The Klimov RD-500 was an unlicensed Soviet copy of the Rolls-Royce Derwent V turbojet that was sold to the Soviet Union in 1947. The Klimov OKB adapted it for Soviet production methods and materials.
107-504: Producing metric drawings and analyzing the materials used in the Derwent V went fairly quickly, but finding a substitute for the high-temperature, creep -resistant Nimonic 80 nickel-chromium alloy was a more difficult challenge. Eventually an alloy that matched Nimonic 80's high-temperature properties was found in KhN 80T, but it was not creep-resistant. The first Derwent V copy, designated as
214-499: A 2 c 0 {\displaystyle B={\frac {9kT}{MG^{2}b^{4}\ln {\frac {r2}{r1}}}}\cdot {\frac {D_{\rm {sol}}}{\varepsilon _{\rm {a}}^{2}c_{0}}}} where k is the Boltzmann constant, and r 1 and r 2 are the internal and external cut-off radii of dislocation stress field. c 0 and D sol are the atomic concentration of the solute and solute diffusivity respectively. D sol also has
321-888: A 3 − a 3 a 3 {\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}} as we consider small deformations, 1 ≫ ε i i ≫ ε i i ⋅ ε j j ≫ ε 11 ⋅ ε 22 ⋅ ε 33 {\displaystyle 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}} therefore
428-1318: A b ) = ( d x + ∂ u x ∂ x d x ) 2 + ( ∂ u y ∂ x d x ) 2 = d x 2 ( 1 + ∂ u x ∂ x ) 2 + d x 2 ( ∂ u y ∂ x ) 2 = d x ( 1 + ∂ u x ∂ x ) 2 + ( ∂ u y ∂ x ) 2 {\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {dx^{2}\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+dx^{2}\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\&=dx~{\sqrt {\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\end{aligned}}} For very small displacement gradients
535-534: A , thus Δ V V 0 = ( 1 + ε 11 + ε 22 + ε 33 + ε 11 ⋅ ε 22 + ε 11 ⋅ ε 33 + ε 22 ⋅ ε 33 + ε 11 ⋅ ε 22 ⋅ ε 33 ) ⋅
642-731: A normal stress will cause a normal strain. Normal strains produce dilations . The normal strain in the x -direction of the rectangular element is defined by ε x = extension original length = l e n g t h ( a b ) − l e n g t h ( A B ) l e n g t h ( A B ) = ∂ u x ∂ x {\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}} Similarly,
749-770: A polymeric material is subjected to an abrupt force, the response can be modeled using the Kelvin–Voigt model . In this model, the material is represented by a Hookean spring and a Newtonian dashpot in parallel. The creep strain is given by the following convolution integral: ε ( t ) = σ C 0 + σ C ∫ 0 ∞ f ( τ ) ( 1 − e − t / τ ) d τ {\displaystyle \varepsilon (t)=\sigma C_{0}+\sigma C\int _{0}^{\infty }f(\tau )\left(1-e^{-t/\tau }\right)\,\mathrm {d} \tau } where σ
856-430: A constant stress that is maintained for a sufficiently long time period. The material responds to the stress with a strain that increases until the material ultimately fails. When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time t 1 at which the stress is relieved, at which time the strain immediately decreases (discontinuity) then continues decreasing gradually to
963-522: A cube with an edge length a , it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions a ⋅ ( 1 + ε 11 ) × a ⋅ ( 1 + ε 22 ) × a ⋅ ( 1 + ε 33 ) {\displaystyle a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})} and V 0 =
1070-409: A dependence on both the attempted jump frequency and the number of nearest neighbor sites and the probability of the sites being vacant. Thus there is a double dependence upon temperature. At higher temperatures the diffusivity increases due to the direct temperature dependence of the equation, the increase in vacancies through Schottky defect formation, and an increase in the average energy of atoms in
1177-475: A distance, r , from a dislocation is given by the Cottrell atmosphere defined as C r = C 0 exp ( − β sin θ r K T ) {\displaystyle C_{r}=C_{0}\exp \left(-{\frac {\beta \sin \theta }{rKT}}\right)} where C 0 is the concentration at r = ∞ and β
SECTION 10
#17327908197261284-423: A function of homologous temperature , shear modulus-normalized stress, and strain rate. Generally, two of these three properties (most commonly temperature and stress) are the axes of the map, while the third is drawn as contours on the map. To populate the map, constitutive equations are found for each deformation mechanism. These are used to solve for the boundaries between each deformation mechanism, as well as
1391-429: A given displacement differs locally from a rigid-body motion. A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the normal strain , and the amount of distortion associated with the sliding of plane layers over each other
1498-428: A material is by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time. Below its critical stress, the viscoelastic creep modulus is independent of the stress applied. A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied stresses are below
1605-519: A material is stressed at a temperature near its melting point. While tungsten requires a temperature in the thousands of degrees before the onset of creep deformation, lead may creep at room temperature, and ice will creep at temperatures below 0 °C (32 °F). Plastics and low-melting-temperature metals, including many solders, can begin to creep at room temperature. Glacier flow is an example of creep processes in ice. The effects of creep deformation generally become noticeable at approximately 35% of
1712-482: A material, neighboring lattice sites or interstitial sites in the crystal structure must be free. A given atom must also overcome the energy barrier to move from its current site (it lies in an energetically favorable potential well ) to the nearby vacant site (another potential well). The general form of the diffusion equation is D = D 0 e E K T {\displaystyle D=D_{0}e^{\frac {E}{KT}}} where D 0 has
