22:
5100:
Inverting the direction of the stress has obviously the opposite effect. By applying an alternating stress, the interstitial atom will keep jumping from one site to the other, in a reversible way, causing dissipation of energy and a producing a so-called Snoek peak. The more atoms take part in this process the more the Snoek peak will be intense. Knowing the energy dissipation of a single event and the height of the Snoek peak can make possible to determine the concentration of atoms involved in the process.
4037:
3528:
4072:
To describe the stress relaxation behaviour, one can also consider another three-parameter model more suited to the stress relaxation experiment, consisting of a
Maxwell unit placed in parallel with a spring. Its differential stressâstrain equation is the same as the other model considered, therefore
3605:
These two mechanical elements can be combined in series or in parallel. In a series combination the stresses are equal, while the strains are additive. Similarly, for a parallel combination of the same elements the strains are equal and the stresses additive. Having said that, the two simplest models
2490:
The loss angle that measures the internal friction can be obtained directly from the plot, since it is the width of the resonance peak at half-maximum. With this and the resonant frequency it is then possible to obtain the primary response functions. By changing the inertia of the sample the resonant
893:
After a creep experiment has been run for a while, when stress is released the elastic spring-back is in general followed by a time dependent decay of the strain. This effect is called the elastic aftereffect or âcreep recoveryâ. The ideal elastic solid returns to zero strain immediately, without any
2637:
Wave propagation methods utilize a wave traveling down the specimen in one direction at a time to avoid any interference effects. If the specimen is long enough and the damping high enough, this can be done by continuous wave propagation. More commonly, for crystalline materials with low damping, a
3740:
Considering the Voigt model, what it lacks is the instantaneous elastic response, characteristic of crystals. To obtain this missing feature, a spring is attached in series with the Voigt model. This is called the Voigt unit. A spring in series with a Voigt unit shows all the characteristics of an
5108:
Grain boundaries in nanocrystalline materials form are significant enough to be responsible for some specific properties of these types of materials. Both their size and structure are important to determine the mechanical effects they have. High resolution microscopy show that material put under
5099:
in one direction parallel to a side of the cube dilates the side while compressing other orthogonal ones. Because of this, the octahedral positions stop being equivalent, and the larger ones will be occupied instead of the smallest ones, making the interstitial atom jump from one to the other.
3323:
In this way though they are correlated in a complicated manner and it is not easy to evaluate one of these functions knowing the other. Hover it is still possible in principle to derive the stress relaxation function from the creep function and vice versa thanks to the
Boltzamann principle.
2666:
Each response function constitutes a complete representation of the anelastic properties of the solid. Therefore, any one of the response functions can be used to completely describe the anelastic behaviour of the solid, and every other response function can be derived from the chosen one.
5067:
Despite being historically uncommon, it has some great utility in solving practical problems regarding industrial production where knowledge and control of the microscopic structure of materials is becoming more and more important. Some of these applications are the following.
3356:
Since from the definition of anelasticity linearity is required, all differential stressâstrain equations of anelasticity must be of first degree. These equations can contain many different constants to the describe the specific solid. The most general one can be written as:
3838:
3849:
443:
dependent reaction, in addition to the elastic one in the material considered. It is also usually a very small fraction of the total response and so, in this sense, the usual meaning of "anelasticity" as "without elasticity" is improper in a physical sense.
1104:
To get information about the behaviour of a material over short periods of time dynamic experiments are needed. In this kind of experiment a periodic stress (or strain) is imposed on the system, and the phase lag of the strain (or stress) is determined.
4343:
3167:
3038:
3363:
3348:) since that is the least number of parameters necessary for a stressâstrain equation describing a simple anelastic solid. This specific basic behaviour is of such importance that a material that exhibits it is called standard anelastic solid.
4239:
2059:
5258:, that is the property of changing size when under a magnetic field, or the opposite case, changing magnetic properties when a mechanical stress is applied. These effects are dependent on the Weiss domains and their ability to re-orient.
3318:
3176:
in the case of anelasticity, and they show that material acts almost as they have a memory of their history of stress and strain. These two of equations imply that there is a relation between the J(t) and M(t). To obtain it the method of
5261:
When a magnetoelastic material is put under stress, the deformation is caused by the sum of the elastic and magnetoelastic ones. The presence of this last one changes the internal friction, by adding an additional dissipation mechanism.
5112:
Using mechanical spectroscopy techniques one can determine whether nanocrystalline metals under thermal treatments change their mechanical behaviour by changing their grain boundaries structure. One example is nanocrystalline aluminium.
1537:
3332:
It is possible to describe anelastic behaviour considering a set of parameters of the material. Since the definition of anelasticity includes linearity and a time dependant stressâstrain relation, it can be described by using a
3601:
To add internal friction to a model, the
Newtonian dashpot is used, represented by a piston moving in an ideally viscous liquid. Its velocity is proportional to the applied force, therefore entirely dissipating work as heat.
2485:
4559:
894:
after-effect, while in the case of anelasticity total recovery takes time, and that is the aftereffect. The linear viscoelastic solid only recovers partially, because the viscous contribution to strain cannot be recovered.
1908:
2909:
707:." The postulate of linearity is used because of its practical usefulness. The theory would become much more complicated otherwise, but in cases of materials under low stress this postulate can be considered true.
3730:
2624:
1805:
1278:
3668:
4973:
A solid described by such function shows a âdiscrete spectrumâ of relaxation processes, or simply a "discrete relaxation spectrum". Each "line" of the spectrum is characterized by a relaxation time
2815:
4432:
981:
809:
4675:. However, taking a sample through a Debye peak by varying the frequency continuously is not possible with the more common resonance methods. It is however possible to plot the peak by varying
3747:
705:
1158:
4032:{\displaystyle J(t)={\frac {\epsilon (t)}{\sigma _{0}}}=J_{\text{R}}-(J_{\text{R}}-J_{\text{U}})e^{-{\frac {t}{\tau _{\sigma }}}}=J_{\text{U}}+\delta (1-e^{-{\frac {t}{\tau _{\sigma }}}}),}
5244:. The boundary zones, called Bloch walls, are about one hundred atoms long, and here the orientation of one domain gradually changes into the one of the adjacent one. Applying an external
647:
4970:
In order to have both elastic and anelastic behaviour, the differential stressâstrain equation must be of the same order in the stress and strain and must start from terms of order zero.
1094:
1046:
2111:
5251:
Crystalline defects tend to anchor the domains, opposing their movement. So, materials can be divided into magnetically soft or hard based on how much the walls are strongly anchored.
4820:
1648:
5050:
4490:
2741:
2550:
2185:
materials. Therefore, subresonant methods are not generally used. Instead, methods where the inertia of the system is considered are used. These can be divided into two categories:
5009:
1435:
4245:
2260:
224:
194:
2179:
1719:
1399:
3523:{\displaystyle a_{0}\sigma +a_{1}{\dot {\sigma }}+a_{2}{\ddot {\sigma }}+\cdot \cdot \cdot =b_{0}\epsilon +b_{1}{\dot {\epsilon }}+b_{2}{\ddot {\epsilon }}+\cdot \cdot \cdot }
589:
517:
258:
4067:
553:
481:
1305:
922:
5220:
5177:
5150:
4633:
1690:
1579:
5222:, especially in areas with less nickel, and when usually martensite formation can only be induced by plastic deformation, around 9% can get formed anyway during cooling.
2372:
2345:
1185:
4746:
2286:
1606:
1351:
4947: may be varied over a wide range simply by changing the temperature. It then becomes possible to treat the dynamic response functions as functions of temperature.
4141:
2404:
2318:
2064:
The ratio of the energy dissipated to the maximum stored energy is called the "specific damping capacityâ. This ratio can be written as a function of the loss angle by
162:
51:
5076:
Unlike other chemical methods of analysis, mechanical spectroscopy is the only technique that can determine the quantity of interstitial elements in a solid solution.
2134:
1962:
1934:
1371:
1325:
888:
4713:
4673:
4582:
2521:
1209:
142:
4945:
4874:
4693:
4653:
4602:
4454:
4381:
4133:
4106:
3596:
3569:
2233:
865:
838:
1443:
4918:
4894:
4843:
1954:
305:
281:
5052:. The standard anelastic solid considered before is just a particular case of a one-line spectrum, that can be also called having a "single relaxation time".
752:
In these experiments a stress applied and held constant while the strain is observed as a function of time. This response function is called creep defined by
2677:
states that every stress applied at a different time deforms the material as it if were the only one. This can be written generally for a series of stresses
3049:
2920:
3843:
The solid whose properties are defined by this equation is called the standard anelastic solid. The solution of this equation for the creep function is:
3533:
For the specific case of anelasticity, which requires the existence of an equilibrium relation, additional restrictions must be placed on this equation.
4959:
Voigt units in series with each other and with a spring. This corresponds to a differential stressâstrain equation which contains all terms up to order
2412:
4498:
5280:
M.S. Blanter, Igor S. Golovin, H. Neuhauser, Hans-Rainer
Sinning, Internal Friction in Metallic Glasses, Springer Series in Materials Science, 2007
3187:
734:. So that the external manifestation of the internal relaxation behaviours is the stress strain relation, which in this case is time dependant.
