Engineering component

Engineering component

1.0 Introduction

The purpose of any engineering component is to safely accomplish what it is designed to achieve. An engineer must consider design criteria when designing a component. An important aspect of design criteria is to optimise the structure of a component for strength and weight purposes. A trade-off is often associated between strength and weight, and a compromise has to be established. An optimised component can reduce both economical and environmental costs. However, to be able to optimise the component, the engineer must be able to determine the internal stress and strain state in the component. From the stress state, it is possible to determine the loads the component will able to withstand in application. Thus, the component can be redesigned; reducing cross sections with relatively low internal stresses and increasing cross sections in areas of relatively high internal stresses. An example of this optimisation is an I-beam.

Many traditional experimental techniques, such as mechanical testing of materials, strain gauging and optical methods are used to evaluate the mechanical properties of materials and determine the stress and strain state. However, information obtained from these tests is used to determine the average mechanical behaviour and thus, these methods are best suited for examining the material behaviour at a macroscopic scale.

Recently, however, there has been a trend towards examining the mechanical properties, elastic-plastic deformation and stress and strain state at the microscopic scale. The reason for the increased research in this area is due to increasingly advanced technologies and the harsh and extreme operating conditions that materials are now subjected to. A notable example is the on-going research in the area of nuclear reactor design where steel materials are frequently subjected to large deformations due to thermal loads (Liu, 2006). Reliable and accurate estimations for the large deformations can be crucial in establishing design criteria and material selection. The traditional continuum mechanics approach, based on the macroscale, is not appropriate for modelling such extreme deformations (Needleman, 2000).

The main drawback of continuum mechanics is the idealisation of the homogenous microstructure of materials, when in reality almost all materials are inhomogeneous at the microscopic level. Continuum mechanics fails to predict material behaviour accurately when the inhomogeneities start to dominate the response of the structure. For large structures; these effects typically become dominant while approaching limit loads, as in the case of a nuclear reactor. At this point the microscopic structure becomes appreciable and, it is over the microscale range that key deformation and fracture processes in a variety of structural materials take place (Needleman, 2000). Therefore, it is more appropriate to adopt another approach to predict the microscopic response of materials.

The alternative to the continuum mechanics approach are numerical models, which obtain the overall macroscopic behaviour of a material from the behaviour of the material constituents on the microscale (e.g., crystal grains with typical size of 10-100 μm). The method carried out for this project is to model the deformation of a polycrystalline sample of steel. A microscopic modelling approach is used, which models the elastic-plastic behaviour of the polycrystalline material. The governing principle of the microscopic model is to divide the continuum (i.e., the polycrystalline aggregate) into a set of subcontinua (i.e., grains). The overall macroscopic properties of the polycrystalline aggregate are dictated by the number of steel grains in the aggregate and the properties of the randomly oriented steel grains. The microscopic response of monocrystal grains is modelled with anisotropic elasticity and crystal plasticity.

The polycrystalline aggregate is modelled using a representative volume element (RVE); containing individual face centered cubic steel grains generated using Abaqus/CAE. Each grain is randomly orientated within the polycrystalline aggregate. As each grain has a different direction, it will also have different properties in the various directions (e.g., loading direction). Thus the grains are directionally dependent and this defines anisotropy in the grains. Crystal plasticity is defined to describe inelastic material behaviour of the grains. The appropriate boundary conditions and loadings are applied to examine the deformation, and the process is carried out using the finite element solver; Abaqus/Standard.

The purpose of the model and the subsequent analysis is to determine the minimum number of grains that are required to give the polycrystalline aggregate an isotropic response. To determine if the polycrystalline aggregate provides an isotropic response, the macroscopic Young's modulus of the aggregate is calculated and compared to experimental Young's moduli for a corresponding sample of steel. It is proposed that a 36 grain polycrystal aggregate structure could be sufficient.

If the 36 grain aggregate is found to be sufficient, this study will determine the lower bound domain of traditional continuum mechanics, below which it is not able to accurately describe polycrystalline material behaviour (Kovač et al., 2003). This analysis is limited to 2-D models and the material parameters are for face centered cubic steel (i.e., austenite). However, the proposed model can, in general, accommodate a variety of grain structures and material models.

