Amplitude-dependent Internal Friction Research Articles

Theory of Nonlinear Damping Due to Dislocation Hysteresis

The nonlinear phenomena of damping are analysed in detail and formulated in two integral transformations; the volume and time integrals. This formulation enables one to examine the relationship between damping and tensile experiments in the microstrain range, and to describe the phenomena consistently in terms of dislocation behavior. An analysis of the amplitude-dependent internal friction is made by the use of the breakaway and friction models under the quasi-static condition. The generalized friction model is presented, which is the most appropriate for the analysis of internal friction under the steady-state vibrations in usual experiments. Several convenient formulae are given for the interpretation on dislocation mechanisms. This approach includes some earlier work such as the Granato-Lücke theory and the Davidenkov-Pisarenko formulation as a special case. In addition, the applicability of the Granato-Lücke theory is critically examined with reference to recent observations of stress-strain hysteresis.

Time and Orientation Dependent Internal Friction in Tin Crystals

The time changes in the amplitude dependent internal friction are studied in β-tin crystals, 99.99% pure, and oriented as the [001] axis is nearly perpendicular to the specimen axis. The measurements are made during the free decay of transverse vibration, at temperatures from 90°C to -196°C. Contrary to the results obtained in the crystals oriented for the [001] single slip, it is found that the internal friction decreases with the time of excitation, and increases with the time of rest after the excitation. At low temperatures, discontinuous change in the strain amplitude occurs when the measurements are made under constant driving force.

Time Dependent Internal Friction in Tin Crystals

The time changes in the amplitude dependent internal friction are studied in β-tin crystals, 99.99% pure, and oriented for the [001] slip. The measurements are made during the free decay of transverse vibration. Comparison of the results with the theory of Granato and Lucke leads to the following conclusions: (a) The length of movable dislocation increases as the amplitude of excitation is increased. (b) The length also increases with the duration of excitation at constant driving force, as log (γ t +1). This is considered to be due to breakaway from the atmosphere of solute atoms. After the breakaway, the diffusion of solute atoms from the atmosphere of the movable dislocation takes place, around room temperature. At low temperatures, the diffusion does not take place, but the brokenaway dislocation moves large distances in the lattice. The amplitude dependent internal friction is in this case caused by the interaction of the movable dislocation with the solute atoms distributed in the lattice.

Internal friction in copper and ?-brasses during plastic deformation

The internal friction Q−1 of annealed copper and α-brass wires containing 10, 20, 30 and 35 at. % of zinc was studied by a torsional oscillation method during plastic deformation. The results are interpreted in terms of two theoretical models ascribing the amplitude-dependent internal friction, observed in the pre-yield stage, to coupling of the cyclic stress with the creep component of the deformation, and the amplitude-independent internal friction at larger, plastic, strains to losses arising from contributions of the torsional stress to the plastic deformation. Up to the maximum tensile strain of 1 % used in the experiments, the influence of zinc content on Q−1 is not pronounced.

微量CおよびNを固溶した鉄の振幅依存内部摩擦

微量CおよびNを固溶した鉄の振幅依存内部摩擦

ENHANCED STRAIN AMPLITUDE DEPENDENCE OF INTERNAL FRICTION IN THE MIXED STATE OF NIOBIUM

Strain amplitude dependence of internal friction was measured as a function of the magnetic state in a niobium single crystal. An enhanced dependence is observed in the mixed state, which is attributed to compression and subsequent breakaway of fluxoids from moving dislocations.

Time-Dependent Internal Friction in Deformed Alkali Halide Crystals

Time-dependent properties of the strain amplitude dependent internal friction in alkali halide crystals were systematically investigated. The measurement technique more reliable than the conventional one was established, and repeatable enhancement and recovery of the internal friction due to external excitation was discovered. A new procedure for the Granato-Lucke plot was derived from recovery of the internal friction. At the same time, this procedure itself shows validity of the Granato-Lucke theory. The enhancement and recovery was interpreted in terms of the diffusion process of pinning elements during breakaway and the counter-diffusion process of pinning elements during resting of the dislocation-motion. The nature of hysteresis phenomena in internal friction measurements in NaCl crystals was also interpreted by the same mechanism.

Amplitude dependent internal friction in polycrystalline specimens

Abstract The Granato-Lucke theory of amplitude dependent internal friction contains a number of parameters, more than can be determined from an experiment. In a polycrystalline specimen all orientations of dislocations towards the external stress are equally probable, and an appropriate integration removes the orientation factors and the fractions of total dislocation length in these orientations. The vibrational entropy is allowed for, and an extension to higher amplitudes is used. In addition, a special experiment allows the determination of the break-away stress of a network length with a single pinning point. Thus the number of parameters is reduced sufficiently. The method is illustrated by an example.

