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partial differential equation
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Series: ASM Technical Books
Publisher: ASM International
Published: 01 February 2005
DOI: 10.31399/asm.tb.chffa.t51040091
EISBN: 978-1-62708-300-3
... operations. As a result, the effect of temperatures on flow stress and metal flow during hot forming are often not considered adequately. Finite Difference Method In the finite difference method, the derivatives in the governing partial differential equations are written in terms of difference...
Abstract
There are numerous approximate methods, both analytical and numerical, for analyzing forging processes. None are perfect because of the assumptions made to simplify the mathematical approach, but all have merit. This chapter discusses the slab, upperbound, and finite element methods, covering basic principles, implementation, and advantages and disadvantages in various applications.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 January 2022
DOI: 10.31399/asm.tb.isceg.t59320049
EISBN: 978-1-62708-332-4
... of safety 5.4.4.1 Stress Analysis Finite Difference Method The finite difference method is a direct method of discretizing (assigning specific data values) using partial differential equations ( Ref 3 ). These equations are known as finite-difference equations. The method uses a simplified...
Abstract
This chapter provides an overview of how the disciplines of design, material, and manufacturing contribute to engineering for functional performance. It describes the interaction of product designers and casting engineers in product development. It discusses the consequences of component failure, uncertainty in data and assumptions, and selection of the factor of safety. The chapter also presents an overview of the functional requirements for product performance and provides an overview of product design development. It also presents a partial list of the different tests that are performed on prototypes and examples of product testing. The chapter describes the requirements of a traceability system.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.ssde.t52310011
EISBN: 978-1-62708-286-0
...(a) would be recorded as Fe /O2 /OH ,Fe2+ /Fe . This cell is an example of a differential aeration corrosion cell, which is discussed later. Processes at a single electrode The Nernst Equation Electrochemical reactions require a transfer of charge; hence, there is a coupling between chemical and electrical...
Abstract
Corrosion is a key subject for more or less all classes of alloys that fall within the broad definition of stainless steels because these alloys were developed with the intention of preventing corrosion. This chapter provides an introduction to the fundamentals of electrochemical theory as it pertains to corrosion resistance of stainless steels. The discussion provides an overview of electrochemical reactions, Faraday's law, the Nernst equation, galvanic versus electrochemical cells, corrosion tendency, and Pourbaix diagrams.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320167
EISBN: 978-1-62708-357-7
... π ∫ 0 ζ exp ( − ζ 2 ) d ζ [Answer] Equation 6.17 can be obtained in various ways, and the conversion of Fick’s second law into an ordinary differential equation using a non-dimensional variable: ζ = z / 2 D t is explained here...
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320259
EISBN: 978-1-62708-357-7
... ) are equal. [Answer] Partially differentiating Eq 9.31 and 9.32 with respect to λ, and finding out the extreme value conditions, then the following relations can be obtained. (Eq 9.33) Maximum growth rate: ( ∂ v / ∂ λ ) = 0 → Δ G min = 4 σ α / β V / λ...
Abstract
This chapter provides a classification of the types of microstructural changes and transformations and then reviews each type. It presents the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation and explains the thermodynamics of eutectic solidification and eutectoid transformation. An appendix covers growth of eutectoid structure in carburized pearlite.
Series: ASM Technical Books
Publisher: ASM International
Published: 01 November 2019
DOI: 10.31399/asm.tb.mfadr7.t91110269
EISBN: 978-1-62708-247-1
... and of the secondary electrons leaving it. The solution of the corresponding differential equation delivers the voltage U c which is building up at the structure. Fig. 7 Equivalent circuit diagram, solution of the corresponding differential equation and graphical representation for different structure...