1819-409: A material. The application of tensile stress opposes the reduction in energy gained by void shrinkage. Thus, a certain magnitude of applied tensile stress is required to offset these shrinkage effects and cause void growth and creep fracture in materials at high temperature. This stress occurs at the sintering limit of the system. The stress tending to shrink voids that must be overcome is related to
1926-552: A more reasonable 7,900 man-hours by November 1948 and declined further still to 4,734 man-hours by 1 March 1949, close to the target of 4,000 man-hours. Production by Factory No. 500 totaled 97 in 1948 and 462 in 1949. Factory No. 16 in Kazan was brought into the program and built 300 engines in 1949. Production was canceled around 1950 in favor of the superior Klimov VK-1 turbojet based on the Rolls-Royce Nene . The RD-500
2033-408: A plastic deformation process and thus sintering can be described as a high temperature creep process. The applied compressive stress during pressing accelerates void shrinkage rates and allows a relation between the steady-state creep power law and densification rate of the material. This phenomenon is observed to be one of the main densification mechanisms in the final stages of sintering, during which
2140-400: A residual strain. Viscoelastic creep data can be presented in one of two ways. Total strain can be plotted as a function of time for a given temperature or temperatures. Below a critical value of applied stress, a material may exhibit linear viscoelasticity. Above this critical stress, the creep rate grows disproportionately faster. The second way of graphically presenting viscoelastic creep in
2247-434: A temperature dependence that makes a determining contribution to Q g . If the cloud of solutes does not form or the dislocations are able to break away from their clouds, glide occurs in a jerky manner where fixed obstacles, formed by dislocations in combination with solutes, are overcome after a certain waiting time with support by thermal activation. The exponent m is greater than 1 in this case. The equations show that
SECTION 20
#17327908197262354-498: A thinning of solute drag atoms as dislocations move. In the secondary, or steady-state, creep, dislocation structure and grain size have reached equilibrium, and therefore strain rate is constant. Equations that yield a strain rate refer to the steady-state strain rate. Stress dependence of this rate depends on the creep mechanism. In tertiary creep, the strain rate exponentially increases with stress. This can be due to necking phenomena, internal cracks, or voids, which all decrease
2461-435: A time-logarithmic creep. Wood is considered as an orthotropic material , exhibiting different mechanical properties in three mutually perpendicular directions. Experiments show that the tangential direction in solid wood tend display a slightly higher creep compliance than in the radial direction. In the longitudinal direction, the creep compliance is relatively low and usually do not show any time-dependency in comparison to
2568-453: A two-dimensional, infinitesimal, rectangular material element with dimensions dx × dy , which, after deformation, takes the form of a rhombus . The deformation is described by the displacement field u . From the geometry of the adjacent figure we have l e n g t h ( A B ) = d x {\displaystyle \mathrm {length} (AB)=dx} and l e n g t h (
2675-402: Is a constant which defines the extent of segregation of the solute. When surrounded by a solute atmosphere, dislocations that attempt to glide under an applied stress are subjected to a back stress exerted on them by the cloud of solute atoms. If the applied stress is sufficiently high, the dislocation may eventually break away from the atmosphere, allowing the dislocation to continue gliding under
2782-400: Is a function of the material's properties, exposure time, exposure temperature and the applied structural load . Depending on the magnitude of the applied stress and its duration, the deformation may become so large that a component can no longer perform its function – for example creep of a turbine blade could cause the blade to contact the casing, resulting in the failure of the blade. Creep
2889-416: Is achieved with stress exponent n = 1, and only when the internal dislocation density prior to testing is exceptionally low. By contrast, Harper–Dorn creep was not observed in polycrystalline Al and single crystal Al when the initial dislocation density was high. However, various conflicting reports demonstrate the uncertainties at very low stress levels. One report by Blum and Maier, claimed that
2996-470: Is activated at higher temperatures, the solute atoms which are "bound" to the dislocations by the misfit can move along with edge dislocations as a "drag" on their motion if the dislocation motion or the creep rate is not too high. The amount of "drag" exerted by the solute atoms on the dislocation is related to the diffusivity of the solute atoms in the metal at that temperature, with a higher diffusivity leading to lower drag and vice versa. The velocity at which
3103-430: Is altered by applied stress: it increases in regions under tension and decreases in regions under compression. So the activation energy for vacancy formation is changed by ± σΩ , where Ω is the atomic volume, the positive value is for compressive regions and negative value is for tensile regions. Since the fractional vacancy concentration is proportional to exp(− Q f ± σΩ / RT ) , where Q f