1844:
4967:
Maxwell units all in parallel with each other and with a spring is also equivalent to a differential stressâstrain equation of the same form.
4073:
the two models are equivalent. The Voigt-type is more convenient in the analysis of creep, while the
Maxwell-type for the stress relaxation.
3741:
anelastic material despite its simplicity. It is differential stressâstrain equation it therefore interesting, and can be calculated to be:
3676:
1834: being called "storage compliance" and "loss compliance" respectively is significant, because calculating the energy stored and the
2823:
2562:
3732:, and since it displays steady viscous creep rather than recoverable creep is yet again not suited to describe an anelastic material.
3340:
To better visualize the anelastic behaviour appropriate mechanical models can be used. The simplest one contains three elements (two
336:
These conditions may be lifted in various combinations to describe different types of behaviour, summarized in the following table:
73:
1727:
1222:
2650:
of the pulse is determined either by the time it takes to reach the end of the sample, or the time it takes to come back after a
3626:
5122:
5248:
makes domains with the same orientations increase in size, until all Bloch walls are removed, and the material is magnetized.
5109:
severe plastic deformation are characterized by significant distortions and dislocations over and near the grain boundaries.
4386:
3833:{\displaystyle J_{\text{R}}\sigma +\tau _{\sigma }J_{\text{U}}{\dot {\sigma }}=\epsilon +\tau _{\sigma }{\dot {\epsilon }}}
5293:
927:
755:
90:
652:
34:
1114:
3536:
Each stressâstrain equation can be accompanied by a mechanical model to help visualizing the behaviour of materials.
718:
changes, the response is known as anelastic relaxation, and in the same formal way can be also described for example
714:
causes a response from the system called thermal relaxation that leads it to a new equilibrium state. In the case of
44:
38:
30:
594:
3670:, allows for no instantaneous deformation, therefore it is not a realistic representation of a crystalline solid.
3611:
2208:
The response of a system in a forced-vibration experiment with a periodic force has a maximum of the displacement
1721:. Two other real response functions can also be introduced by writing the previous equation in another notation:
1051:
1003:
897:
In a stress relaxation experiment the stress Ď is observed as a function of time while keeping a constant strain
746:
2629:
It is a convenient and direct way of measuring the damping, as it is directly related to the internal friction.
2067:
55:
4758:
2136:
gives a measure of the fraction of energy lost per cycle due to anelastic behaviour, and so it is known as the
1611:
5014:
4459:
2746:
2680:
5271:
A.S. Nowick, B.S. Berry, Anelastic
Relaxation in Crystalline Solids, Academic Press, New York and London 1972
3043:
The controlled variable can always be changed, expressing the stress as a function of time in a similar way:
2526:
5230:
2674:
5064:. It is extremely sensitive and can give information not attainable with other experimental methodologies.
4976:
4338:{\displaystyle J_{2}(\omega )=\delta J{\frac {\omega \tau _{\sigma }}{(1+\omega ^{2}\tau _{\sigma }^{2})}}}
319:
5195:
as an example, an anomaly in the distribution of the elements in the alloy can cause a local increase in
5095:
all octahedral positions are the same, having the same probability of being occupied. Applying a certain
1404:
3334:
2238:
284:
200:
170:
94:
5277:
C.M. Zener, Elasticity and anelasticity of metals, University of
Chicago Press, Chicago, Illinois, 1948
2503:
of the free vibrations of a sample. Solving the equation of motion for this case includes the constant
2155:
1695:
1376:
558:
486:
230:
5080:
4846:
4045:
2651:
711:
391:
374:
106:
102:
745:
can be made where either the stress or strain is held constant for a certain time. These are called
522:
450:
1283:
900:
749:, and in this case, anelastic materials exhibit creep, elastic aftereffect, and stress relaxation.
5198:
5155:
5128:
4615:
4234:{\displaystyle J_{1}(\omega )=J_{\text{U}}+{\frac {\delta J}{(1+\omega ^{2}\tau _{\sigma }^{2})}}}
1656:
1545:
4921:
4749:
4356:
for the case of dielectric relaxation phenomena. The width of the peak at half maximum value for
3178:
2350:
2323:
2054:{\displaystyle W=\int _{\omega t=0}^{\pi /2}\sigma d\epsilon ={\frac {1}{2}}J_{1}\sigma _{0}^{2}}
1163:
122:
98:
4721:
2265:
1584:
1330:
2491:
frequency changes, and so can the response functions at different frequencies can be obtained.
2377:
2291:
147:
5254:
In these kind of materials magnetic and elastic phenomena are correlated, like in the case of
4897:
2553:
2190:
2148:
The dynamic response functions can only be measured in an experiment at frequencies below any
2137:
2119:
1916:
1356:
1310:
870:
358:
4698:
4658:
4567:
2506:
1194:
127:
5255:
4955:
The next level of complexity in the description of an anelastic solid is a model containing
3617:
2196:
1936:
is the energy dissipated in a full cycle per unit of volume while the maximum stored energy
1815:
1532:{\displaystyle J^{*}(\omega )={\frac {\epsilon }{\sigma }}=|J|(\omega )e^{-i\phi (\omega )}}
5060:
A technique that measures internal friction and modulus of elasticity is called
Mechanical
4930:
4852:
4678:
4638:
4587:
4439:
4359:
4111:
4084:
3574:
3547:
2211:
843:
5092:
3341:
814:
715:
422:
5245:
5096:
4903:
4879:
4828:
3173:
1939:
1109:
840:
is called the unrelaxed compliance, the equilibrium value is called relaxed compliance
727:
290:
266:
118:
101:
mechanisms but still interprets the anelastic behaviour as a manifestation of internal
5233:
have specific anelastic effects that influence internal friction and dynamic modulus.
3162:{\displaystyle \sigma (t)=\int _{-\infty }^{t}M(t-t'){\frac {d\epsilon (t')}{dt'}}dt'}
3033:{\displaystyle \epsilon (t)=\int _{-\infty }^{t}J(t-t'){\frac {d\sigma (t')}{dt'}}dt'}
2658:
of the pulse is determined by the decrease in amplitude after successive reflections.
5287:
5241:
5237:
1327: is the angle by which the strain lags, called loss angle. For ideal elasticity
318:(1) the strain response to each level of applied stress (or vice versa) has a unique
5274:
R. Montanari, E. Bonetti, Spettroscopia
Meccanica, AIM (2010) ISBN 97-88-88529-87-81
5061:
2480:{\displaystyle {\frac {\omega _{2}-\omega _{1}}{\omega _{\text{r}}}}=Q^{-1}=\phi }
3598:
are not zero, the body is ideally elastic and is modelled by the
Hookean spring.
4554:{\displaystyle \phi \cong \Delta {\frac {\omega \tau }{1+\omega ^{2}\tau ^{2}}}}
2655:
2639:
1835:
1811:
5191:. They can be identified by anomalies in the trend of the modulus. Using steel
5088:
4349:
2643:
742:
719:
311:
1048:, and at a short timescale, when the material behaves as if ideally elastic,
4718:
The basis of why this is possible is that in many cases the relaxation rate
2671:
2262:
the resonant frequency. The resonance equation is simplified in the case of
2149:
1216:
1212:
1188:
731:
329:
3313:{\displaystyle 1=M_{U}J(t)+\int _{0}^{t}J(t-t'){d\sigma (t') \over dt'}dt'}
2642:
whose length is small compared to the specimen. The pulse is produced by a
2235: at a certain frequency of the force. This is known as resonance, and
4584:
can be obtained from the height of such a peak, while the relaxation time
2152:
of the system used. While theoretically easy to do, in practice the angle
5192:
4353:
2647:
2499:
The more common way of obtaining the anelastic response is measuring the
1903:{\displaystyle \Delta W=\oint \sigma d\epsilon =\pi J_{2}\sigma _{0}^{2}}
726:
relaxation. The internal values are coupled to stress and strain through
723:
3345:
2500:
2182:
3725:{\displaystyle \tau {\dot {\sigma }}+\sigma =\tau M{\dot {\epsilon }}}
5188:
5184:
983:
similarly to the creep function, with unrelaxed and relaxed modulus
811:
and characterizes the properties of the solid. The initial value of
105:
processes. It is a behaviour differing (usually very slightly) from
5180:
2904:{\displaystyle \epsilon (t)=\sum _{i=1}^{m}\sigma _{i}J(t-t_{i}')}
447:
The formal definition of linearity is: "If a given stress history
4963:
in both the stress and the strain. Similarly, a model containing
2619:{\displaystyle \delta =\ln \left({\frac {A_{n}}{A_{n+1}}}\right)}
2320:
on frequency is plotted as a Lorentzian curve. If the two values
5084:
3673:
The generalized stressâstrain equation for the Maxwell model is
440:
5117:
Determination of critical points in martensitic transformations
2914:
or in the integral form, is the stress is varied continuously:
5072:
Measurement of quantity of C, N, O and H in solution in metals
15:
1800:{\displaystyle J^{*}(\omega )=J_{1}(\omega )-iJ_{2}(\omega )}
3337:
with terms including stress, strain, and their derivatives.