1.1 Report Structure

This report begins with an overview of the properties of a material at the microscale. The overview explains the differences between material properties at the microscale and the more traditional macroscale.

Once the relevant knowledge of multi-scale modelling is established, information on crystal structures, how they are formed and their influence on mechanical properties is reviewed in chapter 4. In addition, previous studies that are essential in the understanding of what has been done in the area of microscopic modelling are presented. The previous studies demonstrate the most appropriate method of modelling the problem and identify problem areas.

Chapter 5 presents the theory that is necessary for the modelling process. The numerical modelling process in Abaqus requires quite a substantial amount of theoretical analysis to be completed prior to running the various jobs. Complex matrix and vector rotations are addressed in this section. The majority of the theoretical analysis allowed for the creation of MatLab programmes which are available in Appendix A.

The following chapter, chapter 6, presents the study design. This demonstrates the thought process involved in the design for the model, and presents the assumptions proposed for the model. The chapter describes the reasoning for choosing the elements, boundary conditions, loading and modelling analysis.

Chapter 7 is the numerical modelling chapter. This provides a detailed account of how the modelling process was carried out using the computational software package, Abaqus/Standard. A brief description of how the results were obtained is also provided.

The results obtained from the finite element analysis of the deformation of the polycrystal aggregate are presented in chapter 8. The quantitative results are accompanied by visual representations of the deformed polycrystal structure in various stress states. The results are presented in comparison to previous similar studies.

Chapter 9 explains and interprets the main findings of the results chapter. The chapter explains why the results turned out the way they did and how the results compare with previous work done in the area of mesoscopic modelling.

The final chapter of this report presents the conclusions and the recommendations for future work.

2.0 Objectives

The objectives of the report are summarised below, they read as follows:

  • To calculate the Euler rotations of a single polycrystal (i.e., a monocrystal) and apply the results to the polycrystal aggregate;
  • To calculate the fully populated anisotropic stiffness matrix, D;
  • To investigate the microscopic and macroscopic properties of the polycrystal aggregate;
  • To determine if the 36 grain aggregate is sufficient for an isotropic response at the macroscopic scale;
  • To calculate the Schmid factor and hence, the slip systems available in a monocrystal and apply the results to the polycrystalline aggregate;
  • To determine the macroscopic plastic response of the model.

3.0 Background

Traditional solid mechanics treat a material as a continuum and as such, does not take into account the microstructure of the material. Instead, continuum mechanics treats the material as a homogeneous, uniform solid material. For the majority of the time isotropy is also assumed in the material. However, these assumptions are highly dependent on the scale with which the materials are examined. Metals, ceramics and other materials are all built up by a grain structure, which will be discussed at a later stage. The size, structure and orientation of the grains determine the macroscopic mechanical behaviour of the whole material (Nygards, 2002). When examining the properties at the microscopic scale the individual grains are evident. Thus, the material is far from being homogenous and it is appropriate to treat the material as inhomogeneous (i.e., heterogeneous) and anisotropic. These properties need to be addressed to understand why this project was carried out.

3.1 Homogeneity

A homogeneous material is described by Daniel and Ishai (1994) as a material whose properties are the same at every point or are independent of location within the material. Thus, it can be said that a homogeneous material has uniform properties throughout. The concept of homogeneity is associated with the scale with which the material is examined on. Depending on the scale, the material can be determined to be more homogenous or less homogeneous.

This is of relevance to this project as the steel sample could be considered homogeneous when looking at the material with the naked eye (i.e., the macroscopic scale) However, the sample is considered inhomogeneous when viewed under the microscope, which is viewing the sample on the microscale. When viewed under the microscope, it is possible to see the individual steel grains that the material is composed of. In comparison, if the material is viewed on a macroscopic scale, one can only see the whole structure and the geometry, and not the individual grains. The difference is illustrated in figure 3-1, below. It is generally accepted that as the scale increases from microscopic to macroscopic a material can be considered more homogeneous (Daniel and Ishai, 1994).

According to Timoshenko and Goodier (1987) a material can be considered homogeneous as long as the geometrical dimensions defining the material are large in comparison with the dimensions of a single crystal. Therefore the continuum mechanics approach is adequate for many traditional engineering problems, where the length scales are much greater than the inter-atomic distances. However, for relatively small structures, the effects of inhomogeneities may become noticeable already at the level of normal service loads (Needleman, 2000). For this project, where the aim is to determine the macroscopic response of steel from the individual grains, it has been proven that heterogeneity is the correct method to determine the materials response (Kovač et al., 2003).