3210-508: Is an additional condition for strong hardening. Solute drag creep sometimes shows a special phenomenon, over a limited strain rate, which is called the Portevin–Le Chatelier effect . When the applied stress becomes sufficiently large, the dislocations will break away from the solute atoms since dislocation velocity increases with the stress. After breakaway, the stress decreases and the dislocation velocity also decreases, which allows
3317-403: Is applied stress, C 0 is instantaneous creep compliance, C is creep compliance coefficient, τ is retardation time, and f ( τ ) is the distribution of retardation times. When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain. This phenomenon is known as viscoelastic creep. At a time t 0 , a viscoelastic material is loaded with
Klimov RD-500 - Misplaced Pages Continue
3424-408: Is controlled by dislocation movement; namely, since creep can occur by vacancy diffusion (Nabarro–Herring creep, Coble creep), grain boundary sliding, and/or dislocation movement, and since the first two mechanisms are grain-size dependent, Harper–Dorn creep must therefore be dislocation-motion dependent. The same was also confirmed in 1972 by Barrett and co-workers where FeAl 3 precipitates lowered
3531-402: Is controlled by the movement of dislocations . For dislocation creep, Q = Q (self diffusion), 4 ≤ m ≤ 6, and b < 1. Therefore, dislocation creep has a strong dependence on the applied stress and the intrinsic activation energy and a weaker dependence on grain size. As grain size gets smaller, grain boundary area gets larger, so dislocation motion
3638-413: Is crucial for correct functionality to understand the creep deformation behavior of materials. Strain (mechanics) In mechanics , strain is defined as relative deformation , compared to a reference position configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of
3745-561: Is defined as ε E = 1 2 ( l 2 − L 2 l 2 ) = 1 2 ( 1 − 1 λ 2 ) {\displaystyle \varepsilon _{E}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{l^{2}}}\right)={\tfrac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)} The (infinitesimal) strain tensor (symbol ε {\displaystyle {\boldsymbol {\varepsilon }}} )
3852-2838: Is defined as the change in angle between lines AC and AB . Therefore, γ x y = α + β {\displaystyle \gamma _{xy}=\alpha +\beta } From the geometry of the figure, we have tan α = ∂ u y ∂ x d x d x + ∂ u x ∂ x d x = ∂ u y ∂ x 1 + ∂ u x ∂ x tan β = ∂ u x ∂ y d y d y + ∂ u y ∂ y d y = ∂ u x ∂ y 1 + ∂ u y ∂ y {\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}} For small displacement gradients we have ∂ u x ∂ x ≪ 1 ; ∂ u y ∂ y ≪ 1 {\displaystyle {\frac {\partial u_{x}}{\partial x}}\ll 1~;~~{\frac {\partial u_{y}}{\partial y}}\ll 1} For small rotations, i.e. α and β are ≪ 1 we have tan α ≈ α , tan β ≈ β . Therefore, α ≈ ∂ u y ∂ x ; β ≈ ∂ u x ∂ y {\displaystyle \alpha \approx {\frac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\frac {\partial u_{x}}{\partial y}}} thus γ x y = α + β = ∂ u y ∂ x + ∂ u x ∂ y {\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}} By interchanging x and y and u x and u y , it can be shown that γ xy = γ yx . Similarly, for
3959-543: Is defined in the International System of Quantities (ISQ), more specifically in ISO 80000-4 (Mechanics), as a "tensor quantity representing the deformation of matter caused by stress. Strain tensor is symmetric and has three linear strain and three shear strain (Cartesian) components." ISO 80000-4 further defines linear strain as the "quotient of change in length of an object and its length" and shear strain as
4066-588: Is dislocation density (constant for Harper–Dorn creep), D v is the diffusivity through the volume of the material, G is the shear modulus and b is the Burgers vector, σ s , and n is the stress exponent which varies between 1 and 3. Twenty-five years after Harper and Dorn published their work, Mohamed and Ginter made an important contribution in 1982 by evaluating the potential for achieving Harper–Dorn creep in samples of Al using different processing procedures. The experiments showed that Harper–Dorn creep
4173-413: Is essentially nonexistent and all strain is elastic. At low temperatures and high stress, materials experience plastic deformation rather than creep. At high temperatures and low stress, diffusional creep tends to be dominant, while at high temperatures and high stress, dislocation creep tends to be dominant. Deformation mechanism maps provide a visual tool categorizing the dominant deformation mechanism as
4280-765: Is impeded. Some alloys exhibit a very large stress exponent ( m > 10), and this has typically been explained by introducing a "threshold stress," σ th , below which creep can't be measured. The modified power law equation then becomes: d ε d t = A ( σ − σ t h ) m e − Q R ¯ T {\displaystyle {\frac {\mathrm {d} \varepsilon }{\mathrm {d} t}}=A\left(\sigma -\sigma _{\rm {th}}\right)^{m}e^{\frac {-Q}{{\bar {R}}T}}} where A , Q and m can all be explained by conventional mechanisms (so 3 ≤ m ≤ 10), and R
4387-405: Is observed in certain metallic alloys . In these alloys, the creep rate increases during the first stage of creep (Transient creep) before reaching a steady-state value. This phenomenon can be explained by a model associated with solid–solution strengthening. At low temperatures, the solute atoms are immobile and increase the flow stress required to move dislocations. However, at higher temperatures,
