This document provides an overview of crystal plasticity finite element modelling (CPFE). It begins with a recap of important concepts in crystal plasticity including crystalline structures, plasticity occurring through dislocation glide, slip systems, and factors like Schmid factor and critically resolved shear stress that determine when slip occurs. It then discusses why CPFE is needed to model plastic deformation at the crystal level since continuum models do not consider dislocation slip. The key aspects CPFE aims to model are then outlined, including resolving loads onto slip systems, calculating slip and resulting strains, lattice rotation, and dislocation density evolution. Constitutive laws for calculating slip rates are also briefly introduced.
Material remains intact
Original crystal structure is not destroyed
Crystal distortion is extremely localized
Possible mechanisms:
Translational glide (slipping)
Twin glide (twinning)
Sheet metal formability refers to a material's ability to undergo shaping without failures like necking or tearing. Three main factors influence formability: 1) the metal's properties, 2) friction levels during forming, and 3) the equipment used. Common tests to evaluate formability include cupping, tension, bulge, and forming limit diagrams. The earliest test was the Erichsen cupping test which measures the depth a ball can press into a clamped sheet before cracking. The bulge test applies hydraulic pressure to a clamped circular blank to measure its limit before failure from biaxial stretching. Forming limit diagrams map major and minor strains measured from grids printed on sheets after stretching tests.
The document discusses dislocation theory and behavior in different crystal structures. It covers:
- Observation techniques for dislocations like etching and transmission electron microscopy
- Key concepts like Burgers vector, dislocation loops, and dissociation of dislocations into partial dislocations
- Differences in dislocation behavior in FCC, BCC, and HCP lattices including slip systems and interactions between dislocations
- Stress fields and strain energies of dislocations as well as forces acting on dislocations and between dislocations
- Mechanisms of dislocation motion including glide, cross-slip, and climb that enable plastic deformation.
This document discusses toughness and fracture toughness testing. It defines toughness as the energy absorbed by a material until fracture. Common toughness tests include the Charpy and Izod impact tests, which measure the energy absorbed during a high-velocity impact. However, these tests do not provide data needed for designing with cracks and flaws. Fracture toughness is a better property for design, as it indicates the stress required to propagate a preexisting flaw. The document outlines fracture toughness testing methods like compact tension and single edge notch bending specimens. It also discusses factors that influence fracture toughness values like material thickness, grain orientation, and plane strain versus plane stress conditions.
This document discusses various surface coating methods used to improve wear and corrosion resistance of materials. It provides details on several coating techniques including thermal spraying methods like flame spraying, plasma spraying and HVOF. The key points are:
1) Different coating methods like thermal spraying, vapor deposition, mechanical cladding are used to improve surface properties.
2) Thermal spraying techniques like flame spraying, plasma spraying and HVOF are described in detail along with the coating materials, temperatures involved and applications.
3) Characteristics of different coatings like hardness, porosity and adhesion strength obtained from various spraying methods are summarized in tables for comparison.
Measurement of residual stresses in weldmentsN.Prakasan
The document discusses various techniques for measuring residual stresses in weldments. There are two main categories of techniques - stress relaxation techniques and diffraction techniques. Stress relaxation techniques like sectioning, drilling and hole-drilling determine residual stress by measuring strain release when material is removed. Diffraction techniques like X-ray and neutron diffraction measure strain in crystal lattices to determine stress. Ultrasonic techniques also measure wave velocity changes through materials under stress. Each technique has advantages and limitations for measuring surface, subsurface or internal residual stresses in welded structures.
Fatigue is the progressive and localized structural damage that occurs when a material is subjected to fluctuating stresses that are less than the static yield strength of the material. It accounts for about 90% of industrial failures. Fatigue occurs in five stages: cyclic plastic deformation, crack initiation, crack propagation, propagation of macro cracks, and final fracture. It is characterized by beach marks and a rough, brittle fracture surface. Fatigue life can be represented using an S-N curve which plots the maximum stress versus the number of cycles to failure. The fatigue limit or endurance limit is identified as the stress below which a material can undergo any number of stress cycles without failure.
Material remains intact
Original crystal structure is not destroyed
Crystal distortion is extremely localized
Possible mechanisms:
Translational glide (slipping)
Twin glide (twinning)
Sheet metal formability refers to a material's ability to undergo shaping without failures like necking or tearing. Three main factors influence formability: 1) the metal's properties, 2) friction levels during forming, and 3) the equipment used. Common tests to evaluate formability include cupping, tension, bulge, and forming limit diagrams. The earliest test was the Erichsen cupping test which measures the depth a ball can press into a clamped sheet before cracking. The bulge test applies hydraulic pressure to a clamped circular blank to measure its limit before failure from biaxial stretching. Forming limit diagrams map major and minor strains measured from grids printed on sheets after stretching tests.
The document discusses dislocation theory and behavior in different crystal structures. It covers:
- Observation techniques for dislocations like etching and transmission electron microscopy
- Key concepts like Burgers vector, dislocation loops, and dissociation of dislocations into partial dislocations
- Differences in dislocation behavior in FCC, BCC, and HCP lattices including slip systems and interactions between dislocations
- Stress fields and strain energies of dislocations as well as forces acting on dislocations and between dislocations
- Mechanisms of dislocation motion including glide, cross-slip, and climb that enable plastic deformation.
This document discusses toughness and fracture toughness testing. It defines toughness as the energy absorbed by a material until fracture. Common toughness tests include the Charpy and Izod impact tests, which measure the energy absorbed during a high-velocity impact. However, these tests do not provide data needed for designing with cracks and flaws. Fracture toughness is a better property for design, as it indicates the stress required to propagate a preexisting flaw. The document outlines fracture toughness testing methods like compact tension and single edge notch bending specimens. It also discusses factors that influence fracture toughness values like material thickness, grain orientation, and plane strain versus plane stress conditions.
This document discusses various surface coating methods used to improve wear and corrosion resistance of materials. It provides details on several coating techniques including thermal spraying methods like flame spraying, plasma spraying and HVOF. The key points are:
1) Different coating methods like thermal spraying, vapor deposition, mechanical cladding are used to improve surface properties.
2) Thermal spraying techniques like flame spraying, plasma spraying and HVOF are described in detail along with the coating materials, temperatures involved and applications.
3) Characteristics of different coatings like hardness, porosity and adhesion strength obtained from various spraying methods are summarized in tables for comparison.
Measurement of residual stresses in weldmentsN.Prakasan
The document discusses various techniques for measuring residual stresses in weldments. There are two main categories of techniques - stress relaxation techniques and diffraction techniques. Stress relaxation techniques like sectioning, drilling and hole-drilling determine residual stress by measuring strain release when material is removed. Diffraction techniques like X-ray and neutron diffraction measure strain in crystal lattices to determine stress. Ultrasonic techniques also measure wave velocity changes through materials under stress. Each technique has advantages and limitations for measuring surface, subsurface or internal residual stresses in welded structures.
Fatigue is the progressive and localized structural damage that occurs when a material is subjected to fluctuating stresses that are less than the static yield strength of the material. It accounts for about 90% of industrial failures. Fatigue occurs in five stages: cyclic plastic deformation, crack initiation, crack propagation, propagation of macro cracks, and final fracture. It is characterized by beach marks and a rough, brittle fracture surface. Fatigue life can be represented using an S-N curve which plots the maximum stress versus the number of cycles to failure. The fatigue limit or endurance limit is identified as the stress below which a material can undergo any number of stress cycles without failure.
Presentation on Carburizing (Heat Treatment Process).