Amplitude Dependence of Internal Friction and Shear Modulus of Boron Fibers

The internal friction and shear modulus of vapor-deposited boron fibers were measured at room temperature in an inverted torsion pendulum at about 5 Hz. The amplitude dependence was determined from <6×10−7G to 2×10−3G, where G is the shear modulus. Two regions were observed: a relatively amplitude-independent range to about 10−4G and strong amplitude dependence for higher oscillatory stresses. The internal friction and modulus were found to be independent of the number of stress-strain cycles for amplitudes <4×10−4G; irreversible changes started within a few cycles at or above 10−3G. The changes are qualitatively similar to the depinning of dislocations in metals. Absolute values of the shear modulus were determined at a strain amplitude of 2×10−6 and typically were 1.4×1012 dyn/cm2 (2×107 psi).

Investigation of processes occurring during heating of cold-worked steel by measuring the amplitude-dependent internal friction

1. Heating of cold-worked carbon steel induces substantial changes in the amplitude-dependent internal friction characteristics (ɛcr and tan α), which reach extreme values as the temperature is increased. Pinning of dislocations resulting from cold working is highest at heating temperatures of 300–400°C, corresponding to maximumɛcr and minimum tan α. 2. We found that the interaction of dislocations has a strong influence on the mobility of dislocations created in cold working. The more dispersed the secondary phase in the original structure the greater the interaction of dislocations (at the same strain). 3. Heating at the optimal temperature, corresponding to the extreme values ofɛcr and tan α (the minimum dislocation mobility), after cold working produces the highest thermal stability of the structure, leading to more complete retention of the structural imperfections resulting from cold working in subsequent heat treatment.

Thermally Activated Dislocation Depinning in Dilute Copper Alloys

Internal friction and Young's modulus measurements have been performed on a series of dilute copper—silicon, copper—germanium, and copper—tin alloys at temperatures ranging from 4° to 300°K in order to determine the extent of thermally activated dislocation depinning from solute atoms. Specimens are in the form of singel-crystal bars, and are excited piezoelectrically in their fundamental longitudinal mode of vibration at 80 kHz. For temperatures in excess of 180°K, the amplitude-dependent internal friction is characterized by an Arrhenius temperature dependence with an activation energy that varies linearly with the stress amplitude. A low-temperature modification of the Granato—Lücke theory accounts for the temperature dependence of the internal friction below about 120°K; however, the stress dependence is contradictory to that predicted theoretically. In general, thermal depinning is enhanced at high temperatures and stress levels, and in crystals containing long dislocation segment lengths and solute atoms of low pinning strength.

Amplitude-dependent internal friction and induced modulus defect in purified iron

The amplitude-dependent internal friction and associated modulus defect of pure iron were observed at 32 kHz between 77°K and 300°K in a search for features associated with dislocation motion. The samples were variously annealed, cold-worked and subjected to magnetic fields. The amplitude dependence was predominantly magneto-mechanical in nature and therefore attributed to the motion of domain walls. Another method for studying dislocation motion was developed. High amplitude oscillations induced a modulus defect which decayed after the exciting oscillations ceased. The decay was observed, at low amplitudes, as a function of temperature The strain amplitude necessary to induce the modulus defect was 5 × 10 −6 at 300°K and 1.5 × 10 −5 at 77°K in purified iron. The induced defect could be observed at amplitudes at least as low as 10 −7 at 77°K. We attribute the induced modulus defect to the motion of dislocations freed by the excitation oscillation, and conclude that, at 77°K, at least one class of dislocations will undergo further motion when acted on by stresses at least as small as 2 g/mm 2, until immobilized by another process possibly involving kink diffusion.

Frequency Dependence of Dislocation Damping in Single-Crystal Copper

The frequency dependence of the amplitude-independent and amplitude-dependent dislocation damping was measured on the same high-purity copper single crystal in both the kilocycle and megacycle frequency ranges. The dislocation contribution to the elastic modulus was also measured in the kilocycle range. The results are discussed in terms of a theoretical calculation which treats the dislocation as a pinned elastic string which experiences a viscous drag proportional to its velocity. The kilocycle dislocation damping exhibited an approximately linear frequency dependence; however, a somewhat better fit to the data can be made with a relation of the form Δ=αω−1+βω. It is proposed that the term αω−1 is a contribution from the high-frequency side of a low-frequency dislocation damping maximum. The dislocation damping in the megacycle range was proportional to ω−1. The kilocycle and megacycle data are consistent with a curve of the form Δ=K[ωτ/(1+ω2τ2)] (except at the lower kilocycle frequencies measured), as predicted by the string model. The maximum is at 1.7 Mc/sec. Based on the agreement of the data with the theoretical model, the dislocation density is calculated to be 1.1×106 cm−2 and the average loop length to be 5.7×10−4 cm, assuming an exponential loop-length distribution. The modulus defect was found to be independent of frequency, except for a possible small rise with decreasing frequency below 30 kc/sec. The amplitude-dependent internal friction of the specimen in the kilocycle range was found to be frequency dependent in the annealed state and to be frequency independent in the irradiated state. The strain amplitude above which the specimen exhibited amplitude dependence was observed to increase with frequency in the kilocycle range.

