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degrees of freedom

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Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.tb.sfa.t52780171
EISBN: 978-1-62708-268-6
... point and the overall mean, squaring each difference, and adding the squared differences. The total sum of squares represents the total variability of individual data points from the grand mean. The failure analysis team defines the total degrees of freedom, which are the number of independent...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2016
DOI: 10.31399/asm.tb.ascaam.t59190035
EISBN: 978-1-62708-296-9
... , β Cr , γ Cr Al 9 Cr 4 30.8 45.1 45–49.3 wt% 31.3±0.4 at.% … Al 8 Cr 5 Al 8 Cr 5 38.5 54.6 30.4–41 at.% … Equilibrium reactions in the Al-Cr system Table 2.4 Equilibrium reactions in the Al-Cr system Reaction Degrees of freedom Concentration in liquid (L), at.% Cr...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2012
DOI: 10.31399/asm.tb.pdub.t53420001
EISBN: 978-1-62708-310-2
..., or alloy, plus the number of degrees of freedom, F , is equal to the sum of the components, C , of the system plus 2: (Eq 1.5) P + F = C + 2 The phases, P , are the homogeneous parts of a system that, having definite bounding surfaces, are conceivably separable by mechanical means...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 June 2008
DOI: 10.31399/asm.tb.emea.t52240075
EISBN: 978-1-62708-251-8
... where P is the number of phases; F is the number of independent variables, called the degrees of freedom; and C is the number of components in the system. Usually, a component, C , is an alloying element, compound, or solution in the system. The degrees of freedom, F , is the number...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2012
DOI: 10.31399/asm.tb.pdub.t53420015
EISBN: 978-1-62708-310-2
... the phase rule becomes: F = C + 2 − P F = 1 + 2 − 1 = 2 Fig. 2.4 Unary diagram for magnesium with points in one-, two-, and three-phase regions of the diagram. Adapted from: Ref 2.1 That is, there are two degrees of freedom. If both the pressure...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2012
DOI: 10.31399/asm.tb.pdub.t53420191
EISBN: 978-1-62708-310-2
... application of the phase rule, it should be recalled that one degree of freedom has been exercised in the initial construction of the 3-D diagram by electing to draw it at one atmosphere of pressure. 10.1 Space Model of Ternary Systems 10.2 The Gibbs Triangle 10.3 Tie Lines 10.4 Ternary Isomorphous...
Series: ASM Technical Books
Publisher: ASM International
Published: 31 December 2020
DOI: 10.31399/asm.tb.phtbp.t59310001
EISBN: 978-1-62708-326-3
... ) under thermal equilibrium from thermodynamic theory such that: (Eq 3) P = C − F + 2 Where C is number of different elements in the system, and F is the number of degrees of freedom in the system—that is, the number externally controllable independent variables (temperature...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2012
DOI: 10.31399/asm.tb.pdub.t53420263
EISBN: 978-1-62708-310-2
.... This section therefore concentrates on the general principles rather than going into detail concerning the currently available software programs, which, in any case, often contain proprietary code. Essentially, the calculation must be defined so that the number of degrees of freedom is reduced, the Gibbs...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320091
EISBN: 978-1-62708-357-7
... and 4.16 are Gibbs phase rule; f (or f ′) is called the degree of freedom; a system is called a nonvariant system if the degree of freedom = 0 and a monovariant system if the degree of freedom = 1. For example: Nonvariant system: Figure 4.5(a) ternary eutectic point E (L, A, B, C, four-phase...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2012
DOI: 10.31399/asm.tb.pdub.t53420239
EISBN: 978-1-62708-310-2
... degrees of freedom are required to fix temperature and pressure and to vary the composition. Reaction morphologies consisting of two-phase structures (i.e., precipitates or wavy interfaces) are, therefore, thermodynamically forbidden for binary systems, assuming that only volume diffusion takes place...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2016
DOI: 10.31399/asm.tb.ascaam.t59190vii
EISBN: 978-1-62708-296-9
... composition, wt% Si Reaction Degrees of freedom Liquid alloy composition, wt% Si Solid phase composition, wt% Si Temperature ( T ), °C Hypoeutectic 1.65–11.7 L →Al 1 C L Si ( T ) C αAl Si ( T ) T Lα–Al L →Al+Si 0 12.6 1.65 577 Eutectic 11.7 L →Al+Si 0 12.6 1.65 577...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 October 2011
DOI: 10.31399/asm.tb.mnm2.t53060013
EISBN: 978-1-62708-261-7
...—is the Gibb’s phase rule that relates the number of components in a system to the physical state of the system. The number of stable phases ( p ) in a system is defined by the rule: (Eq 2.4) p = c − f + 2 where f is the number of independent variables (called degrees of freedom...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2012
DOI: 10.31399/asm.tb.pdub.t53420073
EISBN: 978-1-62708-310-2
... in adjacent fields in a multicomponent diagram must differ by one. Stable equilibria for binary systems Table 4.2 Stable equilibria for binary systems Number of components Number of phases Degrees of freedom Equilibrium 2 3 0 Invariant 2 2 1 Univariant 2 1 2 Bivariant...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.tb.scm.t52870489
EISBN: 978-1-62708-314-0
... analysis codes and finite element modeling tools are available for detailed composite designs. Modern design is done on workstations with advanced computer-aided design (CAD)/computer-aided manufacturing (CAM) packages. Integrated product definition teams have more freedom with composite construction...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2003
DOI: 10.31399/asm.tb.cfap.t69780028
EISBN: 978-1-62708-281-5
... freedom of rotation for the carbon-carbon single bond. Of course, the greater electron density of the carbon-carbon triple bond does restrict the motion of that bond. Consideration of neopentane shows the resulting reduction of degrees of freedom when substituent hydrogens are replaced by the considerably...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 September 2011
DOI: 10.31399/asm.tb.cfw.t52860035
EISBN: 978-1-62708-338-6
... advantageous and essential to consider all the different machine configurations, from a simple two-axis machine to one with six or more degrees of freedom. Indeed, postprocessing to allow winding by using a standard commercially available six-axis robot is also desirable. When the wealth of different types...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 1995
DOI: 10.31399/asm.tb.sch6.t68200048
EISBN: 978-1-62708-354-6
... to other cast metals and other methods of steel fabrication is also included in this chapter. Design Flexibility Steel castings offer greater freedom to design engineers. The casting process permits the formation of streamlined, intricate, integral parts with strength and rigidity which...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2004
DOI: 10.31399/asm.tb.tt2.t51060137
EISBN: 978-1-62708-355-3
... at the smaller cross section in the gage region. The second situation involves notched specimens. A molded notch may not affect strength to the same degree, or even in the same sense, that would be inferred from stress-concentration theory, because the local flow geometry near the crack tip may enhance...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2018
DOI: 10.31399/asm.tb.aceg.t68410103
EISBN: 978-1-62708-280-8
... in.). Maintain a minimum clearance of about 25 mm (1 in.) between the outside edges of the retainer and the tie bars for quick die changing. Design multiple cavities with slides, recognizing the degrees of freedom as shown in Fig. 6.5 . Fig. 6.5 Die cavities and freedom for slides Selection...
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2003
DOI: 10.31399/asm.tb.cfap.t69780314
EISBN: 978-1-62708-281-5
... ). The pan contains three phases (liquid and gaseous forms of water, and polymer) and thus has but one degree of freedom by Gibb’s phase rule, namely temperature, because this is necessarily the independent variable in a measurement of T g . Results of using this method are given in Table 3 , where...