
268:Fluid mechanics
260:
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172:
170:Solid mechanics
162:
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113:
105:
81:
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28:
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11:
5:
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6706:
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6561:Fluid dynamics
6558:
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6551:
6545:
6543:
6537:
6536:
6534:
6533:
6532:
6531:
6526:
6521:
6519:Bending moment
6510:
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6426:Bresler-Pister
6418:
6413:
6403:
6402:
6401:
6400:
6399:
6397:Concrete creep
6394:
6384:
6379:
6377:hypoelasticity
6374:
6373:
6372:
6367:
6357:
6356:
6355:
6345:
6340:
6335:
6330:
6319:
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6090:External links
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5863:(3): 151–162,
5847:
5821:(2): 100–104,
5805:
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5705:(2): 379–386,
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5422:(3): 275–290.
5402:
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1059:elastic energy
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1024:surface energy
1009:
1008:
997:
994:
989:
982:
978:
952:
948:
927:
911:
910:
907:
896:Griffith crack
892:A. A. Griffith
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850:
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846:
843:
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817:
814:
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790:
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732:tensile stress
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393:Chromatography
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15:
9:
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2:
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6859:Glass physics
6857:
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6834:
6831:
6829:
6828:Eringen Medal
6826:
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6692:Aeroacoustics
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6666:
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6652:Charles's law
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6596:Non-Newtonian
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6582:
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6577:
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6569:
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6556:Fluid statics
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6294:Compatibility
6292:
6290:
6287:
6283:
6280:
6279:
6278:
6275:
6273:
6270:
6266:
6263:
6261:
6260:Cauchy stress
6258:
6257:
6256:
6253:
6252:
6248:
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6241:
6239:
6236:
6232:
6229:
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6211:Navier-Stokes
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5655:"Retardation"
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5319:. CRC Press.
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4915:
4911:
4907:
4904:
4882:
4878:
4874:
4871:
4849:
4845:
4837:
4834:is called as
4821:
4799:
4794:
4790:
4786:
4782:
4776:
4771:
4768:
4764:
4760:
4757:
4735:
4731:
4727:
4722:
4719:
4716:
4713:
4709:
4686:
4683:
4677:
4671:
4668:
4664:
4660:
4651:
4628:
4625:
4621:
4598:
4594:
4580:
4571:
4569:
4565:
4561:
4557:
4550:
4520:
4516:
4512:
4509:
4505:
4500:
4495:
4491:
4482:
4466:
4463:
4458:
4454:
4428:
4425:
4421:
4415:
4411:
4405:
4400:
4397:
4393:
4385:
4384:
4383:
4378:
4371:
4349:
4346:
4342:
4319:
4316:
4312:
4304:
4269:
4266:
4259:
4242:
4238:
4230:
4213:
4209:
4201:
4186:
4179:
4157:
4156:
4155:
4136:
4133:
4129:
4125:
4119:
4116:
4112:
4104:
4101:
4097:
4091:
4087:
4083:
4080:
4067:
4064:
4057:
4047:
4043:
4031:
4027:
4023:
4020:
4017:
4013:
4001:
3997:
3994:
3987:
3986:
3985:
3982:
3972:
3968:
3965:
3961:
3957:
3956:James R. Rice
3951:
3941:
3939:
3935:
3932:
3926:
3916:
3912:
3908:
3903:
3890:
3887:
3886:
3885:
3879:
3876:
3875:
3874:
3867:
3863:
3859:
3850:
3848:
3843:
3840:
3832:
3828:
3827:
3820:
3811:
3809:
3803:
3786:
3782:
3759:
3755:
3732:
3728:
3705:
3701:
3678:
3674:
3651:
3647:
3624:
3620:
3590:
3585:
3581:
3577:
3574:
3568:
3563:
3559:
3553:
3548:
3544:
3536:
3535:
3534:
3530:
3528:
3517:
3512:
3505:
3501:
3500:
3493:
3491:
3487:
3480:
3476:
3472:
3468:
3464:
3426:
3412:
3407:
3403:
3394:
3390:
3386:
3383:
3353:
3339:
3334:
3330:
3315:
3310:
3305:
3301:
3297:
3294:
3287:
3286:
3285:
3283:
3278:
3276:
3272:
3268:
3264:
3243:
3238:
3230:
3215:
3201:
3196:
3193:
3188:
3183:
3175:
3160:
3146:
3141:
3138:
3131:
3130:
3129:
3127:
3126:
3120:
3115:
3087:
3082:
3078:
3075:
3069:
3065:
3062:
3057:
3053:
3048:
3045:
3040:
3036:
3029:
3028:
3027:
3013:
3010:
2990:
2970:
2944:
2941:
2936:
2933:
2930:
2925:
2921:
2913:
2912:
2911:
2909:
2893:
2885:
2867:
2863:
2853:
2839:
2836:
2833:
2813:
2810:
2790:
2782:
2766:
2744:
2741:
2737:
2714:
2710:
2687:
2683:
2679:
2674:
2670:
2660:
2646:
2624:
2620:
2599:
2577:
2573:
2563:
2528:
2524:
2520:
2517:
2503:
2499:
2495:
2465:
2461:
2457:
2449:
2444:
2439:
2435:
2426:
2406:
2403:
2398:
2395:
2390:
2386:
2378:
2377:
2376:
2374:
2369:
2324:
2321:
2317:
2296:
2274:
2271:
2267:
2246:
2226:
2218:
2200:
2197:
2193:
2166:
2158:
2155:
2151:
2143:
2134:
2131:
2128:
2113:
2109:
2097:
2093:
2088:
2085:
2081:
2073:
2072:
2071:
2055:
2051:
2039:
2029:
2013:
1986:
1978:
1975:close to the
1974:
1956:
1944:
1941:
1936:
1932:
1928:
1925:
1903:
1891:
1888:
1885:
1882:
1879:
1876:
1853:
1844:
1832:
1826:
1814:
1809:
1802:
1798:
1790:
1789:
1788:
1785:
1769:
1765:
1744:
1719:
1715:
1711:
1708:
1705:
1702:
1699:
1692:
1691:
1690:
1684:
1681:
1680:
1679:
1676:
1674:
1670:
1666:
1661:
1657:
1652:
1650:
1646:
1642:
1626:
1623:
1618:
1611:
1607:
1598:
1594:
1590:
1584:
1583:
1573:
1564:
1562:
1556:
1540:
1536:
1515:
1489:
1485:
1480:
1475:
1471:
1467:
1461:
1456:
1452:
1444:
1443:
1442:
1439:
1437:
1416:
1412:
1408:
1405:
1394:
1390:
1374:
1354:
1334:
1308:
1304:
1299:
1295:
1291:
1285:
1282:
1275:
1274:
1273:
1259:
1250:
1234:
1221:
1218:
1215:
1187:
1184:
1181:
1161:
1141:
1112:
1100:
1097:
1094:
1082:
1079:
1072:
1071:
1067:
1063:
1060:
1056:
1053:
1049:
1048:
1047:
1033:
1025:
1020:
1018:
1017:thermodynamic
1014:
995:
992:
987:
980:
976:
968:
967:
966:
950:
946:
925:
915:
908:
905:
901:
900:
899:
897:
893:
873:
864:
847:
844:
841:
838:
835:
834:
833:
830:
828:
824:
813:
811:
807:
791:
780:
776:
772:
767:
765:
761:
757:
749:
746:
743:
739:
736:
733:
729:
726:
725:
724:
722:
706:
699:
695:
691:
687:
683:
677:
675:
671:
667:
663:
652:
647:
645:
640:
638:
633:
632:
630:
629:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
547:
540:
539:
528:
525:
523:
520:
518:
515:
514:
513:
512:
509:
506:
505:
500:
497:
495:
492:
490:
487:
486:
485:
484:
479:
474:
473:
464:
461:
460:
454:
451:
449:
446:
444:
441:
439:
436:
434:
433:Charles's law
431:
429:
426:
424:
421:
420:
418:
417:
414:
411:
410:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
380:
378:
377:
374:
371:
370:
364:
361:
357:
354:
350:
347:
342:
341:non-Newtonian
339:
335:
331:
330:
329:
326:
324:
321:
317:
314:
312:
309:
307:
304:
300:
297:
295:
292:
288:
285:
284:
282:
281:
278:
275:
274:
269:
264:
263:
255:
252:
250:
247:
243:
240:
239:
238:
235:
233:
230:
228:
227:Compatibility
225:
221:
218:
216:
215:Finite strain
213:
212:
211:
208:
206:
203:
201:
198:
196:
193:
189:
186:
185:
184:
181:
179:
176:
175:
171:
166:
165:
154:
151:
150:
149:
148:
144:
143:
138:
135:
133:
130:
128:
125:
124:
123:
122:
119:Conservations
118:
117:
109:
108:
104:
85:
82:
77:
74:
68:
65:
62:
59:
52:
51:
48:
45:
44:
40:
39:
32:
26:
22:
6715:Smart fluids
6608:Pascal's law
6475:
6440:Frictionless
6289:Large strain
6277:Small strain
6080:
6065:
6052:
6036:
6032:Bockrath, G.
