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phenomenological constitutive equation
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Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005433
EISBN: 978-1-62708-196-2
... Abstract This article presents a mechanical description of superplasticity and discusses constitutive equations that are essential for simulating superplastic forming processes, applicable to structural superplasticity. It presents the phenomenological constitutive equations of superplasticity...
Abstract
This article presents a mechanical description of superplasticity and discusses constitutive equations that are essential for simulating superplastic forming processes, applicable to structural superplasticity. It presents the phenomenological constitutive equations of superplasticity and classical physical constitutive equations. The article also reviews the accommodation mechanisms that are divided into two major groups, namely, diffusional accommodation and accommodation by dislocations.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005404
EISBN: 978-1-62708-196-2
... materials using expressions known as constitutive equations that relate the dependence of stress, temperature, and microstructure on deformation. The article reviews the characteristics of creep deformation and mechanisms of creep, such as power-law creep, low temperature creep, power-law breakdown...
Abstract
This article, to develop an understanding of the underlying mechanisms governing deformation at elevated temperatures, discusses the phenomenological effects resulting from temperature-induced thermodynamic and kinetic changes. It describes the deformation behavior of engineering materials using expressions known as constitutive equations that relate the dependence of stress, temperature, and microstructure on deformation. The article reviews the characteristics of creep deformation and mechanisms of creep, such as power-law creep, low temperature creep, power-law breakdown, diffusional creep, twinning during creep deformation, and deformation mechanism maps. It discusses the creep-strengthening mechanisms for most structural engineering components. The article provides a description of the microstructural modeling of creep in engineering alloys.
Series: ASM Handbook
Volume: 24A
Publisher: ASM International
Published: 30 June 2023
DOI: 10.31399/asm.hb.v24A.a0006988
EISBN: 978-1-62708-439-0
... is treated as a homogeneous material, and the process-induced microstructural features that contribute to the overall material response are accounted for implicitly within a phenomenological constitutive model. Typically, the phenomenological constitutive model comprises an anisotropic yield criterion...
Abstract
This article focuses specifically on material modeling applied to structure-property predictions. It provides general guidelines and considerations in terms of modeling the salient material features that ultimately impact the mechanical performance of parts produced by additive manufacturing (AM). Two of the primary ingredients needed to predict structure-property relationships via material modeling include a geometrical representation of the microstructural features of interest (e.g., grain structure and void defects) and a suitable constitutive model describing the material behavior, both of which can be scale and resource dependent. The article also presents modeling challenges to predict various aspects of (process-) structure-property relationships in AM.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005425
EISBN: 978-1-62708-196-2
... of: Deformation and strain hardening, such as Schmid's law and the Hollomon equation Kinetics of recrystallization (such as the Avrami equation), grain growth (such as the Beck equation), and precipitation/phase transformation Ductility for solid-state processes Similarly, phenomenological models...
Abstract
This article provides a brief historical perspective, a classification of metallurgical processes, basic model development efforts, and an overview of the potential future directions for the modeling of metals processing. It describes the classification of material behavior models, which can be grouped broadly into three classes: statistical, phenomenological, and mechanistic models. The article also presents an overview of the potential directions for the modeling of metals processing.
Series: ASM Handbook
Volume: 14B
Publisher: ASM International
Published: 01 January 2006
DOI: 10.31399/asm.hb.v14b.a0005133
EISBN: 978-1-62708-186-3
..., and hence, one needs evolution equations for them to complete the constitutive equation. In the event that explicit evolution equations for microstructural parameters cannot be determined, approximate phenomenological constitutive relations can be obtained by scaling experimentally determined stress-versus...
Abstract
Compared to cold-formed parts, age-formed parts have lower residual stresses and consequently better stress corrosion resistance. This article addresses the technical issues that arise in the investigations of creep in precipitate-strengthened materials. The issues addressed help in developing alloys and tempers particularly suited for the age-forming process. The different steps involved in the program for predicting the final part shape are discussed. These basic steps involve developing mechanical tests to study creep at low temperatures and low stresses, describing low-temperature creep in terms of a constitutive model, and then using the constitutive model in a process model or finite element analysis to predict the final part shape.