1273:{\displaystyle \epsilon =\epsilon _{0}a^{i(\omega t-\phi )}}
2523:
called logarithmic decrement. Its value is constant and is
1401:
is complex. This quantity is called the complex compliance
5240:, each one possessing a spontaneous and randomly directed
3663:{\displaystyle J\sigma =\epsilon +\tau {\dot {\epsilon }}}
1653:
This way two real dynamic response functions are defined,
307:
is called the modulus of compliance (or just compliance).
4456:
may also be expressed as a Debye peak, in the case where
325:(2) the equilibrium response is achieved instantaneously;
97:. Its formal definition does not include the physical or
2638:
pulse propagation method is used. This method employs a
2181:
is difficult to measure when very small, for example in
439:
Anelasticity is therefore by the existence of a part of
3606:
that combine more than one element are the following:
1608:, is called the absolute dynamic compliance, given by
1219:. Then the strain is periodic with the same frequency
5201:
5158:
5131:
5017:
4979:
4933:
4927:
Therefore, where this equation applies, the quantity
4906:
4882:
4855:
4831:
4761:
4724:
4701:
4681:
4661:
4641:
4618:
4590:
4570:
4501:
4462:
4442:
4427:{\displaystyle \Delta (\log _{10}\omega \tau )=1.144}
4389:
4362:
4248:
4144:
4114:
4087:
4048:
3852:
3750:
3679:
3629:
3577:
3550:
3366:
3190:
3052:
2923:
2826:
2749:
2683:
2565:
2529:
2509:
2415:
2380:
2353:
2326:
2294:
2268:
2241:
2214:
2158:
2122:
2070:
1965:
1942:
1919:
1847:
1730:
1698:
1659:
1614:
1587:
1548:
1446:
1407:
1379:
1359:
1333:
1313:
1286:
1225:
1197:
1166:
1117:
1054:
1006:
930:
903:
873:
846:
817:
758:
655:
597:
561:
525:
489:
453:
293:
269:
233:
203:
173:
150:
130:
710:
In general, the change of an external variable of a
3181:can be used, or they can be related implicitly by:
3172:These integral expressions are a generalization of
2556:of the ratio of successive vibrations' amplitudes:
1838:in a cycle of vibration gives following equations:
976:{\displaystyle M(t)\equiv \sigma (t)/\epsilon _{0}}
804:{\displaystyle J(t)\equiv \epsilon (t)/\sigma _{0}}
5214:
5171:
5144:
5044:
5003:
4939:
4912:
4888:
4868:
4837:
4814:
4740:
4707:
4687:
4667:
4647:
4627:
4596:
4576:
4553:
4484:
4448:
4426:
4375:
4337:
4233:
4127:
4100:
4077:Dynamic properties of the standard anelastic solid
4069:is called the relaxation time at constant stress.
4061:
4031:
3832:
3724:
3662:
3590:
3563:
3522:
3312:
3161:
3032:
2903:
2809:
2735:
2618:
2544:
2515:
2479:
2398:
2366:
2339:
2312:
2280:
2254:
2227:
2173:
2128:
2105:
2053:
1948:
1928:
1902:
1799:
1713:
1684:
1642:
1600:
1573:
1531:
1429:
1393:
1365:
1345:
1319:
1299:
1272:
1203:
1179:
1152:
1088:
1040:
975:
916:
882:
859:
832:
803:
699:
641:
583:
547:
511:
475:
299:
275:
252:
218:
188:
156:
136:
5226:Magnetoelastic effects in ferromagnetic materials
5104:Structural stability in nanocrystalline materials
700:{\displaystyle \epsilon _{1}(t)+\epsilon _{2}(t)}
5121:Mechanical spectroscopy allows to determine the
1153:{\displaystyle \sigma =\sigma _{0}a^{i\omega t}}
43:but its sources remain unclear because it lacks
2817:. In this situation, the total strain will be:
93:that describes their behaviour when undergoing
5236:A non-magnetized ferromagnetic material forms
5087:'s, interstitial atoms position themselves in
4612:The dynamic properties plotted as function of
4604:from the frequency at which the peak occurs.
3610:a spring and dashpot in parallel, called the
642:{\displaystyle \sigma _{1}(t)+\sigma _{2}(t)}
117:Considering first an ideal elastic material,
8:
1814:part is called "storage compliance" and the
890:is called the relaxation of the compliance.
3623:The Voigt model, described by the equation
3616:a spring and dashpot in series, called the
1373: is in general not zero, so the ratio
1089:{\displaystyle M_{\text{U}}=1/J_{\text{U}}}
1041:{\displaystyle M_{\text{R}}=1/J_{\text{R}}}
2106:{\displaystyle \Delta W/W=2\pi \tan \phi }
924:and defining a stress relaxation function
5206:
5200:
5163:
5157:
5136:
5130:
5030:
5025:
5016:
4989:
4984:
4978:
4932:
4905:
4881:
4860:
4854:
4830:
4815:{\displaystyle \tau ^{-1}=v_{0}e^{-Q/kT}}
4799:
4792:
4782:
4766:
4760:
4729:
4723:
4700:
4680:
4660:
4640:
4617:
4589:
4569:
4542:
4532:
4511:
4500:
4476:
4461:
4441:
4400:
4388:
4367:
4361:
4323:
4318:
4308:
4287:
4277:
4253:
4247:
4219:
4214:
4204:
4180:
4171:
4149:
4143:
4119:
4113:
4092:
4086:
4053:
4047:
4013:
4004:
4000:
3975:
3958:
3949:
3945:
3932:
3919:
3903:
3888:
3868:
3851:
3819:
3818:
3812:
3788:
3787:
3781:
3771:
3755:
3749:
3711:
3710:
3684:
3683:
3678:
3649:
3648:
3628:
3582:
3576:
3555:
3549:
3497:
3496:
3490:
3472:
3471:
3465:
3449:
3419:
3418:
3412:
3394:
3393:
3387:
3371:
3365:
3260:
3231:
3226:
3201:
3189:
3109:
3080:
3072:
3051:
2980:
2951:
2943:
2922:
2889:
2867:
2857:
2846:
2825:
2798:
2770:
2754:
2748:
2688:
2682:
2598:
2588:
2582:
2564:
2528:
2508:
2462:
2447:
2436:
2423:
2416:
2414:
2390:
2385:
2379:
2358:
2352:
2331:
2325:
2304:
2299:
2293:
2267:
2246:
2240:
2219:
2213:
2157:
2121:
2077:
2069:
2045:
2040:
2030:
2016:
1994:
1990:
1976:
1964:
1941:
1918:
1894:
1889:
1879:
1846:
1782:
1757:
1735:
1729:
1697:
1668:
1660:
1658:
1643:{\displaystyle \epsilon _{0}/\sigma _{0}}
1634:
1625:
1619:
1613:
1592:
1586:
1557:
1549:
1547:
1508:
1490:
1482:
1469:
1451:
1445:
1412:
1406:
1383:
1378:
1358:
1332:
1312:
1291:
1285:
1246:
1236:
1224:
1196:
1171:
1165:
1138:
1128:
1116:
1100:Dynamic response functions and loss angle
1080:
1071:
1059:
1053:
1032:
1023:
1011:
1005:
967:
958:
929:
908:
902:
872:
851:
845:
816:
795:
786:
757:
682:
660:
654:
624:
602:
596:
566:
560:
530:
524:
494:
488:
458:
452:
314:that define the ideal elastic behaviour:
292:
268:
240:
232:
202:
172:
149:
129:
74:Learn how and when to remove this message
5045:{\displaystyle \delta J_{\sigma }^{(i)}}
4485:{\displaystyle \delta J\ll J_{\text{U}}}
2810:{\displaystyle t_{1}',t_{2}',...,t_{m}'}
2736:{\displaystyle \sigma _{i}(i=1,2,...,m)}
2189:methods employing resonant systems at a
338:
4608:Dynamic properties as functions of time
4436:The equation for the internal friction
2545:{\displaystyle \delta \simeq \pi \phi }
2288:. In this case the dependence of
287:(or just modulus) while its reciprocal
5004:{\displaystyle \tau _{\sigma }^{(i)}}
3544:In the case where only the constants
2743:that are applied at successive times
2144:Resonant and wave propagation methods
7:
5056:Mechanical spectroscopy applications
3352:Differential stressâstrain equations
5179:in martensitic transformations for
2406:falls to half maximum value, then:
1430:{\displaystyle J^{\star }(\omega )}
4571:
4508:
4390:
3076:
2947:
2646:at one end of the sample, and the
2255:{\displaystyle \omega _{\text{r}}}
2071:
1920:
1848:
1307: is the strain amplitude and
219:{\displaystyle \epsilon =J\sigma }
189:{\displaystyle \sigma =M\epsilon }
14:
2662:Boltzmann superposition principle
2374: are the ones at which
2174:{\displaystyle \varphi (\omega )}
1714:{\displaystyle \varphi (\omega )}
1394:{\displaystyle \epsilon /\sigma }
344:Unique equilibrium relationship
2193:(forced vibration or free decay)
584:{\displaystyle \epsilon _{2}(t)}
512:{\displaystyle \epsilon _{1}(t)}
253:{\displaystyle M={\frac {1}{J}}}
20:
4081:The dynamic response functions
4062:{\displaystyle \tau _{\sigma }}
2116:This shows that the loss angle
1108:The stress can be written as a
5081:body centered cubic structures
5037:
5031:
4996:
4990:
4415:
4393:
4329:
4295:
4265:
4259:
4225:
4191:
4161:
4155:
4023:
3987:
3938:
3912:
3880:
3874:
3862:
3856:
3280:
3269:
3257:
3240:
3216:
3210:
3129:
3118:
3106:
3089:
3062:
3056:
3000:
2989:
2977:
2960:
2933:
2927:
2898:
2876:
2836:
2830:
2730:
2694:
2168:
2162:
1794:
1788:
1769:
1763:
1747:
1741:
1708:
1702:
1679:
1673:
1669:
1661:
1568:
1562:
1558:
1550:
1524:
1518:
1501:
1495:
1491:
1483:
1463:
1457:
1424:
1418:
1265:
1250:
955:
949:
940:
934:
827:
821:
783:
777:
768:
762:
694:
688:
672:
666:
636:
630:
614:
608:
578:
572:
548:{\displaystyle \sigma _{2}(t)}
542:
536:
506:
500:
476:{\displaystyle \sigma _{1}(t)}
470:
464:
1:
4876: is a frequency factor,
4655: constant while varying
4635: are considered keeping
1956:per unit volume is given by:
1818:is called "loss compliance".