6024:
6023:Lawn, B.R.,
6015:
6003:
5996:
5989:
5956:
5950:
5942:
5937:
5929:
5912:
5900:. Retrieved
5860:
5856:
5850:
5818:
5814:
5808:
5798:, retrieved
5768:
5764:
5751:
5740:, retrieved
5702:
5698:
5667:. Retrieved
5658:
5649:
5637:. Retrieved
5628:
5619:
5610:
5590:. Retrieved
5562:
5558:
5534:
5529:
5521:
5505:
5468:
5464:
5455:
5444:. Retrieved
5419:
5415:
5405:
5394:. Retrieved
5377:
5373:
5363:
5344:
5316:
5310:
5272:Peridynamics
5204:atomic scale
5201:
5093:
4991:
4982:
4969:
4968:
4835:
4585:
4555:
4552:
4481:plane stress
4376:
4369:
4367:
4153:
3983:
3953:
3937:
3933:
3928:
3913:
3909:
3905:
3883:
3871:
3844:
3836:
3825:
3804:
3610:
3531:
3523:
3510:
3503:
3497:
3494:
3478:
3470:
3462:
3460:
3438:plane strain
3365:plane stress
3282:mode I crack
3279:
3274:
3270:
3266:
3262:
3260:
3123:
3121:
3117:
2962:
2907:
2854:
2661:
2592:is the mode
2564:
2427:
2424:
2370:
2184:
2041:
1868:
1786:
1736:
1688:
1677:
1653:
1640:
1586:
1580:
1579:
1557:
1507:
1440:
1435:
1393:plane strain
1326:
1251:
1133:
1050:Compute the
1021:
1010:
916:
912:
895:
889:
831:
819:
768:
755:
753:
747:
742:shear stress
737:
727:
693:
689:
682:plastic zone
681:
678:
661:
660:
508:Smart fluids
453:Graham's law
359:
352:
337:
323:Pascal's law
319:
302:
290:
253:
145:Inequalities
6730:Ferrofluids
6647:Boyle's law
6333:Hooke's law
6272:Deformation
6249:Definitions
6002:Green, D.,
5688:Rice, J. R.
5669:20 November
5639:20 November
5565:: 134–143.
4959:micrometers
3826:Schenectady
3814:Limitations
2070:following:
1665:dissipation
1645:G. R. Irwin
1272:, becomes:
1066:free energy
527:Ferrofluids
428:Boyle's law
200:Hooke's law
178:Deformation
6848:Categories
6785:Gay-Lussac
6752:Scientists
6642:Atmosphere
6507:Structures
6481:J-integral
6445:Frictional
6406:Plasticity
6343:Orthotropy
6323:Elasticity
6226:Archimedes
6221:Poiseuille
6175:Vibrations
6144:Topics in
5800:2022-06-08
5771:: 55–129,
5742:2008-05-31
5592:2017-11-01
5446:2021-12-18
5396:2021-12-18
5303:References
5242:Earthquake
4560:Barenblatt
3971:J-integral
3950:J-integral
3944:J-integral
816:Motivation
771:similitude
760:J-integral
580:Gay-Lussac
543:Scientists
443:Fick's law
423:Atmosphere
242:frictional
195:Plasticity
183:Elasticity
6760:Bernoulli
6742:Rheometer
6737:Rheometry
6679:Acoustics
6591:Newtonian
6586:Viscosity
6216:Bernoulli
6170:Acoustics
6153:Divisions
5997:Fail Safe
5843:136484892
5715:CiteSeerX
5436:115306909
5182:λ
5162:δ
5122:τ
5102:σ
5078:δ
5059:λ
5019:τ
4999:σ
4791:σ
4787:π
4732:σ
4710:σ
4684:π
4652:σ
4595:σ
4570:in 1968.
4517:ν
4513:−
4496:∗
4459:∗
4416:∗
4343:ε
4313:σ
4290:Γ
4165:Γ
4130:ε
4113:σ
4098:ε
4088:∫
4055:∂
4040:∂
4024:−
4006:Γ
4002:∫
3958:(then at
3783:σ
3729:σ
3702:σ
3648:σ
3582:σ
3578:π
3391:ν
3387:−
3228:∂
3213:∂
3197:−
3173:∂
3158:∂
3076:π
3066:
2942:π
2937:σ
2680:≥
2647:ν
2525:ν
2521:−
2404:π
2399:σ
2247:θ
2194:σ
2167:θ
2132:π
2082:σ
1929:≈
1886:γ
1880:≈
1845:π
1799:σ
1745:γ
1709:γ
1608:σ
1472:σ
1468:π
1413:ν
1409:−
1335:σ
1296:σ
1292:π
1216:γ
1162:γ
1113:π
1101:γ
993:≈
977:σ
947:σ
789:Δ
666:mechanics
620:Truesdell
550:Bernoulli
499:Rheometer
494:Rheometry
334:Newtonian
328:Viscosity
78:φ
66:−
6702:Rheology
6613:Pressure
6581:Buoyancy
6365:Hugoniot
6013:(1962).