Abstract
This chapter provides a comprehensive overview over all phenomena related to Voltage Contrast (VC) mechanisms in SEM and FIB. The multiple advantages, possibilities, and limits of active and passive VC failure localization are systemized and discussed. The knowledge of all facts influencing the VC generation (capacitance, leakage, doping, and circuitry) is very helpful for successful failure localization.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2018
DOI: 10.31399/asm.tb.aceg.t68410151
EISBN: 978-1-62708-280-8
... mechanical properties benefit from these types of inserts. Figure 7.6 shows a concept of a tilt-poured differential carrier fed through a ring feeder on the top flange. The pinion end, which has a mass concentration, is cooled intensely to create partial solidification during pouring and the initial...
Abstract
This chapter discusses the various factors pertinent to gravity permanent mold (GPM) castings, along with their advantages, limitations, and significance. The discussion covers the geometric factors, process and manufacturing elements, gating practices, and feeding principles of and pouring systems in GPM. The influences of mold coatings on GPM and low pressure permanent mold castings are described. The chapter also discusses various processes involved in the engineering of core boxes and cooling of GPM for casting integrity and cycle time control. It provides information on some of the processes involved in post-casting operations, namely de-coring and de-gating. The key design aspects for consideration in water quenching during the T6 heat treatment are reviewed. The chapter also provides information on two critical cycle events important in engineering at the manufacturing facility: tipper cycle planning and table or cell cycle planning.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.ssde.t52310057
EISBN: 978-1-62708-286-0
... υ X V m = N C r 0 3 2 X V m Differentiating Eq 31 with time, equating to Eq 3 , and separating variables results in: (Eq 32) X d X = N O s D O υ N B O d t = N O s D O...
Abstract
Stainless steel retains strength and has excellent oxidation resistance from room temperature to nearly 1000 deg C relative to competitive materials. This chapter focuses on the high-temperature oxidation of stainless steel by oxygen or water vapor. It begins by discussing the thermodynamic conditions and electrochemical nature of oxidation and providing information on transient oxidation. This is followed by a description of Wagner's theory for metal oxidation. The volatile nature of Cr 2 O 3 is then reviewed. The chapter further discusses the causes and preventive measures of spalling and cracking of oxide scale. It ends with a section providing information on oxidation behaviors under less-oxidizing atmospheres.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 30 September 2023
DOI: 10.31399/asm.tb.stmflw.t59390064
EISBN: 978-1-62708-459-8
..., and the role of elastic deflection and ultrasonic vibration. adsorption boundary lubrication elastohydrodynamic lubrication frictional instabilities hydrodynamic lubrication lubricant entrainment lubrication regimes mixed lubrication plastohydrodynamic lubrication Reynolds equation Stribeck...
Abstract
This chapter reviews the basic theory of lubrication in the context of metal forming applications. It discusses the rheological properties of fluids and their effect on fluid-film thickness at pressures, temperatures, and loading conditions typical of metal working processes. It describes the three lubrication regimes (boundary, mixed, and hydrodynamic) of a Stribeck curve and the forces that maintain surface separation. It also discusses mixed, elastohydrodynamic, plastohydrodynamic, and solid or semi-solid lubrication, the effects of starvation and frictional instabilities, and the role of elastic deflection and ultrasonic vibration.
Book Chapter
Book: Principles of Brazing
Series: ASM Technical Books
Publisher: ASM International
Published: 01 August 2005
DOI: 10.31399/asm.tb.pb.t51230105
EISBN: 978-1-62708-351-5
... All chemical reactions are reversible, including oxidation reactions. In general, the oxidation of any metal can be described by an equation of the form: (Eq 3.1) n M + [ m / 2 ] O 2 ↔ M n O m The reaction will proceed spontaneously in either direction...
Abstract
This chapter discusses joining atmospheres that are used for brazing, along with their advantages and disadvantages. It discusses the processes, advantages, and disadvantages of chemical fluxing, self-fluxing, and fluxless brazing. Information on stop-off compounds that are considered as the antithesis of fluxes is also provided.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320055
EISBN: 978-1-62708-357-7
... components increase by a factor φ, then so does the value of V , the following equation can be obtained: (Eq 3.34) φ ⋅ V ( m A , m B ) = V ( φ m A , φ m B ) Differentiate both sides of Eq 3.34 by φ and then replace φ = 1 to obtain (Eq 3.35...