3.2 Isotropy

A material is defined as isotropic when its properties are the same in all directions or are independent of the orientation of reference axes (Daniel & Ishai, 1994). The mechanical properties are not a function of the orientation at point in the body. The condition of isotropy requires a generalisation of material constituents. It is perfectly adequate when studying traditional engineering materials using continuum mechanics, at the macroscale. In reality, however, this cannot always be assumed to be the case.

When a material is observed at a microscopic scale it is inhomogeneous and contains many grains. Each grain is orientated at a different angle with respect to each other and as such, the properties will depend on the grain orientation. It has been proven that the macroscopic properties of a polycrystalline sample, which consists of individual grains of different shapes, sizes and orientations, are affected by the properties of the individual grains (Kamaya, 2009). Figure 3-2 illustrates the grain structure of a polycrystalline material.

It has been shown by Kamaya (2009) that the elastic deformation of a single crystal does not demonstrate isotropy, but in fact, demonstrates anisotropy and this depends on the orientation of the crystal. However, the macroscopic behaviour of a polycrystalline aggregate can be considered isotropic, in terms of elastic deformation, when there are a sufficient number of grains within the aggregate. Studies by Mullen et al., (1997) and Nygårds (2003) demonstrated that Young's modulus of a microstructure consisting of a small number of grains is dependent on the crystal orientations of the individual grains, in addition to the anisotropic elasticity of a single crystal. One of the aims of this project is to determine if the 36-grain aggregate is sufficient to give a macroscopic isotropic response.

3.3 Anisotropy

A material is anisotropic when its properties at a point, within a material, vary with direction or depend on the orientation of the reference axes (Daniel & Ishai, 1994). Therefore, an anisotropic material has mechanical properties that are different in all directions, at a point in the body. There are no planes of symmetry in an anisotropic body, unlike an isotropic specimen, which has an infinite number of planes.

When studying the microscopic response of polycrystal aggregates, results of Kamaya (2009), show that the macroscopic properties of the aggregate are determined by the properties of the individual grains. Studies by Kovač et al., (2003) used anisotropic elasticity to model the elasticity of the crystal grains.These studies were modelling the extreme deformations of steel and when the steel undergoes such deformations the orientations of the crystals in the steel material are influential parameters. At this point, the elastic and plastic properties of the metal differ, depending on the direction. Hence, anisotropic elasticity is the observed type elastic of response. In addition, anisotropy is the correct elastic type to use in Abaqus, when modelling the polycrystalline sample. Examples of the extreme deformation that steel undergoes are when it is used as a material in the pressure boundaryin a nuclear fission reactor, Liu (2006), and when it undergoes a process such as rolling(Timoshenko & Goodier, 1987). In both of these examples, it is required to understand the microstructure and the effect it has on the macroscopic properties.

Anisotropy elasticity is simply an extension of Hooke's law. It is slightly more complicated, and is presented in section 5.2.1, chapter 5.

3.4 Micromechanical Modelling

The engineering response of materials and small scale structures essentially results from the physics and mechanics of their underlying microstructures. Elastic, plastic and fracture properties are strongly governed by small scale processes, observed on the microscale. At this micromechanical level, multi- phase structures, voids, grains and interfaces play an intrinsic role (Eindhoven University of Technology, 2004).

It has been established that a polycrystalline aggregate is a heterogeneous material at the microscale because it consists of clearly distinguishable constituents (i.e., grains), that have different mechanical properties. At the microscale, continuum mechanics is not an adequate method of examining the mechanical properties, because modelling an object as a continuum assumes that there are no discontinuities in the material and that the microstructure is completely homogeneous. An alternative approach, known as micromechanical modelling is instead used. Micromechanics are described as the study of interactions of the constituents on the microscopic level (Daniel & Ishai, 1994). It deals with the state of deformation and stress in the constituents. The micromechanical approach is particularly relevant to this project because the elastic and plastic response of the steel aggregate is strongly influenced by the local characteristics. These local characteristics allow for the prediction of the average behaviour as a function of the constituent properties and local conditions (Daniel & Ishai, 1994). Thus, the micromechanical approach for the polycrystalline aggregate can be compared against the macromechanical response which is achieved from experimental data.