Presented To,
Engr. Ubaid-ur-Rehman Ghouri, Department of Industrial & Manufacturing Engineering, UET Lahore (RCET Campus).
Presented By,
Muhammad Zeeshan
Zahid Mehmood
Ali Iqbal
Muhammad Waqas
The document discusses plasticity theory and yield criteria. It introduces Hooke's law and its limitations under large strains. Generalized Hooke's law is presented for isotropic and anisotropic materials. Common stress-strain curves are shown including elastic-plastic and strain hardening responses. Several yield criteria are covered, including maximum principal stress, Tresca, and von Mises criteria. The von Mises criterion uses a second invariant of stress to predict yielding of ductile materials. An example compares predictions of yielding using Tresca and von Mises criteria for a given stress state in aluminum.
This document discusses friction, friction measurement techniques, wear, and wear measurement techniques. It describes four common methods for measuring friction: the weight ratio method, spring balance method, tilted plane method, and clamping method. It also outlines six wear measurement techniques: the dry sand rubber wheel method, pin on drum method, linear tribo machine method, block on ring method, pin on disc method, and block on disc method. The key applications and standards for each technique are provided.
This document provides an overview of manufacturing processes and riser design concepts. It discusses solidification of castings, functions of risers, types of risers, and methods for riser design including the Chvorinov rule, modulus method, and NRL method. Examples are provided to demonstrate how to calculate riser dimensions using these methods based on properties of the casting such as volume, surface area, and solidification time.
The document discusses various aspects of solidification processes for pure metals and alloys. It covers topics such as solidification curves, grain structure formation, mushy zone formation in alloys, segregation of elements, shrinkage during solidification, and directional solidification techniques. It also discusses the functions and design of gating systems, including elements like pouring basins, sprues, runners, gates, and risers.
This presentation is by Flt Lt Dinesh Gupta, Associate Professor (Mechanical Engineering) NIET, Alwar (Rajasthan). It covers topic on Fluctuating Stresses related to Machine Design subject.
The document discusses fatigue failure and fatigue analysis. It begins by explaining that fatigue failure starts with a crack, usually at a stress concentration, which then propagates until sudden fracture. It then provides examples of fatigue failures and discusses different fatigue analysis methods. The key points are:
- Fatigue failure results from repeated or fluctuating stresses that are lower than the material's ultimate strength.
- It can be analyzed using stress-life, strain-life, or fracture mechanics methods, with stress-life most common for high-cycle fatigue.
- The stress-life approach estimates fatigue strength (Sf) based on stress levels and uses modifying factors to account for real-world differences from test specimens.
Dispersion Hardening:
Hard particles:
Mixed with matrix powder
Consolidated
Processed by powder metallurgy techniques
Second phase – Very little solubility (Even at elevated temp.)
No coherency
So thermally Stable at very high temp.
Resists :
Grain growth
Over aging
Recrystallization
Mobility of dislocation
Different from particle Metallic Composites (Volume Fraction is 3 to 4% max.) (Does not affect stiffness)
Examples : Al2O3 in Al or Cu, ThO2 in Ni
This document provides an introduction to fracture mechanics. It discusses different types of brittle and ductile fracture, modes of failure, energy release rate and crack resistance, crack growth, stress intensity factor, and the J-integral. It also mentions a case study on liberty ship failures and provides references for further reading. The key topics covered are the assumptions of fracture mechanics, using energy-based approaches like compliance and strain energy to analyze crack growth, and stress intensity factors which characterize how potent a crack is under different loading conditions.
This document summarizes the metal forming process of rolling. It describes how rolling works by passing metal between rolls, subjecting it to compressive and shear stresses. It discusses different types of rolling mills and explains how hot and cold rolling differ, with hot rolling reducing size at high temperatures and cold rolling providing better surface finish. The document also outlines defects that can occur during rolling such as surface irregularities, inclusions, and edge cracking or center splitting.
Creep is the time-dependent deformation of a material under constant load at high temperatures. It occurs when a material is loaded below its yield strength and the load is maintained for an extended period of time, resulting in plastic deformation that increases over time. Creep can cause failure through rupture or excessive plastic deformation beyond a certain limit. The rate of creep deformation and time to rupture depend on factors like temperature, stress, and material microstructure. Creep becomes significant engineering issue at temperatures over 40% of the material's melting point.
- Dendritic crystal growth occurs when a liquid-solid interface moves into a supercooled liquid. Heat is removed from the interface into both the solid and liquid.
- Undercooling of the liquid allows the formation of spikes at the interface that grow faster than the surrounding interface. This leads to the formation of branched, tree-like dendritic structures.
- Secondary and tertiary branches can form from primary branches/spikes. The branching occurs due to temperature gradients that arise from the release of heat at the interface.
This document discusses metal forming processes. It defines forming and shaping, and provides examples of each. Metal forming involves plastic deformation of material under large external forces to change its shape. The document classifies metal forming processes as cold working, hot working, or warm working based on the temperature of the material. It also discusses properties important for metal forming like ductility and strength. Rolling, forging, extrusion, drawing, and press working are provided as examples of metal forming processes.
This document discusses surface topography and its importance in engineering applications. It defines surface topography as the small deviations of a surface from being perfectly flat, including roughness and waviness. Surface finish is important as it influences functions like lubrication, wear resistance, and friction. The document discusses different measurement methods for surface roughness like profilometry and atomic force microscopy. It defines parameters like arithmetic mean value and root mean square average that are used to analyze measured surface roughness data. The importance of considering factors like required precision and cost when specifying a surface roughness for an engineering application is also highlighted.
- Impact tests are used to determine a material's impact energy, toughness, and tendency to fracture in a brittle manner. They are important for selecting materials that may experience sudden loading like collisions.
- Common impact tests include the Charpy and Izod tests, which involve striking a notched sample with a falling pendulum. The Charpy test uses a simply supported beam setup while the Izod uses a cantilever.
- Factors like yield strength, ductility, temperature, and strain rate can influence a material's impact performance and whether it fractures in a brittle or ductile manner. Many materials exhibit a ductile to brittle transition around a specific temperature.
This document discusses various methods for producing metal powders, including mechanical, atomization, electrochemical, and chemical methods. Mechanical methods include chopping, abrasion, milling and the cold stream process. Atomization methods include gas, water, centrifugal atomization and using a rotating electrode. Factors that influence particle size and shape from atomization are also covered. Electrochemical production involves electrolysis of molten metals. Chemical methods decompose metal compounds with heat or catalysts. The document provides details on the principles, equipment used, advantages and limitations of each production method.
Types of stresses and theories of failure (machine design & industrial drafti...Digvijaysinh Gohil
This document summarizes different types of stresses and theories of failure in mechanical components. It discusses eight types of stresses: tensile, compressive, bending, direct shear, torsional shear, bearing pressure, crushing, and contact stresses. It then explains three main theories of failure - maximum principal stress theory, maximum shear stress theory, and distortion energy theory - and their applications based on the material properties.
Powder metallurgy is a process that involves producing metal powders and compacting and sintering them to form finished parts. It allows for complex alloy compositions and near-net shape manufacturing, avoiding costly machining. The key steps are powder production, blending/mixing, compaction into a green compact, sintering to bond particles, and optional finishing. It offers advantages over casting and machining for net shape precision parts in large volumes.
