Lecture 21 Pressure stress elastic moduli and density A very broad classification of the states of matter is Solid liquid gas We all know examples of these states so for example at room temperature steel is a solid water is liquid and oxygen is a gas We also know the basic differences between these states of matter Solids do not flow they resist a shear force Liquids flow but resist compression or expansion Gases expand and fill any container they are put in During the next 3 weeks we will be studying the properties of solids liquids and gases There are many subdivisions within the broad classes above for example it is possible to make glasses which have an atomic structure like a liquid but which do not flow at least on the time scale of a human being These materials are called amorphous solids There are also many different types of crystal structures for solids and the particular crystal structure plays a big role in determining the properties of the solid e g in determining whether the material is a conductor or an insulator As we change from a solid to a liquid or from a liquid to a gas properties such as density elastic moduli pressure volume etc usually change Before we can talk about these changes we need to introduce each of these properties more precisely Pressure and stress To get started we need to understand pressure and stress which are better for describing the mechanical behavior of solids Pressure is related to force through P P ressure F orce area 1 Stress is also force area but now the force is applied in different ways Uniaxial tensile stress is a stress which tries to stretch a object and the shear stress tries to shear it We then have U niaxial stress T ensile F orce Area 2 Shear stress Shear F orce Area 3 and Note that these forces are applied in a manner that ensures that the material sample does not translate or rotate The unit for pressure or stress is the 1 Pascal Pa which is equal to N m2 Consequences of applying pressure or stress When we apply a compressive pressure P to a material it usually gets smaller In technical terms we say that pressure or compression leads a volume change If its initial volume is V then its volume change is V Every material is characterized by its compressibility and the variable which is used to describe this is called the bulk modulus B Squishy materials have a low bulk modulus and are easy to compress while hard materials have a large bulk modulus The precise relation between these quantities is P B V V 4 Notice that the V is negative because the volume decreases Also since V V is dimensionless the bulk modulus has the same units as the pressure The bulk modulus is one of the so called elastic moduli and the reason for this can be understood by considering the case of applying uniaxial tension to a piece of material In that case we have Ftensile L Y A L0 5 In this case the piece of material of length L0 extends by a distance L when a tensile stress is applied to it Now notice that this looks like the external force required to stretch an elastic spring F kx 6 By comparing the two equations we can see that k Y A L0 is the effective spring constant of a material of cross section A and of length L0 When a shear stress is applied to a piece of material it leads to a shear distortion x h Fshear x S 7 A h where S is the shear modulus Density 2 Finally we define the density which is the mass per unit volume M V 8 The density of water at 40 C is 103 kg m3 and the specific gravity of a material is the ratio of the density of the material to the density of water at 40 C 3
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