Klimov RD-500 - Misplaced Pages Continue
4494-430: Is related to the diffusion coefficient of atoms along the grain boundary, Q = Q (grain boundary diffusion), m = 1, and b = 3. Because Q (grain boundary diffusion) is less than Q (self diffusion), Coble creep occurs at lower temperatures than Nabarro–Herring creep. Coble creep is still temperature dependent, as the temperature increases so does the grain boundary diffusion. However, since
4601-413: Is related to the diffusion coefficient of atoms through the lattice, Q = Q (self diffusion), m = 1, and b = 2. Therefore, Nabarro–Herring creep has a weak stress dependence and a moderate grain size dependence, with the creep rate decreasing as the grain size is increased. Nabarro–Herring creep is strongly temperature dependent. For lattice diffusion of atoms to occur in
4708-471: Is resistant to elevated temperatures and has other desirable properties, but is notoriously vulnerable to cold-flow cut-through failures caused by creep. In steam turbine power plants, pipes carry steam at high temperatures (566 °C, 1,051 °F) and pressures (above 24.1 MPa, 3,500 psi). In jet engines, temperatures can reach up to 1,400 °C (2,550 °F) and initiate creep deformation in even advanced-design coated turbine blades. Hence, it
4815-399: Is the engineering normal strain , L is the original length of the fiber and l is the final length of the fiber. The true shear strain is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The engineering shear strain is defined as the tangent of that angle, and is equal to
4922-411: Is the gas constant . The creep increases with increasing applied stress, since the applied stress tends to drive the dislocation past the barrier, and make the dislocation get into a lower energy state after bypassing the obstacle, which means that the dislocation is inclined to pass the obstacle. In other words, part of the work required to overcome the energy barrier of passing an obstacle is provided by
5029-431: Is the identity tensor . The displacement of a body may be expressed in the form x = F ( X ) , where X is the reference position of material points of the body; displacement has units of length and does not distinguish between rigid body motions (translations and rotations) and deformations (changes in shape and size) of the body. The spatial derivative of a uniform translation is zero, thus strains measure how much
5136-456: Is the shear strain , within a deforming body. This could be applied by elongation, shortening, or volume changes, or angular distortion. The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the normal strain , which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other,
5243-484: Is the apparent activation energy for glide and B 0 is a constant. The parameter B in the above equation was derived by Cottrell and Jaswon for interaction between solute atoms and dislocations on the basis of the relative atomic size misfit ε a of solutes to be B = 9 k T M G 2 b 4 ln r 2 r 1 ⋅ D s o l ε
5350-470: Is the creep strain, C is a constant dependent on the material and the particular creep mechanism, m and b are exponents dependent on the creep mechanism, Q is the activation energy of the creep mechanism, σ is the applied stress, d is the grain size of the material, k is the Boltzmann constant , and T is the absolute temperature . At high stresses (relative to the shear modulus ), creep
5457-422: Is the densification rate, ρ is the density, P e is the pressure applied, n describes the exponent of strain rate behavior, and A is a mechanism-dependent constant. A and n are from the following form of the general steady-state creep equation, ε ˙ = A σ n {\displaystyle {\dot {\varepsilon }}=A\sigma ^{n}} where ε̇
SECTION 50
#17327908197265564-641: Is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path. The Green strain is defined as: ε G = 1 2 ( l 2 − L 2 L 2 ) = 1 2 ( λ 2 − 1 ) {\displaystyle \varepsilon _{G}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{L^{2}}}\right)={\tfrac {1}{2}}(\lambda ^{2}-1)} The Euler-Almansi strain
5671-490: Is the relative variation of the volume, as arising from dilation or compression ; it is the first strain invariant or trace of the tensor: δ = Δ V V 0 = I 1 = ε 11 + ε 22 + ε 33 {\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}} Actually, if we consider
5778-758: Is the strain rate, and σ is the tensile stress. For the purposes of this mechanism, the constant A comes from the following expression, where A ′ is a dimensionless, experimental constant, μ is the shear modulus, b is the Burgers vector, k is the Boltzmann constant, T is absolute temperature, D 0 is the diffusion coefficient, and Q is the diffusion activation energy: A = A ′ D 0 μ b k T exp ( − Q k T ) {\displaystyle A=A'{\frac {D_{0}\mu b}{kT}}\exp \left(-{\frac {Q}{kT}}\right)} Creep can occur in polymers and metals which are considered viscoelastic materials. When
5885-409: Is the tendency of a solid material to undergo slow deformation while subject to persistent mechanical stresses . It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increases as they near their melting point. The rate of deformation
5992-450: Is the vacancy-formation energy, the vacancy concentration is higher in tensile regions than in compressive regions, leading to a net flow of vacancies from the regions under tension to the regions under compression, and this is equivalent to a net atom diffusion in the opposite direction, which causes the creep deformation: the grain elongates in the tensile stress axis and contracts in the compressive stress axis. In Nabarro–Herring creep, k
