The document discusses phase transformations in solids, focusing on diffusionless martensitic transformations. It describes the mechanisms of martensite formation, including the Bain distortion, lattice-invariant shear, and rigid body rotation that constitute the complete transformation. It discusses nucleation and growth of martensite plates and laths, and how alloying elements and pressure can affect the martensitic transformation.
The document contains slides from a course on masonry structures. It discusses the lateral strength and behavior of unreinforced masonry (URM) shear walls. It covers design criteria for URM shear walls including allowable flexural and shear stresses. An example problem is presented to calculate the maximum base shear capacity of a URM shear wall according to codes.
1) There are two approaches to describing grain boundaries mechanically: a continuous approach using dislocation density tensors and a discrete approach considering arrays of dislocations.
2) The continuous Frank-Bilby equation relates the Burgers vector density necessary to achieve compatibility at the interface between crystals to the affine transformations relating their lattices.
3) The discrete Read-Shockley model links the misorientation angle and spacing of edge dislocations forming low-angle symmetrical tilt grain boundaries. It predicts the intergranular energy as a function of angle.
This document discusses two approaches to describing the mechanical behavior of grain boundaries: the continuous Frank-Bilby approach and the discrete Read-Shockley approach. It focuses on the discrete approach, explaining Bollmann's model of intrinsic dislocations that form periodic networks at grain boundaries according to the misorientation angle between crystals. Primary intrinsic dislocations account for the deviation from a single crystal structure and have Burgers vectors of the crystal lattice. Their spacing decreases with increasing misorientation angle according to the Read-Shockley formula. Examples are provided for low-angle tilt and twist grain boundaries.
Predictive model of moment of resistance for rectangular reinforced concrete ...Alexander Decker
This document presents a predictive model for calculating the moment of resistance (MR) for rectangular reinforced concrete sections. The model is developed based on stress-strain analysis of a singly reinforced concrete beam. The governing equation relates MR to the concrete compressive strength (fcu), breadth (b), and effective depth (d). Simulation results show that MR increases with larger b and d values. MR also increases at a higher rate with greater d due to its quadratic relationship in the equation, whereas the increase is linear with b. The model allows accurate selection of section dimensions for structural design based on required resistance.
The document discusses limitations of analyzing masonry structures on a storey-by-storey basis and provides an overview of macroelement modeling approaches. It notes that storey-mechanism analysis makes assumptions about boundary conditions that may not accurately capture the behavior of coupling elements. Global analysis is needed to understand stresses in these elements. It also summarizes characteristics of several macroelement models, including multi-fan, PEFV, TREMURI, and SAM models, that can better model the behavior of entire masonry buildings through use of macro-elements representing portions of the structure.
what is static and kinematic indeterminacy of structure NIROB KR DAS
This document discusses the concepts of determinacy and indeterminacy in structural analysis. It defines a statically determinate structure as one where the static equilibrium equations are sufficient to determine internal forces and reactions. Simply supported beams and cantilever beams are provided as examples. It then defines a statically indeterminate structure as one where the equilibrium equations are insufficient, and provides formulas to calculate the degrees of static indeterminacy for beams, frames, and trusses. The document also discusses kinematic determinacy and indeterminacy, and provides similar formulas for calculating degrees of kinematic indeterminacy for various structural elements.
1. Determine the reinforcement ratio ρ.
2. Calculate the modular ratio n based on concrete and steel properties.
3. Use an iterative process to locate the neutral axis depth kd by solving for the parameter k.
4. With k determined, calculate the moment arm j.
5. Compute the moment capacity as Mallow = R * b * d^2, where R is the resisting stress block parameter dependent on k and j.
This document discusses seismic design and assessment of masonry structures. It begins by showing how historical masonry buildings are typically assembled over time in a piecemeal fashion. It then outlines various damage mechanisms seen in masonry structures during earthquakes, such as out-of-plane instability of walls, overturning of facades, and damage from thrust forces from the roof. The document proposes using limit analysis and modeling vulnerable subsystems as rigid bodies to evaluate the static threshold for collapse. It provides examples of applying the principle of virtual work to determine static thresholds for simple mechanisms.
The document contains slides from a course on masonry structures. It discusses the lateral strength and behavior of unreinforced masonry (URM) shear walls. It covers design criteria for URM shear walls including allowable flexural and shear stresses. An example problem is presented to calculate the maximum base shear capacity of a URM shear wall according to codes.
1) There are two approaches to describing grain boundaries mechanically: a continuous approach using dislocation density tensors and a discrete approach considering arrays of dislocations.
2) The continuous Frank-Bilby equation relates the Burgers vector density necessary to achieve compatibility at the interface between crystals to the affine transformations relating their lattices.
3) The discrete Read-Shockley model links the misorientation angle and spacing of edge dislocations forming low-angle symmetrical tilt grain boundaries. It predicts the intergranular energy as a function of angle.
This document discusses two approaches to describing the mechanical behavior of grain boundaries: the continuous Frank-Bilby approach and the discrete Read-Shockley approach. It focuses on the discrete approach, explaining Bollmann's model of intrinsic dislocations that form periodic networks at grain boundaries according to the misorientation angle between crystals. Primary intrinsic dislocations account for the deviation from a single crystal structure and have Burgers vectors of the crystal lattice. Their spacing decreases with increasing misorientation angle according to the Read-Shockley formula. Examples are provided for low-angle tilt and twist grain boundaries.
Predictive model of moment of resistance for rectangular reinforced concrete ...Alexander Decker
This document presents a predictive model for calculating the moment of resistance (MR) for rectangular reinforced concrete sections. The model is developed based on stress-strain analysis of a singly reinforced concrete beam. The governing equation relates MR to the concrete compressive strength (fcu), breadth (b), and effective depth (d). Simulation results show that MR increases with larger b and d values. MR also increases at a higher rate with greater d due to its quadratic relationship in the equation, whereas the increase is linear with b. The model allows accurate selection of section dimensions for structural design based on required resistance.
The document discusses limitations of analyzing masonry structures on a storey-by-storey basis and provides an overview of macroelement modeling approaches. It notes that storey-mechanism analysis makes assumptions about boundary conditions that may not accurately capture the behavior of coupling elements. Global analysis is needed to understand stresses in these elements. It also summarizes characteristics of several macroelement models, including multi-fan, PEFV, TREMURI, and SAM models, that can better model the behavior of entire masonry buildings through use of macro-elements representing portions of the structure.
what is static and kinematic indeterminacy of structure NIROB KR DAS
This document discusses the concepts of determinacy and indeterminacy in structural analysis. It defines a statically determinate structure as one where the static equilibrium equations are sufficient to determine internal forces and reactions. Simply supported beams and cantilever beams are provided as examples. It then defines a statically indeterminate structure as one where the equilibrium equations are insufficient, and provides formulas to calculate the degrees of static indeterminacy for beams, frames, and trusses. The document also discusses kinematic determinacy and indeterminacy, and provides similar formulas for calculating degrees of kinematic indeterminacy for various structural elements.