This document discusses dislocations and mechanisms of plastic deformation in metals. It explains that plastic deformation occurs due to the movement of dislocations in crystals. Dislocations can move through slip and climb, leading to cumulative plastic deformation. Their movement is influenced by interactions with other dislocations and defects. Two main mechanisms of plastic deformation in metals are slip and twinning. Slip involves the sliding of crystal blocks along slip planes and directions, while twinning involves a symmetrical rearrangement of a crystal portion.
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.
Presentation on Carburizing (Heat Treatment Process).
Presented To,
Engr. Ubaid-ur-Rehman Ghouri, Department of Industrial & Manufacturing Engineering, UET Lahore (RCET Campus).
Presented By,
Muhammad Zeeshan
Zahid Mehmood
Ali Iqbal
Muhammad Waqas
The document discusses plasticity theory and yield criteria. It introduces Hooke's law and its limitations under large strains. Generalized Hooke's law is presented for isotropic and anisotropic materials. Common stress-strain curves are shown including elastic-plastic and strain hardening responses. Several yield criteria are covered, including maximum principal stress, Tresca, and von Mises criteria. The von Mises criterion uses a second invariant of stress to predict yielding of ductile materials. An example compares predictions of yielding using Tresca and von Mises criteria for a given stress state in aluminum.
This document discusses friction, friction measurement techniques, wear, and wear measurement techniques. It describes four common methods for measuring friction: the weight ratio method, spring balance method, tilted plane method, and clamping method. It also outlines six wear measurement techniques: the dry sand rubber wheel method, pin on drum method, linear tribo machine method, block on ring method, pin on disc method, and block on disc method. The key applications and standards for each technique are provided.
This document provides an overview of manufacturing processes and riser design concepts. It discusses solidification of castings, functions of risers, types of risers, and methods for riser design including the Chvorinov rule, modulus method, and NRL method. Examples are provided to demonstrate how to calculate riser dimensions using these methods based on properties of the casting such as volume, surface area, and solidification time.
The document discusses various aspects of solidification processes for pure metals and alloys. It covers topics such as solidification curves, grain structure formation, mushy zone formation in alloys, segregation of elements, shrinkage during solidification, and directional solidification techniques. It also discusses the functions and design of gating systems, including elements like pouring basins, sprues, runners, gates, and risers.
This presentation is by Flt Lt Dinesh Gupta, Associate Professor (Mechanical Engineering) NIET, Alwar (Rajasthan). It covers topic on Fluctuating Stresses related to Machine Design subject.
The document discusses fatigue failure and fatigue analysis. It begins by explaining that fatigue failure starts with a crack, usually at a stress concentration, which then propagates until sudden fracture. It then provides examples of fatigue failures and discusses different fatigue analysis methods. The key points are:
- Fatigue failure results from repeated or fluctuating stresses that are lower than the material's ultimate strength.
- It can be analyzed using stress-life, strain-life, or fracture mechanics methods, with stress-life most common for high-cycle fatigue.
- The stress-life approach estimates fatigue strength (Sf) based on stress levels and uses modifying factors to account for real-world differences from test specimens.
Dispersion Hardening:
Hard particles:
Mixed with matrix powder
Consolidated
Processed by powder metallurgy techniques
Second phase – Very little solubility (Even at elevated temp.)
No coherency
So thermally Stable at very high temp.
Resists :
Grain growth
Over aging
Recrystallization
Mobility of dislocation
Different from particle Metallic Composites (Volume Fraction is 3 to 4% max.) (Does not affect stiffness)
Examples : Al2O3 in Al or Cu, ThO2 in Ni
This document provides an introduction to fracture mechanics. It discusses different types of brittle and ductile fracture, modes of failure, energy release rate and crack resistance, crack growth, stress intensity factor, and the J-integral. It also mentions a case study on liberty ship failures and provides references for further reading. The key topics covered are the assumptions of fracture mechanics, using energy-based approaches like compliance and strain energy to analyze crack growth, and stress intensity factors which characterize how potent a crack is under different loading conditions.
This document summarizes the metal forming process of rolling. It describes how rolling works by passing metal between rolls, subjecting it to compressive and shear stresses. It discusses different types of rolling mills and explains how hot and cold rolling differ, with hot rolling reducing size at high temperatures and cold rolling providing better surface finish. The document also outlines defects that can occur during rolling such as surface irregularities, inclusions, and edge cracking or center splitting.
Creep is the time-dependent deformation of a material under constant load at high temperatures. It occurs when a material is loaded below its yield strength and the load is maintained for an extended period of time, resulting in plastic deformation that increases over time. Creep can cause failure through rupture or excessive plastic deformation beyond a certain limit. The rate of creep deformation and time to rupture depend on factors like temperature, stress, and material microstructure. Creep becomes significant engineering issue at temperatures over 40% of the material's melting point.
- Dendritic crystal growth occurs when a liquid-solid interface moves into a supercooled liquid. Heat is removed from the interface into both the solid and liquid.
- Undercooling of the liquid allows the formation of spikes at the interface that grow faster than the surrounding interface. This leads to the formation of branched, tree-like dendritic structures.
- Secondary and tertiary branches can form from primary branches/spikes. The branching occurs due to temperature gradients that arise from the release of heat at the interface.
This document discusses metal forming processes. It defines forming and shaping, and provides examples of each. Metal forming involves plastic deformation of material under large external forces to change its shape. The document classifies metal forming processes as cold working, hot working, or warm working based on the temperature of the material. It also discusses properties important for metal forming like ductility and strength. Rolling, forging, extrusion, drawing, and press working are provided as examples of metal forming processes.
This document discusses surface topography and its importance in engineering applications. It defines surface topography as the small deviations of a surface from being perfectly flat, including roughness and waviness. Surface finish is important as it influences functions like lubrication, wear resistance, and friction. The document discusses different measurement methods for surface roughness like profilometry and atomic force microscopy. It defines parameters like arithmetic mean value and root mean square average that are used to analyze measured surface roughness data. The importance of considering factors like required precision and cost when specifying a surface roughness for an engineering application is also highlighted.
- Impact tests are used to determine a material's impact energy, toughness, and tendency to fracture in a brittle manner. They are important for selecting materials that may experience sudden loading like collisions.
- Common impact tests include the Charpy and Izod tests, which involve striking a notched sample with a falling pendulum. The Charpy test uses a simply supported beam setup while the Izod uses a cantilever.
- Factors like yield strength, ductility, temperature, and strain rate can influence a material's impact performance and whether it fractures in a brittle or ductile manner. Many materials exhibit a ductile to brittle transition around a specific temperature.
This document discusses various methods for producing metal powders, including mechanical, atomization, electrochemical, and chemical methods. Mechanical methods include chopping, abrasion, milling and the cold stream process. Atomization methods include gas, water, centrifugal atomization and using a rotating electrode. Factors that influence particle size and shape from atomization are also covered. Electrochemical production involves electrolysis of molten metals. Chemical methods decompose metal compounds with heat or catalysts. The document provides details on the principles, equipment used, advantages and limitations of each production method.
Types of stresses and theories of failure (machine design & industrial drafti...Digvijaysinh Gohil
This document summarizes different types of stresses and theories of failure in mechanical components. It discusses eight types of stresses: tensile, compressive, bending, direct shear, torsional shear, bearing pressure, crushing, and contact stresses. It then explains three main theories of failure - maximum principal stress theory, maximum shear stress theory, and distortion energy theory - and their applications based on the material properties.
Powder metallurgy is a process that involves producing metal powders and compacting and sintering them to form finished parts. It allows for complex alloy compositions and near-net shape manufacturing, avoiding costly machining. The key steps are powder production, blending/mixing, compaction into a green compact, sintering to bond particles, and optional finishing. It offers advantages over casting and machining for net shape precision parts in large volumes.