6099-529: Is used in the analysis of materials that exhibit large deformations, such as elastomers , which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios. The logarithmic strain ε , also called, true strain or Hencky strain . Considering an incremental strain (Ludwik) δ ε = δ l l {\displaystyle \delta \varepsilon ={\frac {\delta l}{l}}}
6206-432: Is usually of concern to engineers and metallurgists when evaluating components that operate under high stresses or high temperatures. Creep is a deformation mechanism that may or may not constitute a failure mode . For example, moderate creep in concrete is sometimes welcomed because it relieves tensile stresses that might otherwise lead to cracking. Unlike brittle fracture , creep deformation does not occur suddenly upon
6313-1017: The Kelvin–Voigt model with a Hookean spring dashpot but with metals, the creep can be represented by plastic deformation mechanisms such as dislocation glide, climb and grain boundary sliding. Understanding the mechanisms behind creep in metals is becoming increasingly more important for reliability and material lifetime as the operating temperatures for applications involving metals rise. Unlike polymers, in which creep deformation can occur at very low temperatures, creep for metals typically occur at high temperatures. Key examples would be scenarios in which these metal components like intermetallic or refractory metals are subject to high temperatures and mechanical loads like turbine blades, engine components and other structural elements. Refractory metals , such as tungsten, molybdenum, and niobium, are known for their exceptional mechanical properties at high temperatures, proving to be useful materials in aerospace, defense and electronics industries. Although mostly due to
6420-511: The calcium silicate hydrates (C-S-H) in the hardened Portland cement paste (which is the binder of mineral aggregates ), is fundamentally different from the creep of metals as well as polymers . Unlike the creep of metals, it occurs at all stress levels and, within the service stress range, is linearly dependent on the stress if the pore water content is constant. Unlike the creep of polymers and metals, it exhibits multi-months aging, caused by chemical hardening due to hydration which stiffens
6527-439: The crystallite grain boundaries is used to slow the rate of Coble creep . Creep can cause gradual cut-through of wire insulation, especially when stress is concentrated by pressing insulated wire against a sharp edge or corner. Special creep-resistant insulations such as Kynar ( polyvinylidene fluoride ) are used in wire wrap applications to resist cut-through due to the sharp corners of wire wrap terminals. Teflon insulation
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#17327908197266634-413: The shear strain , radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions. If there is an increase in length of the material line, the normal strain is called tensile strain ; otherwise, if there is reduction or compression in the length of the material line, it is called compressive strain . Depending on
6741-461: The spatial derivative of displacement : ε ≐ ∂ ∂ X ( x − X ) = F ′ − I , {\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},} where I
6848-758: The yz - and xz -planes, we have γ y z = γ z y = ∂ u y ∂ z + ∂ u z ∂ y , γ z x = γ x z = ∂ u z ∂ x + ∂ u x ∂ z {\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}} The volumetric strain, also called bulk strain,
6955-1925: The "quotient of parallel displacement of two surfaces of a layer and the thickness of the layer". Thus, strains are classified as either normal or shear . A normal strain is perpendicular to the face of an element, and a shear strain is parallel to it. These definitions are consistent with those of normal stress and shear stress . The strain tensor can then be expressed in terms of normal and shear components as: ε _ _ = [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ ε x x 1 2 γ x y 1 2 γ x z 1 2 γ y x ε y y 1 2 γ y z 1 2 γ z x 1 2 γ z y ε z z ] {\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{bmatrix}}} Consider
7062-442: The KhN 80T alloy resulted in dangerous elongation of the turbine blades. Up to 40% of the early production RD-500s had to be individually adjusted before delivery and the service life of the engine never approached the 100 hours demonstrated in the acceptance test. The Soviets had enormous problems building the engines to standard, as demonstrated in the 20,000 man-hours required to build a single engine in 1947. This figure dropped to
7169-530: The RD-500 ( Reaktivnyy Dvigatel ' — jet engine) after Factory No. 500 where the engine was first produced, was being tested on 31 December 1947, but problems cropped up quickly. Combustion was uneven and this cracked the combustion chambers. This may have had something to do with the modifications made by the Soviets to the fuel, speed, and starter systems. But these problems were resolved by September 1948 when
7276-425: The action of the applied stress. The maximum force (per unit length) that the atmosphere of solute atoms can exert on the dislocation is given by Cottrell and Jaswon F m a x L = C 0 β 2 b k T {\displaystyle {\frac {F_{\rm {max}}}{L}}={\frac {C_{0}\beta ^{2}}{bkT}}} When the diffusion of solute atoms
7383-462: The amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories: In each of these theories the strain is then defined differently. The engineering strain is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g., elastomers and polymers, subjected to large deformations,