1. Determine the reinforcement ratio ρ.
2. Calculate the modular ratio n based on concrete and steel properties.
3. Use an iterative process to locate the neutral axis depth kd by solving for the parameter k.
4. With k determined, calculate the moment arm j.
5. Compute the moment capacity as Mallow = R * b * d^2, where R is the resisting stress block parameter dependent on k and j.
This document discusses seismic design and assessment of masonry structures. It begins by showing how historical masonry buildings are typically assembled over time in a piecemeal fashion. It then outlines various damage mechanisms seen in masonry structures during earthquakes, such as out-of-plane instability of walls, overturning of facades, and damage from thrust forces from the roof. The document proposes using limit analysis and modeling vulnerable subsystems as rigid bodies to evaluate the static threshold for collapse. It provides examples of applying the principle of virtual work to determine static thresholds for simple mechanisms.
Analytical solution of bending stress equation for symmetric and asymmetric i...Alexander Decker
This document presents an analytical method for determining the bending stress equation for symmetric and
asymmetric involute gear teeth with and without profile correction. Key steps include:
1) Deriving analytical expressions for geometric properties of involute gear tooth profiles, such as tooth loading angle,
radius, and involute function.
2) Formulating an expression for tooth loading angle based on the applied force position along the tooth profile.
3) Extending the analysis to account for profile correction by incorporating corrected pressure angle, pitch radius,
and loading angle into the equations.
The analytical method avoids previous trial graphical methods and provides a more accurate and efficient way to
solve the bending stress
This document provides an overview of modeling approaches for seismic design and assessment of masonry structures, including:
- Vertical structures can be modeled as cantilever walls, equivalent frames with varying degrees of coupling between floors/piers.
- Equivalent frame models are more realistic and require defining floor/spandrel stiffness. Rigid offsets can limit horizontal deformation.
- Refined 2D/3D finite element models may be needed for complex geometries or nonlinear analysis, but are not usually practical.
- Linear static analysis uses equivalent static loads distributed by storey based on vibration mode. Nonlinear static pushover analyzes failure by increasing loads until a mechanism forms.
This document discusses the out-of-plane seismic response of unreinforced masonry walls. It covers several topics: mechanisms of out-of-plane failure including parapet failure and overturning; the seismic load path and how ground motion is transmitted; important issues in evaluating out-of-plane response such as strength, displacement capacity, and dynamic response; and methods for assessing out-of-plane flexural strength including tensile strength of masonry and arching action. Slides show examples of damage from past earthquakes and diagrams illustrating failure mechanisms and load paths.
The document presents design charts for estimating the deflection of a thin circular elastic plate resting on a Pasternak foundation. The charts show deflection values for different nondimensional values of modulus of subgrade reaction and shear modulus. The charts were developed using a nondimensional expression for deflection derived through a strain energy approach. The analysis considers the tensionless characteristics of the Pasternak foundation model and the potential for lift-off of the plate from the surface.
The document summarizes some macroelement models for unreinforced masonry (URM) structures, including:
1) The SAM model which uses simplified strength criteria and constitutive rules to model flexural and shear failure of URM elements.
2) A nonlinear equivalent frame model that represents URM walls as piers and spandrels with rigid offsets and uses force-deformation relationships to model flexural, shear, and rocking behavior.
3) A comparison showing similar force-displacement responses between a 3D storey mechanism model and the nonlinear frame model for a 2-story URM building.
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This research focuses on simulating crack propagation in fluid-saturated porous materials undergoing large deformations. A two-scale numerical model is used, with the micro-scale governing fluid flow through the deformable porous material, and the macro-scale using finite elements coupled to the local momentum and mass balances from the micro-scale. Micro-cracks are modeled using cohesive zone laws, where the traction decreases with crack opening. The resulting equations are nonlinear and solved iteratively using Newton-Raphson and Crank-Nicholson schemes. The goal is to extend the approach to fully capture large material deformations with small crack openings, using hyperelastic constitutive models.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document provides instructions and questions for a final examination in electromagnetic field theory. It consists of 5 questions testing concepts such as electric and magnetic fields, Maxwell's equations, boundary conditions, wave propagation, and vector calculus identities. The examination is for a course taught in the 2009/2010 semester and covers topics including electrostatics, magnetostatics, and time-varying fields. Students have 2 hours and 30 minutes to answer 4 out of the 5 questions.
The document summarizes the presentation given by Vamsi Krishna Rentala on stress fields around dislocations. It first defines dislocations and the three main types: edge, screw, and mixed. It then describes how stress fields are produced by dislocations using linear elasticity theory. Simple models are used to illustrate the stress fields around screw and edge dislocations. Key results presented include the diverging stresses near dislocations and variations in stress type (tensile vs. compressive) above and below the slip plane for an edge dislocation. The document concludes by noting mixed dislocations contain both edge and screw components and summarizing some properties of stress fields.
Dislocations are line defects in crystals that represent disrupted planes of atoms. They allow plastic deformation via slip along crystallographic planes and directions.
A dislocation is characterized by its Burgers vector, which represents the lattice displacement caused by the dislocation and determines the direction of slip. The Burgers vector connects one lattice position to another.
Dislocations lower the theoretical shear strength of crystals by several orders of magnitude, enabling plasticity. Their motion through glide and climb allows crystals to deform plastically under stress.
This document provides an overview of masonry structures and materials. It discusses the mechanical behavior of masonry walls, arches, vaults and domes. Traditional masonry construction techniques are compared to modern methods. Various masonry elements like walls, columns and beams are examined. Finally, common masonry materials like fired clay units are described in terms of their manufacturing, properties and testing standards. The document serves as teaching material for a course on seismic design and assessment of masonry structures.
1) The document discusses analyzing statically indeterminate structures when supports settle or temperatures change, which can induce stresses.
2) The force method is extended to include compatibility equations that account for support displacements and temperature effects.
3) Two examples demonstrate calculating reactions in beams due to support settlements and temperature differences using the modified force method equations.
Surface roughness effect on the performance of a magnetic fluid based porous ...Alexander Decker
This document summarizes a study analyzing the performance of a porous rough secant shaped slider bearing using a magnetic fluid lubricant. The authors develop stochastic equations to model the pressure distribution and account for the random surface roughness. They present dimensionless equations for the load carrying capacity and friction. Graphs show that the magnetic fluid increases load capacity and decreases friction compared to conventional lubricants. However, increased porosity and surface roughness negatively impact performance. The positive effects of magnetization can reduce these negative impacts, especially for negatively skewed roughness and negative variance. The study concludes magnetic fluids improve bearing performance and can support loads even without fluid flow.
The document discusses rock mass properties and the Hoek-Brown failure criterion for estimating the strength of jointed rock masses. It presents the generalized Hoek-Brown criterion equation and describes how to determine the intact rock properties of uniaxial compressive strength (σci) and the Hoek-Brown constant (mi) from triaxial test data or estimates. It also discusses estimating the Geological Strength Index (GSI) of the rock mass.