This document discusses dislocations and mechanisms of plastic deformation in metals. It explains that plastic deformation occurs due to the movement of dislocations in crystals. Dislocations can move through slip and climb, leading to cumulative plastic deformation. Their movement is influenced by interactions with other dislocations and defects. Two main mechanisms of plastic deformation in metals are slip and twinning. Slip involves the sliding of crystal blocks along slip planes and directions, while twinning involves a symmetrical rearrangement of a crystal portion.
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.
This document discusses dislocations and strengthening mechanisms in metals. It begins by explaining how dislocations allow plastic deformation through slip and describes the slip systems in FCC and BCC crystals. It then discusses three main mechanisms for strengthening metals: reducing grain size, solid solution strengthening, and strain hardening. Reducing grain size increases the number of grain boundaries that impede dislocation motion. Solid solution strengthening involves alloying with impurity atoms that distort the lattice and impede dislocations. Strain hardening occurs through plastic deformation, which increases dislocation density and causes dislocations to impede each other. The document concludes by discussing recovery, recrystallization, and grain growth processes in metals after plastic deformation.
The document discusses various topics related to materials science and metallurgy including:
1. Crystal structures such as body centered cubic (BCC), face centered cubic (FCC), and hexagonal close packed (HCP) structures. It defines unit cell, calculates rank and packing density for each structure.
2. Miller indices - a system to designate crystal planes and directions using integer indices. Examples are provided to illustrate how Miller indices are determined.
3. Imperfections in crystal structures including point defects, line defects, surface defects, and bulk defects caused by processing and fabrication. Mechanisms of plastic deformation through slip and differences between single crystal and polycrystalline materials are described.
This document discusses plastic deformation in metals caused by the motion of dislocations. There are two main types of dislocations - edge and screw. Dislocations normally move under shear stress, allowing permanent deformation. Slip and twinning are two modes of plastic deformation that involve the motion of dislocations on specific crystallographic planes and directions. Strengthening methods like work hardening, solid solution strengthening, grain refinement, and precipitation hardening make it harder for dislocations to move by introducing barriers to their motion. This increases the strength of metals.
Biomaterials and biosciences biometals.pptxKoustavGhosh26
This document provides an introduction to materials, including:
1. It discusses the evolution of materials from the Stone Age to today's Silicon Age and how materials drive modern society.
2. It explains that materials science studies the relationship between structure and properties of materials, while materials engineering designs materials for specific properties and applications.
3. It briefly introduces common materials like metals, ceramics, polymers, and composites, describing their basic structures and properties.
Point defects are defects that occur at a single lattice point and are not extended in space. The main types are vacancies, interstitials, and substitutions. Line defects include edge, screw, and mixed dislocations. Grain boundaries are interfaces between crystalline grains. Volume defects are 3D aggregates of atoms or vacancies that manifest as pores and cracks.
The section will cover the behaviour of materials by introducing the stress-strain curve. The concepts of elastic and plastic deformation will be covered. This will then lead to a discussion of the micro-structure of materials and a physical explanation of what is happening to a polycrystalline material as it is loaded to failure.
The document describes various topics relating to plastic deformation in materials including dislocation motion, slip systems, strengthening mechanisms, and the effects of processing. It defines slip systems for FCC metals as occurring on {111} planes in <110> directions, allowing for 12 total slip systems. Strengthening mechanisms discussed include reducing grain size, solid solution strengthening, precipitation strengthening, and strain hardening. Solid solution strengthening is achieved through lattice strain interactions between solute atoms and dislocations. Strain hardening occurs as dislocation density increases during plastic deformation, causing dislocations to interact and impede further motion. Processing such as cold working was also described as increasing yield strength through dislocation generation and accumulation.
This document discusses various types of defects in crystalline solids including point defects like Schottky defects and line defects like dislocations. It describes Schottky defects as a pair of cation and anion vacancies that can occur in ionic crystals like alkali halides. It also discusses the different types of dislocations including edge dislocations where an incomplete plane of atoms results in regions of compression and tension, and screw dislocations where atoms are displaced in two perpendicular planes forming a spiral ramp. The document outlines how the magnitude and direction of displacement caused by defects is defined by the Burgers vector.
The document summarizes research using crystal plasticity finite element modeling (CPFEM) to predict scatter in the mechanical properties of a nickel-based superalloy due to grain anisotropy. CPFEM can model the effects of individual grain orientations and how they influence deformation at the microscale. The research aims to use CPFEM to simulate different grain configurations and predict the resulting scatter in properties, which is difficult to determine experimentally due to having only a few large grains in cast superalloy test specimens. Validation of the CPFEM model will be done using digital image correlation to experimentally measure deformation patterns.
The document summarizes key concepts from Chapter 7 of the textbook "Introduction to Materials Science" related to strengthening mechanisms in materials. It discusses how plastic deformation occurs through the motion of dislocations in materials and different ways to strengthen materials by impeding dislocation motion, such as reducing grain size, alloying, and increasing dislocation density through strain hardening. It also covers recovery, recrystallization and grain growth processes in materials after plastic deformation.
This document discusses solid state physics and crystal structures. It begins by defining solid state physics as explaining the properties of solid materials by analyzing the interactions between atomic nuclei and electrons. It then discusses different types of solids including single crystals, polycrystalline materials, and amorphous solids. Single crystals have long-range periodic atomic order, while polycrystalline materials are made of many small crystals joined together and amorphous solids lack long-range order. The document goes on to describe crystal structures including crystal lattices, unit cells, and common crystal systems such as cubic, hexagonal, and orthorhombic. It provides examples of crystal structures including sodium chloride and its cubic lattice structure.
This document discusses solid state physics and crystal structures. It begins by defining solid state physics as explaining the properties of solid materials by analyzing the interactions between atomic nuclei and electrons. It then discusses different types of solids including single crystals, polycrystalline materials, and amorphous solids. Single crystals have long-range periodic atomic order, while polycrystalline materials are made of many small crystals joined together and amorphous solids lack long-range order. The document goes on to describe crystal structures including crystal lattices, unit cells, and common crystal systems such as cubic, hexagonal, and orthorhombic. It provides examples of crystal structures including sodium chloride and its cubic lattice structure.
1. Plastic deformation in metals occurs through the movement of line defects called dislocations. Dislocations can move more easily in metals compared to ceramics and covalent solids due to metals having non-directional bonding and close-packed crystal structures.
2. The critical resolved shear stress is the minimum shear stress required to initiate slip and plastic deformation. Strengthening mechanisms like reducing grain size, solid solution strengthening, precipitation strengthening, and strain hardening increase the critical resolved shear stress.
3. Recovery and recrystallization processes allow deformed metals to reduce their dislocation density and form new defect-free grains, allowing deformed metals to soften after deformation.
The document discusses the size effects of nanoparticles including their physical properties, shapes, and applications. It states that nanoparticles less than 100 nm exhibit size-dependent properties not seen in bulk materials, such as higher strength. The properties of nanoparticles can change with temperature and pressure due to changes in crystal structure. Their large surface area to volume ratio gives nanoparticles additional properties like improved catalytic activity.
The document describes a crystal plasticity study of tin (Sn) using finite element modeling of nanoindentation simulations. Single crystal tin has an anisotropic diamond cubic structure that influences its deformation behavior depending on crystal orientation. The simulations examine nanoindentation on tin crystals oriented in the [001] and [100] directions. Results show differences in displacement topography and varying activity of slip systems under each orientation. The simulations provide insight into tin's anisotropic deformation properties at the crystal level.