4.0 Literature Review

This chapter outlines work completed by other people that relates to the assumptions, values and designs within this report. Information discussed in this section is of relevance to the area of modelling the microscopic response of polycrystals. Presented in this literature review is information, methods and results from other authors that may be of interest to understanding the overall study of this project.

4.1 Basic Structure of Materials

In order to gain an understanding of the properties of materials, the structure of materials needs to be understood. The properties of materials are influenced by what occurs at the microscopic level and as such it is important to know how materials behave at this level. This section gives a brief description of atomic structures, with a consideration of the bonding that occurs between particles in solids and their arrangement in crystalline structures, hence; leading to a review of the basic structure of metals. The topics reviewed will be limited in their relevance to steel, as it is the material this project is focused on.

4.1.1 The Atomic Structure

All known matter is made up of three basic quantities; electrons, protons and neutrons. These are known as sub-atomic particles and together they form the atom. The protons are positively charged and together with the neutral neutrons, reside in the “core” of the atom known as the nucleus. The negatively charged electrons orbit the nucleus in orbital clouds (Higgins, 1994). According to Bohr (1913) the electrons in the orbitals have a certain amount of energy associated with them and as such, are said to be quantised. These electrons can absorb and emit energy. However, the electrons can only absorb/emit energy in certain discrete packets, termed quanta. This means the electrons have to move to energy levels that are equal to the energy that the electrons possess. This movement of electrons is a property which leads to chemical bonding.

4.1.2 Intra-Molecular Bonding

Intra-molecular bonding, or primary bonding, as it is commonly referred to, is the chemical bonding that binds individual atoms together. It is based on the Octet Rule whereby valence electrons are transferred or shared between atoms to satisfy a stable configuration(Higgins, 1994). If the stable configuration is satisfied, the result is the formation of a chemical bond containing ions. If an atom receives a negatively charged electron it becomes negatively charged (i.e., an anion). If an atom looses an electron it becomes positively charged (i.e., a cation). All metals are held together by primary bonds, and this is one of the reasons for high moduli. The most common primary bonds are; ionic, covalent and metallic.

4.1.3 The Metallic Bond

The metallic bond is of significant relevance to this project as it the type of bonding that exists in metals and in particular, the austenitic steel specimen that is examined and modelled on Abaqus. In general, metals have relatively high melting and boiling points and are good conductors of heat and electricity. Moreover, they possess elasticity and plasticity, which are properties studied in this project. All of these properties are related to the nature of the metallic bond. Atoms in a metal have valence electrons in their orbitals. As these valence electrons are not bound to any particular atom, the atoms form ions. The atoms of a metal are thus arranged such that their ions conform to some regular crystal pattern, similar to ionic structures, whilst the valence electrons move throughout the structure in a “sea of de-localised electrons” (Shackelford, 1988). This is illustrated in the figure 4-1.

Although the individual electrons are bound to the metallic atoms much less strongly than those present in an ionic bond, the shared electrons do bind the metallic ions very tightly into a crystal lattice structure. This is due to the presence of the electrostatic forces of attraction between the positively charged nucleus of the ions and the de-localised electrons. The lattice structure and the metallic bond are very important factors when studying elasticity and plasticity on a microscale. They are responsible for the movement of dislocations which is discussed in more detail at a later stage.

4.1.4 The Crystal Lattice

The crystal lattice structure is an extremely important atomic structure in the study of materials and their properties. At the atomic scale, traditional engineering materials are crystalline. This means that they are made up of many lattices repeating throughout the structure (Higgins, 1994). It is important to understand the origin and formation of this structure as elasticity and plasticity of the steel sample, studied in this project, is explained using the lattice.

The lattice structure is formed due to the nature of the metallic bond. The electrostatic forces between the positive ion cores and the negative de-localised electrons form the metallic bond. This, in turn, repeats throughout the structure giving a long range order pattern. The ions in a given material are arranged in a 3-dimensional, grid-like pattern to form a lattice structure as seen in figure 4-2.