This document provides instructions and questions for a structural design exam. It consists of 4 questions. Students must answer question 1 and any other two questions. Question 1 involves calculating bending moments, designing reinforcement, and determining shear capacity for concrete beams. Question 2 involves checking the adequacy of steel sections and designing a bolt connection. Question 3 uses force methods to determine reactions and draws shear and bending moment diagrams. Question 4 analyzes a frame under vertical and lateral loads to determine reactions and internal forces at specific points. The document also includes relevant design formulas and appendices on load combinations, bending moment coefficients, and steel design strengths.
This document discusses different types of dislocations that occur in crystalline materials including edge dislocations, screw dislocations, and mixed dislocations. It describes how dislocations move through the crystal lattice during plastic deformation from the application of stress. It also covers characteristics of dislocations like lattice strain, slip systems, and deformation mechanisms in both single crystals and polycrystalline materials including twinning.
This section discusses the calculation of crack width in prestressed concrete flexural members (Type 3). It describes the factors that influence crack width and outlines the method specified in the Indian code IS:456-2000. The crack width is calculated at the point of maximum estimated width, usually in the soffit equidistant from two longitudinal bars. The calculation involves determining the shortest distance from the calculation point to the nearest bar, the neutral axis depth, and the average strain at the calculation point from a sectional analysis considering tension stiffening.
Edge Extraction with an Anisotropic Vector Field using Divergence MapCSCJournals
The aim of this work is the extraction of edges by a deformable contour procedure, using an external force field derived from an anisotropic flow, with different external and initial conditions. By evaluating the divergence of the force field, we have generated a divergence map associated with it in order to analyze the field convergence. As we know, the divergence measures the intensity of convergence or divergence of a vector field at a given point, so by means level curves of the divergence map, we have automatically selected an initial contour for the deformation process. The initial curve must include areas from which the vector field diverges pushing it towards the edges. Furthermore the divergence map brings out the presence of curves pointing to the most significant geometric parts of boundaries corresponding to high curvature values, in this way it will result better defined the geometrical shape of the extracted object.
Seismic design and construction of retaining wallAhmedEwis13
This document discusses seismic design considerations for retaining walls. It describes the common types of retaining walls, including gravity, cantilever, reinforced soil, and anchored bulkhead walls. Static lateral earth pressures are calculated using Rankine and Coulomb theories, with the Mononobe-Okabe method extending Coulomb theory to account for seismic inertial forces. Dynamic response of retaining walls is complex, with wall movement, pressures, and permanent displacements dependent on the response of the wall, backfill soil, and foundation soil to ground shaking.
The document discusses several topics related to vertical curves and superelevation design for roads, including:
1. Equations for parabolic vertical curves and methods for designing vertical curves to connect lines with different grades, including passing a curve through a fixed point.
2. Minimum length requirements for vertical curves based on sight distance standards from AASHTO to ensure safety.
3. Design of unequal tangent vertical curves where the curves on each side have different lengths.
4. Considerations for pavement cross-slope or "crown" and superelevation rates for horizontal curves based on design speed, road classification, climate and other factors.
The ideal, perfectly regular crystal structures in which atoms are arranged in a regular way does not exist in actual situations. In actual cases, the regular arrangements of atoms disrupted . These disruptions are known as Crystal imperfections or crystal defects
Analytical solution of bending stress equation for symmetric and asymmetric i...Alexander Decker
This document presents an analytical method for determining the bending stress equation for symmetric and
asymmetric involute gear teeth with and without profile correction. Key steps include:
1) Deriving analytical expressions for geometric properties of involute gear tooth profiles, such as tooth loading angle,
radius, and involute function.
2) Formulating an expression for tooth loading angle based on the applied force position along the tooth profile.
3) Extending the analysis to account for profile correction by incorporating corrected pressure angle, pitch radius,
and loading angle into the equations.
The analytical method avoids previous trial graphical methods and provides a more accurate and efficient way to
solve the bending stress
This document provides an overview of modeling approaches for seismic design and assessment of masonry structures, including:
- Vertical structures can be modeled as cantilever walls, equivalent frames with varying degrees of coupling between floors/piers.
- Equivalent frame models are more realistic and require defining floor/spandrel stiffness. Rigid offsets can limit horizontal deformation.
- Refined 2D/3D finite element models may be needed for complex geometries or nonlinear analysis, but are not usually practical.
- Linear static analysis uses equivalent static loads distributed by storey based on vibration mode. Nonlinear static pushover analyzes failure by increasing loads until a mechanism forms.
This document discusses the out-of-plane seismic response of unreinforced masonry walls. It covers several topics: mechanisms of out-of-plane failure including parapet failure and overturning; the seismic load path and how ground motion is transmitted; important issues in evaluating out-of-plane response such as strength, displacement capacity, and dynamic response; and methods for assessing out-of-plane flexural strength including tensile strength of masonry and arching action. Slides show examples of damage from past earthquakes and diagrams illustrating failure mechanisms and load paths.
The document presents design charts for estimating the deflection of a thin circular elastic plate resting on a Pasternak foundation. The charts show deflection values for different nondimensional values of modulus of subgrade reaction and shear modulus. The charts were developed using a nondimensional expression for deflection derived through a strain energy approach. The analysis considers the tensionless characteristics of the Pasternak foundation model and the potential for lift-off of the plate from the surface.
The document summarizes some macroelement models for unreinforced masonry (URM) structures, including:
1) The SAM model which uses simplified strength criteria and constitutive rules to model flexural and shear failure of URM elements.
2) A nonlinear equivalent frame model that represents URM walls as piers and spandrels with rigid offsets and uses force-deformation relationships to model flexural, shear, and rocking behavior.
3) A comparison showing similar force-displacement responses between a 3D storey mechanism model and the nonlinear frame model for a 2-story URM building.
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This research focuses on simulating crack propagation in fluid-saturated porous materials undergoing large deformations. A two-scale numerical model is used, with the micro-scale governing fluid flow through the deformable porous material, and the macro-scale using finite elements coupled to the local momentum and mass balances from the micro-scale. Micro-cracks are modeled using cohesive zone laws, where the traction decreases with crack opening. The resulting equations are nonlinear and solved iteratively using Newton-Raphson and Crank-Nicholson schemes. The goal is to extend the approach to fully capture large material deformations with small crack openings, using hyperelastic constitutive models.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document provides instructions and questions for a final examination in electromagnetic field theory. It consists of 5 questions testing concepts such as electric and magnetic fields, Maxwell's equations, boundary conditions, wave propagation, and vector calculus identities. The examination is for a course taught in the 2009/2010 semester and covers topics including electrostatics, magnetostatics, and time-varying fields. Students have 2 hours and 30 minutes to answer 4 out of the 5 questions.
The document summarizes the presentation given by Vamsi Krishna Rentala on stress fields around dislocations. It first defines dislocations and the three main types: edge, screw, and mixed. It then describes how stress fields are produced by dislocations using linear elasticity theory. Simple models are used to illustrate the stress fields around screw and edge dislocations. Key results presented include the diverging stresses near dislocations and variations in stress type (tensile vs. compressive) above and below the slip plane for an edge dislocation. The document concludes by noting mixed dislocations contain both edge and screw components and summarizing some properties of stress fields.