Book: Composites
Series: ASM Handbook
Volume: 21
Publisher: ASM International
Published: 01 January 2001
DOI: 10.31399/asm.hb.v21.a0003402
EISBN: 978-1-62708-195-5
... are the permeabilities of fiber network in thickness and in-plane directions, respectively. Experimental Validation Almost all process models that are developed involve one or more assumptions, and many of them use constitutive or phenomenological equations to represent material behavior. Thus, it is important...
Abstract
This article provides information on the classification of various composites manufacturing processes based on similar transport processes. The composites manufacturing processes can be grouped into three categories: short-fiber suspension methods, squeeze flow methods, and porous media methods. The article presents an overview of the modeling philosophy and approach that is useful in describing composite manufacturing processes.
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005502
EISBN: 978-1-62708-197-9
... Abstract This article presents the governing equations and methodologies to model the press and sinter powder metallurgy, including continuum, micromechanical, multiparticle, and molecular dynamics approaches. It describes the constitutive relation for compaction and sintering. The article...
Abstract
This article presents the governing equations and methodologies to model the press and sinter powder metallurgy, including continuum, micromechanical, multiparticle, and molecular dynamics approaches. It describes the constitutive relation for compaction and sintering. The article discusses the experimental determination of material properties and simulation verification for compaction and sintering. It reviews the use of modeling and simulation of press and sinter powder metallurgy, including gravitational distorting in sintering, compaction optimization, sintering optimization, and coupled press and sinter optimization.
Book: Powder Metallurgy
Series: ASM Handbook
Volume: 7
Publisher: ASM International
Published: 30 September 2015
DOI: 10.31399/asm.hb.v07.a0006033
EISBN: 978-1-62708-175-7
... and constitutive relations, a system of partial differential equations is created that includes the initial and boundary conditions. These must be integrated with structure and property models so that final compact properties can be predicted. Because the constitutive relations for compaction and sintering...
Abstract
This article discusses continuum modeling, which is the most relevant approach in modeling grain growth, densification, and deformation during sintering. Continuum plasticity models are frequently used to describe the mechanical response of metal powders during compaction. The article illustrates the typical procedure for computer simulation for press and sinter process. It describes the procedure to obtain the material properties based on the generalized Shima-Oyane model. The article presents a wide variety of tests, accounting for data on the grain growth, densification, and distortion where these data help in the development of a constitutive model for sintering simulation. Finally, the article provides information on the simulation approaches used to optimize die compaction and sintering.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005413
EISBN: 978-1-62708-196-2
.... A systematic search for other algebraic approximations to the data (for many materials) has led to the conclusion that the phenomenological equation for the reduced activation energy Δ G /μ b 3 ≡ g : (Eq 16) ( Δ G μ b 3 ) 1 / 2 ≡ g 1 / 2 ∝ 1 − ( σ / μ...
Abstract
This article focuses on the analyzing and modeling of stress-strain behavior of polycrystals of pure face-centered cubic (fcc) metals in the range of temperatures and strain rates where diffusion is not important. It presents a phenomenological description of stress-strain behavior and provides information on the physical background, alternative interpretations, and directions of research. The quantitative description of strain hardening of fcc polycrystals is provided. The article also discusses the modeling of stress-strain behavior in body-centered cubic metals, hexagonal metals, stage IV work hardening, and the various classes of single-phase alloys.
Book Chapter
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005431
EISBN: 978-1-62708-196-2
... on steady-state equations of mechanical equilibrium and constitutive equations for mechanical flow. The complete set of equations serves as the foundation for continuum mechanics models of bulk working processes. A model for each case is developed by imposing appropriate boundary conditions and initial...