Thus, the central feature of any crystalline structure is that it is regular, organised and repeating. There is a base structural unit present in every crystal material called a unit cell. The key feature of the unit cell is that it contains a full description of the structure as a whole. The complete structure is simply a repeated stacking of adjacent unit cells, face to face, throughout the whole structure in three-dimensional space.

The description of a crystal structure in terms of a unit cell means all crystal structures can be described by the unit cell shape and the way in which the atoms are stacked within a given unit cell. There are seven known unit cell shapes. Of concern to this project is the cubic crystal system, as that is the unit cell shape in steel. Furthermore, there exists fourteen possible arrangements for the stacking of atoms in a given unit cell, known as Bravais lattices (Shakelford, 2004). The atoms in the steel specimen studied in this project are arranged in a face-centered cubic Bravais lattice. This unit cell is commonly referred to as austenite. This unit cell repeats, in all directions, throughout the entire material to form individual grains of FCC steel. This study seeks to determine the response of the individual grains and determine if 36 grains of FCC steel are sufficient for the polycrystal aggregate to obtain an isotropic response. An FCC unit cell, removed from the crystal lattice of the steel specimen, is illustrated in figure 4-3.

4.1.5 Polycrystalline Structure

Almost all metals encountered in general engineering are polycrystalline. Instead of being large single-crystal structures, they are composed of many small crystals known as crystallites of grains. A typical grain might be about 0.3mm in diameter. Within this grain there could be approximately 1018 unit cells whose face would be parallel to one another just as in the case of a monocrystal (Keyser, 1986). However, where one grain meets another adjacent, neighbouring grain there will be a mismatching of unit cells because of the fact that each grain has a slightly different orientation that its neighbouring grains (see chapter 3, section 3.1). The region of mismatch is known as a grain boundary.

It has been suggested that the properties of metallic materials are not directional (i.e. isotropic) as long as the orientations of the grains are random and that there are a sufficient number of grains in the polycrystalline aggregate (Keyser, 1986). If this is the case; the properties measured represent the average directional properties of the randomly orientated grains of which the metal is composed of.

Similar studies to this have modelled the polycrystalline sample using a representative volume element and used Voronoi tessellation to generate the random grain structure within said aggregate (Kovač et al., 2003). This project will follow a similar procedure where the steel sample will be modelled using a RVE. Studies by Kamaya (2009) have shown that the macroscopic properties of a polycrystal are affected by the properties of the monocrystal (i.e., individual grain). He states “the elastic deformation of a single crystal exhibits anisotropy in metals and depends on the orientation of the crystal”. His study is also in agreement with Keyser (1986) as the macroscopic properties are regarded as isotropic and homogeneous when the materials have random crystal orientations. Kamaya (2009) states the above assumption does not hold true when the aggregate does not consist of a sufficient number of grains. He states that “Young's modulus of a microstructure consisting of a small number of grains is dependent on the crystal orientations of individual grains in addition to the elasticity of a single crystal”. Therefore it is one of the primary objectives of this study to determine the properties of the 36 grains and determine their effect on the macroscopic properties.

4.1

4.2 Elastic Deformation of Materials

When a material is subjected to an externally applied load it will deform by elastic deformation or plastic deformation or a combination of the two. In general, elastic deformation is a prerequisite for plastic deformation; however, there are some exceptions. The type and magnitude of the loading has a significant influence on the type of deformation a material undergoes. An elastic material will return to its original shape, once the external load has been removed. This condition holds true up to a certain point; the elastic limit or the limit of proportionality. Up until this point, Hooke's law can be applied and thus a material exhibits elastic deformation.

Elastic deformation occurs when an external force is applied to a material. The external force causes the ions in the metallic bonds to move small distances from their positions of equilibrium within the lattice structure. As the forces of attraction in the metallic bond are quite strong, a significant force is required to break the bonds of all the atoms in a crystal plane. This force leads to permanent, plastic deformation.

4.2.1 Anisotropic Elasticity

The elasticity of a material is expressed by the mechanical property, Young's modulus. The Young's modulus of a material is a measure of the stiffness of an isotropic, homogeneous material. Young's modulus is a measured on the macroscopic scale using the continuum mechanics approach. Studies by Kamaya (2009) have proved that this is correct to determine the overall elastic response of the polycrystalline aggregate. However, the individual grains within the aggregate exhibit anisotropic elasticity. Kovač et al., (2003) stated in their paper on modelling elastic-plastic behavior of polycrystalline grain structures of steels at the microscopic level that “Each crystal grain in a polycrystalline material is assumed to behave as a randomly oriented anisotropic continuum”. This is because at the microscopic scale the material is not homogeneous but, hetereogeneous and the mechanical properties of the grains vary with direction because of the orientation. Thus the elasic moduli are anisotropic within grains and crystals.