Abstract
This chapter explains the idea of solution theory and the nature of mixed materials. The chapter considers approximation of free energy by the regular solution model and sublattice model. It discusses chemical potential and nonrandom distribution based on the interactions between solute atoms.
Series: ASM Technical Books
Publisher: ASM International
Published: 01 September 2011
DOI: 10.31399/asm.tb.cfw.t52860115
EISBN: 978-1-62708-338-6
.... The local α (previously defined) can be better defined taking into account friction on the dome (a much more complicated numerical solution—three nonlinear partial differential equations). This equation is solved down to the inflection point (2/ a = tan 2 (α)) when the local α equals 54.74...
Abstract
The necessity of developing the lightest-weight structures with sufficient strength was the driving factor for the development of filament-wound composite pressure vessels. This chapter presents a brief history of the development of rocket motor cases (RMCs), followed by a comparison of the advantages of composites over metals for RMCs. A discussion on a typical design, analysis, and manufacturing operation follows. The chapter introduces the basic design approach and shows some sizing techniques along with example calculations. It discusses the processes involved in the testing of the composite pressure vessel.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2015
DOI: 10.31399/asm.tb.cpi2.t55030349
EISBN: 978-1-62708-282-2
... corrosivity. The charge-transfer controlled reaction involving carbonic acid and carbon steel (or iron) can be represented in terms of the concentration or partial pressure of dissolved CO 2 in the medium to arrive at a corrosion rate equation that incorporates the order of the reaction and an exponential...
Abstract
This chapter examines methods of internal corrosion prediction for multiphase pipelines and details methodologies to perform internal corrosion direct assessment for natural gas pipelines. Further, real-time monitoring techniques for assessing actual corrosion at critical locations are discussed.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.ssde.t52310027
EISBN: 978-1-62708-286-0
... or a mixture of hydrofluoric and nitric acids. Environmental Variables Influencing Uniform Corrosion The corrosion of stainless steels is usually the result of contact with an electrolyte, allowing a complex set of partial electrochemical reactions, which may occur sequentially or concurrently...
Abstract
This chapter explores the behavior of stainless steel in media that promote corrosion. The forms of corrosion covered are uniform corrosion, atmospheric corrosion, localized corrosion, pitting corrosion, crevice corrosion, and grain boundary corrosion. The chapter discusses the influence of material and environmental variables on stress-corrosion cracking (SCC) and the mechanisms proposed for SCC in stainless steel, comparing the mechanism of SCC with hydrogen embrittlement. In addition, it provides information on biocorrosion and microbiologically induced corrosion in ambient aqueous environments.
Book Chapter
Book: Principles of Soldering
Series: ASM Technical Books
Publisher: ASM International
Published: 01 April 2004
DOI: 10.31399/asm.tb.ps.t62440001
EISBN: 978-1-62708-352-2
... for a brief historical background of solders and brazes, respectively). The type of metallurgical reaction between a filler and parent metal is sometimes used to differentiate soldering from brazing. Solders usually react to form intermetallic phases, that is, compounds of the constituent elements...
Abstract
Soldering and brazing represent one of several types of methods for joining solid materials. These methods may be classified as mechanical fastening, adhesive bonding, soldering and brazing, welding, and solid-state joining. This chapter summarizes the principal characteristics of these joining methods. It presents a comparison between solders and brazes. Further details on pressure welding and diffusion bonding are also provided. Key parameters of soldering are discussed, including surface energy and surface tension, wetting and contact angle, fluid flow, filler spreading characteristics, surface roughness of components, dissolution of parent materials and intermetallic growth, significance of the joint gap, and the strength of metals. The chapter also examines the principal aspects related to the design and application of soldering processes.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.tb.scm.t52870201
EISBN: 978-1-62708-314-0
... depends on the viscosity as predicted from the viscosity submodel, which in turn is dependent on the resin kinetics (kinetics submodel), which in turn is dependent on heat (heat transfer submodel). When using any model, it is important to understand (1) the mathematics of the equations that make up...