Abstract
Several methods are developed for the numerical solution of partial differential equations, namely, meshed-solution methods such as the finite-element method (FEM), finite-difference method, and boundary-element method; and numerical algorithms consisting of so-called meshed-solution methods. This article introduces the methods of so-called meshed solutions, with an emphasis on the FEM. It presents some basic differential equations that are used to model the responses of structures, components, processes, or systems with emphasis on continuum mechanics. The article provides an outline on the mathematical principles of solving differential equations. It also reviews linear structural problems to illustrate the concept of the FEMs.
Series: ASM Handbook Archive
Volume: 11
Publisher: ASM International
Published: 01 January 2002
DOI: 10.31399/asm.hb.v11.a0003558
EISBN: 978-1-62708-180-1
... and to assess the significance of various parameters, including materials, which can be changed to resolve the wear problem. Modeling is an essential element in resolving wear failures. The characterizations of the tribosystem and the wear situation provide the basis for models. Simple phenomenological...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005415
EISBN: 978-1-62708-196-2
... microstructure activation energy critical nucleus configuration phase-field kinetic equations chemical free energy kinetic coefficients Fundamentals The collection of a hierarchy of structural and chemical nonuniformities (imperfections or defects) in solids constitutes materials crostructure...
Abstract
This article discusses the fundamental aspects of phase-field microstructure modeling. It describes the evolution of microstructure modeling, including nucleation, growth, and coarsening. The article reviews two approaches used in the modeling nucleation of microstructure: the Langevin force approach and explicit nucleation algorithm. Calculation of activation energy and critical nucleus configuration is discussed. The article presents the deterministic phase-field kinetic equations for modeling growth and coarsening of microstructure. It also describes the material-specific model inputs, chemical free energy and kinetic coefficients, for phase-field microstructure modeling. The article provides four examples that illustrate some aspects of phase-field modeling.
Series: ASM Handbook
Volume: 14A
Publisher: ASM International
Published: 01 January 2005
DOI: 10.31399/asm.hb.v14a.a0009002
EISBN: 978-1-62708-185-6
... ). In this technique, each grain is divided into a number of elements. Each of the elements has its own constitutive equation depending on its prior history (e.g., level of hardening or softening) and instantaneous crystallographic orientation. With these local constitutive equations, the FEM analysis...
Abstract
This article reviews the general aspects of microstructure evolution during thermomechanical processing. The effect of thermomechanical processing on microstructure evolution is summarized to provide insight into the aspect of process design. The article provides information on hot working and key processes that control microstructure evolution: dynamic recovery, static recovery, recrystallization, and grain growth. Some of the key phenomenological descriptions of plastic flow and microstructure evolution are also summarized. The article concludes with a discussion on the modeling of microstructure evolution.
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005619
EISBN: 978-1-62708-174-0
... matrix, ü is the nodal acceleration vector, u is the nodal velocity vector, { u } is the nodal displacement vector, and { L } is the applied load vector. The basic constitutive equations between stress (σ) and strain (ε) are as follows: (Eq 2) { σ } = [ D ] { ε el...
Abstract
Ultrasonic welding (UW), as a solid-state joining process, uses an ultrasonic energy source and pressure to induce oscillating shears between the faying surfaces to produce metallurgical bonds between a wide range of metal sheets and wires. This article reviews the models of the ultrasonic welding with an emphasis on governing equations, material behavior, and heat generation of the process. It discusses the resulting factors, namely, vibration, friction, temperature, and plastic deformation as well as the bonding strength and its mechanism.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002476
EISBN: 978-1-62708-194-8
... on formulating the basic equations for system reliability.) After the following discussion the reader should be cognizant that the expression given in Eq 11 represents the probability of failure for a specified set of boundary conditions. If the boundary conditions are modified in any way, Eq 11 is no longer...
Abstract
Brittle materials, such as ceramics, intermetallics, and graphites, are increasingly being used in the fabrication of lightweight components. This article focuses on the design methodologies and characterization of certain material properties. It describes the fundamental concepts and models associated with performing time-independent and time-dependent reliability analyses for brittle materials exhibiting scatter in ultimate strength. The article discusses the two-parameter and three-parameter Weibull distribution for representing the underlying probability density function for tensile strength. It reviews life prediction reliability models used for predicting the life of a component with complex geometry and loading. The article outlines reliability algorithms and presents several applications to illustrate the utilization of these reliability algorithms in structural applications.