The linear elastic model is valid for small; strains normally less than 5% (Systèmes, 1998). According to (Hill & Rice, 1972) when using the small strain approximation there are certain assumptions for the mechanical properties of an elastic-plastic crystal at the macroscopic level that are adhered to. The distortion of the lattice is elastic, the crystal deforms by simple shears relative to specific lattice planes and directions and such ‘slip systems' are active only when the corresponding shear stresses attain critical values. This means the elasticity is modelled as linear elastic and that it is unaffected by slip. The onset of plastic deformation is brought about by the slip mechanism.

For this project the crystal grains are modelled with anisotropic elasticity, which is in agreement with the literature reviewed. To determine if 36 grains are sufficient to give the polycrystal specimen an isotropic, homogeneous response at the macroscopic level the average Young's modulus of the individual crystal grains is compared to the Young's modulus calculated for the aggregate sample using continuum mechanics. Previous research by Kamaya (2009) has shown that, when the number of grains is large enough, the Young's modulus of the polycrystalline body correlates well with the averaged Young's modulus of all grains.

4.3 Plastic Deformation of Materials

Plastic deformation occurs beyond the elastic limit of a material and it is a permanent deformation, meaning the material will not return to its original shape. A mechanical process known as yielding can be seen when a material undergoes plastic deformation. The reasons materials deform plastically are due to lattice imperfections which lead to a movement known as slip. A perfect crystal, with every atom of the same type in the correct lattice position, does not exist. All crystals have some defects. In fact, defects contribute to the mechanical properties of metals, and as such, can be advantageous. Defects can be introduced to a material through alloying, for example. A crystalline defect means there is an irregularity in the lattice structure. In existence, there are three different types of defects; point defects, linear defects and planar defects.

4.3.1 Point Defect

Point defects are structural irregularities in the crystal lattice. They can be classified as vacant, interstitial or substitutional point defect. These defects are illustrated in the lattice in figure 4-4 below. If a vacant point defect exists there will be one or more atoms missing from their site(s) in the lattice. An interstitial defect occurs when an extra atom has crowded its way into an interstitial void in the crystal lattice structure. Finally, a substitutional defect occurs when an atom of a different type than the lattice atoms is introduced and has replaced one of the bulk atoms in the lattice.

4.3.2 Linear Defects

Linear defects are of particular relevance to this study, because they have a significant influence on plastic deformation. Linear defects are commonly referred to as dislocations and are defined as a one-dimensional defect around which some of the atoms in a plane are misaligned (Callister, 2000). There are two types of known dislocations; edge dislocations and screw dislocations.

An edge dislocation, illustrated in figure 4-5, occurs when an extra half plane of atoms is introduced in the lattice. An edge dislocation is focused around the line that is defined along the end of the extra half plane of atoms. This is known as the dislocation line. Within the region around the line there is localised lattice distortion caused by the deformation of the metallic bonds.

If a load is applied to the exterior of the material minor atomic movements will occur causing the dislocation line to move through the lattice. It is important to note that the applied force must produce a stress exceeding the elastic limit for bond breakage to occur; otherwise there will just be bond stretching and vibration within the lattice. The movement of a dislocation is best explained in the following figure, figure 4-6. The load, in the form of a shearing stress, is applied and the dislocation moves a small amount at a time, breaking and forming bonds as it does so. The movement of the dislocation across the plane eventually causes the top half of the crystal to move with respect to the bottom half, forming what is known as a slip step, which can be seen on the exterior of a crystal. The mechanism by which a dislocation moves is known as slip.