Crystals consist of periodically repeating patterns of atoms or molecules arranged in unit cells. Common crystal structures include cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic. Defects in crystals such as dislocations and grain boundaries influence properties like strength and ductility. Dislocations are line defects associated with plastic deformation that allow slip to occur in crystals. Motion of dislocations during plastic deformation leads to changes in shape without changing chemical properties.
Similar to Introduction to Crystal Plasticity Modelling (20)
Adaptive synchronous sliding control for a robot manipulator based on neural ...IJECEIAES
Robot manipulators have become important equipment in production lines, medical fields, and transportation. Improving the quality of trajectory tracking for
robot hands is always an attractive topic in the research community. This is a
challenging problem because robot manipulators are complex nonlinear systems
and are often subject to fluctuations in loads and external disturbances. This
article proposes an adaptive synchronous sliding control scheme to improve trajectory tracking performance for a robot manipulator. The proposed controller
ensures that the positions of the joints track the desired trajectory, synchronize
the errors, and significantly reduces chattering. First, the synchronous tracking
errors and synchronous sliding surfaces are presented. Second, the synchronous
tracking error dynamics are determined. Third, a robust adaptive control law is
designed,the unknown components of the model are estimated online by the neural network, and the parameters of the switching elements are selected by fuzzy
logic. The built algorithm ensures that the tracking and approximation errors
are ultimately uniformly bounded (UUB). Finally, the effectiveness of the constructed algorithm is demonstrated through simulation and experimental results.
Simulation and experimental results show that the proposed controller is effective with small synchronous tracking errors, and the chattering phenomenon is
significantly reduced.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
1. Crystal Plasticity Finite Element Modelling
Crystal Plasticity
Finite Element
Modelling
CDT Lecture – Nov 2018
Suchandrima Das
suchandrima90@gmail.com
3. Crystal Plasticity Finite Element Modelling
Crystalline solids: Quick Recap
Crystal: Highly regular arrangement of atoms
• how do we describe them ? - lattices
• how can we study their structures ?
many ways, for example x-ray diffraction.
1-D Lattice
Crystal structures repeat in 3D
Motif Single atom OR groups of
atoms (unit cell)
Space Lattice + Motif = Crystal
Structure
https://www.seas.upenn.edu/~chem101/sschem/solidstatechem.html/
5. Crystal Plasticity Finite Element Modelling
Plasticity
Linear elastic – return to original shape upon removal of load
Plastic Beyond yield point
Permanent deformation
Energy dissipated
Path dependent
Due to dislocation generation
and movement
6. Crystal Plasticity Finite Element Modelling
How does plastic deformation occur?
Not a rigid body slip
phenomenon
Part slipped Part not yet
slipped
Dislocation Line:
Line defect
7. Crystal Plasticity Finite Element Modelling
1 plane of atoms slides over adjacent plane by defect motion (dislocations).
Adapted from Fig. 7.1, Callister 7e.
Note: Process is isochoric
Dislocations: Key Points
8. Crystal Plasticity Finite Element Modelling
Plastic deformation
G. Dieter, Mechanical Metallurgy, 3rd Edition, McGraw-Hill, 1986.
Energy barrier
ΔE minimized growing slipped area by
advancing DISLOCATION
Dislocations: Key Points
Caterpillar hump extra half plane
of atoms
Subsequent bond breaking and
bond forming
Note: Process is isochoric
9. Crystal Plasticity Finite Element Modelling
Dislocation Recap
Material displaces by ‘b’
Key points to remember:
• Dislocations are line defects
• Plastic deformation occurs by dislocation glide
• No volume change involved
10. Crystal Plasticity Finite Element Modelling
** https://www.mm.ethz.ch/teaching.html
** Micro Meso Macro
Bulk Many single
crystals + GB
Multiscale Modelling
CPFE lies in meso-scale region
11. Crystal Plasticity Finite Element Modelling
Polycrystalline Grain Structure
CPFE needs information about the orientation of the crystal
Electron Backscatter Diffraction
Orientation Sample relative to the crystal
Electrons diffracted from samples
create diffraction patterns.
Positions of bands orientation
12. Crystal Plasticity Finite Element Modelling
Crystallographic Orientation: EBSD Intro
Important to acquire information about the orientation of each grain in the polycrystal
Electron Backscatter Diffraction
Electrons leaving
sample
Some exit at Bragg
condition
Bragg condition
spacing of lattice planes
Electrons diffract
Kikuchi bands
Bands phase &
orientation
13. Crystal Plasticity Finite Element Modelling
Crystallographic Orientation: EBSD Intro
EBSD gives us the Euler angles
Euler Angles 3 angles through which the crystal must be rotated in order to relate the
crystallographic axes with the sheet reference axes
1 2 3
4
15. Crystal Plasticity Finite Element Modelling
Crystal plasticity
Crystalline
materials
Thermal &
Mechanical
Behavior
Microstructure
Critical Role played by:
Texture & orientation of Single crystals
Total area of GBs
EXAMPLE
Hall-Petch relationship
Smaller the grain Stronger the
polycrystal. Why?
Effect cannot be captured by
macroscopic continuum model.
16. Crystal Plasticity Finite Element Modelling
Hall-Petch effect & Dislocations: Key Points
So how does Hall-Petch effect come by?
GB offer resistance to dislocation motion. Why?
https://en.wikipedia.org/wiki/Grain_bo
undary_strengthening#/
pileup_force>
GB_Repulsive
force ;
deformation
continues
Pile-up
increases,
pileup_force
increases.
Dislocations
pile-up.
Orientation
mismatch :
hinders
dislocation
motion.
GB_Repulsive
force
GB –
boundary
between 2
grains of diff
orientation
Smaller grain Lesser pile-up
More difficult
to move
dislocation
across GB
Stronger
material
17. Crystal Plasticity Finite Element Modelling
Polycrystal plasticity
Single crystal
plasticity
Homogenization
Polycrystal
plasticity
Assumption: GB do not play an active role in plastic deformation other than providing a
constraint on plastic strain of neighboring grains (G.I.Taylor 1938).
Assumption confirmed by exp at ambient temp. At high temp. GB sliding & diffusion
creep
Each single crystal elastically & plastically
anisotropic
Aggregation of many grains Isotropic
Assumption: lengthscale ‘l’ over which fields of
interest (e.g. strain) varies is << than macroscopic
lengthscale ‘L’.
‘Effective homogeneous
material’ relaxed
continuity requirements
19. Crystal Plasticity Finite Element Modelling
Single Crystal Slip systems
Stretching single crystal Slip on crystallographic planes along specific directionsStretching single crystal Slip on crystallographic planes along specific directions
Remember: There is an energy
barrier to dislocation slip.
Inherent resistance to
deformation by crystal lattice
Peierls stress is the shear stress required to move a dislocation through
a crystal lattice at 0 K. Peierls found
𝜏 = 𝐺exp −
2𝜋𝑤
𝑏
Large w and small b will reduce 𝜏
20. Crystal Plasticity Finite Element Modelling
Single Crystal Slip systems
Width measure of degree of disruption a dislocation creates wrt perfect lattice
𝜏 = 𝐺exp −
2𝜋𝑤
𝑏
Large w and small b will reduce 𝜏
Therefore, slip most likely to occur in close-packed planes in close-packed directions.