Dislocations are line defects in crystals that represent disrupted planes of atoms. They allow plastic deformation via slip along crystallographic planes and directions.
A dislocation is characterized by its Burgers vector, which represents the lattice displacement caused by the dislocation and determines the direction of slip. The Burgers vector connects one lattice position to another.
Dislocations lower the theoretical shear strength of crystals by several orders of magnitude, enabling plasticity. Their motion through glide and climb allows crystals to deform plastically under stress.
This document provides an overview of masonry structures and materials. It discusses the mechanical behavior of masonry walls, arches, vaults and domes. Traditional masonry construction techniques are compared to modern methods. Various masonry elements like walls, columns and beams are examined. Finally, common masonry materials like fired clay units are described in terms of their manufacturing, properties and testing standards. The document serves as teaching material for a course on seismic design and assessment of masonry structures.
1) The document discusses analyzing statically indeterminate structures when supports settle or temperatures change, which can induce stresses.
2) The force method is extended to include compatibility equations that account for support displacements and temperature effects.
3) Two examples demonstrate calculating reactions in beams due to support settlements and temperature differences using the modified force method equations.
Surface roughness effect on the performance of a magnetic fluid based porous ...Alexander Decker
This document summarizes a study analyzing the performance of a porous rough secant shaped slider bearing using a magnetic fluid lubricant. The authors develop stochastic equations to model the pressure distribution and account for the random surface roughness. They present dimensionless equations for the load carrying capacity and friction. Graphs show that the magnetic fluid increases load capacity and decreases friction compared to conventional lubricants. However, increased porosity and surface roughness negatively impact performance. The positive effects of magnetization can reduce these negative impacts, especially for negatively skewed roughness and negative variance. The study concludes magnetic fluids improve bearing performance and can support loads even without fluid flow.
The document discusses rock mass properties and the Hoek-Brown failure criterion for estimating the strength of jointed rock masses. It presents the generalized Hoek-Brown criterion equation and describes how to determine the intact rock properties of uniaxial compressive strength (σci) and the Hoek-Brown constant (mi) from triaxial test data or estimates. It also discusses estimating the Geological Strength Index (GSI) of the rock mass.
This document provides instructions and questions for a structural design exam. It consists of 4 questions. Students must answer question 1 and any other two questions. Question 1 involves calculating bending moments, designing reinforcement, and determining shear capacity for concrete beams. Question 2 involves checking the adequacy of steel sections and designing a bolt connection. Question 3 uses force methods to determine reactions and draws shear and bending moment diagrams. Question 4 analyzes a frame under vertical and lateral loads to determine reactions and internal forces at specific points. The document also includes relevant design formulas and appendices on load combinations, bending moment coefficients, and steel design strengths.
This document discusses different types of dislocations that occur in crystalline materials including edge dislocations, screw dislocations, and mixed dislocations. It describes how dislocations move through the crystal lattice during plastic deformation from the application of stress. It also covers characteristics of dislocations like lattice strain, slip systems, and deformation mechanisms in both single crystals and polycrystalline materials including twinning.
This section discusses the calculation of crack width in prestressed concrete flexural members (Type 3). It describes the factors that influence crack width and outlines the method specified in the Indian code IS:456-2000. The crack width is calculated at the point of maximum estimated width, usually in the soffit equidistant from two longitudinal bars. The calculation involves determining the shortest distance from the calculation point to the nearest bar, the neutral axis depth, and the average strain at the calculation point from a sectional analysis considering tension stiffening.
Edge Extraction with an Anisotropic Vector Field using Divergence MapCSCJournals
The aim of this work is the extraction of edges by a deformable contour procedure, using an external force field derived from an anisotropic flow, with different external and initial conditions. By evaluating the divergence of the force field, we have generated a divergence map associated with it in order to analyze the field convergence. As we know, the divergence measures the intensity of convergence or divergence of a vector field at a given point, so by means level curves of the divergence map, we have automatically selected an initial contour for the deformation process. The initial curve must include areas from which the vector field diverges pushing it towards the edges. Furthermore the divergence map brings out the presence of curves pointing to the most significant geometric parts of boundaries corresponding to high curvature values, in this way it will result better defined the geometrical shape of the extracted object.
Seismic design and construction of retaining wallAhmedEwis13
This document discusses seismic design considerations for retaining walls. It describes the common types of retaining walls, including gravity, cantilever, reinforced soil, and anchored bulkhead walls. Static lateral earth pressures are calculated using Rankine and Coulomb theories, with the Mononobe-Okabe method extending Coulomb theory to account for seismic inertial forces. Dynamic response of retaining walls is complex, with wall movement, pressures, and permanent displacements dependent on the response of the wall, backfill soil, and foundation soil to ground shaking.
The document discusses several topics related to vertical curves and superelevation design for roads, including:
1. Equations for parabolic vertical curves and methods for designing vertical curves to connect lines with different grades, including passing a curve through a fixed point.
2. Minimum length requirements for vertical curves based on sight distance standards from AASHTO to ensure safety.
3. Design of unequal tangent vertical curves where the curves on each side have different lengths.
4. Considerations for pavement cross-slope or "crown" and superelevation rates for horizontal curves based on design speed, road classification, climate and other factors.
The ideal, perfectly regular crystal structures in which atoms are arranged in a regular way does not exist in actual situations. In actual cases, the regular arrangements of atoms disrupted . These disruptions are known as Crystal imperfections or crystal defects
1. The document discusses methods for analyzing slope failures in rock masses, including kinematic analysis using stereographic projections.
2. Kinematic analysis involves plotting the orientation of discontinuities like joints, bedding planes, and faults on a stereonet to determine potential failure mechanisms like planar, wedge, and toppling failures.
3. The main types of failures are analyzed based on the dip and dip direction of discontinuities relative to the slope face. Planar failures occur when discontinuities daylight on the slope face, while wedge and toppling failures have specific conditions related to the intersection of two discontinuities or the dip of discontinuities into the slope.
This document introduces new associated curves called k-principal direction curves and kN slant helices for spatial curves. It defines k-principal direction curves as integral curves of the k-th principle normal vector of the curve. A curve is a kN slant helix if its k-principal direction curve has constant geodesic curvature. The document establishes properties of the Frenet frame and curvature formulas for k-principal direction curves. It explores using these new curves to characterize different types of spatial curves.
Types of armature winding of dc generator manoharpitchai
This document discusses the constructional features of DC machines, including:
- The function of the commutator and brushes is to convert the alternating voltage induced in the armature coils into direct current by means of mechanical rectification.
- Armature windings can be of two types - lap winding or wave winding. Lap winding connects coil ends to consecutive commutator segments, while wave winding connects ends approximately two pole pitches apart.
- The number of parallel paths in the armature is equal to the number of poles for lap winding, and is equal to 2 for wave winding. Understanding armature windings and parallel paths is important for analyzing DC machine performance.