1300:{\displaystyle \epsilon _{0}}
917:{\displaystyle \epsilon _{0}}
649:will give rise to the strain
121:defines the relation between
5215:{\displaystyle M_{\text{s}}}
5172:{\displaystyle M_{\text{f}}}
5145:{\displaystyle M_{\text{s}}}
4628:{\displaystyle \omega \tau }
4352:since were first derived by
1685:{\displaystyle |J|(\omega )}
1574:{\displaystyle |J|(\omega )}
4748: is expressible by an
4348:These are often called the
2367:{\displaystyle \omega _{2}}
2340:{\displaystyle \omega _{1}}
1180:{\displaystyle \sigma _{0}}
5310:
4741:{\displaystyle \tau ^{-1}}
2281:{\displaystyle \phi \ll 1}
1601:{\displaystyle J^{\star }}
1346:{\displaystyle \varphi =0}
346:(complete recoverability)
2399:{\displaystyle x_{0}^{2}}
2313:{\displaystyle x_{0}^{2}}
1353:. For the anelastic case
738:Static response functions
157:{\displaystyle \epsilon }
113:Definition and elasticity
5091:sites. In an undeformed
4564:The relaxation strength
3736:Standard anelastic solid
2129:{\displaystyle \varphi }
1929:{\displaystyle \Delta W}
1581:, the absolute value of
1366:{\displaystyle \varphi }
1320:{\displaystyle \varphi }
883:{\displaystyle \delta J}
29:This article includes a
5231:Ferromagnetic materials
4708:{\displaystyle \omega }
4668:{\displaystyle \omega }
4577:{\displaystyle \Delta }
3612:Voigt (or Kelvin) model
2675:superposition principle
2516:{\displaystyle \delta }
1204:{\displaystyle \omega }
137:{\displaystyle \sigma }
58:more precise citations.
5216:
5173:
5152:and martensite finish
5146:
5046:
5005:
4941:
4914:
4890:
4870:
4839:
4816:
4742:
4709:
4689:
4669:
4649:
4629:
4598:
4578:
4555:
4486:
4450:
4428:
4377:
4339:
4235:
4129:
4102:
4063:
4033:
3834:
3726:
3664:
3592:
3565:
3524:
3314:
3163:
3034:
2905:
2862:
2811:
2737:
2620:
2546:
2517:
2481:
2400:
2368:
2341:
2314:
2282:
2256:
2229:
2175:
2130:
2107:
2055:
1950:
1930:
1904:
1801:
1715:
1686:
1644:
1602:
1575:
1533:
1431:
1395:
1367:
1347:
1321:
1301:
1274:
1205:
1181:
1154:
1090:
1042:
977:
918:
884:
861:
834:
805:
701:
643:
585:
549:
513:
477:
301:
277:
254:
220:
190:
158:
138:
5217:
5174:
5147:
5047:
5006:
4942:
4940:{\displaystyle \tau }
4915:
4891:
4871:
4869:{\displaystyle v_{o}}
4840:
4817:
4743:
4710:
4690:
4688:{\displaystyle \tau }
4670:
4650:
4648:{\displaystyle \tau }
4630:
4599:
4597:{\displaystyle \tau }
4579:
4556:
4487:
4451:
4449:{\displaystyle \phi }
4429:
4378:
4376:{\displaystyle J_{2}}
4340:
4236:
4130:
4128:{\displaystyle J_{2}}
4103:
4101:{\displaystyle J_{1}}
4064:
4034:
3835:
3727:
3665:
3593:
3591:{\displaystyle b_{0}}
3566:
3564:{\displaystyle a_{0}}
3525:
3335:differential equation
3315:
3164:
3035:
2906:
2842:
2812:
2738:
2621:
2547:
2518:
2482:
2401:
2369:
2342:
2315:
2283:
2257:
2230:
2228:{\displaystyle x_{0}}
2176:
2131:
2108:
2056:
1951:
1931:
1905:
1802:
1716:
1687:
1645:
1603:
1576:
1534:
1432:
1396:
1368:
1348:
1322:
1302:
1275:
1206:
1182:
1155:
1091:
1043:
978:
919:
885:
867:and their difference
862:
860:{\displaystyle J_{R}}
835:
806:
702:
644:
586:
550:
514:
478:
302:
285:modulus of elasticity
278:
255:
221:
191:
159:
139:
91:property of materials
5294:Elasticity (physics)
5199:
5156:
5129:
5015:
4977:
4931:
4904:
4880:
4853:
4847:absolute temperature
4829:
4759:
4722:
4699:
4695: while keeping
4679:
4659:
4639:
4616:
4588:
4568:
4499:
4460:
4440:
4387:
4360:
4246:
4142:
4112:
4085:
4046:
3850:
3748:
3677:
3627:
3575:
3548:
3364:
3188:
3050:
2921:
2824:
2747:
2681:
2563:
2552:. It represents the
2527:
2507:
2413:
2378:
2351:
2324:
2292:
2266:
2239:
2212:
2156:
2120:
2068:
1963:
1940:
1917:
1845:
1728:
1696:
1657:
1612:
1585:
1546:
1444:
1405:
1377:
1357:
1331:
1311:
1284:
1223:
1195:
1164:
1115:
1052:
1004:
928:
901:
871:
844:
833:{\displaystyle J(t)}
815:
756:
712:thermodynamic system
653:
595:
559:
523:
487:
483:produces the strain
451:
375:Nonlinear elasticity
328:(3) the response is
291:
267:
231:
201:
171:
148:
128:
5041:
5000:
4328:
4224:
3236:
3085:
2956:
2897:
2806:
2778:
2762:
2395:
2309:
2050:
2003:
1899:
5212:
5169:
5142:
5042:
5021:
5011:, and a magnitude
5001:
4980:
4937:
4922:Boltzmann constant
4910:
4886:
4866:
4835:
4812:
4750:Arrhenius equation
4738:
4705:
4685:
4665:
4645:
4625:
4594:
4574:
4551:
4482:
4446:
4424:
4373:
4335:
4314:
4231:
4210:
4125:
4098:
4059:
4029:
3830:
3722:
3660:
3588:
3561:
3520:
3310:
3222:
3179:Laplace transforms
3159:
3068:
3030:
2939:
2901:
2885:
2807:
2794:
2766:
2750:
2733:
2616:
2542:
2513:
2477:
2396:
2381:
2364:
2337:
2310:
2295:
2278:
2252:
2225:
2171:
2126:
2103:
2051:
2036:
1972:
1946:
1926:
1900:
1885:
1797:
1711:
1682:
1640:
1598:
1571:
1529:
1427:
1391:
1363:
1343:
1317:
1297:
1270:
1201:
1177:
1150:
1086:
1038:
973:
914:
880:
857:
830:
801:
730:processes such as
697:
639:
591:, then the stress
581:
545:
519:, and if a stress
509:
473:
297:
273:
250:
216:
186:
154:
134:
31:list of references
5209:
5166:
5139:
5125:martensite start
4913:{\displaystyle k}
4898:activation energy
4889:{\displaystyle Q}
4838:{\displaystyle T}
4549:
4479:
4333:
4229:
4174:
4019:
3978:
3964:
3935:
3922:
3906:
3894:
3827:
3796:
3784:
3758:
3719:
3692:
3657:
3540:Mechanical models
3505:
3480:
3427:
3402:
3328:Mechanical models
3297:
3146:
3017:
2610:
2554:natural logarithm
2453:
2450:
2249:
2204:Forced vibrations
2191:natural frequency
2140:of the material.