7490-434: The application of a compressive stress. For typical descriptions of creep, it is assumed that the applied tensile stress exceeds the sintering limit. Creep also explains one of several contributions to densification during metal powder sintering by hot pressing. A main aspect of densification is the shape change of the powder particles. Since this change involves permanent deformation of crystalline solids, it can be considered
7597-442: The application of stress. Instead, strain accumulates as a result of long-term stress. Therefore, creep is a "time-dependent" deformation. Creep or cold flow is of great concern in plastics. Blocking agents are chemicals used to prevent or inhibit cold flow. Otherwise rolled or stacked sheets stick together. The temperature range in which creep deformation occurs depends on the material. Creep deformation generally occurs when
7704-420: The applied stress and the remainder by thermal energy. Nabarro–Herring (NH) creep is a form of diffusion creep , while dislocation glide creep does not involve atomic diffusion. Nabarro–Herring creep dominates at high temperatures and low stresses. As shown in the figure on the right, the lateral sides of the crystal are subjected to tensile stress and the horizontal sides to compressive stress. The atomic volume
7811-792: The applied stress is not enough for a moving dislocation to overcome the obstacle on its way via dislocation glide alone, the dislocation could climb to a parallel slip plane by diffusional processes, and the dislocation can glide on the new plane. This process repeats itself each time when the dislocation encounters an obstacle. The creep rate could be written as: d ε d t = A C G D L M ( σ Ω k T ) 4.5 {\displaystyle {\frac {\mathrm {d} \varepsilon }{\mathrm {d} t}}={\frac {A_{\rm {CG}}D_{\rm {L}}}{\sqrt {M}}}\left({\frac {\sigma \Omega }{kT}}\right)^{4.5}} where A CG includes details of
7918-485: The body and on whether the metric tensor or its dual is considered. Strain has dimension of a length ratio , with SI base units of meter per meter (m/m). Hence strains are dimensionless and are usually expressed as a decimal fraction or a percentage . Parts-per notation is also used, e.g., parts per million or parts per billion (sometimes called "microstrains" and "nanostrains", respectively), corresponding to μm /m and nm /m. Strain can be formulated as
8025-619: The case of a material line element or fiber axially loaded, its elongation gives rise to an engineering normal strain or engineering extensional strain e , which equals the relative elongation or the change in length Δ L per unit of the original length L of the line element or fibers (in meters per meter). The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have e = Δ L L = l − L L {\displaystyle e={\frac {\Delta L}{L}}={\frac {l-L}{L}}} , where e
8132-416: The creep curves obtained in the very high purity material exhibited regular and periodic accelerations. They also found that the creep behavior no longer follows a stress exponent of n = 1 when the tests are extended to very high strains of >0.1 but instead there is evidence for a stress exponent of n > 2. At high temperatures, it is energetically favorable for voids to shrink in
8239-755: The creep rates by 2 orders of magnitude compared to highly pure Al, thus, indicating Harper–Dorn creep to be a dislocation based mechanism. Harper–Dorn creep is typically overwhelmed by other creep mechanisms in most situations, and is therefore not observed in most systems. The phenomenological equation which describes Harper–Dorn creep is d ε d t = ρ 0 D v G b 3 k T ( σ s n G ) {\displaystyle {\frac {\mathrm {d} \varepsilon }{\mathrm {d} t}}=\rho _{0}{\frac {D_{\rm {v}}Gb^{3}}{kT}}\left({\frac {\sigma _{\rm {s}}^{n}}{G}}\right)} where ρ 0
8346-614: The creep resistance of a polymer. Both polymers and metals can creep. Polymers experience significant creep at temperatures above around −200 °C (−330 °F); however, there are three main differences between polymeric and metallic creep. In metals, creep is not linearly viscoelastic, it is not recoverable, and it is only present at high temperatures. Polymers show creep basically in two different ways. At typical work loads (5% up to 50%) ultra-high-molecular-weight polyethylene (Spectra, Dyneema ) will show time-linear creep, whereas polyester or aramids ( Twaron , Kevlar ) will show
8453-438: The cross-sectional area and increase the true stress on the region, further accelerating deformation and leading to fracture. Depending on the temperature and stress, different deformation mechanisms are activated. Though there are generally many deformation mechanisms active at all times, usually one mechanism is dominant, accounting for almost all deformation. Various mechanisms are: At low temperatures and low stress, creep
8560-615: The densification rate (assuming gas-free pores) can be explained by: ρ ˙ = 3 A 2 ρ ( 1 − ρ ) ( 1 − ( 1 − ρ ) 1 n ) n ( 3 2 P e n ) n {\displaystyle {\dot {\rho }}={\frac {3A}{2}}{\frac {\rho (1-\rho )}{\left(1-(1-\rho )^{\frac {1}{n}}\right)^{n}}}\left({\frac {3}{2}}{\frac {P_{\rm {e}}}{n}}\right)^{n}} in which ρ̇
8667-561: The dislocation loop geometry, D L is the lattice diffusivity, M is the number of dislocation sources per unit volume, σ is the applied stress, and Ω is the atomic volume. The exponent m for dislocation climb-glide creep is 4.5 if M is independent of stress and this value of m is consistent with results from considerable experimental studies. Harper–Dorn creep is a climb-controlled dislocation mechanism at low stresses that has been observed in aluminum, lead, and tin systems, in addition to nonmetal systems such as ceramics and ice. It
