1、Continuous MediaZheng JiangChongqing UniversityApril-13-2016Outline1 Objectives2 Conservation And Continuity Equations3 Constitutive Equations4 Boundary and Initial Conditions1 ObjectivesnIntroduce the equations of conservation of mass, solute , momentum and energy.nIntroduce the principal equations
2、 for materials behaviornDefine the boundary conditions and initial conditions.n介绍质量守恒。n介绍材料特性（材料力学行为）的主要公式n解释边界条件和初始条件nThe equations of electromagnetism will not be covered here; we will discuss conservation of mass, of momentum, of energy and of solute. nNext, we will cover the most important const
3、itutive equations for material behavior that connect, for example: stress to strain or to velocity in deformation or flow problems; or enthalpy and heat flux to temperature in heat transfer calculations; or, finally, the flux of solute to concentration in diffusion.The derivation of these equations
4、will be done in the perspective of subsequent applications that may involve two different scales. 方程的推导将运用在两个尺度At the macroscopic level, the conservation and constitutive equations allow modeling, or even optimization, of an industrial process or simulation of the behavior of a sample undergoing mec
5、hanical testing.在宏观尺度，守恒和本构方程允许建模,甚至优化of一个工业过程或行为的模拟of进行机械测试的样本At the microscopic scale, the equations can be used to describe the formation and evolution of the microstructure (dendrites, spherulites, lamellae, or fibers, etc.), the interaction between a fiber and the matrix in composite materials,
6、 or the deformation of a crystal lattice around a dislocation.在微观层面,方程可以用来描述微观结构的形成和演化(树突,球晶、薄片或纤维,等等),纤维和基质之间的相互作用in复合材料or 一个位错周围的晶格的变形。HomogenizationRollingAnnealCasting 1mEngine Block 1-10 mmMacrostructureGrainsMacroporosityPropertiesHigh-cycle fatigueDuctility 100-500 mMicrostructureEutectic Pha
7、seMicroporosityIntermetallicPropertiesYield strengthTensile strengthHigh-cycle fatigueLow-cycle fatigueThermal GrowthDuctility10-100 Atomatic StructureCrystal StructureInterface StructurePropertiesThermal GrowthYield strength3-100 nmNanostructurePrecipitatePropertiesYield strengthThermal GrowthTensi
8、le strengthLow-cycle fatigueDuctility Based on processing flowchart Based on the metallurgical length scalesFigure 1 The different scales appearing in materials science. A turbine blade (a), solidified in a ceramic mold (investment casting), measures a few dozen centimeters. It is composed of grains
9、 which are clearly visible after chemical etching (scale: a few millimeters), which themselves are made up of dendrites which are spaced at a few dozen to hundreds of microns (b). In (c) is shown a schematic at atomic scale of the transformation from a liquid to a solid for a metal alloy during the
10、formation of such a blade by precision casting.1.2 Conservation and Continuity EquationsFigure 2 In the continuous casting of aluminum (a)，liquid metal is injected from a nozzle through a distribution bag which filters out inclusions and oxide skin debris. The metal cools on contact with a mold, whi
11、ch itself is cooled by circulating water and water spraying. Solid metal is extracted continuously by a jack attached to a bottom block. Convection in the liquid (indicated by arrows) can transport growing grains of aluminum in the mushy region (b) The environment is not homogeneous as it is compose
12、d of at least two phase, namely solid and liquid. The phenomena occurs at multiscale. It can be described by four conservation equations (mass, momentum, energy and solute) The type of behavior (elastic, plastic, viscous, etc.) and also the values of thermo-mechanical properties of the material (spe
13、cific mass, viscosity, elastic modulus, strain-hardening coefficient, thermal conductivity, etc.) enter into the constitutive equations for the different phases of the materials. NoticeFigure 3 A few areas of materials science where modeling plays an important role: polymer injection (a), water diff
14、usion in concrete(b)， deformation of a test specimen under tension (c)1.2.1 Definition“Nothing is lost, nothing is created, everything is transformed” - Lavoisiers principleFigure 4 Diagram of the calculation domain n and its boundary on (a). The outgoing normal vector n and the tangent vector are a
15、lso shown. In three dimensions (b) ，there two tangent vectors. In (c)，the volume element V is shown with the velocities in the media at each face for the derivation of the conservation equations.的计算范围和边界?是图a，向外的法向量和切向量也如图示；在b中的三维图里，有两个切向量；c中，体积元素V 和它在介质中每个表面的速度一起显示出来，用于本构方程的推导。 In the first, called