2138:internal friction
2024:
1949:{\displaystyle W}
1836:energy dissipated
1477:
1083:
1062:
1035:
1014:
437:
436:
300:{\displaystyle J}
276:{\displaystyle M}
248:
107:elastic behaviour
84:
83:
76:
5301:
5256:magnetostriction
5221:
5219:
5218:
5213:
5211:
5210:
5207:
5178:
5176:
5175:
5170:
5168:
5167:
5164:
5151:
5149:
5148:
5143:
5141:
5140:
5137:
5051:
5049:
5048:
5043:
5040:
5029:
5010:
5008:
5007:
5002:
4999:
4988:
4951:Discrete spectra
4946:
4944:
4943:
4938:
4919:
4917:
4916:
4911:
4895:
4893:
4892:
4887:
4875:
4873:
4872:
4867:
4865:
4864:
4844:
4842:
4841:
4836:
4821:
4819:
4818:
4813:
4811:
4810:
4803:
4787:
4786:
4774:
4773:
4747:
4745:
4744:
4739:
4737:
4736:
4715: constant.
4714:
4712:
4711:
4706:
4694:
4692:
4691:
4686:
4674:
4672:
4671:
4666:
4654:
4652:
4651:
4646:
4634:
4632:
4631:
4626:
4603:
4601:
4600:
4595:
4583:
4581:
4580:
4575:
4560:
4558:
4557:
4552:
4550:
4548:
4547:
4546:
4537:
4536:
4520:
4512:
4491:
4489:
4488:
4483:
4481:
4480:
4477:
4455:
4453:
4452:
4447:
4433:
4431:
4430:
4425:
4405:
4404:
4382:
4380:
4379:
4374:
4372:
4371:
4344:
4342:
4341:
4336:
4334:
4332:
4327:
4322:
4313:
4312:
4293:
4292:
4291:
4278:
4258:
4257:
4240:
4238:
4237:
4232:
4230:
4228:
4223:
4218:
4209:
4208:
4189:
4181:
4176:
4175:
4172:
4154:
4153:
4134:
4132:
4131:
4126:
4124:
4123:
4107:
4105:
4104:
4099:
4097:
4096:
4068:
4066:
4065:
4060:
4058:
4057:
4038:
4036:
4035:
4030:
4022:
4021:
4020:
4018:
4017:
4005:
3980:
3979:
3976:
3967:
3966:
3965:
3963:
3962:
3950:
3937:
3936:
3933:
3924:
3923:
3920:
3908:
3907:
3904:
3895:
3893:
3892:
3883:
3869:
3839:
3837:
3836:
3831:
3829:
3828:
3820:
3817:
3816:
3798:
3797:
3789:
3786:
3785:
3782:
3776:
3775:
3760:
3759:
3756:
3731:
3729:
3728:
3723:
3721:
3720:
3712:
3694:
3693:
3685:
3669:
3667:
3666:
3661:
3659:
3658:
3650:
3597:
3595:
3594:
3589:
3587:
3586:
3570:
3568:
3567:
3562:
3560:
3559:
3529:
3527:
3526:
3521:
3507:
3506:
3498:
3495:
3494:
3482:
3481:
3473:
3470:
3469:
3454:
3453:
3429:
3428:
3420:
3417:
3416:
3404:
3403:
3395:
3392:
3391:
3376:
3375:
3319:
3317:
3316:
3311:
3309:
3298:
3296:
3295:
3283:
3279:
3261:
3256:
3235:
3230:
3206:
3205:
3168:
3166:
3165:
3160:
3158:
3147:
3145:
3144:
3132:
3128:
3110:
3105:
3084:
3079:
3039:
3037:
3036:
3031:
3029:
3018:
3016:
3015:
3003:
2999:
2981:
2976:
2955:
2950:
2910:
2908:
2907:
2902:
2893:
2872:
2871:
2861:
2856:
2816:
2814:
2813:
2808:
2802:
2774:
2758:
2742:
2740:
2739:
2734:
2693:
2692:
2654:at the end. The
2633:Wave propagation
2625:
2623:
2622:
2617:
2615:
2611:
2609:
2608:
2593:
2592:
2583:
2551:
2549:
2548:
2543:
2522:
2520:
2519:
2514:
2486:
2484:
2483:
2478:
2470:
2469:
2454:
2452:
2451:
2448:
2442:
2441:
2440:
2428:
2427:
2417:
2405:
2403:
2402:
2397:
2394:
2389:
2373:
2371:
2370:
2365:
2363:
2362:
2346:
2344:
2343:
2338:
2336:
2335:
2319:
2317:
2316:
2311:
2308:
2303:
2287:
2285:
2284:
2279:
2261:
2259:
2258:
2253:
2251:
2250:
2247:
2234:
2232:
2231:
2226:
2224:
2223:
2197:wave propagation
2180:
2178:
2177:
2172:
2135:
2133:
2132:
2127:
2112:
2110:
2109:
2104:
2081:
2060:
2058:
2057:
2052:
2049:
2044:
2035:
2034:
2025:
2017:
2002:
1998:
1989:
1955:
1953:
1952:
1947:
1935:
1933:
1932:
1927:
1909:
1907:
1906:
1901:
1898:
1893:
1884:
1883:
1806:
1804:
1803:
1798:
1787:
1786:
1762:
1761:
1740:
1739:
1720:
1718:
1717:
1712:
1691:
1689:
1688:
1683:
1672:
1664:
1649:
1647:
1646:
1641:
1639:
1638:
1629:
1624:
1623:
1607:
1605:
1604:
1599:
1597:
1596:
1580:
1578:
1577:
1572:
1561:
1553:
1538:
1536:
1535:
1530:
1528:
1527:
1494:
1486:
1478:
1470:
1456:
1455:
1436:
1434:
1433:
1428:
1417:
1416:
1400:
1398:
1397:
1392:
1387:
1372:
1370:
1369:
1364:
1352:
1350:
1349:
1344:
1326:
1324:
1323:
1318:
1306:
1304:
1303:
1298:
1296:
1295:
1279:
1277:
1276:
1271:
1269:
1268:
1241:
1240:
1210:
1208:
1207:
1202:
1186:
1184:
1183:
1178:
1176:
1175:
1159:
1157:
1156:
1151:
1149:
1148:
1133:
1132:
1095:
1093:
1092:
1087:
1085:
1084:
1081:
1075:
1064:
1063:
1060:
1047:
1045:
1044:
1039:
1037:
1036:
1033:
1027:
1016:
1015:
1012:
1000:At equilibrium,
982:
980:
979:
974:
972:
971:
962:
923:
921:
920:
915:
913:
912:
889:
887:
886:
881:
866:
864:
863:
858:
856:
855:
839:
837:
836:
831:
810:
808:
807:
802:
800:
799:
790:
706:
704:
703:
698:
687:
686:
665:
664:
648:
646:
645:
640:
629:
628:
607:
606:
590:
588:
587:
582:
571:
570:
554:
552:
551:
546:
535:
534:
518:
516:
515:
510:
499:
498:
482:
480:
479:
474:
463:
462:
359:Ideal elasticity
339:
310:There are three
306:
304:
303:
298:
282:
280:
279:
274:
259:
257:
256:
251:
249:
241:
225:
223:
222:
217:
195:
193:
192:
187:
163:
161:
160:
155:
143:
141:
140:
135:
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
5309:
5308:
5304:
5303:
5302:
5300:
5299:
5298:
5284:
5283:
5268:
5228:
5202:
5197:
5196:
5159:
5154:
5153:
5132:
5127:
5126:
5123:critical points
5119:
5106:
5074:
5058:
5013:
5012:
4975:
4974:
4953:
4929:
4928:
4902:
4901:
4878:
4877:
4856:
4851:
4850:
4827:
4826:
4788:
4778:
4762:
4757:
4756:
4725:
4720:
4719:
4697:
4696:
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4657:
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4497:
4496:
4472:
4458:
4457:
4438:
4437:
4396:
4385:
4384:
4363:
4358:
4357:
4350:Debye equations
4304:
4294:
4283:
4279:
4249:
4244:
4243:
4200:
4190:
4182:
4167:
4145:
4140:
4139:
4115:
4110:
4109:
4088:
4083:
4082:
4079:
4049:
4044:
4043:
4009:
3996:
3971:
3954:
3941:
3928:
3915:
3899:
3884:
3870:
3848:
3847:
3808:
3777:
3767:
3751:
3746:
3745:
3738:
3675:
3674:
3625:
3624:
3578:
3573:
3572:
3551:
3546:
3545:
3542:
3486:
3461:
3445:
3408:
3383:
3367:
3362:
3361:
3354:
3330:
3302:
3288:
3284:
3272:
3262:
3249:
3197:
3186:
3185:
3151:
3137:
3133:
3121:
3111:
3098:
3048:
3047:
3022:
3008:
3004:
2992:
2982:
2969:
2919:
2918:
2863:
2822:
2821:
2745:
2744:
2684:
2679:
2678:
2664:
2635:
2594:
2584:
2578:
2561:
2560:
2525:
2524:
2505:
2504:
2497:
2495:Free vibrations
2458:
2443:
2432:
2419:
2418:
2411:
2410:
2376:
2375:
2354:
2349:
2348:
2327:
2322:
2321:
2290:
2289:
2264:
2263:
2242:
2237:
2236:
2215:
2210:
2209:
2206:
2154:
2153:
2146:
2118:
2117:
2066:
2065:
2026:
1961:
1960:
1938:
1937:
1915:
1914:
1875:
1843:
1842:
1833:
1826:
1778:
1753:
1731:
1726:
1725:
1694:
1693:
1655:
1654:
1630:
1615:
1610:
1609:
1588:
1583:
1582:
1544:
1543:
1504:
1447:
1442:
1441:
1408:
1403:
1402:
1375:
1374:
1355:
1354:
1329:
1328:
1309:
1308:
1287:
1282:
1281:
1242:
1232:
1221:
1220:
1193:
1192:
1167:
1162:
1161:
1134:
1124:
1113:
1112:
1102:
1076:
1055:
1050:
1049:
1028:
1007:
1002:
1001:
996:
989:
963:
926:
925:
904:
899:
898:
869:
868:
847:
842:
841:
813:
812:
791:
754:
753:
740:
678:
656:
651:
650:
620:
598:
593:
592:
562:
557:
556:
526:
521:
520:
490:
485:
484:
454:
449:
448:
423:viscoelasticity
289:
288:
265:
264:
229:
228:
199:
198:
169:
168:
146:
145:
126:
125:
115:
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
5307:
5305:
5297:
5296:
5286:
5285:
5282:
5281:
5278:
5275:
5272:
5267:
5264:
5246:magnetic field
5227:
5224:
5205:
5162:
5135:
5118:
5115:
5105:
5102:
5097:tensile stress
5073:
5070:
5057:
5054:
5039:
5036:
5033:
5028:
5024:
5020:
4998:
4995:
4992:
4987:
4983:
4952:
4949:
4936:
4909:
4885:
4863:
4859:
4834:
4823:
4822:
4809:
4806:
4802:
4798:
4795:
4791:
4785:
4781:
4777:
4772:
4769:
4765:
4735:
4732:
4728:
4704:
4684:
4664:
4644:
4624:
4621:
4609:
4606:
4593:
4573:
4562:
4561:
4545:
4541:
4535:
4531:
4527:
4524:
4519:
4516:
4510:
4507:
4504:
4475:
4471:
4468:
4465:
4445:
4423:
4420:
4417:
4414:
4411:
4408:
4403:
4399:
4395:
4392:
4370:
4366:
4346:
4345:
4331:
4326:
4321:
4317:
4311:
4307:
4303:
4300:
4297:
4290:
4286:
4282:
4276:
4273:
4270:
4267:
4264:
4261:
4256:
4252:
4241:
4227:
4222:
4217:
4213:
4207:
4203:
4199:
4196:
4193:
4188:
4185:
4179:
4170:
4166:
4163:
4160:
4157:
4152:
4148:
4122:
4118:
4095:
4091:
4078:
4075:
4056:
4052:
4040:
4039:
4028:
4025:
4016:
4012:
4008:
4003:
3999:
3995:
3992:
3989:
3986:
3983:
3974:
3970:
3961:
3957:
3953:
3948:
3944:
3940:
3931:
3927:
3918:
3914:
3911:
3902:
3898:
3891:
3887:
3882:
3879:
3876:
3873:
3867:
3864:
3861:
3858:
3855:
3841:
3840:
3826:
3823:
3815:
3811:
3807:
3804:
3801:
3795:
3792:
3780:
3774:
3770:
3766:
3763:
3754:
3737:
3734:
3718:
3715:
3709:
3706:
3703:
3700:
3697:
3691:
3688:
3682:
3656:
3653:
3647:
3644:
3641:
3638:
3635:
3632:
3621:
3620:
3614:
3585:
3581:
3558:
3554:
3541:
3538:
3531:
3530:
3519:
3516:
3513:
3510:
3504:
3501:
3493:
3489:
3485:
3479:
3476:
3468:
3464:
3460:
3457:
3452:
3448:
3444:
3441:
3438:
3435:
3432:
3426:
3423:
3415:
3411:
3407:
3401:
3398:
3390:
3386:
3382:
3379:
3374:
3370:
3353:
3350:
3329:
3326:
3321:
3320:
3308:
3305:
3301:
3294:
3291:
3287:
3282:
3278:
3275:
3271:
3268:
3265:
3259:
3255:
3252:
3248:
3245:
3242:
3239:
3234:
3229:
3225:
3221:
3218:
3215:
3212:
3209:
3204:
3200:
3196:
3193:
3170:
3169:
3157:
3154:
3150:
3143:
3140:
3136:
3131:
3127:
3124:
3120:
3117:
3114:
3108:
3104:
3101:
3097:
3094:
3091:
3088:
3083:
3078:
3075:
3071:
3067:
3064:
3061:
3058:
3055:
3041:
3040:
3028:
3025:
3021:
3014:
3011:
3007:
3002:
2998:
2995:
2991:
2988:
2985:
2979:
2975:
2972:
2968:
2965:
2962:
2959:
2954:
2949:
2946:
2942:
2938:
2935:
2932:
2929:
2926:
2912:
2911:
2900:
2896:
2892:
2888:
2884:
2881:
2878:
2875:
2870:
2866:
2860:
2855:
2852:
2849:
2845:
2841:
2838:
2835:
2832:
2829:
2805:
2801:
2797:
2793:
2790:
2787:
2784:
2781:
2777:
2773:
2769:
2765:
2761:
2757:
2753:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2691:
2687:
2663:
2660:
2634:
2631:
2627:
2626:
2614:
2607:
2604:
2601:
2597:
2591:
2587:
2581:
2577:
2574:
2571:
2568:
2541:
2538:
2535:
2532:
2512:
2496:
2493:
2488:
2487:
2476:
2473:
2468:
2465:
2461:
2457:
2446:
2439:
2435:
2431:
2426:
2422:
2393:
2388:
2384:
2361:
2357:
2334:
2330:
2307:
2302:
2298:
2277:
2274:
2271:
2245:
2222:
2218:
2205:
2202:
2201:
2200:
2194:
2170:
2167:
2164:
2161:
2145:
2142:
2125:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2080:
2076:
2073:
2062:
2061:
2048:
2043:
2039:
2033:
2029:
2023:
2020:
2015:
2012:
2009:
2006:
2001:
1997:
1993:
1988:
1985:
1982:
1979:
1975:
1971:
1968:
1945:
1925:
1922:
1911:
1910:
1897:
1892:
1888:
1882:
1878:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1831:
1824:
1816:imaginary part
1808:
1807:
1796:
1793:
1790:
1785:
1781:
1777:
1774:
1771:
1768:
1765:
1760:
1756:
1752:
1749:
1746:
1743:
1738:
1734:
1710:
1707:
1704:
1701:
1681:
1678:
1675:
1671:
1667:
1663:
1637:
1633:
1628:
1622:
1618:
1595:
1591:
1570:
1567:
1564:
1560:
1556:
1552:
1540:
1539:
1526:
1523:
1520:
1517:
1514:
1511:
1507:
1503:
1500:
1497:
1493:
1489:
1485:
1481:
1476:
1473:
1468:
1465:
1462:
1459:
1454:
1450:
1426:
1423:
1420:
1415:
1411:
1390:
1386:
1382:
1362:
1342:
1339:
1336:
1316:
1294:
1290:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1245:
1239:
1235:
1231:
1228:
1200:
1174:
1170:
1147:
1144:
1141:
1137:
1131:
1127:
1123:
1120:
1110:complex number
1101:
1098:
1079:
1074:
1070:
1067:
1058:
1031:
1026:
1022:
1019:
1010:
994:
987:
970:
966:
961:
957:
954:
951:
948:
945:
942:
939:
936:
933:
911:
907:
879:
876:
854:
850:
829:
826:
823:
820:
798:
794:
789:
785:
782:
779:
776:
773:
770:
767:
764:
761:
739:
736:
696:
693:
690:
685:
681:
677:
674:
671:
668:
663:
659:
638:
635:
632:
627:
623:
619:
616:
613:
610:
605:
601:
580:
577:
574:
569:
565:
555:gives rise to
544:
541:
538:
533:
529:
508:
505:
502:
497:
493:
472:
469:
466:
461:
457:
435:
434:
431:
428:
425:
418:
417:
414:
411:
408:
404:
403:
400:
397:
394:
390:Instantaneous
387:
386:
383:
380:
377:
371:
370:
367:
364:
361:
355:
354:
351:
350:Instantaneous
348:
342:
334:
333:
326:
323:
296:
283:is called the
272:
261:
260:
247:
244:
239:
236:
226:
215:
212:
209:
206:
196:
185:
182:
179:
176:
153:
133:
114:
111:
82:
81:
39:external links
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
5306:
5295:
5292:
5291:
5289:
5279:
5276:
5273:
5270:
5269:
5265:
5263:
5259:
5257:
5252:
5249:
5247:
5243:
5242:magnetization
5239:
5238:Weiss domains
5234:
5232:
5225:
5223:
5203:
5194:
5190:
5186:
5182:
5160:
5133:
5124:
5116:
5114:
5110:
5103:
5101:
5098:
5094:
5090:
5086:
5082:
5077:
5071:
5069:
5065:
5063:
5055:
5053:
5034:
5026:
5022:
5018:
4993:
4985:
4981:
4971:
4968:
4966:
4962:
4958:
4950:
4948:
4934:
4925:
4923:
4907:
4899:
4883:
4861:
4857:
4848:
4832:
4807:
4804:
4800:
4796:
4793:
4789:
4783:
4779:
4775:
4770:
4767:
4763:
4755:
4754:
4753:
4751:
4733:
4730:
4726:
4716:
4702:
4682:
4662:
4642:
4622:
4619:
4607:
4605:
4591:
4543:
4539:
4533:
4529:
4525:
4522:
4517:
4514:
4505:
4502:
4495:
4494:
4493:
4473:
4469:
4466:
4463:
4443:
4434:
4421:
4418:
4412:
4409:
4406:
4401:
4397:
4368:
4364:
4355:
4351:
4324:
4319:
4315:
4309:
4305:
4301:
4298:
4288:
4284:
4280:
4274:
4271:
4268:
4262:
4254:
4250:
4242:
4220:
4215:
4211:
4205:
4201:
4197:
4194:
4186:
4183:
4177:
4168:
4164:
4158:
4150:
4146:
4138:
4137:
4136:
4120:
4116:
4093:
4089:
4076:
4074:
4070:
4054:
4050:
4026:
4014:
4010:
4006:
4001:
3997:
3993:
3990:
3984:
3981:
3972:
3968:
3959:
3955:
3951:
3946:
3942:
3929:
3925:
3916:
3909:
3900:
3896:
3889:
3885:
3877:
3871:
3865:
3859:
3853:
3846:
3845:
3844:
3824:
3821:
3813:
3809:
3805:
3802:
3799:
3793:
3790:
3778:
3772:
3768:
3764:
3761:
3752:
3744:
3743:
3742:
3735:
3733:
3716:
3713:
3707:
3704:
3701:
3698:
3695:
3689:
3686:
3680:
3671:
3654:
3651:
3645:
3642:
3639:
3636:
3633:
3630:
3619:
3618:Maxwell model
3615:
3613:
3609:
3608:
3607:
3603:
3599:
3583:
3579:
3556:
3552:
3539:
3537:
3534:
3517:
3514:
3511:
3508:
3502:
3499:
3491:
3487:
3483:
3477:
3474:
3466:
3462:
3458:
3455:
3450:
3446:
3442:
3439:
3436:
3433:
3430:
3424:
3421:
3413:
3409:
3405:
3399:
3396:
3388:
3384:
3380:
3377:
3372:
3368:
3360:
3359:
3358:
3351:
3349:
3347:
3343:
3338:
3336:
3327:
3325:
3306:
3303:
3299:
3292:
3289:
3285:
3276:
3273:
3266:
3263:
3253:
3250:
3246:
3243:
3237:
3232:
3227:
3223:
3219:
3213:
3207:
3202:
3198:
3194:
3191:
3184:
3183:
3182:
3180:
3175:
3155:
3152:
3148:
3141:
3138:
3134:
3125:
3122:
3115:
3112:
3102:
3099:
3095:
3092:
3086:
3081:
3073:
3069:
3065:
3059:
3053:
3046:
3045:
3044:
3026:
3023:
3019:
3012:
3009:
3005:
2996:
2993:
2986:
2983:
2973:
2970:
2966:
2963:
2957:
2952:
2944:
2940:
2936:
2930:
2924:
2917:
2916:
2915:
2894:
2890:
2886:
2882:
2879:
2873:
2868:
2864:
2858:
2853:
2850:
2847:
2843:
2839:
2833:
2827:
2820:
2819:
2818:
2803:
2799:
2795:
2791:
2788:
2785:
2782:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2689:
2685:
2676:
2673:
2668:
2661:
2659:
2657:
2653:
2649:
2645:
2641:
2632:
2630:
2612:
2605:
2602:
2599:
2595:
2589:
2585:
2579:
2575:
2572:
2569:
2566:
2559:
2558:
2557:
2555:
2539:
2536:
2533:
2530:
2510:
2502:
2494:
2492:
2474:
2471:
2466:
2463:
2459:
2455:
2444:
2437:
2433:
2429:
2424:
2420:
2409:
2408:
2407:
2391:
2386:
2382:
2359:
2355:
2332:
2328:
2305:
2300:
2296:
2275:
2272:
2269:
2243:
2220:
2216:
2203:
2198:
2195:
2192:
2188:
2187:
2186:
2184:
2165:
2159:
2151:
2143:
2141:
2139:
2123:
2114:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2078:
2074:
2046:
2041:
2037:
2031:
2027:
2021:
2018:
2013:
2010:
2007:
2004:
1999:
1995:
1991:
1986:
1983:
1980:
1977:
1973:
1969:
1966:
1959:
1958:
1957:
1943:
1923:
1895:
1890:
1886:
1880:
1876:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1841:
1840:
1839:
1837:
1830:
1823:
1819:
1817:
1813:
1791:
1783:
1779:
1775:
1772:
1766:
1758:
1754:
1750:
1744:
1736:
1732:
1724:
1723:
1722:
1705:
1699:
1676:
1665:
1651:
1635:
1631:
1626:
1620:
1616:
1593:
1589:
1565:
1554:
1521:
1515:
1512:
1509:
1505:
1498:
1487:
1479:
1474:
1471:
1466:
1460:
1452:
1448:
1440:
1439:
1438:
1421:
1413:
1409:
1388:
1384:
1380:
1360:
1340:
1337:
1334:
1314:
1292:
1288:
1262:
1259:
1256:
1253:
1247:
1243:
1237:
1233:
1229:
1226:
1218:
1214:
1198:
1190:
1172:
1168:
1145:
1142:
1139:
1135:
1129:
1125:
1121:
1118:
1111:
1106:
1099:
1097:
1077:
1072:
1068:
1065:
1056:
1029:
1024:
1020:
1017:
1008:
998:
993:
986:
968:
964:
959:
952:
946:
943:
937:
931:
909:
905:
895:
891:
877:
874:
852:
848:
824:
818:
796:
792:
787:
780:
774:
771:
765:
759:
750:
748:
744:
737:
735:
733:
729:
725:
721:
717:
713:
708:
691:
683:
679:
675:
669:
661:
657:
633:
625:
621:
617:
611:
603:
599:
575:
567:
563:
539:
531:
527:
503:
495:
491:
467:
459:
455:
445:
442:
432:
429:
426:
424:
420:
419:
415:
412:
409:
407:Anelasticity
406:
405:
401:
398:
395:
393:
389:
388:
384:
381:
378:
376:
373:
372:
368:
365:
362:
360:
357:
356:
352:
349:
347:
343:
341:
340:
337:
331:
327:
324:
321:
317:
316:
315:
313:
308:
294:
286:
270:
263:The constant
245:
242:
237:
234:
227:
213:
210:
207:
204:
197:
183:
180:
177:
174:
167:
166:
165:
151:
131:
124:
120:
112:
110:
108:
104:
100:
96:
92:
88:
78:
75:
67:
57:
53:
47:
46:
40:
36:
32:
27:
18:
17:
5260:
5253:
5250:
5235:
5229:
5120:
5111:
5107:
5078:
5075:
5066:
5062:Spectroscopy
5059:
4972:
4969:
4964:
4960:
4956:
4954:
4926:
4824:
4717:
4611:
4563:
4435:
4383:is given by
4347:
4080:
4071:
4041:
3842:
3739:
3672:
3622:
3604:
3600:
3543:
3535:
3532:
3355:
3339:
3331:
3322:
3171:
3042:
2913:
2669:
2665:
2636:
2628:
2498:
2489:
2207:
2147:
2115:
2063:
1912:
1828:
1821:
1820:
1809:
1652:
1541:
1107:
1103:
1096:also holds.