8774-415: The dislocations glide can be approximated by a power law of the form v = B σ ∗ m B = B 0 exp ( − Q g R T ) {\displaystyle v=B{\sigma ^{*}}^{m}B=B_{0}\exp \left({\frac {-Q_{\rm {g}}}{RT}}\right)} where m is the effective stress exponent, Q
8881-488: The engine passed its 100-hour State acceptance test. RD-500 was a close copy of the Derwent with a single-stage centrifugal compressor , nine combustion chambers , and a single-stage turbine . It matched the Derwent's thrust of 15.9 kN (3,570 lb f ) and was only 13.7 kg (30 lb) heavier. The main problem with the engine in service was with its turbine blades, 30% of which failed inspection due to recrystallization after casting. The poor creep resistance of
8988-425: The engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%; thus other more complex definitions of strain are required, such as stretch , logarithmic strain , Green strain , and Almansi strain . Engineering strain , also known as Cauchy strain , is expressed as the ratio of total deformation to the initial dimension of the material body on which forces are applied. In
9095-488: The experimental evidence for Harper–Dorn creep is not fully convincing. They argued that the necessary condition for Harper–Dorn creep is not fulfilled in Al with 99.99% purity and the steady-state stress exponent n of the creep rate is always much larger than 1. The subsequent work conducted by Ginter et al. confirmed that Harper–Dorn creep was attained in Al with 99.9995% purity but not in Al with 99.99% purity and, in addition,
9202-440: The filament coil between its supports increases with time due to the weight of the filament itself. If too much deformation occurs, the adjacent turns of the coil touch one another, causing local overheating, which quickly leads to failure of the filament. The coil geometry and supports are therefore designed to limit the stresses caused by the weight of the filament, and a special tungsten alloy with small amounts of oxygen trapped in
9309-401: The formula. [REDACTED] A strain field associated with a displacement is defined, at any point, by the change in length of the tangent vectors representing the speeds of arbitrarily parametrized curves passing through that point. A basic geometric result, due to Fréchet , von Neumann and Jordan , states that, if the lengths of the tangent vectors fulfil the axioms of a norm and
9416-477: The grains. To heal this, grain-boundary sliding occurs. The diffusional creep rate and the grain boundary sliding rate must be balanced if there are no voids or cracks remaining. When grain-boundary sliding can not accommodate the incompatibility, grain-boundary voids are generated, which is related to the initiation of creep fracture. Solute drag creep is one of the mechanisms for power-law creep (PLC), involving both dislocation and diffusional flow. Solute drag creep
9523-487: The hardening effect of solutes is strong if the factor B in the power-law equation is low so that the dislocations move slowly and the diffusivity D sol is low. Also, solute atoms with both high concentration in the matrix and strong interaction with dislocations are strong gardeners. Since misfit strain of solute atoms is one of the ways they interact with dislocations, it follows that solute atoms with large atomic misfit are strong gardeners. A low diffusivity D sol
9630-415: The length of deformation at its maximum divided by the perpendicular length in the plane of force application, which sometimes makes it easier to calculate. The stretch ratio or extension ratio (symbol λ) is an alternative measure related to the extensional or normal strain of an axially loaded differential line element. It is defined as the ratio between the final length l and the initial length L of
9737-815: The logarithmic strain is obtained by integrating this incremental strain: ∫ δ ε = ∫ L l δ l l ε = ln ( l L ) = ln ( λ ) = ln ( 1 + e ) = e − e 2 2 + e 3 3 − ⋯ {\displaystyle {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \end{aligned}}} where e
9844-425: The material line. λ = l L {\displaystyle \lambda ={\frac {l}{L}}} The extension ratio λ is related to the engineering strain e by e = λ − 1 {\displaystyle e=\lambda -1} This equation implies that when the normal strain is zero, so that there is no deformation, the stretch ratio is equal to unity. The stretch ratio
9951-478: The material's critical stress value. Additionally, the molecular weight of the polymer of interest is known to affect its creep behavior. The effect of increasing molecular weight tends to promote secondary bonding between polymer chains and thus make the polymer more creep resistant. Similarly, aromatic polymers are even more creep resistant due to the added stiffness from the rings. Both molecular weight and aromatic rings add to polymers' thermal stability, increasing
10058-479: The material. Nabarro–Herring creep dominates at very high temperatures relative to a material's melting temperature. Coble creep is the second form of diffusion-controlled creep. In Coble creep the atoms diffuse along grain boundaries to elongate the grains along the stress axis. This causes Coble creep to have a stronger grain size dependence than Nabarro–Herring creep, thus, Coble creep will be more important in materials composed of very fine grains. For Coble creep k