5902:13 April
5791:archived
5733:archived
5690:(1968),
5663:Archived
5633:Archived
5583:Archived
5579:51902898
5440:Archived
5390:Archived
5218:See also
4974:concrete
3490:mode III
2729:becomes
2215:are the
1973:polymers
748:Mode III
721:G. Irwin
674:fracture
478:Rheology
383:Adhesion
363:Pressure
349:Buoyancy
294:Dynamics
132:Momentum
21:Fracture
6775:Charles
6618:Liquids
6514:Bending
6471:Fatigue
5865:Bibcode
5823:Bibcode
5707:Bibcode
5473:Bibcode
5248:Fatigue
5031:/ √(1+{
4382:to it:
3967:plastic
3938:R-curve
3931:Irwin's
3919:R-curve
3527:notches
3486:mode II
3465:is the
3026:, by:
1971:. For
1656:plastic
1647:at the
1593:ductile
1589:brittle
1387:is the
1327:where
1013:elastic
823:failure
779:fatigue
762:or the
738:Mode II
686:elastic
565:Charles
373:Liquids
287:Statics
232:Bending
6820:Awards
6810:Stokes
6805:Navier
6800:Newton
6795:Pascal
6770:Cauchy
6669:Plasma
6549:Fluids
6328:linear
6316:Solids
6255:Stress
6231:Pascal
6072:
6044:
5841:
5783:
5717:
5577:
5434:
5351:
5323:
5224:AFGROW
5094:where
4564:Willis
4446:where
4154:where
3477:, and
3461:where
3261:where
2963:where
2565:where
2185:where
2148:
1830:
1737:where
1669:energy
1225:
1191:
1134:where
728:Mode I
615:Stokes
610:Pascal
600:Navier
595:Newton
585:Graham
560:Cauchy
463:Plasma
358:
356:Mixing
351:
336:
318:
301:
289:
277:Fluids
210:Strain
205:Stress
188:linear
137:Energy
6790:Hooke
6780:Euler
6765:Boyle
6635:Gases
6392:Creep
6304:Solid
5897:(PDF)
5839:S2CID
5794:(PDF)
5761:(PDF)
5736:(PDF)
5695:(PDF)
5586:(PDF)
5575:S2CID
5555:(PDF)
5432:S2CID
4302:, and
3824:S.S.
3808:graph
1597:steel
904:glass
590:Hooke
570:Euler
555:Boyle
413:Gases
6306:and
6195:Laws
6070:ISBN
6042:ISBN
5904:2013
5781:ISBN
5671:2016
5641:2016
5349:ISBN
5321:ISBN
5090:}),
4908:>
4875:<
4656:fail
4568:Rice
4483:and
4479:for
4334:and
4076:with
3896:CTOD
3822:The
2425:and
1945:1000
1673:heat
1660:load
1208:and
694:LEFM
605:Noll
575:Fick
127:Mass
112:Laws
6370:JWL
5962:doi
5873:doi
5831:doi
5773:doi
5725:doi
5567:doi
5563:149
5481:doi
5469:221
5424:doi
5420:108
5382:doi
3774:to
3473:is
3063:sec
2826:or
2348:MPa
2009:J/m
1952:J/m
1899:J/m
1671:as
1667:of
1508:If
1230:J/m
1195:GPa
6850::
6064:,
5960:.
5921:^
5885:^
5871:,
5861:15
5859:,
5837:,
5829:,
5817:,
5789:,
5779:,
5767:,
5763:,
5731:,
5723:,
5713:,
5703:35
5701:,
5697:,
5679:^
5657:.
5627:.
5601:^
5581:.
5573:.
5561:.
5557:.
5542:^
5513:^
5495:^
5479:,
5467:,
5438:.
5430:.
5418:.
5414:.
5388:.
5378:71
5376:.
5372:.
5335:^
5011:=
4380:Ic
4373:Ic
3979:Ic
3975:Ic
3514:Ic
3507:Ic
3469:,
3142::=
2886:,
2852::
2368:.