21. Crystal Plasticity Finite Element Modelling
Single Crystal Slip systems
Stretching single crystal Slip on crystallographic planes along specific directions
Slip Directions: Shortest Lattice repeat vectors
Slip Planes: Planes with highest in-plane density
Stretching single crystal Slip on crystallographic planes along specific directions
X
Y
Z
(111)
½[𝟏ഥ𝟏 𝟎]
FCC
Z
X
Y
(110)
½[ഥ𝟏𝟏𝟏]
BCC
Q. How many slip systems in BCC & FCC?
22. Crystal Plasticity Finite Element Modelling
Schmid Factor
https://www.doitpoms.ac.uk/tlplib/slip/slip_geometry.php
Angle between Ԧ𝒍 and 𝑺 = λ
Angle between Ԧ𝒍 and 𝒏 = Φ
cos λ = Ԧ𝒍. 𝑺
cos Φ = Ԧ𝒍. 𝒏
τ Shear stress on plane 𝒏 along 𝑺
γ Shear strain on plane 𝒏 along 𝑺
𝜏 = 𝑷. 𝑚
𝒎 = 𝒄𝒐𝒔 𝝀 𝒄𝒐𝒔 𝜱 = Schmid Factor
How much normal stress can be
effectively transferred on to the slip
plane as shear stress?
23. Crystal Plasticity Finite Element Modelling
Slip occurs when the shear stress acting in the slip direction on the slip plane reaches
some critical value CRSS
https://www.doitpoms.ac.uk/tlplib/slip/slip_geometry.php
Critically Resolved Shear Stress
𝜏 𝐶 = 𝝈 𝑦. 𝑚
𝝈 𝑦 =
𝜏 𝐶
𝑚
𝑚 Depends on relative orientation of loading axis w.r.t slip
systems
Hence, we expect 𝝈 𝑦 of single crystal to be orientation-
dependent (but not on tension vs. compression)
FCC obeys this law well
BCC shows tension compression asymmetry
𝝉 𝑹 > 𝝉 𝑪
For deformation
Yield stress and Schmid Factor
24. Crystal Plasticity Finite Element Modelling
Taylor Factor: Quick Look
Further Info: https://nptel.ac.in/courses/113108054/module4/lecture17.pdf
https://inis.iaea.org/collection/NCLCollectionStore/_Public/29/000/29000976.pdf
𝝈 𝑦 =
𝜏 𝐶
𝑚
Macroscopic yield stress Schmid factor of single grain
Different yield stress for applied stress along different directions
Active slip system one with smallest m
Single Crystals
• Crystal cannot change its shape freely Constraints from surrounding crystals
• Grains must fit together without voids after deformation
• Several slip systems should be activated in a grain, or in a part of grain
• Slip from 5 independent slip systems is generally required to accommodate the five
independent strain components for plastic deformation (Taylor 1938)
• Active combination combination of 5 systems which minimizes accumulated slip
• Uniaxial tension in X direction 𝑀 =
σ 𝑖 ሶ𝛾 𝑖
ሶ𝜀11
𝑃 ; 𝝈 𝑦 =
𝜏 𝐶
𝑀
• Assumption: Strain and strain rate in any grain = Average strain
Poly Crystals
25. Crystal Plasticity Finite Element Modelling
Plasticity
So we know now that dislocations gliding bring about plastic deformation.
Then how do they cause hardening?
Due to dislocation generation
and movement
26. Crystal Plasticity Finite Element Modelling
Dislocation Hardening
material
hardens
slip rate
reduced
more
obstacles
created
Dislocation
density
increases
progressive
plastic
deformation
Dislocations multiply and entangle with one another
Distance of separation between them reduces.
They restrict each others motion. Dislocation motion becomes more difficult
27. Crystal Plasticity Finite Element Modelling
Dislocation Hardening
Dislocations
SSD
Inherently
present. No
material is perfect
GND
To accommodate
strain gradient
28. Crystal Plasticity Finite Element Modelling
SSDs & GNDs
Extra half plane of atoms. So there is lattice distortion
around the dislocation; Bonds are bent or strained.
Dislocations Generate strain field around them
No effective strain
field SSDs
Effective strain field
GNDs
If you zoom in and just see one dislocation, is it a SSD or GND?
30. Crystal Plasticity Finite Element Modelling
What do we know so far?
• Plastic deformation Permanent deformation & path dependent
• Dislocations are line defects
• Plastic deformation occurs by dislocation glide
• Dislocations glide on slip systems when the 𝝉 𝑹 > 𝝉 𝑪
• Dislocations bring about hardening.
Jiang J, Zhang T, Dunne FPE, Britton TB. 2016 Deformation compatibility in a single crystalline Ni superalloy. Proc.R.Soc.A 472: 20150690
31. Crystal Plasticity Finite Element Modelling
What do we want to know from CPFE?
Remember Continuum macroscopic model does not consider
dislocation slip. Primarily we want CPFE to consider this.
Given
• Slip systems; Crystal
• Elastic properties
• Orientation
• Apply macroscopic
loading
CPFE
• Resolve load into assigned slip
systems
• Calculate slip if there is any
• Resulting plastic strain
• Lattice rotation
• Dislocation density
UMAT
32. Crystal Plasticity Finite Element Modelling
Remember: Schmid Factor
https://www.doitpoms.ac.uk/tlplib/slip/slip_geometry.php
Angle between Ԧ𝒍 and 𝑺 = λ
Angle between Ԧ𝒍 and 𝒏 = Φ
cos λ = Ԧ𝒍. 𝑺
cos Φ = Ԧ𝒍. 𝒏
τ Shear stress on plane 𝒏 along 𝑺
γ Shear strain on plane 𝒏 along 𝑺
𝜏 = 𝑷. 𝑚
𝒎 = 𝒄𝒐𝒔 𝝀 𝒄𝒐𝒔 𝜱 = Schmid Factor
How much normal stress can be
effectively transferred on to the slip
plane as shear stress?
33. Crystal Plasticity Finite Element Modelling
What do we want to know from CPFE?
Remember Continuum macroscopic model does not consider
dislocation slip. Primarily we want CPFE to consider this.
Given
• Slip systems; Crystal
• Elastic properties
• Orientation
• Apply macroscopic
loading
CPFE
• Resolve load into assigned slip
systems
• Calculate slip if there is any
• Resulting plastic strain
• Lattice rotation
• Dislocation density
UMAT
34. Crystal Plasticity Finite Element Modelling
Slip occurs when the shear stress acting in the slip direction on the slip plane reaches
some critical value CRSS
Critically Resolved Shear Stress
𝝉 𝑹 > 𝝉 𝑪
For deformation
35. Crystal Plasticity Finite Element Modelling
Crystallographic Slip 𝜷 𝒑
Crystallographic slip rate ሶ𝛽 𝑝
𝜆
Crystallographic slip
ሶ𝛽 𝑝
𝜆
∆𝑡 = ∆𝛽 𝑝
𝜆
Summed over all slip systems
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡 = ∆𝛽 𝑝
Summed over time
𝛽 𝑝
𝑡+∆𝑡
= 𝛽 𝑝
𝑡
+
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡
In FE We progress from time 𝑡 to 𝑡 +
∆𝑡
For each time increment ∆𝑡, it is useful
to develop the formulation in terms of
rate. Some formulations like
viscoplasticity are rate dependent.
Plasticity is an incremental process.
Instead of dealing with increments in
slip, we consider slip rate.
