Search Results for close-form solution

This appendix presents a close-form solution to determine the stress distribution around a hole of any shape or size in a strip of any material of any width. It also compares the close-form equation to classical solutions and the results of finite element analysis, demonstrating near perfect matches in each case.

This chapter deals with the effects of fatigue in rotating shafts subjected to elastic and plastic strains associated with bending stresses. It begins with a review of the basic approach to treating low-cycle fatigue in bending, explaining that the assumption that stress is proportional to strain is incorrect due to plastic flow, causing considerable discrepancy between measured and calculated stresses. Data plots of the axial and bending fatigue characteristics of a 4130 steel help illustrate the problem. A closed-form solution is then presented and used to analyze the effects of flexural bending on solid as well as hollow rectangular and round bars. The chapter also discusses the difference in the treatment of a rotating shaft in which all surface elements undergo the same stress and strain and a nonrotating shaft in which a few surface elements carry most of the load. The difference, as explained, is due to the volumetric effect of stress in fatigue.

Large-scale yielding at the crack tip and time-dependent crack growth mechanisms, such as stress relaxation due to creep, are nonlinear behaviors requiring nonlinear analysis methods. This chapter presents two such methods, one based on elastic-plastic fracture mechanics, the other on time-dependent fracture mechanics. It also introduces two new fracture indices, the J-integral for handling large-scale yielding and the C*-integral for creep crack growth, providing close-form and handbook solutions for each.

This chapter demonstrates the versatility of the strain-range partitioning method and its application to creep-fatigue problems involving complex loading histories. It begins with a derivation showing that it is possible to assess the damage of hysteresis loops combining two or more strain ranges using generic loops based on fundamental data. It then explains how to treat problems involving sequential loading with both healing and damage cycles and presents a general solution for combining two loops with arbitrary amounts of the four strain-range components. The chapter also derives closed-form equations that account for interactions among any number of adjacent loops and can be used, through successive application, to analyze any loading history.

The building block of all matter, including metals, is the atom. This chapter initially provides information on atomic bonding and the crystal structure of metals and alloys, followed by a description of three crystal lattice structures of metals: face-centered cubic, hexagonal close-packed, and body-centered cubic. It then describes the four main divisions of crystal defects, namely point defects, line defects, planar defects, and volume defects. The chapter provides information on grain boundaries of metals, processes involved in atomic diffusion, and key properties of a solid solution. It also explains the aspects of a phase diagram that shows what phase or phases are present in the alloy under conditions of thermal equilibrium. Finally, a discussion on the applications of equilibrium phase diagrams is presented.

This chapter introduces many of the key concepts on which metallurgy is based. It begins with an overview of the atomic nature of matter and the forces that link atoms together in crystal lattice structures. It discusses the types of imperfections (or defects) that occur in the crystal structure of metals and their role in mechanical deformation, annealing, precipitation, and diffusion. It describes the concept of solid solutions and the effect of temperature on solubility and phase transformations. The chapter also discusses the formation of solidification structures, the use of equilibrium phase diagrams, the role of enthalpy and Gibb’s free energy in chemical reactions, and a method for determining phase compositions along the solidus and liquidus lines.

This chapter discusses the similarities and differences of forging and forming processes used in the production of wrought superalloy parts. Although forming is rarely concerned with microstructure, forging processes are often designed with microstructure in mind. Besides shaping, the objectives of forging may include grain refinement, control of second-phase morphology, controlled grain flow, and the achievement of specific microstructures and properties. The chapter explains how these objectives can be met by managing work energy via temperature and deformation control. It also discusses the forgeability of alloys, addresses problems and practical issues, and describes the forging of gas turbine disks. On the topic of forming, the chapter discusses the processes involved, the role of alloying elements, and the effect of alloy condition on formability. It addresses practical concerns such as forming speed, rolling direction, rerolling, and heat treating precipitation-hardened alloys. It presents several application examples involving carbide-hardened cobalt-base and other superalloys, and it concludes with a discussion on superplasticity and its adaptation to commercial forging and forming operations.

The solidification process has a major influence on the microstructure and mechanical properties of metal casting as well as wrought products. This appendix covers the fundamentals of solidification. It discusses the formation of solidification structures, the characteristics of planar, cellular, and dendritic growth, the basic freezing sequence for an alloy casting, and the variations in cooling rate, heat flow, and grain morphology in different areas of the mold. It also describes the types of segregation that occur during freezing, the effect of solidification rate on secondary dendrite arm spacing, and the factors that contribute to porosity and shrinkage.

This chapter reviews the fundamentals of stress, strain, and deformation and demonstrates some of the tools and techniques used to analyze how materials and structures respond to tension, compression, bending, and shear. It begins with an overview of the behavior of perfectly elastic and plastic materials and viscous substances. It then describes the stress-strain response of two- and three-dimensional solids, explaining how to determine principle stresses and strains using Mohr’s circle and how to derive equivalent stress and strain using the von Mises relationship. It then goes on to analyze the stress state of load-bearing members, pressurized tubes, and pin-loaded lugs, accounting for the effect of geometric discontinuities, such as cutouts, fillets, and holes, as well as cracks. It also explains how finite element methods are used to solve problems involving complex geometric and loading conditions.