The document summarizes research analyzing 3D stress intensity factors (SIF) for arrays of inner radial lunular or crescentic cracks in thin and thick-walled spherical pressure vessels. Finite element analysis was used to evaluate SIF distributions along crack fronts for various crack configurations, sphere geometries, crack depths, and ellipticities. The results provide insight into how SIF is affected by these parameters to better predict fatigue life and fracture of spherical pressure vessels.
This document discusses plane frame analysis using the finite element method. It begins by introducing the learning objectives of deriving the stiffness matrix for a 2D beam element and analyzing rigid plane frames using the direct stiffness method. It then derives the local stiffness matrix for a generally oriented 2D beam element. The document provides an example problem of analyzing a rigid plane frame with four elements subjected to loads at two nodes. It shows the derivation of the element stiffness matrices, assembly of the global stiffness matrix, application of boundary conditions, and solution for nodal displacements and rotations.
The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
The document discusses the concept of curves of intersection that occur when two solids penetrate or intersect each other. It provides the following key points:
- When two solids intersect, their surfaces meet at a common curve called the curve of intersection. This curve remains common to both solids.
- Curves of intersection show the exact and maximum surface contact between two intersecting solids. They are important when objects need to be joined together with strong, leak-proof joints.
- Several examples of actual intersecting objects from industry are shown, with their curves of intersection indicated.
- Step-by-step solutions are provided for generating curves of intersection between various geometric solids, including cylinders, pr
The document discusses the concept of curves of intersection that occur when two solids penetrate or intersect each other. It provides the following key points:
- When two solids intersect, their surfaces meet at a common curve called the curve of intersection. This curve remains common to both solids.
- Curves of intersection show the exact and maximum surface contact between two intersecting solids. They are important when objects need to be joined together with strong, leak-proof joints.
- Several examples of actual intersecting objects from industry are shown, with their curves of intersection indicated.
- Step-by-step solutions are provided for generating curves of intersection between various geometric solids, including cylinders, pr
This document discusses DC electric machines. It begins by introducing electric machines as devices that continuously convert electrical energy to mechanical energy or vice versa through electromechanical energy conversion. It then describes the basic principles of DC motors and generators, including how motion in a magnetic field induces voltage and how current in a magnetic field produces force. The document provides equations for induced voltage and electromagnetic force. It uses a simple loop generator model to illustrate how alternating current is produced and how a commutator converts it to direct current for a load. Finally, it discusses components and windings of practical DC generators.
1. The document discusses various types of two-dimensional defects in crystalline solids including stacking faults, twinning, and grain boundaries.
2. The construction of a general grain boundary is described in 5 steps: starting with two copies of a lattice, rotating one copy, overlaying the lattices, defining the boundary position and direction, and removing one lattice on both sides.
3. Grain boundaries can be characterized by their tilt and twist and specific rotation angles like 26.57 degrees may result in more stable coincidence site lattice structures due to their geometry.
This presentation discusses deformation bands and kink bands in metals. Deformation bands are irregularly shaped regions of different crystallographic orientation that form in plastically deformed metals due to non-uniform deformation. Kink bands form in hexagonal close packed crystals under compression when slip is difficult. Kink bands accommodate stress by a localized region abruptly tilting into a new orientation, shortening the crystal. Factors like density, modulus, and cohesion influence kink band formation. Both deformation bands and kink bands are common inexperience incompatibilities in crystal structure during plastic deformation.
Circular Motion JEE Advanced Important Questions
A large sphere of radius R is moving with velocity v horizontally. A small sphere ‘B’ of mass m starts sliding
from top of the sphere downwards. Let at point C it looses contact with the large sphere. Let the velocity of
‘B’ with respect to large sphere is u, then which of the following statement follows.
A bead of mass ‘m’ is released from rest at A move along the fixed smooth circular
track as shown in figure. The ratio of magnitudes of centripetal force and normal reaction
by the track on the bead at any point P.
A string is wrapped around a cylinder of radius R. If the cylinder is released from rest,
the velocity of the cylinder after it has moves h distance is.
A point P moves along a circle of radius r with constant speed v. Its angular velocity about any fixed point on
the circle will be.
Find the magnitude and direction of the force acting on a particle of mass m during its motion in the xy plane
according to the law x = a sin t and y = b cos t, where a, b and are constants.
An athlete completes one round of a circular track of radius R in 40 second. What will
be his displacement at the end of 2 minute 20 second.
In the aboveProblem, what is the ratio of the displacement to the distance, when the athlete has covered 3/
4 th of the circular track?
A particle P is moving in a circle of radius ‘r’ with a uniform speed v. C is the centre of the circle and AB is a
diameter. When passing through B the angular velocity of P about A and C are in the ratio.
Two beads A and B of equal mass m are connected by a light inextensible cord. They are constrained to
move on a frictionless ring in vertical plane. The blocks are released from rest as shown in figure. The tension
in the cord just after the release is.
A plastic circular disc of radius R is placed on a thin oil film, spread over a flat horizontal surface. The torque
required to spin the disc about its central vertical axis with a constant angular velocity is proportional to.
Inside a hollow uniform sphere of mass M, a uniform rod of length R 2 is released from
the state of rest. The mass of the rod is same as that of the sphere. If the inner radius of the
hollow sphere is R then find out its horizontal displacement of sphere with respect to earth
in the time in which the rod becomes horizontal.
A circular uniform hoop of mass m and radius R rests flat on a horizontal frictionless surface. A bullet, also of
mass m and moving with a velocity v, strikes the hoop and gets embedded in it. The thickness of the hoop is
much smaller than R. The angular velocity with which the system rotates after the bullet strikes the hoop is.
The document discusses reinforcement in two-way slabs and footing design. It describes two types of shear failure in slabs: one-way shear and two-way shear. One-way shear results in inclined cracking and pull-out of negative reinforcement from the slab. Two-way shear can result in either inclined cracking or the slab sliding down the column. The critical perimeter for two-way shear is located at d/2 from the column face, where d is the effective depth of the slab. Formulas are provided to calculate the nominal shear resistance Vn of slabs under two-way shear with negligible moment transfer.
The document discusses calculating and interpreting the gradient of a straight line segment.
1. The gradient of a line segment is a measure of how steep the line is, and can be calculated as the change in y over the change in x between two points on the line.
2. Parallel lines have the same gradient, while perpendicular lines have gradients that are negative reciprocals of each other.
3. Examples are provided to demonstrate calculating gradients of lines and using gradients to determine whether lines are parallel or perpendicular.
Question-no.docx
Chapter7
Question no’s: 2,3,4,5,6,8,10,13,14,15,17,18,19,20,21,27,28,29,31,32,33,36
Chapter 8
Question no’s: 1,2,3,4,6,7,9,13,14,15,19,20,21,22,24,26,28,29,30
ch7.pdf
Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge
7.13 Problems
7.1 Finding Charge From Potential
The potential in a spherical region r < R is '(x, y, z) = '0(z/R)
3. Find a volume charge density
Ω(r, µ) in the region r < R and a surface charge density æ(µ) on the surface r = R which together
produce this potential. Express your answers in terms of elementary trigonometric functions.
