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1Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesMicrostructure-Properties: ILecture 4A:Mathematical Descriptions of Properties;Anisotropic Properties27-301Fall, 2007A. D. RollettMicrostructurePropertiesProcessingPerformance2Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesObjectives• The objective of this lecture is to explain themeaning of a material property and how todescribe the anisotropy of material properties.• Students are reminded of the properties oftensors and how to work examples ofanisotropic properties, including the effect ofcrystal symmetry.• A central topic is that of the effect of rotations,expressed as axis transformations, andsymmetry operators.3Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesMathematical Descriptions• [From L1] Some properties have rigorous physical definitions, with welldefined values, such as elastic modulus, thermal conductivity. Thereis a straightforward physical basis for the property and they can bepredicted based on a knowledge of the component atoms and atomicstructure. Nevertheless, most such properties are sensitive tomicrostructure. We discuss in this lecture these “Analytical Properties”since we can use analytical expressions of the type: Response = Property × Stimuluse.g. Stress = Modulus × Strain• Mathematical descriptions of analytical properties are straightforwardbecause of their linear nature.• Mathematics, or a type of mathematics provides a quantitativeframework. It is always necessary, however, to make acorrespondence between mathematical variables and physicalquantities.• In group theory one might say that there is a set of mathematicaloperations & parameters, and a set of physical quantities andprocesses: if the mathematics is a good description, then the two setsare isomorphous.4Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesApplication example: motion sensors• Many motion sensors have pyroelectricmaterials such as lead titanate (PbTiO3) andtriglycine sulphate ((NH2CH2COOH)3H2SO4).• The principle of operation is that variableheat input (temperature, stimulus) to thematerial results in variation in the electricpolarization (response).• Di = pi ∆T [C/m2]http://www.fuji-piezo.com/Pyro3.htm#High%20Sensitivity5Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesApplication example: quartz oscillators• Piezoelectric quartz crystals are commonly used for frequencycontrol in watches and clocks. Despite having small values ofthe piezoelectric coefficients, quartz has positive aspects of lowlosses and the availability of orientations with negligibletemperature sensitivity. The property of piezoelectricity relatesstrain to electric field, or polarization to stress.• εij = dijkEk• PZT, lead zirconium titanate PbZr1-xTixO3, is another commonlyused piezoelectric material.6Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesTopics• Response = Property x Stimulus• Linear Properties• Axis transformations (changing the coordinatesystem)• Scalars, vectors and tensors• Neumann’s Principle• Crystal Symmetry• Examples7Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesMath of Microstructure-Property Relationships• In order to describe properties, we must first relate aresponse to a stimulus with a property.• A stimulus is something that one does to a material,e.g. apply a load.• A response is something that is the result of applyinga stimulus, e.g. if you apply a load (stress), thematerial will change shape (strain).• The material property is the connection between thestimulus and the response.8Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesStimulus → Property→Response• Mathematical framework for this approach?• The Property is equivalent to a function, P, and the{stimulus, F, response, R} are variables. Thestimulus is also called a field because in many cases,the stimulus is actually an applied electrical ormagnetic field (or pressure, or force of some kind).• The response is a function of the field so we insertthe symbol P to designate the material property:R = R(F)≡R = P(F)9Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesScalar, Linear Properties• In many instances, bothstimulus and response arescalar quantities, meaning thatyou only need one number toprescribe them, so the propertyis also scalar.ModulusTemperature• To further simplify, some properties are linear, whichmeans that the response is linearly proportional to thestimulus: R = P × F. However, the property is generallydependent on other variables.• Example: elastic stiffness in tension/compression as afunction of temperature: σ = EYoungs(T) × ε ≡ R = P(T) × F.10Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesScalar, non-linear properties• Unfortunately not all properties are linear!• What do we do? In many cases, it useful to expandabout a known point (Taylor series).• The response function (property) is expanded aboutthe zero field value, assuming that it is a smoothfunction and therefore differentiable according to therules of calculus. ! R = P F( )= P0+11!"R"FF = 0F +12!"2R"F2F = 0F2+ K1n!"nR"FnF = 0Fn11Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesScalar, non-linear properties, contd.• In the previous expression,the state of the material atzero field is defined by R0which is sometimes zero(e.g. elastic strain in theabsence of applied stress)and sometimes non-zero(e.g. in ferromagneticmaterials in the absence ofan external magnetic field).• Example: magnetization ofiron-3%Si alloy, used fortransformers [Chen].12Intro.LinearProps..Non-Lin.MagntsmAnisotrpySymmetryRotationTensorsExamplesExample: magnetization• Magnetization, or B-H curve, in a ferromagneticmaterial measures the extent to which the atomicscale magnetic moments (atomic magnets, if youlike) are aligned.• The stimulus is the applied magnetic field, H,measured in Oersteds (Oe). The response is theInduction, B, measured in kilo-Gauss (kG).• As shown in the plot, the magnetization is a non-linear function of the applied field. Even moreinteresting is the hysteresis that occurs when youreverse the stimulus. For alternating directions offield, this means that energy is dissipated in thematerial during each


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CMU MSE 27301 - L4A_aniso_props_17Sep07

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