16、the Lagrangian, analogous to traditional mechanics, we follow the material element through its movements (For solid deformation);拉格朗日，类似于经典力学，我们遵从材料元素的运动 Secondly, Eulerian, the reference frame is fixed and we watch what happens at a point as a function of time. (For fluid mechanics) 欧拉，修正了参考坐标系，并且我
17、们能观察到随时欧拉，修正了参考坐标系，并且我们能观察到随时间发生了什么间发生了什么 Two possible approached to describe conservation (of mass, energy, momentum, etc) As long as the domain V is small, we can make the assumption that, on each face, the values considered are constant. V足够小时，我们能够假设，在每个面（所取的足够小时，我们能够假设，在每个面（所取的V ）上，值是恒定不变的。）上，值是
18、恒定不变的。1.2.2 Equation of conservation of massThis sum includes only the two contributions expressing that the mass variation inside the element must be due to transfer of mass across the faces by the velocity field v. There is no diffusion, nor production of mass. The mass variation inside the elemen
19、t is given by: The total quantity of material leaving the volume V is given by: The integral (1.2) can be manipulated as: There is neither loss nor creation of material : Stationary case: Incompressible:1.2.3 Conservation of solute As for the mass sum derived above, a temporal variation of the quant
20、ity in solution in the volume element as well as a transport term containing the velocity field will appear in the sum. Two new contributions need to be taken into account. The first is a diffusive term扩散项扩散项 associated with concentration gradients. The second is a source term (or sink) for chemical
21、 reactions.is the number of moles that appear or disappear locally perunit time and unit volume. We obtain the local equation for the conservation of solute. For mass specific concentration.1.2.4 Conservation of momentumFigure 5 The surface forces, T, and the gravity force, g, acting on a small volu
22、me element V. Taking the sum of the contributions to the forces acting on the surface and on the volume, we obtain for the component x: The momentum flux in the x direction entering or leaving the domain is given by: We obtain for the x component of the conservation of momentum: All three directiona
23、l components of the momentum can be expressed as follows: For quasi-static problem:1.2.5 Displacement, strain, strain rate Over a period of time, displacements of the material occur due to the applied stress and volume forces. In Lagrangian coordinates, the movement of the material is described by t
24、he set of trajectories of all the points in the material: ,where xo is the initial position of a point at time t = 0 The displacement u(xo,t) of a point xo at time t is naturally defined as the difference between its position at time t and its initial position:Figure 7 Forging of a solid to obtain a
25、 complex form (a) and two-dimensional representation of the displacements (b) for such a processTo simplify the expression, it is necessary at this stage to switch to the coordinate notation (x1,x2,x3) in place of (x, y, z).Where is the Kronecker delta functionThis equation defines the gradient tens
26、or of the transformation F = I + Grad u.The result is a symmetric second order tensor, a 3 x 3 matrix called the right Cauchy-Green strain tensor, or Cauchys dilatation tensor,Returbing to the (x, y, z) coordinates, the symmetric strain tensor is written:Figure 8 Illustration of the different elemen
27、tary strain components for a parallelepiped.Where designates the velocity vector associated with the displacement vector u .This velocity is simply that of a point in Lagrangian coordinates: Engineering strainTheengineering normal strainorengineering extensional strainornominal straineof a material
28、line element or fiber axially loaded is expressed as the change in length Lper unit of the original lengthLof the line element or fibers. The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have e is the engineering normal strain, L is the
29、 original length of the fiber and is l is the final length of the fiber. Stretch strainThe extension ratio is approximately related to the engineering strain byThis equation implies that the normal strain is zero, so that there is no deformation when the stretch is equal to unity.The extension ratio
30、 is approximately related to the engineering strain byThetrue strain, (although nothing is particularly true about it compared to other valid definitions of strain). Considering an incremental strain.the true strain is obtained by integrating this incremental strain:whereeis the engineering strain.