7.2 A Periodic Array of Charged Rings
Let the z-axis be the symmetry axis for an infinite number of identical rings, each with charge
Q and radius R. There is one ring in each of the planes z = 0, z = ±b, z = ±2b, etc. Exploit
the Fourier expansion in Example 1.6 to find the potential everywhere in space. Check that your
solution makes sense in the limit that the cylindrical variable Ω ¿ R, b. Hint: If IÆ(y) and KÆ are
modified Bessel functions,
I
0
Æ(y)KÆ(y) ° IÆ(y)K0Æ(y) = 1/y.
7.3 Two Electrostatic Theorems
Use the orthogonality properties of the spherical harmonics to prove the following for a function
'(r) which satisfies Laplace’s equation in and on an origin-centered spherical surface S of radius
R:
(a)
R
S
dS '(r) = 4ºR2'(0)
(b)
Z
S
dSz'(r) =
4º
3
R
4 @'
@z
ØØØØ
r=0
7.4 Make a Field Inside a Sphere
Find the volume charge density Ω and surface charge density æ which much be placed in and on a
sphere of radius R to produce a field inside the sphere of
E = °2V0
xy
R3
x̂ +
V0
R3
(y
2 ° x2)ŷ ° V0
R
ẑ.
There is no other charge anywhere. Express your answer in terms of trigonometric functions of µ
and ¡.
7.5 Green’s Formula
Let n̂ be the normal to an equipotential surface at a point P . If R1 and R2 are the principal
radii of curvature of the surface at P . A formula due to George Green relates normal derivatives
(@/@n ¥ n̂ · r) of the potential '(r) (which satisfies Laplace’s equation) at the equipotential surface
to the mean curvature of that equipotential surface ∑ = 1
2
(R°11 + R
°1
2 ):
@2'
@n2
+ 2∑
@'
@n
= 0.
Derive Green’s equation by direct manipulation of Laplace’s equation.
7.6 The Channeltron
c∞2009 Andrew Zangwill 278
Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge
The parallel plates of a channeltron are segmented into conducting strips of width b so the po-
tential can be fixed on the strips at staggered values. We model this using infinite-area plates, a
finite portion of which is shown below. Find the potential '(x, y) between the plates and sketch
representative field lines and equipotentials. Note the orientation of the x and y axes.
1 1
02 2
x
y
d
b
7.7 The Calculable Capacitor
The figure below shows a circle which has been divided into two pairs of segments with equal
arc length by a horizontal bisector and a vertical line. The positive x-axis bisects the segment
labe ...
This document presents a mathematical analysis of the vibration response of the vertical support members (columns) of a portal frame bridge subjected to a moving concentrated load. The analysis involves:
1) Constructing free body diagrams of the portal frame and its members to determine the forces and moments acting on them due to the moving load.
2) Performing a quantitative force analysis to derive equations representing the time-varying forces acting on the two vertical support columns as the load moves across the bridge.
3) Discussing different approaches to model the vibration response of the columns considering distributed/lumped mass-spring-damper systems and different loading patterns including random, harmonic, and single load cases.
The
Point defects like vacancies and interstitials, line defects like dislocations, and area defects like grain boundaries arise in solids. Defects can affect material properties and their concentration and type can be controlled. For example, temperature controls the equilibrium vacancy concentration and grain boundaries impede dislocation motion, influencing plastic deformation. Defects may be desirable or undesirable depending on the specific material property and application.
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The document discusses inventory management and logistics costs in global sourcing processes. It provides insights into global sourcing strategies used by multinational corporations. Some key challenges to global sourcing mentioned include sustainability concerns, developing expertise in local emerging markets, and developing the necessary human resource capabilities to support global sourcing efforts. Factors influencing global sourcing decisions include product characteristics, organizational characteristics, and country characteristics. The total cost of ownership model can help evaluate sourcing and logistics strategies based on these factors.
Hershey's 1999 implementation of an ERP system was not successful and led to financial losses. The ERP go-live was delayed by 3 months until July 1999 due to integration issues. This caused delivery delays of up to 15 days, loss of credibility with suppliers who found alternatives, and a 25% increase in excess inventory. The key reasons for failure included an unrealistic schedule with no buffer for testing, implementing during the peak season, integrating 3 different vendor systems, lack of IT leadership, insufficient preparation, and overloading employees during training. Lessons included allowing adequate time for implementation, thorough testing and integration, avoiding peak seasons, and establishing proper IT management.
The document provides an overview of the Indian agriculture and tractor industry. It notes that India has the highest arable land globally at 47% of total land. The average farm size is small at less than 2 hectares. There is a shortage of agricultural labor due to increasing wages and urbanization, increasing the need for mechanization. Tractors occupy nearly 90% of the Indian agricultural machinery market, which is expected to grow steadily due to factors like low tractor penetration, availability of credits, and opportunities in new markets. The largest players in the industry are Mahindra, TAFE, Escorts and ITL.
This document provides an overview of operations management concepts related to inventory control. It begins with an outline of topics to be covered, including inventory control, material requirements planning, aggregate planning, scheduling, quality management, and lean manufacturing. The document then focuses on inventory, discussing the goals of inventory management, different types of inventory costs, and inventory classification systems. It also introduces concepts like economic order quantity, reorder points, safety stock, and how to determine when to reorder.
(1) A group of hikers encountered a sick Indian sadhu while trekking in Nepal and each person helped in small ways but no one took full responsibility for ensuring his recovery. (2) They justified leaving him due to their own goals and stresses, though one member argued they failed their ethical duty. (3) The author later realized they ignored an ethical dilemma and should have acted heroically rather than just permissibly.
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The document describes an existing manual sales order system and a proposed automated system using ERP software. The proposed system allows: 1) automatic conversion of sales quotes to orders; 2) more accurate estimated delivery dates using order promising; 3) combined invoicing for quantity-based discounts; and 4) automated picking and shipping tied to bin locations. This streamlines the sales order process.
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Peter Loscher takes over as CEO of Siemens in 2007 during a turbulent time with compliance issues. He implements structural changes including dismantling the complex hierarchy and centralized decision making structure. The changes establish three main sectors each led by a CEO, simplify financial reporting to four main categories, and develop a "right of way" model to better connect regional and global operations. However, tensions remain between regional and global priorities and disconnect with employees unaccustomed to an outsider as CEO. Suggested solutions include further streamlining bureaucracy, re-educating employees, and focusing products on sustainability megatrends.
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Strategy, not tactics, should drive aggregate planning. The research found that most manufacturers follow a chase or modified chase strategy to deal with seasonal demand fluctuations. While a mixed strategy is theoretically cheaper, no firms in the study adopted it. The research also found that aggregate planning is often flawed and not truly strategy-driven. Manufacturers need to select the appropriate production strategy to tackle seasonal demands for long-term viability.