4.3.3 Crystal Plasticity

Plastic deformation is a permanent deformation which occurs due to the movement of dislocations by the slip mechanism. Cubic metals, such as FCC steel deform predominantly by plastic shear. Slip in a crystal occurs on certain crystal planes in certain directions, namely; slip planes and slip directions, respectively. The slip direction in a metal crystal is the direction with the highest linear density of lattice points, or the shortest distance between lattice points. The slip planes are planes that have high wide interplanar spacing's and, hence, high planar densities (Van Vlack, 1990). For FCC materials the most densely packed plane of atoms are those of the {111} family and within each (111) plane there are three possible directions of slip. The combination of a slip plane and a slip direction is termed a slip system. There are four sets of planes of this type within a FCC structure, and coupled with the three possible directions, gives an austenitic material twelve slip systems. Therefore, irrespective of the direction relative to the crystal, if a direct stress is applied there will be resolved shear stress acting on several slip planes and at least one slip system will be activated so that plastic deformation can occur (Vernon, 1992).

Crystal plasticity is a method developed to study a material's heterogeneous plastic deformation based on modelling plastic slip on different slip systems within the crystal (Liu, 2006). It has been established by previous studies that isotropic plasticity is not an appropriate method for modelling the plastic deformation of an inhomogeneous material because it does not take into account any local anisotropic behaviour, which invariably, is the case when studying the microstructures of metals. The crystal plasticity is proposed because it is able to explain anisotropic behaviour observed in experiments. Therefore, the plastic deformation of the polycrystalline aggregate will be modelled with crystal plasticity which corresponds to the work done by Liu (2006) and Kovač et al., (2004). In these methods it is assumed that the plastic deformation is a result of crystalline slip only and the crystalline slip is driven by resolved shear stress.

5.0 Theory

This chapter derives, describes and explains the equations and corresponding theory that is necessary for the modelling and simulation of the deformation of the polycrystal sample. The micromechanical modelling process is quite complicated and a substantial amount of theory has to be established as a prerequisite to the numerical modelling process on Abaqus. This chapter features the governing equations for finite element analysis in addition to the complimentary equations that were essential to correctly model anisotropic elasticity and crystal plasticity. This chapter is divided into three sections; the first section describes the finite element procedure. The second section describes Euler rotations, which are necessary to fully describe anisotropy in crystals. Finally, the third section deals with the equations for the macroscopic response of the polycrystal aggregate, the anisotropic elasticity of the microscopic grains and the crystal plasticity of the grains.

5.1 The Finite Element Method

The finite element method is a numerical technique for solving the field equations of a body. In the context of material analysis, which is the scope of this project, the finite element method is used to calculate the displacements and stresses of the polycrystalline aggregate, or the individual grains. In addition, the material properties can be determined from the finite element results. The governing equation of the finite element method, known as the global matrix equation is given below (Kwon, 2005);

Ku=f (5.1)

Where: [K] = Element stiffness matrix

{u} = Nodal displacement matrix

{f} = Resultant force matrix

Abaqus calculates the displacements at the nodes of a given finite element. These displacements can be then used to determine the overall displacement of a given structure, consisting of many finite elements. In addition, the nodal displacements are necessary to calculate the strain and hence, the stress. The strain is calculated from the following equation;

ε=∂u∂x (5.2)

The stress can then be determined from the constitutive equation (i.e., the stress-strain relationship). This is given in the following equation;

σ=C{ε} (5.3)

Where: {σ} = Global stress vector

[C] = Stiffness matrix, and depends on the problem being analysed

{ε} = Global strain vector

5.2 Linear Elasticity

When a material undergoes elastic deformation it will return to its original shape once the applied load has been removed. This condition holds true up until the elastic limit of a given material is exceeded. Within the elastic range, Hooke's law applies; the deformation is proportional to the force applied. This can be extended to include the stress and strain state where the strain is proportional to the stress, and this relationship is linear up until the limit of proportionality is breached. The generalised Hooke's law can be expressed as follows;

σij=Cijkl.εkl

Where: σij = 2nd Order Stress Tensor

εkl = 2nd Order Strain Tensor

Cijkl = 4th Order Elasticity Tensor (i.e., stiffness matrix)

i, j, k, l = 1, 2, 3, indices

5.2.1 Anisotropic Elasticity

The state of stress at a point in a general continuum can be represented by nine stress components acting n the sides of an elemental cube as seen in figure 5-1.