Rate of slip is calculated
So we need to find ሶ𝜷 𝒑
𝝀
to find 𝛽 𝑝
36. Crystal Plasticity Finite Element Modelling
Finding ሶ𝜷 𝑝
Constitutive Slip Law is used to define how much slip will take place
ሶ𝛽 𝑝 𝜆
∝ 𝑓 𝜏, 𝜏 𝐶
Can be a simple power law
1. Phenomenological: Commonly Used Power Law
ሶ𝛽 𝑝 𝜆
= K
𝜏
𝜏 𝐶
𝑛
for 𝜏 > 𝜏 𝐶
38. Crystal Plasticity Finite Element Modelling
What do we want to know from CPFE?
Remember Continuum macroscopic model does not consider
dislocation slip. Primarily we want CPFE to consider this.
Given
• Slip systems; Crystal
• Elastic properties
• Orientation
• Apply macroscopic
loading
CPFE
• Resolve load into assigned slip
systems
• Calculate slip if there is any
• Resulting plastic strain
• Lattice rotation
• Dislocation density
UMAT
39. Crystal Plasticity Finite Element Modelling
How to find strain, rotation and dislo density?
ሶ𝜷 𝑝 or 𝜷 𝒑 F L
Strain
Rotation
Total lattice
deformation
Gradient GND
40. Crystal Plasticity Finite Element Modelling
Deformation Gradient
U determines deformation:
• Translation
• Rigid Body rotation
• Stretch/Shape change
• Or a combination
Second-order tensor
Maps the undeformed state to the deformed state of a sample
𝑭 =
𝜕𝒙
𝜕𝑿
= 𝑰 +
𝜕𝒖
𝜕𝑿
= 𝑰 + 𝜷
Representation Analogous to
strain for small deformations
Displacement gradient
Infinitesimal line element dX becomes dx
𝑭 Contains all the information about
stretch, rotation, translation.
41. Crystal Plasticity Finite Element Modelling
F.P.E.Dunne: Lecture – The Crystal Approach 2014
Examples of Deformation
43. Crystal Plasticity Finite Element Modelling
Deformation Gradient Calculation: Examples
F.P.E.Dunne: Lecture – The Crystal Approach 2014
44. Crystal Plasticity Finite Element Modelling
Deformation Gradient Calculation: Examples
F.P.E.Dunne: Lecture – The Crystal Approach 2014
𝑭 Contains all the information about stretch, rotation, translation.
How can we extract strain and rotation from it?
45. Crystal Plasticity Finite Element Modelling
How to find strain, rotation and dislo density?
ሶ𝜷 𝑝 or 𝜷 𝒑 F L
Strain
Rotation
Total lattice
deformation
Gradient GND
So we know what is 𝑭
Is this only computed for CPFE?
Any FE implementation undergoing plastic deformation will compute 𝑭
So what does CPFE do in this context?
46. Crystal Plasticity Finite Element Modelling
Splitting F
Introduce an intermediate imaginary
configuration dp
𝑑𝒑 = 𝑭 𝑝
𝑑𝑿
𝑑𝒙 = 𝑭 𝑒
𝑑𝒑
𝑑𝒑 = 𝑭 𝑝 𝑑𝑿 = 𝑭 𝑒−1
𝑑𝒙
𝑭 =
𝝏𝒙
𝝏𝑿
= 𝑭 𝒆 𝑭 𝒑
𝑭 can be split into the elastic (𝑭 𝑒
) and the plastic (𝑭 𝑝
) component
CPFE considers plastic deformation by dislocation slip and computes the resulting 𝑭 𝒑
47. Crystal Plasticity Finite Element Modelling
Plastic Deformation Gradient
How can we calculate 𝑭 𝒑
?
𝑭 𝑃
=
𝜆
𝛽 𝑝 𝜆
𝒔 𝜆
⨂𝒎 𝜆
crystallographic slip 𝜷 𝑝
s Slip direction
M Slip normal
𝜆 A particular Slip system
Just like we consider slip rate, we will also consider the rate of 𝑭 𝑃
i.e. ሶ𝑭 𝑝
ሶ𝑭 𝑝 =
𝜆
ሶ𝛽 𝑝 𝜆
𝒔 𝜆⨂𝒎 𝜆
Remember Any FE implementation undergoing plastic deformation will compute 𝑭
CPFE will compute 𝑭 𝑷
& ሶ𝑭 𝒑 by considering dislocation slip.
Knowing 𝑭 and 𝑭 𝑷
, you can then compute 𝑭 𝒆
48. Crystal Plasticity Finite Element Modelling
How to find strain, rotation and dislo density?
ሶ𝜷 𝑝 or 𝜷 𝒑 F L
Strain
Rotation
Total lattice
deformation
Gradient GND
So we know what is 𝑭 and how we can compute its elastic and plastic components.
Now what is L?
We have discussed so far rate quantities like ሶ𝜷 𝑝
& ሶ𝑭 𝑝. L is a form used to represent
ሶ𝑭 𝑝 .
Simply, 𝑳 =
𝜕𝒗
𝜕𝒙
= ሶ𝑭𝑭−1 (theoretically spatial rate of change of velocity).
FE implementation will compute L. Once again we need CPFE to compute its plastic
component. So how to get strain and rotation from L?
50. Crystal Plasticity Finite Element Modelling
Equations to complete the picture
We know the slip rate from the slip law
51. Crystal Plasticity Finite Element Modelling
Finding ሶ𝜷 𝑝
Constitutive Slip Law is used to define how much slip will take place
ሶ𝛽 𝑝 𝜆
∝ 𝑓 𝜏, 𝜏 𝐶
Can be a simple power law
1. Phenomenological: Commonly Used Power Law
ሶ𝛽 𝑝 𝜆
= K
𝜏
𝜏 𝐶
𝑛
for 𝜏 > 𝜏 𝐶
52. Crystal Plasticity Finite Element Modelling
Crystallographic Slip 𝜷 𝒑
Crystallographic slip rate ሶ𝛽 𝑝
𝜆
Crystallographic slip
ሶ𝛽 𝑝
𝜆
∆𝑡 = ∆𝛽 𝑝
𝜆
Summed over all slip systems
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡 = 𝛽 𝑝
Summed over time
𝛽 𝑝
𝑡+∆𝑡
= 𝛽 𝑝
𝑡
+
λ=1
𝑛
ሶ𝛽 𝑝
𝜆
∆𝑡
In FE We progress from time 𝑡 to 𝑡 +
∆𝑡
For each time increment ∆𝑡, it is useful
to develop the formulation in terms of
rate. Some formulations like
viscoplasticity are rate dependent.
Plasticity is an incremental process.
Instead of dealing with increments in
slip, we consider slip rate.
Rate of slip is calculated
So we need to find ሶ𝜷 𝒑
𝝀
to find 𝛽 𝑝
53. Crystal Plasticity Finite Element Modelling
How to find strain, rotation and dislo density?
ሶ𝜷 𝑝 or 𝜷 𝒑 F L
Strain
Rotation
Total lattice
deformation
Gradient GND
54. Crystal Plasticity Finite Element Modelling
Equations to complete the picture
𝑳 𝑝
≅ ሶ𝑭 𝑝
𝑳 𝑒
= 𝑳-𝑭 𝑒
𝑳 𝑝
𝑭 𝑒−1
𝑭 𝑃
=
𝜆
𝛽 𝑝 𝜆
𝒔 𝜆
⨂𝒎 𝜆
ሶ𝑭 𝑝 =
𝜆
ሶ𝛽 𝑝 𝜆
𝒔 𝜆
⨂𝒎 𝜆
We will know F from FE, so we can find 𝑭 𝒆
𝑭 =
𝝏𝒙
𝝏𝑿
= 𝑭 𝒆
𝑭 𝒑
We also know L from FE. We then find 𝑳 𝑝
& 𝑳 𝑒
We then find 𝑫 𝑝
𝑾 𝑝
𝑫 𝑒
𝑾 𝑒
56. Crystal Plasticity Finite Element Modelling
How to find strain, rotation and dislo density?