999:
991:
984:
896:
892:
751:
747:quasi-static
741:
709:
446:
438:
345:
335:
309:
262:
116:
87:Anelasticity
86:
85:
70:
61:
50:Please help
42:
3174:Hooke's law
2656:attenuation
2640:wave packet
2183:crystalline
743:Experiments
320:equilibrium
144:and strain
119:Hooke's law
95:deformation
56:introducing
5266:References
5183:and other
5089:octahedral
2652:reflection
2644:transducer
1810:where the
720:dielectric
716:mechanical
392:plasticity
312:postulates
103:relaxation
64:March 2021
5027:σ
5019:δ
4986:σ
4982:τ
4935:τ
4794:−
4768:−
4764:τ
4731:−
4727:τ
4703:ω
4683:τ
4663:ω
4643:τ
4623:τ
4620:ω
4592:τ
4572:Δ
4540:τ
4530:ω
4518:τ
4515:ω
4509:Δ
4506:≅
4503:ϕ
4470:≪
4464:δ
4444:ϕ
4413:τ
4410:ω
4407:
4391:Δ
4320:σ
4316:τ
4306:ω
4289:σ
4285:τ
4281:ω
4272:δ
4263:ω
4216:σ
4212:τ
4202:ω
4184:δ
4159:ω
4055:σ
4051:τ
4015:σ
4011:τ
4002:−
3994:−
3985:δ
3960:σ
3956:τ
3947:−
3926:−
3910:−
3886:σ
3872:ϵ
3825:˙
3822:ϵ
3814:σ
3810:τ
3803:ϵ
3794:˙
3791:σ
3773:σ
3769:τ
3762:σ
3717:˙
3714:ϵ
3705:τ
3699:σ
3690:˙
3687:σ
3681:τ
3655:˙
3652:ϵ
3646:τ
3640:ϵ
3634:σ
3518:⋅
3515:⋅
3512:⋅
3503:¨
3500:ϵ
3478:˙
3475:ϵ
3456:ϵ
3440:⋅
3437:⋅
3434:⋅
3425:¨
3422:σ
3400:˙
3397:σ
3378:σ
3267:σ
3247:−
3224:∫
3116:ϵ
3096:−
3077:∞
3074:−
3070:∫
3054:σ
2987:σ
2967:−
2948:∞
2945:−
2941:∫
2925:ϵ
2883:−
2865:σ
2844:∑
2828:ϵ
2686:σ
2672:Boltzmann
2576:
2567:δ
2540:ϕ
2537:π
2534:≃
2531:δ
2511:δ
2475:ϕ
2464:−
2445:ω
2434:ω
2430:−
2421:ω
2356:ω
2329:ω
2273:≪
2270:ϕ
2244:ω
2166:ω
2160:φ
2150:resonance
2124:φ
2101:ϕ
2098:
2092:π
2072:Δ
2038:σ
2011:ϵ
2005:σ
1992:π
1978:ω
1974:∫
1921:Δ
1887:σ
1873:π
1867:ϵ
1861:σ
1858:∮
1849:Δ
1792:ω
1773:−
1767:ω
1745:ω
1737:∗
1706:ω
1700:φ
1677:ω
1632:σ
1617:ϵ
1594:⋆
1566:ω
1522:ω
1516:ϕ
1510:−
1499:ω
1475:σ
1472:ϵ
1461:ω
1453:∗
1422:ω
1414:⋆
1389:σ
1381:ϵ
1361:φ
1335:φ
1315:φ
1289:ϵ
1263:ϕ
1260:−
1254:ω
1234:ϵ
1227:ϵ
1217:vibration
1213:frequency
1199:ω
1189:amplitude
1169:σ
1143:ω
1126:σ
1119:σ
965:ϵ
947:σ
944:≡
906:ϵ
875:δ
793:σ
775:ϵ
772:≡
732:diffusion
680:ϵ
658:ϵ
622:σ
600:σ
564:ϵ
528:σ
492:ϵ
456:σ
214:σ
205:ϵ
184:ϵ
175:σ
152:ϵ
132:σ
99:atomistic
5288:Category
5193:AISI 304
4354:P. Debye
3307:′
3293:′
3277:′
3254:′
3156:′
3142:′
3126:′
3103:′
3027:′
3013:′
2997:′
2974:′
2895:′
2804:′
2776:′
2760:′
2648:velocity
1437:. Thus,
724:magnetic
5093:lattice
5083:, like
4920:is the
4896:is the
4845:is the
4135:, are:
3346:dashpot
3342:springs
2501:damping
2199:methods
1187:is the
728:kinetic
421:Linear
353:Linear
52:improve
5189:alloys
5185:metals
4825:where
4042:where
3344:and a
1913:where
1542:where
1280:where
1160:where
330:linear
322:value;
123:stress
5181:steel
4422:1.144
89:is a
37:, or
5187:and
5085:iron
4492:as:
4108:and
3571:and
2670:The
2347:and
1827:and
1812:real
1692:and
1211:the
1191:and
990:and
441:time
433:Yes
416:Yes
410:Yes
399:Yes
382:Yes
379:Yes
369:Yes
366:Yes
363:Yes
164:as:
5079:In
4398:log
2095:tan
1215:of
722:or
430:No
427:No
413:No
402:No
396:No
385:No
5290::
4924:.
4900:,
4849:,
4752::
4402:10
2573:ln
2113:.
1650:.
997:.
109:.
41:,
33:,
5208:s
5204:M
5165:f
5161:M
5138:s
5134:M
5038:)
5035:i
5032:(
5023:J
4997:)
4994:i
4991:(
4965:n
4961:n
4957:n
4908:k
4884:Q
4862:o
4858:v
4833:T
4808:T
4805:k
4801:/
4797:Q
4790:e
4784:0
4780:v
4776:=
4771:1
4734:1
4544:2
4534:2
4526:+
4523:1
4478:U
4474:J
4467:J
4419:=
4416:)
4394:(
4369:2
4365:J
4330:)
4325:2
4310:2
4302:+
4299:1
4296:(
4275:J
4269:=
4266:)
4260:(
4255:2
4251:J
4226:)
4221:2
4206:2
4198:+
4195:1
4192:(
4187:J
4178:+
4173:U
4169:J
4165:=
4162:)
4156:(
4151:1
4147:J
4121:2
4117:J
4094:1
4090:J
4027:,
4024:)
4007:t
3998:e
3991:1
3988:(
3982:+
3977:U
3973:J
3969:=
3952:t
3943:e
3939:)
3934:U
3930:J
3921:R
3917:J
3913:(
3905:R
3901:J
3897:=
3890:0
3881:)
3878:t
3875:(
3866:=
3863:)
3860:t
3857:(
3854:J
3806:+
3800:=
3783:U
3779:J
3765:+
3757:R
3753:J
3708:M
3702:=
3696:+
3643:+
3637:=
3631:J
3584:0
3580:b
3557:0
3553:a
3509:+
3492:2
3488:b
3484:+
3467:1
3463:b
3459:+
3451:0
3447:b
3443:=
3431:+
3414:2
3410:a
3406:+
3389:1
3385:a
3381:+
3373:0
3369:a
3304:t
3300:d
3290:t
3286:d
3281:)
3274:t
3270:(
3264:d
3258:)
3251:t
3244:t
3241:(
3238:J
3233:t
3228:0
3220:+
3217:)
3214:t
3211:(
3208:J
3203:U
3199:M
3195:=
3192:1
3153:t
3149:d
3139:t
3135:d
3130:)
3123:t
3119:(
3113:d
3107:)
3100:t
3093:t
3090:(
3087:M
3082:t
3066:=
3063:)
3060:t
3057:(
3024:t
3020:d
3010:t
3006:d
3001:)
2994:t
2990:(
2984:d
2978:)
2971:t
2964:t
2961:(
2958:J
2953:t
2937:=
2934:)
2931:t
2928:(
2899:)
2891:i
2887:t
2880:t
2877:(
2874:J
2869:i
2859:m
2854:1
2851:=
2848:i
2840:=
2837:)
2834:t
2831:(
2800:m
2796:t
2792:,
2789:.
2786:.
2783:.
2780:,
2772:2
2768:t
2764:,
2756:1
2752:t
2731:)
2728:m
2725:,
2722:.
2719:.
2716:.
2713:,
2710:2
2707:,
2704:1
2701:=
2698:i
2695:(
2690:i
2613:)
2606:1
2603:+
2600:n
2596:A
2590:n
2586:A
2580:(
2570:=
2472:=
2467:1
2460:Q
2456:=
2449:r
2438:1
2425:2
2392:2
2387:0
2383:x
2360:2
2333:1
2306:2
2301:0
2297:x
2276:1
2248:r
2221:0
2217:x
2169:)
2163:(
2089:2
2086:=
2083:W
2079:/
2075:W
2047:2
2042:0
2032:1
2028:J
2022:2
2019:1
2014:=
2008:d
2000:2
1996:/
1987:0
1984:=
1981:t
1970:=
1967:W
1944:W
1924:W
1896:2
1891:0
1881:2
1877:J
1870:=
1864:d
1855:=
1852:W
1832:2
1829:J
1825:1
1822:J
1795:)
1789:(
1784:2
1780:J
1776:i
1770:)
1764:(
1759:1
1755:J
1751:=
1748:)
1742:(
1733:J
1709:)
1703:(
1680:)
1674:(
1670:|
1666:J
1662:|
1636:0
1627:/
1621:0
1590:J
1569:)
1563:(
1559:|
1555:J
1551:|
1525:)
1519:(
1513:i
1506:e
1502:)
1496:(
1492:|
1488:J
1484:|
1480:=
1467:=
1464:)
1458:(
1449:J
1425:)
1419:(
1410:J
1385:/
1341:0
1338:=
1293:0
1266:)
1257:t
1251:(
1248:i
1244:a
1238:0
1230:=
1173:0
1146:t
1140:i
1136:a
1130:0
1122:=
1082:U
1078:J
1073:/
1069:1
1066:=
1061:U
1057:M
1034:R
1030:J
1025:/
1021:1
1018:=
1013:R
1009:M
995:R
992:M
988:U
985:M
969:0
960:/
956:)
953:t
950:(
941:)
938:t
935:(
932:M
910:0
878:J
853:R
849:J
828:)
825:t
822:(
819:J
797:0
788:/
784:)
781:t
778:(
769:)
766:t
763:(
760:J
695:)
692:t
689:(
684:2
676:+
673:)
670:t
667:(
662:1
637:)
634:t
631:(
626:2
618:+
615:)
612:t
609:(
604:1
579:)
576:t
573:(
568:2
543:)
540:t
537:(
532:2
507:)
504:t
501:(
496:1
471:)
468:t
465:(
460:1
332:.
295:J
271:M
246:J
243:1
238:=
235:M
211:J
208:=
181:M
178:=
77:)
71:(
66:)
62:(
48:.
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