Abstract
This chapter provides an overview of the tools and techniques, as well as some of the underlying theory, that have proven useful for process modeling and simulation. It begins by presenting the framework of a thermoset cure model that accounts for kinetics, viscosity, heat transfer, flow, voids, and residual stress. It then discusses each variable in detail, explaining how it affects the cure process, how it is measured, and how it can be expressed mathematically in the form of a simple model. The discussions throughout the chapter are supported by numerous images, diagrams, and data plots.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 January 2015
DOI: 10.31399/asm.tb.spsp2.t54410099
EISBN: 978-1-62708-265-5
... and incomplete bainite transformation. Source: Ref 6.6 , 6.7 Bainite Transformation Start Temperatures The temperature at which bainite transformation starts is referred to as the B S temperature, and several empirical equations that show the effect of alloying elements on B S have been determined...
Abstract
Bainite is an intermediate temperature transformation product of austenite. This chapter describes the conditions under which bainite is likely to form. It discusses the effects of alloying on bainitic transformation, the difference between upper and lower bainite, and the influence of solute drag on bainite formation mechanisms. It also discusses the development of ferrite-carbide bainites and their effect on toughness, hardness, and ductility.
Book Chapter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2004
DOI: 10.31399/asm.tb.tt2.t51060251
EISBN: 978-1-62708-355-3
... impacts one end of the specimen. If ( x , t ) denotes the displacement of any point in the x direction and assuming purely uniaxial motion—that is, neglecting radial inertia effects—the equation of motion is: (Eq 1) ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 where...
Abstract
High strain rate tensile testing is used to understand the response of materials to dynamic loading. The behavior of materials under high strain rate tensile loads may differ considerably from that observed in conventional tensile tests. This chapter discusses the processes involved in determining strain rate effects in tension by conventional tensile tests and covers expanding ring tests, flat plate impact tests, split-Hopkinson pressure bar tests, and rotating wheel tests.
Book Chapter
Book: Principles of Brazing
Series: ASM Technical Books
Publisher: ASM International
Published: 01 August 2005
DOI: 10.31399/asm.tb.pb.t51230001
EISBN: 978-1-62708-351-5
... is sometimes used to differentiate brazing from soldering. Brazes mostly alloy with the parent materials to form solid solutions, which are mixtures of the constituents on an atomic scale. By contrast, solders usually react to form intermetallic phases, that is, compounds of the constituent elements that have...
Abstract
Brazing and soldering jointly represent one of several methods for joining solid materials. This chapter summarizes the principal characteristics of the various joining methods. It then discusses key parameters of brazing including surface energy and tension, wetting and contact angle, fluid flow, filler spreading characteristics, surface roughness of components, dissolution of parent materials, new phase formations, significance of the joint gap, and the strength of metals. The chapter also describes issues in processing aspects that must be considered when designing a joint, and the health, safety, and environmental aspects of brazing.
Series: ASM Technical Books
Publisher: ASM International
Published: 01 June 1983
DOI: 10.31399/asm.tb.mlt.t62860001
EISBN: 978-1-62708-348-5
... in a material’s properties. These differential equations also provide useful relationships for computing propagations of known errors. For example, B and ν determined by either resonance or static stress-strain are highly inaccurate compared with pulse-method results. In the pulse case, 1% measurement...
Abstract
Many scientific-technological advances depend critically on solid-state elastic properties, their magnitudes, and their responses to variables like stress and temperature. This chapter provides the definitions and descriptions of elastic constants and emphasizes five aspects of engineering-material solid-state elastic constants: general properties; interrelationships; relationships, especially thermodynamic to other physical properties; changes during cooling from ambient to near-zero temperature; and near-zero-temperature behavior.
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