ሶ𝜷 𝑝 or 𝜷 𝒑 F L
Strain
Rotation
Total lattice
deformation
Gradient GND
We know:
𝑭 𝑃
𝑭 𝑒
𝑳 𝑃
𝑳 𝑒
𝑫 𝑃
𝑫 𝑒
57. Crystal Plasticity Finite Element Modelling
Equations to complete the picture
∆𝜺 𝑝 = 𝑫 𝑝
∆𝑡
∆𝜺 𝑒 = 𝑫 𝑒
∆𝑡
∆𝝎 𝑒 = sym 𝑳 𝑒 ∆𝑡
Jiang J, Zhang T, Dunne FPE, Britton TB. 2016 Deformation compatibility in a single crystalline Ni superalloy. Proc.R.Soc.A 472: 20150690
Continuum Rotation: Bonds at the
atomic level remain unchanged.
No necessary change in lattice
orientation.
58. Crystal Plasticity Finite Element Modelling
What do we want to know from CPFE?
Remember Continuum macroscopic model does not consider
dislocation slip. Primarily we want CPFE to consider this.
Given
• Slip systems; Crystal
• Elastic properties
• Orientation
• Apply macroscopic
loading
CPFE
• Resolve load into assigned slip
systems
• Calculate slip if there is any
• Resulting plastic strain
• Lattice rotation
• Dislocation density
UMAT
59. Crystal Plasticity Finite Element Modelling
Dislocation Density calculation
GND can be calculated from Lattice deformation.
Lattice
rotation
Lattice
strain
Total lattice
deformation
Gradient
GND
SSD value not evolved. But can be.
61. Crystal Plasticity Finite Element Modelling
What do we want to know from CPFE?
Remember Continuum macroscopic model does not consider
dislocation slip. Primarily we want CPFE to consider this.
Given
• Slip systems; Crystal
• Elastic properties
• Orientation
• Apply macroscopic
loading
CPFE
• Resolve load into assigned slip
systems
• Calculate slip if there is any
• Resulting plastic strain
• Lattice rotation
• Dislocation density
UMAT
62. Crystal Plasticity Finite Element Modelling
Key Points to Remember
• Dislocations are line defects
• Plastic deformation occurs by dislocation glide
• No volume change involved
• Dislocations bring about hardening too
• FE is an incremental process: Deformation computed for every Δt
63. Crystal Plasticity Finite Element Modelling
FE computes
total
deformation
Calls CPFE
UMAT
Compute
∆𝜺 𝑒 ∆𝜺 𝑝 &
∆𝝎 𝑒
Compute
stress
increment
from ∆𝜺 𝑒
Elastic and plastic
deformation
quantities are
sent back to FE.
F & L
Start from time = 0, deformation has occurred for Δt
Consider
dislocation slip,
computes slip
rate, 𝐹 𝑝
& 𝐿 𝑝
Next time
increment
begins
Key Points to Remember
65. Crystal Plasticity Finite Element Modelling
Y
ZX
[100]
[110]
[1-10]
5 µm
5 µm
• Substantial Increase in Pile Up
• Slip localization
• Higher max load
0
30
60
90
0 200 400 600
LoadonElement(mN)
Depth (nm)
Load vs Depth
Implanted
Unimplanted
Nano-Indentation SEM
[010]
4.2 µm radius; 500 nm deep
Unimp
He-imp
S. Das et al., Scr. Mater. 146 (2018) 335–339.
http://linkinghub.elsevier.com/retrieve/pii/S1359646217307145
66. Crystal Plasticity Finite Element Modelling
𝛾ሶ 𝛼
= 𝜌 𝑔 𝜈(𝑏 𝛼 )2
exp−
∆𝐹 𝛼
𝑘𝑇
sinh
(𝜏 𝛼
− 𝜏 𝑐
𝛼
)𝛾0∆𝑉 𝛼
𝑘𝑇
Slip-systems – {110}<111>
• 12 edge type
• 4 screw type
𝑙 =
1
)𝛹(𝜌 𝐺𝑁𝐷 + 𝜌 𝑆𝑆𝐷
𝑉 spacing between the pinning
dislocations 𝑙 .
𝑳 𝑝
=
𝜆
ሶ𝛾 𝛼
𝒔 𝛼
ໆ 𝒏 𝛼
≅ ሶ𝑭 𝑝
Fitted to
experiment
Slip Law for slip system α
Example of Nano-indentation in Tungsten
https://www.sciencedirect.com/science/article/pii/S0749641918300068?via%3Dihub
S. Das et al., Int. J. Plast. 109 (2018) 18–42.
67. Crystal Plasticity Finite Element Modelling
Surface Profile Prediction
What happens without CPFE?
S. Das et al., arXiv Prepr. arXiv1901.00745 (2018)
http://arxiv.org/abs/1901.00745
68. Crystal Plasticity Finite Element Modelling
Surface Profile Prediction for different
orientations
https://ora.ox.ac.uk/objects/uuid:ea8aa247-fd6a-4b2d-a4be-04aa229da828
69. Crystal Plasticity Finite Element Modelling
Lattice rotation
Das, S. et al. Int. J. Plast. https://doi.org/10.1016/j.ijplas.2018.05.001
70. Crystal Plasticity Finite Element Modelling
Lattice strain
Das, S. et al. Int. J. Plast. https://doi.org/10.1016/j.ijplas.2018.05.001
73. Crystal Plasticity Finite Element Modelling
Strain and Rotation from F
𝑭 =
𝜕𝒙
𝜕𝑿
=
𝜕(𝑿 + 𝒖)
𝜕𝑿
= 𝑰 +
𝜕𝒖
𝜕𝑿
= 𝑰 + 𝜷
𝑰 =
𝟏 𝟎 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
𝜷
Symmetric
1
2
𝜷 + 𝜷 𝑇
Strain
Asymmetric
1
2
𝜷 − 𝜷 𝑇
Rotation
For small deformations only
But we want elastic and plastic components of strain. So to get that we decompose
F into elastic and plastic parts.
74. Crystal Plasticity Finite Element Modelling
Dislocation Hardening
ሶ𝛽 𝑝
𝜆
= 𝜌 𝑔 𝜈 𝑏 𝜆 2
exp −
∆𝐹 𝜆
𝑘𝑇
sinh
sgn(𝜏 𝜆
)( 𝜏 𝜆
− 𝜏 𝑐
𝜆
)𝑉 𝜆
𝑘𝑇
𝑉 depends on the spacing between the pinning dislocations
𝑙 . Decrease mean free path of slip 𝑙
𝑙 =
1
𝛹(𝜌 𝐺𝑁𝐷 + 𝜌 𝑆𝑆𝐷)
GNDs can also be introduced in the form of increasing CRSS
ሶ𝛽 𝑝 𝜆
= K
𝜏
𝜏 𝐶
𝑛
for 𝜏 > 𝜏 𝐶
𝜏 𝐶
𝑡+∆𝑡
= 𝜏 𝐶
𝑡
+ 𝐺𝑏 𝛹(𝜌 𝐺𝑁𝐷 + 𝜌 𝑆𝑆𝐷)
1. Phenomenological: Commonly Used Power Law
2. Physically Based Slip Law