This chapter discusses the metallurgical changes that occur and the improvements that can be achieved in superalloys through solid-solution hardening, precipitation hardening, and dispersion strengthening. It also explains how further improvements can be achieved through the control of grain structure, as in columnar-grained alloys, or by the elimination of grain boundaries as with single-crystal superalloys.

This chapter describes the structures, phases, and phase transformations observed in metals and alloys as they solidify and cool to lower temperatures. It begins with a review of the solidification process, covering nucleation, grain growth, and the factors that influence grain morphology. It then discusses the concept of solid solutions, the difference between substitutional and interstitial solid solubility, the effect of alloying elements, and the development of intermetallic phases. The chapter also covers the construction and use of binary and ternary phase diagrams and describes the helpful information they contain.

Titanium alloys are classified according to the amount of alpha and beta phase material retained in their structures at room temperature. This chapter discusses the metallurgy, composition, processing, and properties of titanium and its alloys. It provides information on melting, forging, casting, heat treating, and secondary fabrication. It also discusses the advantages and disadvantages of titanium and its alloys in various applications.

This chapter reviews the basic chemistry of beryllium metals and compounds, including beryllium hydroxide, beryllium carbonates, beryllium fluoride, and beryllium chloride. It discusses the uses as well as application challenges of various forms of beryllium and includes information on their chemical properties and reactions.

The microstructure of superalloys is highly complex, with a large number of dispersed intermetallics and other phases that modify alloy behavior through their composition, morphology, and distribution. This chapter provides an overview of the most notable phases, including the matrix phase and geometrically and topologically close-packed phases, and describes how superalloy microstructure can be modified via heat treatments and directional solidification. It also discusses the role of carbides, borides, oxides, and nitrides and the detrimental effects of sulfocarbides.

This chapter addresses the basic concepts important to understanding corrosion of metals. It begins with an overview of the three types of behaviors that a metal exhibits when immersed in an environment and of the four requirements of a corrosion cell. The chapter then covers the important characteristics of metals with respect to corrosion, namely the metallurgical characteristics, the inherent tendency to corrode, and the tendency to form insoluble corrosion products. The important characteristics of aqueous solutions with respect to corrosion are then addressed. The characteristics include: conductivity of the solution, acidity and alkalinity, oxidizing power, degree of ionization, and solubility in the solution. These characteristics, in combination with the characteristics of the metal, will determine the corrosion behavior of a metal/environment combination. The chapter concludes with a section on the determination of corrosion rates and corrosion rate allowances.

Almost all metals and alloys are produced from liquids by solidification. For both castings and wrought products, the solidification process has a major influence on both the microstructure and mechanical properties of the final product. This chapter discusses the three zones that a metal cast into a mold can have: a chill zone, a zone containing columnar grains, and a center-equiaxed grain zone. Since the way in which alloys partition on freezing, it follows that all castings are segregated to different categories. The different types of segregation discussed include normal, gravity, micro, and inverse. The chapter also provides information on grain refinement and secondary dendrite arm spacing and porosity and shrinkage in castings. It concludes with a brief overview of six of the most important casting processes in industries: sand casting, plaster mold casting, evaporative pattern casting, investment casting, permanent mold casting, and die casting.

Titanium is a lightweight metal used in a growing number of applications for its strength, toughness, stiffness, corrosion resistance, biocompatibility, and high-temperature operating characteristics. This chapter discusses the applications, metallurgy, properties, compositions, and grades of commercially pure titanium and alpha and near-alpha, alpha-beta, and beta titanium alloys. It describes primary and secondary fabrication processes, including melting, forging, forming, heat treating, casting, machining, and joining as well as powder metallurgy and direct metal deposition. It also compares and contrasts the properties of wrought, cast, and powder metal titanium products and discusses corrosion behaviors.

This chapter describes the phases and constituents present in iron-carbon steels in near-equilibrium conditions. It explains how to use phase diagrams to predict and manage the development of ferrite, austenite, cementite, and pearlite through controlled cooling. It discusses the transformations, grain structure, and properties associated with each phase and identifies the primary stabilizing elements. It includes several micrographs revealing various microstructural features and describes the processing route by which they were achieved. It explains how to estimate the volume fraction of iron-carbon phases in equilibrium and how to determine the amount of each phase that must be present to reach a desired composition. The chapter also discusses the phases associated with hypo- and hyper-eutectoid steels and presents more than a dozen micrographs, identifying important structural features along with cooling conditions and sample preparation procedures.

This chapter provides practical information on the forming and forging processes used to manufacture titanium parts, including die forging, precision die forging, hot and cold forming, superplastic forming, and deep drawing. It explains how process variables such as temperature, pressure, and strain rate influence microstructure and properties and provides recommended ranges for commonly formed and forged titanium alloys.