2219:,
2028:.
1563:.
1528:≥
1438:.
1188:62
766:.
676:.
6137:e
6130:t
6123:v
6048:.
5985:.
5968:.
5964::
5906:.
5880:.
5875::
5867::
5833::
5825::
5819:8
5775::
5769:7
5746:.
5727::
5709::
5673:.
5643:.
5595:.
5569::
5490:.
5483::
5475::
5449:.
5426::
5399:.
5384::
5357:.
5329:.
5142:d
5051:/
5039:d
4943:t
4939:a
4916:t
4912:a
4905:a
4883:t
4879:a
4872:a
4850:t
4846:a
4822:a
4800:2
4795:Y
4783:/
4777:2
4772:c
4769:I
4765:K
4761:=
4758:a
4736:Y
4728:=
4723:l
4720:i
4717:a
4714:f
4687:a
4678:/
4672:c
4669:I
4665:K
4661:=
4629:c
4626:I
4622:K
4599:Y
4521:2
4510:1
4506:E
4501:=
4492:E
4467:E
4464:=
4455:E
4429:c
4426:I
4422:J
4412:E
4406:=
4401:c
4398:I
4394:K
4377:J
4370:K
4350:j
4347:i
4320:j
4317:i
4270:s
4267:d
4243:i
4239:u
4214:i
4210:T
4187:w
4137:j
4134:i
4126:d
4120:j
4117:i
4105:j
4102:i
4092:0
4084:=
4081:w
4071:)
4068:s
4065:d
4058:x
4048:i
4044:u
4032:i
4028:T
4021:y
4018:d
4014:w
4011:(
3998:=
3995:J
3787:Y
3760:C
3756:K
3733:Y
3706:Y
3679:c
3675:K
3652:Y
3625:C
3621:K
3591:2
3586:Y
3575:2
3569:2
3564:C
3560:K
3554:=
3549:p
3545:r
3511:K
3504:G
3482:I
3479:K
3471:ν
3463:E
3427:E
3413:2
3408:I
3404:K
3400:)
3395:2
3384:1
3381:(
3354:E
3340:2
3335:I
3331:K
3316:{
3311:=
3306:I
3302:G
3298:=
3295:G
3275:u
3271:P
3267:a
3263:U
3244:u
3239:]
3231:a
3216:U
3202:[
3194:=
3189:P
3184:]
3176:a
3161:U
3147:[
3139:G
3088:)
3083:W
3079:a
3070:(
3058:=
3054:)
3049:W
3046:a
3041:(
3037:Y
3014:a
3011:2
2991:W
2971:Y
2945:a
2934:Y
2931:=
2926:I
2922:K
2894:Y
2868:I
2864:K
2840:I
2837:I
2834:I
2814:I
2811:I
2791:I
2767:I
2745:c
2742:I
2738:K
2715:c
2711:K
2688:c
2684:K
2675:I
2671:K
2625:c
2621:K
2600:I
2578:I
2574:K
2529:2
2518:1
2504:c
2500:G
2496:E
2466:c
2462:G
2458:E
2450:{
2445:=
2440:c
2436:K
2407:a
2396:=
2391:I
2387:K
2354:m
2325:j
2322:i
2318:f
2297:K
2275:j
2272:i
2268:f
2227:r
2201:j
2198:i
2170:)
2164:(
2159:j
2156:i
2152:f
2144:)
2135:r
2129:2
2114:I
2110:K
2098:(
2094:=
2089:j
2086:i
2056:I
2052:K
2014:2
1987:G
1957:2
1942:=
1937:p
1933:G
1926:G
1904:2
1892:2
1889:=
1883:2
1877:G
1854:.
1833:G
1827:E
1815:=
1810:a
1803:f
1770:p
1766:G
1720:p
1716:G
1712:+
1706:2
1703:=
1700:G
1641:γ
1627:C
1624:=
1619:a
1612:f
1541:c
1537:G
1516:G
1490:E
1486:a
1481:2
1476:f
1462:=
1457:c
1453:G
1422:)
1417:2
1406:1
1403:(
1375:E
1355:a
1309:E
1305:a
1300:2
1286:=
1283:G
1260:G
1235:2
1222:1
1219:=
1185:=
1182:E
1142:E
1098:E
1095:2
1083:=
1080:C
1034:C
996:C
988:a
981:f
951:f
926:a
918:(
874:a
792:K
707:K
692:(
650:e
643:t
636:v
360:·
353:·
343:)
338:·
332:(
320:·
303:·
291:·
86:x
83:d
75:d
69:D
63:=
60:J
27:.
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