Series: ASM Handbook
Volume: 14B
Publisher: ASM International
Published: 01 January 2006
DOI: 10.31399/asm.hb.v14b.a0005168
EISBN: 978-1-62708-186-3
... Abstract This article discusses the numerical simulation of the forming of aluminum alloy sheet metals. The macroscopic and microscopic aspects of the plastic behavior of aluminum alloys are reviewed. The article presents constitutive equations suitable for the description of aluminum alloy...
Abstract
This article discusses the numerical simulation of the forming of aluminum alloy sheet metals. The macroscopic and microscopic aspects of the plastic behavior of aluminum alloys are reviewed. The article presents constitutive equations suitable for the description of aluminum alloy sheets. It explains testing procedures and analysis methods that are used to measure the relevant data needed to identify the material coefficients. The article describes the various formulations of finite element methods used in sheet metal forming process simulations. Stress-integration procedures for both continuum and crystal-plasticity mechanics are also discussed. The article also provides various examples that illustrate the simulation of aluminum sheet forming.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002482
EISBN: 978-1-62708-194-8
... the process under consideration Mathematical equations describing the physical process Data needed to solve the equations Numerical algorithm to solve the equations given the boundary conditions and the constitutive behavior Availability of the software to provide answers One of the most...
Abstract
Manufacturing processes typically involve the reshaping of materials from one form to another under a set of processing conditions. This article discusses the two classification schemes of modeling for manufacturing processes, namely, on-line or off-line models and empirical, mechanistic, or deterministic models along with their important considerations. It describes the various aspects of modeling of deformation processes, casting operations, and fusion welding processes, with examples.
Series: ASM Handbook
Volume: 14B
Publisher: ASM International
Published: 01 January 2006
DOI: 10.31399/asm.hb.v14b.a0005183
EISBN: 978-1-62708-186-3
... Abstract Constitutive relations for metal-working include elements of behavior at ambient temperature as well as high-temperature response. This article presents the equations for the strain hardening and strain-rate-sensitive flow, with alternate sections on empirically determined properties...
Abstract
Constitutive relations for metal-working include elements of behavior at ambient temperature as well as high-temperature response. This article presents the equations for the strain hardening and strain-rate-sensitive flow, with alternate sections on empirically determined properties, followed by models of constitutive behavior. These models include the isothermal constitutive model and the physical model for superplastic flow. A formal description of the superposition of the operative mechanisms for dynamic recovery at hot-working strain rates is also provided. The article describes creep mechanisms that are useful for illustrating the strong stress dependence of dislocation and diffusional flow.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005409
EISBN: 978-1-62708-196-2
... alloys, with their corresponding equations. Microstructure modeling thermomechanical processing titanium alloys beta titanium alloys spheroidization coarsening particle growth phase decomposition heat treatment alpha-beta titanium alloys TITANIUM AND TITANIUM ALLOYS are widely used...
Abstract
This article focuses on the modeling of microstructure evolution during thermomechanical processing in the two-phase field for alpha/beta and beta titanium alloys. It also discusses the mechanisms of spheroidization, the coarsening, particle growth, and phase decomposition in titanium alloys, with their corresponding equations.
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005509
EISBN: 978-1-62708-197-9
... deformations, approximately 15 to 20% linear. The initial density is approximately 60 to 65% of the theoretical density, which is usually reached by the end of the process. From a mathematical point of view, it means that the constitutive equations describing the material flow will be nonlinear...
Abstract
This article discusses the two major applications of hot isostatic pressing (HIP), such as healing of inherent internal defects in castings and welds, and consolidation of powder materials. It describes the design principles of the HIP tooling, as well as the problems associated with mathematical modeling of HIP. The article presents an example for the modeling process of the HIP. It reviews the numerical modeling and tooling design of a casing component demonstration.
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