Strain amplitude dependent internal friction in beryllium

Investigation of the low frequency internal friction of polycrystalline beryllium has revealed a significant strain amplitude dependency. Measurements between −166°C and 721°C at 1 c/s in both high purity and commercial grade beryllium show that amplitude dependent internal friction varies greatly as a function of temperature and purity. In the high strain amplitude region, amplitude dependent internal friction exhibits the same type of damping that Granato and Lucke have proposed as being associated with dislocation hysteresis. Activation energies were obtained which suggest that, at the lower temperatures, damping is governed by a dislocation-impurity atmosphere interaction effect, while at higher temperatures a diffusion controlled process predominates. Other observations that have been noted in beryllium are the effects of small and large amounts of strain and deformation on the recovery and rate of recovery of internal friction in high purity beryllium. Thus, good evidence was obtained for the dislocation origin of amplitude dependent internal friction in beryllium.

Dislocation Damping in Rutile Single Crystals

Rutile (TiO2) crystals exhibited amplitude dependent internal friction behavior characteristic of that associated with dislocation damping. The generation of excess nonstoichiometric defects decreased damping, particularly for the (1|01) [1|01|] extended dislocation system. An increase in test temperature decreased damping, suggesting that ionized defects may interact more strongly with dislocations than the defects characteristic of lower temperatures.

Calculation of the amplitude dependence of internal friction from measurements of torsional damping

In some materials the internal friction increases with the amplitude of the applied stress or strain. The damping of a specimen in torsion is then an average value over the specimen. A method is presented by which the dependence of internal friction on stress amplitude may be calculated from the average values measured in torsion. The method has general applicability but is developed for the particular case of the torsion of a flat parallelepipedal specimen. The results of the analysis for a circular cylindrical specimen are also given. The method is compared with earlier analyses and demonstrated by reference to results obtained on a specimen of annealed aluminium.

Dislocation Kink Chain Model Versus String Model

The Peierls stress for dislocation motion causes characteristic nonlinearities of the micro-stress-strain law of dislocations. Contrary to the string model, the nonlinear stress-strain law of kinked dislocations can be the source of amplitude-dependent internal friction and modulus defect at amplitudes which are below or comparable to the breakaway stress from pinning points. Under the assumption that double-kink generation can be neglected, in terms of a bias stress experiment, the following points have been found to be the essential difference between the dislocation string model and the kink model: (1) The anelastic strain as measured by low-amplitude oscillations increases with bias stress for the string model, whereas it decreases for the kink model, due to the exhaustion of geometric kink motion. (2) The restoring force of kinked dislocations is one to three orders of magnitude more sensitive to bias stress than is the restoring force of dislocation strings. The conditions for measurable changes of the restoring force are, for the kink model, ε> (10−1 b/L) (sin φ+5kT/Gb2a), and for the string model ε> 10−1 πb/L (ε is the bias stress in units of shear modulus G, L is the line length, φ is the average angle against close-packed direction, b is the Burgers vector, a is the lattice constant, and T is the temperature).

Damping Behaviour of Some Al-Cu Alloys

The effect of solute concentration, heat treatment and ageing on the strain amplitude dependence of internal friction in Al-rich Al-Cu alloys containing 0.5, 1.0 and 2.0 wt. % Cu has been investigated. The internal friction of quenched alloys is generally found to decay for all the cases during room temperature ageing, except for Al-2% Cu alloy, where it increases temporarily after an initial decrease at the beginning of measurement, and then decreases. The results are discussed in terms of possible clustering, handling strain and dislocation climb, and also in the light of the dislocation damping model of Granato and Lücke.

Dose Dependence of the Dislocation Breakaway Stress in Neutron-Irradiated Copper as Measured by Amplitude-Dependent Internal Friction

Measurements of the strain-amplitude dependence of the internal friction in copper have been made at 100°C as a function of fast neutron dose. Although it is found that the Granato-Lücke theory of amplitude-dependent internal friction does not describe the results obtained, the dose dependence of the dislocation breakaway stress can be deduced by other means. After taking into account the variation with neutron dose of the mean free dislocation length, it is found that the breakaway stress is proportional to the cube root of the integrated flux in the dose range from 1011 to 1014 neutrons/cm2 (E>0.6 MeV). Extrapolation of the breakaway stress results to doses greater than 1019 neutrons/cm2 shows essential agreement with earlier neutron-hardening results. The results are discussed in terms of a source-hardening mechanism.

Mechanical damping in carbon and graphite at low temperature

The mechanical damping of polycrystalline graphite, pyrolytic graphite, and glassy carbon, was investigated, using the longitudinal resonance-bar technique in the temperature range from 300 down to 20°K as an extension of the previous work. (1) The presence of the so-called Bordoni peaks in internal friction at 50 and 75°K was reconfirmed for polycrystalline graphite. In addition, a broad peak was found at around 220°K. Two weak peaks were found in internal friction for pyrolytic graphite at around 75 and 215°K, and some structure was observed in the curve for glassy carbon, but not sufficient for drawing any definite conclusion. It was also shown that the strain amplitude dependence of internal friction does not become appreciable until a pyrolytic graphite is heat treated well above 3000°C.