10165-573: The melting point (in Kelvin) for metals and at 45% of melting point for ceramics. Creep behavior can be split into three main stages. In primary, or transient, creep, the strain rate is a function of time. In Class M materials, which include most pure materials, primary strain rate decreases over time. This can be due to increasing dislocation density , or it can be due to evolving grain size . In class A materials, which have large amounts of solid solution hardening, strain rate increases over time due to
10272-559: The microstructure, and multi-year aging, caused by long-term relaxation of self-equilibrated microstresses in the nanoporous microstructure of the C-S-H. If concrete is fully cured, creep effectively ceases. Creep in metals primarily manifests as movement in their microstructures. While polymers and metals share some similarities in creep, the behavior of creep in metals displays a different mechanical response and must be modeled differently. For example, with polymers, creep can be modeled using
10379-461: The normal strain in the y - and z -directions becomes ε y = ∂ u y ∂ y , ε z = ∂ u z ∂ z {\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}} The engineering shear strain ( γ xy )
10486-576: The number of nearest neighbors is effectively limited along the interface of the grains, and thermal generation of vacancies along the boundaries is less prevalent, the temperature dependence is not as strong as in Nabarro–Herring creep. It also exhibits the same linear dependence on stress as Nabarro–Herring creep. Generally, the diffusional creep rate should be the sum of Nabarro–Herring creep rate and Coble creep rate. Diffusional creep leads to grain-boundary separation, that is, voids or cracks form between
10593-468: The other directions. It has also been shown that there is a substantial difference in viscoelastic properties of wood depending on loading modality (creep in compression or tension). Studies have shown that certain Poisson's ratios gradually go from positive to negative values during the duration of the compression creep test, which does not occur in tension. The creep of concrete, which originates from
10700-800: The reduced yield strength at higher temperatures, the collapse of the World Trade Center was due in part to creep from increased temperature. The creep rate of hot pressure-loaded components in a nuclear reactor at power can be a significant design constraint, since the creep rate is enhanced by the flux of energetic particles. Creep in epoxy anchor adhesive was blamed for the Big Dig tunnel ceiling collapse in Boston , Massachusetts that occurred in July 2006. The design of tungsten light bulb filaments attempts to reduce creep deformation. Sagging of
10807-400: The solute atoms are more mobile and may form atmospheres and clouds surrounding the dislocations. This is especially likely if the solute atom has a large misfit in the matrix. The solutes are attracted by the dislocation stress fields and are able to relieve the elastic stress fields of existing dislocations. Thus the solutes become bound to the dislocations. The concentration of solute, C , at
10914-477: The solute atoms to approach and reach the previously departed dislocations again, leading to a stress increase. The process repeats itself when the next local stress maximum is obtained. So repetitive local stress maxima and minima could be detected during solute drag creep. Dislocation climb-glide creep is observed in materials at high temperature. The initial creep rate is larger than the steady-state creep rate. Climb-glide creep could be illustrated as follows: when
11021-673: The squares of the derivative of u y {\displaystyle u_{y}} and u x {\displaystyle u_{x}} are negligible and we have l e n g t h ( a b ) ≈ d x ( 1 + ∂ u x ∂ x ) = d x + ∂ u x ∂ x d x {\displaystyle \mathrm {length} (ab)\approx dx\left(1+{\frac {\partial u_{x}}{\partial x}}\right)=dx+{\frac {\partial u_{x}}{\partial x}}dx} For an isotropic material that obeys Hooke's law ,
11128-478: The strain rate contours. Deformation mechanism maps can be used to compare different strengthening mechanisms, as well as compare different types of materials. d ε d t = C σ m d b e − Q k T {\displaystyle {\frac {\mathrm {d} \varepsilon }{\mathrm {d} t}}={\frac {C\sigma ^{m}}{d^{b}}}e^{\frac {-Q}{kT}}} where ε
11235-567: The surface energy and surface area-volume ratio of the voids. For a general void with surface energy γ and principle radii of curvature of r 1 and r 2 , the sintering limit stress is σ s i n t = γ r 1 + γ r 2 {\displaystyle \sigma _{\rm {sint}}={\frac {\gamma }{r_{1}}}+{\frac {\gamma }{r_{2}}}} Below this critical stress, voids will tend to shrink rather than grow. Additional void shrinkage effects will also result from
11342-440: Was first observed by Harper and Dorn in 1957. It is characterized by two principal phenomena: a power-law relationship between the steady-state strain rate and applied stress at a constant temperature which is weaker than the natural power-law of creep, and an independent relationship between the steady-state strain rate and grain size for a provided temperature and applied stress. The latter observation implies that Harper–Dorn creep
11449-976: Was used in a number of early Soviet jet fighters including the Lavochkin La-15 , the Yakovlev Yak-25 , and the Yakovlev Yak-30 , but only the Yakovlev Yak-23 , Yakovlev Yak-25 and Lavochkin La-15 were accepted for service. The RD-500 was copied and developed further in the People's Republic of China (PRC) at the Shenyang Aircraft Development Office PF-1A. Data from Kay, Turbojet Related lists Creep (deformation) In materials science , creep (sometimes called cold flow )
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