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3) Higher display resolutions like 720p and 1080p were preferred over lower resolutions like 480p.
4) A regression equation was developed to predict preferences based on feature levels.
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The document discusses the martensitic transformation that occurs in some solids. It describes how martensite forms through a diffusionless, shear-based mechanism below a critical temperature called Ms. The transformation results in a crystal structure change from face-centered cubic to body-centered cubic or hexagonal close-packed. Martensite plates nucleate and grow rapidly through the parent austenite grains. The Bain correspondence is described as a model where the austenite lattice distorts into the martensite lattice through homogeneous expansion and contraction, maintaining the positions of neighboring atoms.
3. Shear transformation
Exp. Martensite can be generated by shear on γ
Both shears are possible
and identical to Bain
distortion if disregarded
the rigid body rotation. 3
4. Shear transformation
Shear of cooperative movements of atoms can
be in different planes rather than (111)γ plane,
depending on alloy composition and
transformation temp.
Shear does not have to act along the same
direction on every parallel atomic plane.
4
5. Shear transformation
Greninger and Troiano (1949) found that
Observed shear plane in Fe-22% Ni-0.8% C
was not the {111}γ plane and the shear angle
was 10.45°, not 19.5° as predicted by shear
mechanism.
Theysuggested that another shear had to be
added in order to complete the mechanism.
5
6. Double shear transformation
The first shear isa macroscopic shear
that contributes the shape change and
change in crystal structure.
The second shear is a microscopic shear.
Invariant plane
Bain distortion has no invariant plane
Lattice-invariant shear with Bain distortion
6
7. Invariant plane
During the martensitic transformation
The interface should be an invariant plane
Undistorted and unrotated plane
Any deformation on the invariant plane will
be termed an invariant plane strain.
7
8. From the Bain distortion
α lattice with bcc can be generated from
an fcc γ lattice by
Compression about 20% along
one principle axis and
a simultaneous uniform
expansion about 12% along
the other two axes perpendicular to
the first principle axis
8
9. Bain distortion of a sphere
Due to the Bain distortion
A unit sphere of the parent crystal
transforms into an oblate spheroid of the
product crystal
Contraction about 20% along the one
principle axis
Expansion about 12% along the other
two axes perpendicular to the first
principle axis
9
11. Bain distortion of a sphere
Due to the lattice deformation x12 + x2 + x3 = 1
2 2
Vectors OA’ and OB’
represent the final
position of vectors
Vectors OA and OB
represent the initial
position of the same
vectors
unchanged in
( x1 ) + ( x2 ) + ( x3 ) = 1
' 2 ' 2 ' 2
( 1.12 ) ( 1.12 ) ( 11 )
2 2 2
magnitude 0.80
12. Bain distortion of a sphere
Vectors unchanged in magnitude during
the lattice deformation
Corresponding to the
cones AOB and COD and
the cones A’OB’ and C’OD’
These vectors are termed
unextended lines.
A homogeneous strain would
result in an undistorted plane
of contact between the initial sphere of
austenite and the ellipsoid of martensite.
12
13. Bain distortion of a sphere
Allother vectors not involved in the cones
A’OB’ and C’OD’ would be
changed in magnitude.
Bain distortion would result
in no undistorted plane.
Hence, there is no invariant plane.
Very difficult to obtain a coherent planar
interface between the parent and the product
crystals only by the Bain distortion. 13
14. Bain distortion of a sphere
Therefore,Bain distortion
has no invariant plane.
14
15. Lattice-invariant shear
Lattice-invariant shear
must be of such
magnitude so as to produce
an undistorted plane
when combined with
the Bain distortion.
Consider slip or twinning
Must not make any
change in crystal structure.
15
16. Lattice-invariant shear
Graphical analysis of a simple shear of
slip or twinning of a unit sphere
Shear on an equatorial
plane K1 as the shear plane
d as the shear direction
α as shear angle Slip
16
17. Lattice-invariant shear
As a result of shear on K1
Any vector in the plane AK B is
2
transformed into a vector in
the plane AK’2B, which is
unchanged with length
although rotated relatively
to its original position.
The plane AK B is the initial Slip
2
position of a plane AK’2B,
which remains undistorted as
a result of the shear. 17
18. Lattice-invariant shear
As a result ofshear on K1
The relative positions of
the planes AK2B and AK’2B
depend on the amount of
shear involved.
The shear plane itself
remains undistorted
after shear. Slip
Vectors that remain invariant in length
(unextended lines) to this shear operation
are define as potential habit planes. 18
19. Lattice-invariant shear
As a result ofshear on K1
The relative positions of
the planes AK2B and AK’2B
depend on the amount of
shear involved.
The shear plane itself
remains undistorted
after shear. Slip
Vectors that remain invariant in length
(unextended lines) to this shear operation
are define as potential habit planes. 19
20. Lattice-invariant shear
When initial sphere → ellipsoid
by lattice deformation using
Bain distortion is distorted by
simple shear into another ellipsoid
+
and the lattice is left invariant,
The simple shear is termed
a lattice-invariant shear.
shear
20
22. Stereographic representation
of the Bain distortion
Any vector lying on the initial
cone AOB with a semiapex
of φ moves radially onto the
final cone A’OB’ with a
semiapex of φ’.
Vectors in the cones of
unextended lines do not
change their length,
but only the angle ∆φ.
22
23. Stereographic representation
of the lattice-invariant shear
An unextended line C
moves to the final position
along the circumference
of the great circle
defined by d*
(dash line).
23
24. Stereographic representation
of the lattice-invariant shear
Vectors in K’2 plane do not
change their length due to
shear, and the line OC’ in
the plane represents the final
position of an unextended line.
Line OC in K2 plane
represents the
initial position
of OC’.
24
25. Requirement for habit plane
Both Bain distortion and lattice
invariant shear provide an undistorted
plane for the habit plane.
Additional requirement is that the habit
plane be unrotated.
A rigid body rotation must be able to
return the undistorted plane to its
original position before
transformation.
25
27. Bain distortion with slip #1
Vectors b and c are defined
by the intersections of the
initial Bain cone with K1 plane
1.Apply a complementary shear
Vectors b and c become b’ and c’ and still lie
in the K1 plane and remain unchanged in
both direction and magnitude.
They are invariant lines.
27
28. Bain distortion with slip #1
Vectors b and c are defined
by the intersections of the
initial Bain cone with K1 plane
2.Apply a Bain distortion
Vectors b’ and c’ become b’’ and c’’ lie on the
initial and final Bain cones, respectively,
without changing their magnitude.
28
29. Bain distortion with slip #1
Complementary shear
b and c to b’ and c’
Bain distortion
b’ and c’ to b’’ and c’’
Angle btw b and c ≠ angle btw b” and c”
Appropriate rotation cannot be applied to
return b” and c” to initial positions of b and c.
Plane defined by b and c cannot be an invariant
plane. 29
30. Bain distortion with slip #2
To obtainan invariant plane,
must have other extended lines
Ifassumed to know
the shear angle α,
vectors a and d obtained from the intersections
of the K2 plane change to a’ and d’ along the
great circles.