In a similar fashion, the state of deformation can be represented by nine strain components. In an anisotropic material, linear elasticity is represented by the generalised Hooke's law as follows;

σ11σ22σ33σ23σ31σ12σ32σ13σ21=C1111C1122C1133C1123C1131C1112C1132C1113C1121C2211C2222C2233C2223C2231C2212C2232C2213C2221C3311C3322C3333C3323C3331C3312C3332C3313C3321C2311C2322C2333C2323C2331C2312C2332C2313C2321C3111C3122C3133C3123C3131C3112C3132C3313C3121C1211C1222C1233C1223C1231C1212C1232C1213C1221C3211C3222C3233C3223C3231C3212C3232C3213C3221C1311C1322C1333C1323C1331C1312C1332C1313C1321C2111C2122C2133C2123C2131C2112C2132C2113C2121 ε11ε22ε33ε23ε31ε12ε32ε13ε21 (5.5)

Where; equation 5.5 can be expressed in indicial notation as shown in equation 5.4, in the previous section, and Cijkl are the stiffness components. From equation 5.5, it can be seen that the stiffness matrix is a 9x9 matrix, and thus, in a general case, it would require 81 elastic constants (i.e., stiffness components) to fully characterise anisotropic elasticity in a material. However there exists symmetry of the stress and strain tensors;

σij=σji

εij=εji

This symmetry, demonstrated in equation 5.6 reduces the number of elastic constants to 36. This is seen in the following constitutive equation for an anisotropic body;

σ11σ22σ33σ23σ31σ12=C1111C1122C1133C1123C1131C1112C2211C2222C2233C2223C2231C2212C3311C3322C3333C3323C3331C3312C2311C2322C2333C2323C2331C2312C3111C3122C3133C3121C3131C3112C1211C1222C1233C1223C1231C1212ε11ε22ε33ε23ε31ε12 (5.7)

Contracted notationis a convenient method of representing tensor notation and thus, equation 5.7 can be represented as follows;

σ1σ2σ3σ4σ5σ6=C11C12C13C14C15C16C21C22C23C24C25C26C31C32C33C34C35C36C41C42C43C44C45C46C51C52C53C54C55C56C61C62C63C64C65C66ε1ε2ε3ε4ε5ε6 (5.8)

Equation 5.8 can be represented in indicial notation;

σi=Cijεj

Thus the stiffness matrix has 36 independent constants, as seen by the presence of a 6x6 matrix in equations 5.7 and 5.8. However, less than 36 of the constants are actually independent for elastic materials when the strain energy is considered (Jones, 1999). The work per unit volume is expressed as;

W=12Cijεiεj (5.10)

The constitutive equation in equation 5.9 can be obtained by differentiating equation 5.10;

σi=∂W∂εi=Cijεj (5.11)

By differentiating again, the following expression is obtained;

Cij=∂2W∂εi∂εj (5.12)

In a similar manner, by reversing the order of differentiation;

Cji=∂2W∂εj∂εi (5.13)

Therefore, from equations 5.12 and 5.13, it can be seen that the order of differentiation of W is immaterial and the following is proved;

Cij=Cji (5.14)

Equation 5.14 proves that the stiffness matrix is symmetric. The stress-strain relationship in equations 5.7 and 5.8 is expressed in terms of 21 independent stiffness components. Thus, to define anisotropic elasticity in a material, 21 elastic constants are required. Anisotropic elasticity is the most complicated form of elasticity, however, other forms of elasticity such as orthotropic and isotropic can be defined by the constitutive relationship, equation 5.8, albeit with less elastic constants in the stiffness matrix. For example, an isotropic material needs only 2 independent constants in the stiffness matrix (Jones, 1999).

5.3Euler's Rotations for Crystal Orientations

Euler angles and rotations are generally used to describe the orientation of a rigid body in three dimensional Euclidean space. In materials science, and for the purpose of this project, the random orientations of the crystal grains on the microscale are described using Euler angles. Thus, the orientation of a monocrystal in the polycrystalline aggregate can be described by three intrinsic rotations about a single axis. According to (Mathworld, 2010); “Euler angles can be used to represent the spatial orientation of any chosen reference frame as a composition of rotations from a reference frame of reference”.

For this particular project the local frame (i.e., the material crystal frame) is the reference frame that is being rotated. The reference frame of reference is the global frame (i.e., the macroscopic material frame), about which the rotations occur.

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