31、The true strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path. True strainThe Green strain is defined as: Green strain Almansi strainThe Almansi strain is defined as: Normal strainConsider
32、 a two-dimensional infinitesimal rectangular material element with dimensions , which after deformation, takes the form of a rhombus. From the geometry of the adjacent figure we haveFor very small displacement gradients the squares of the derivatives are negligible and we haveThe normal strain in th
33、e x direction of the rectangular element is defined bySimilarly, the normal strain in the y direction, and z direction becomes Shear strainThe engineering shear strain ( ) is defined as the change in angle between lines and , thereforeFrom the geometry of the figure, we haveFor small displacement gr
34、adients we haveFor small rotations, i.e. and are , we have tan , tan , therefore By interchanging x and y and ux and uy, it can be shown that , similarly, for the y-z plane and z-x plane, we haveGeneral three-dimensional body with an 8-node three-dimensional element1.2.6 Virtual powerThe body is loc
35、ated in the fixed (stationary) coordinate system X,Y,Z Considering the body surface area, the body is supported on the area Su with prescribed displacements USu and is subjected to surface forces fsf (forces per unit surface area) on the surface area Sf.The body is subjected to externally applied bo
36、dy forces fB (forces per unit volume) and concentrated loads Rc (where i denotes the point of load application). We introduce the forces Rc as separate quantities, although each such force could also be considered surface tractions fsf over a very small area.Ingeneral,theexternallyappliedforceshavet
37、hreecomponentscorrespondingtotheX,Y,Zcoordinateaxes:U = USu on the surface area. The strains corresponding to U are C is the stress-strain material matrix and the vector denotes given initial stresses1.2.7 Conservation of EnergyIn general, the first law of thermodynamics tells us that the variation
38、of total energy of a domain under consideration is due to the mechanical power of the external forces, Pmech, and the caloric power applied, Pcal The total energy is composed of the kinetic energy, Eb and the internal energy, Ej. Putting all the terms together:The sum of the first three terms on the
39、 right hand side of the equation is the deformation power.1.2.8 Unified form of the conservation equationsThe conservation equations derived above are similar in that they all contain a temporal variation term, an advective transport term, and some have a diffusion term and a source term. They can b
40、e reduced to a single general equation:1.3 Constitutive EquationsIn physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities.1.3.1 Constitutive equations for mass1.3.2 Constitutive equations for soluteFigure 9 Diagram of diffusion accord
41、ing to Ficks first law relating the chemical flux to the concentration gradientThe flux can generally be expressed in terms of the gradient of the concentration by Ficks first law:where Dj is the diffusion coefficient.1.3.3 Constitutive equations for EnergyConsidering the pressure constant, which is
42、 a good approximation for condensed matter, the specific enthalpy of a phase, H, is given by:where K is the thermal conductivity of the material.The diffusive heat flux, jT, is given by an equation similar to Ficks firstlaw (Fouriers law):where E is the electrical field (Voltage), jE the electrical
43、current density, and E electrical resistivity.In the case of chemical reactions, the heat source term becomes a sum over all the chemical species:The source term introduced by a phase transformation will be treated in detail in chapter 5. Strictly speaking, the latent heat per unit mass, L/, associa
44、ted with a phase transformation ，is not a volumetric source term as it is produced at the moving interface /.1.3.4 Constitutive equations for Materials: quasi-static caseIn the simplest case, that of linear elasticity, we try to relate the stress tensor to the strain (x, y, z). Being symmetric, (ij
45、= ji and ij=ji), these two tensors only have six independent components. In the coordinates (x, y, z)，Hookes law relating the six components takes the form:The matrix Del is the elasticity matrix.where E is the elastic modulus and vp is Poissons coefficient.Assuming that the material is stretched or
46、 compressed along the axial direction：A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poissons ratio of . The green cube is unstrained, the red is expanded in the x direction by due to tension, and contracted in the y and z directio
47、ns by L due to tension, and contracted in the y and z directions by L.The relative change of volume V/V of a cube due to the stretch of the material can now be calculated. Using V=L3 and V+ V = (L+L)(L-L)2(x + x)1-2 = x1-2 + (1-2)x-2 x - 2(1-2) x-2-1 x2 The problem encountered in materials science i
48、s that most of the time thematerial does not have linear elastic behavior; rather, it can undergo plastic strain, pl. In addition to this component, others of a thermal nature ，th， or associated with phase transformations，tr(volume changes), can contribute to the local total strain, , given by (1.36
49、). In general, one has:1.4 Boundary and Initial Conditions1.4.1 GeneralitiesIn a non stationary problem, the first term of the equation requires the specification of the values of the field (x, t = 0) at every point in the domain at time t = 0. This is the initial condition.In the same equation, the
50、re are two integrals on the boundary of the domain: the first involves the normal velocity component, vn = vn, and corresponds to the transport of the quantity across the surface, whereas the second is related to the flux entering or leaving the domain at the surface by diffusion, j = jn. n. It can