Bain distortion,
vectors a’ and d’ become a” and d”, respectively
30
31. Bain distortion with slip #2
Through the transformation of
the complementary shear and
the Bain distortion
Sequences of a→a’→a”
and sequences d→d’→d”
reveal no change in length
However, angle btw a & d ≠ angle btw a” & d”
Plane defined by a and d cannot be an invariant
plane. 31
32. Complete transformation
process
Possible invariant planes will
depend on the choice of
combination of b or c
with a or d such as
Vectors a and b
Vectors a and c
Vectors b and d
Vectors c and d
32
33. Complete transformation
process
If theinvariant plane is the
plane defined by vectors a & c
Angle btw a & c = angle btw a’’ & c’’
Let the axis required for rotation
be at point u
Determine the amount of rotation
stereographically by intersection
of a great circle bisecting a-a”
with another great circle bisecting c-c” 33
34. Complete transformation
process
Once a” and c” coincide simultaneously
with a and c, respectively
Angle btw a & c = angle btw a’’ & c’’
Therefore, orientation relationship btw γ plane
(defined by the vectors a and c) and α’ plane
(defined by the vectors a” and c”) can be
determined for a specific variant of the Bain
distortion (B), lattice invariant shear (P), and
rotation operation (R).
T = BPR
34
36. Bain distortion with twinning
Twinned martensite can take place by having
alternate regions in the parent phase undergo
the lattice deformation along different
contraction axes, which are initially at right
angles to each other.
In the first region, contraction occurs along
the x3 [ 001] f axis.
In the adjacent region, contraction direction
can be either x1 [100] f or x2 [ 010] f axis.
Two rigid body rotations are also involved in
the twinning analysis. 36
37. Nucleation and growth
It only takes about 10-5 to 10-7 seconds for a plate
of martensite to grow to its full size.
The nucleation during the martensitic
transformation is extremely difficult to study
experimentally.
Average number of martensite is as large as 104
nuclei/mm3
Number of martensite nuclei can be
increased by increasing ∆T prior to Ms.
It is too small in term of number of
nucleation sites for homogeneous nucleation.
37
38. Nucleation and growth
Less likely to occur by homogeneous nucleation
process, but heterogeneous.
Surfaces and grain boundaries are not
significantly contributing to nucleation.
Most likely types of defect that could produce
the observed density of martensite nuclei are
dislocations (> 105 dislocation/mm2).
C. Zener (1948): movement of partial dislocations
during twinning could generate a thin bcc region
of lattice from an fcc region.
38
39. Nucleation and growth
Dissociation of a dislocation
into 2 partials is favorable
→ lower strain energy.
r r r
To generate b1 = b2 + b3
bcc structure, a a a
[ 110] = [ 211] + 121
the requirements are that all 2 6 6
green atoms move (shear)
a
forward by 12 [ 211] and an
additional dilatation
to correct lattice spacings. 39
40. Nucleation and growth
Growth of lath martensite with dimension
a > b >> c growing on a {111}γ planes
Thickening mechanism would involve the
nucleation and glide of transformation
dislocations moving on discrete ledges
behind the growing front.
Due to large misfit between
bct and fcc lattice,
dislocations could be
self-nucleated at the
lath interface as the lath moves forward.
40
41. Nucleation and growth
In medium and high carbon steels,
Morphology of martensite turns to change
from a lath to a plate-like shape.
As carbon concentration decreases,
Decrease lath structure
Decrease martensitic temperature
Increase twinning
Increase retained austenite
Depending on compositions, the habit plane
changes from {111}γ → {225}γ → {259}γ
41
42. Effect of pressure to martensite
As pressure increases
In Fe unary system, the equilibrium
temperature decreases
In Fe-C binary system, the phase region
around γ phase shifts to the left and
downward.
Similar to adding austenite stabilizer
42
43. Effect of alloying element to
martensite
Each alloying element will effect the martensitic
transformation differently.
If initially Hγ = Hα
When adding C
The ē of C will decrease Hα and cause α to
be less stable.
∆H = Hγ – Hα > 0, stabilize the γ
When adding X
Increase Hα and ∆H < 0, stabilize the α
43
44. Effect of external stress to
martensite
As martensite prefers to nucleate and grow
along the dislocation
Expected that an externally applied shear
stress will assist and accelerate the
generation of dislocations and hence the
growth of martensite.
An external shear stress can aid martensite
nucleation if the external elastic strain
components play as a part of the Bain strain.
This can also help by raising the M
s
temperature. 44
45. Effect of external stress to
martensite
Once the plastic deformation occurs
There is an upper limit value of M that the
s
stress can be applied.
The limit temp. of M is called M (highest
s d
temperature that stress helps to form
martensite)
Too much plastic deformation will
suppress the transformation.
45
46. Effect of external stress to
martensite
If a tensile
stress is applied
M temperature can be suppressed to lower
s
temperature
Transformation may be reversed from α’ →
γ
Presence of large magnetic field may favor the
formation of the ferromagnetic phase and
therefore raise Ms temp.
46
47. Effect of external stress to
martensite
Plastic deformation of γ before transformation
will assist on increasing number of nucleation
sites.
Once the transformation occurs
Result in very fine plate size of martensite
(Called the ausforming process)
Combined effect of very fine martensite plates,
1
2
solution hardening of carbon, and 3dislocation
hardening
Very high strength ausformed steel 47
48. Shape-memory alloys (SMA)
Unique property of some alloys
After being deformed at one temperature,
they recover the original undeformed shape
when heated to a higher temperature.
48
49. Shape-memory alloys (SMA)
Unique property of some alloys
After being deformed at one temperature,
they recover the original undeformed shape
when heated to a higher temperature.
Fundamental to the shape-memory effect
(SME) is the occurrence of a martensitic phase
transformation and its subsequent reversal.
Alloys: Ni-Ti (called NiTiNOL), Ni-Al, Fe-Pt,
Cu-Al-Ni, Cu-Au-Zn, Cu-Zn-(Al,Ga,Sn,Si),
Ni-Mn-Ga 49
50. SMA
Common characteristics
Atomicordering transformation from
ordered parent phase to ordered martensite
phase
Thermoelastic martensitic transformation
that is crystallographic reversible
Martensite
phase that forms in a self-
accommodating manner (slip or twinning)
50
51. SMA
Typical plot of property changes versus temp.
A hysteresis is usually on the order of 20°C
51
52. One-way SMA
Sample is cooled from above Af to
below Mf → martensite forms
Sample has no shape change
Sample is deformed below Mf
Sample remains deformed
until heated.
Begin shape recovery at A and complete at A
s f
No shape change when cooled below Mf
Deforming the 52
martensite again will reactivate SME
53. Two-way SMA
Sample is cooled from above Af to
below Mf → martensite forms
Sample has no shape change
Sample is deformed below Mf
Sample remains deformed
until heated.
Begin shape recovery at A and complete at A
s f
Returnsto the deformed shape when cooled
below Mf 53