Materials PHYA2

MATERIALS DENSITY, SPRINGS, STRESS AND STRAIN Topics 11, pp.162–173

Studying materials in terms of their –bulk properties –and tensile strength Materials scientists then go on to link these bulk properties to the atomic/ molecular level, but we don’t get that far in this course.

Density Density = mass per unit volume ρ = m/V Units kg m –3 (in Chemistry g cm –3. Factor different?) How do you find the density of (i) regular solids, (ii) irregular solids, (iii) liquids, (iv) gases? Now do Summary Q1, p.163

Measuring density Displacement methods to find irregular volumes Density of alloys: –mass of A, m A = ρ A V A, mass of B, m B = ρ B V B Total volume = V Total density = ρ A V A + ρ B V B V Use this method for layers, eg: Have a go at summary Q4

Hooke’s Law The force needed to stretch a spring is directly proportional to the extension of the spring from its natural length. F = kΔL k is Hooke’s constant for the spring = spring constant = stiffness How can you measure k?

Hooke’s Law A graph of force vs extension is…? a straight line through the origin i.e., directly proportional Gradient = k extension force Note: We have exchanged usual arrangement of axes for dependent/independent variables. Why?

Hooke’s Law What mass would you need to hang from a spring of stiffness (k) 49 Nm –1 to achieve an extension of 0.6 m? What is the extension of a spring with spring constant of 120 Nm –1 if a mass of 40 g is added?

Spring combinations Two springs in parallel, equal extension What is the new effective stiffness k? Force on P: Force on Q: Total Force: So

Spring combinations Two spring in series What is the new effective stiffness k? Both springs loaded with force W Extension of P: Extension of Q: Total extension: So

Now try Summary Q2 and Q3, p.166

Energy stored in a spring Elastic PE is stored in a stretched (or compressed) spring Energy stored = work done Work to extend length by  L is area under graph extension force LL F

Some jargon to learn Tensile deformation: stretching Compressive deformation: squashing Stiff: hard to deform Flexible: easy to deform Elastic: returns to original shape when force is removed Plastic: permanently deformed when force is removed

Some more jargon to learn… Plastic materials are also: –Ductile: can be drawn into wires –Malleable: can be hammered into sheets Malleable materials are tough: –They give way gradually and need a lot of energy to break them The opposite of tough is brittle: –These materials do not deform plastically but crack or shatter suddenly

Deformation of solids How much a sample deforms depends on: –The material –The force applied –The dimensions of the sample

Stress and strain To remove the dependency on dimensions it is useful to define: Tensile stress (  ): –Tension (force) per cross-sectional area –Units Nm –2 or Pa (like pressure) Tensile Strain (  ): –Fractional extension –No units (a ratio) Hooke’s Law: 

Young’s Modulus We can now define a material property similar to stiffness but independent of geometry: The Young Modulus unit: Pa E is given by the gradient of the elastic part of a stress-strain graph

Measuring Young’s Modulus Measure extension of wire for different loads What else do you need to measure? –How are you going to measure them? Obtain a value for E Cu and estimate its uncertainty

Measuring Young’s Modulus Reminder of reading micrometer scales What is the uncertainty in the measurement?

Stress-strain curve for steel Elastic deformation (gradient = E) Limit of proportionality Elastic limit Yield point Ultimate tensile strength fracture Onset of work hardening

Examples Material UTS E Aluminium 80 Mpa 71 GPa Gold 120 MPa 71 GPa Mild steel 460 MPa210 GPa Steel piano wire3 000 MPa210 GPa Lead 15 MPa 18 GPa Bone 140 MPa345 GPa Concrete ~4 MPa 14 GPa Glass ~100 MPa 71 GPa LD polythene 13 MPa 0.18 GPa

Exam questions [8], June ‘07

Different ways of deforming Describe how these materials behave under stress as shown in the graph. What might they be? Have a go at Summary Q4, p.169 (careful with axes!)

The diagram shows tensile stress-strain curves for three different materials X, Y and Z. For each material named below, state which curve is typical of the material, giving the reasoning behind your choice. (a)copper (b)glass (c)hard steel

Loading and Unloading a.On unloading, the elastic deformation is reversed, but plastic deformation remains b.Unloading curve below loading, remains elastic (“hysteresis”). c.Mainly plastic deformation.

Strain energy We do work to stretch a metal wire stretched within its elastic limit –This is stored as elastic energy –The amount is equal to the area under the load-extension graph –When the stress is released all the stored energy can be recovered Beyond the elastic limit, some energy cannot be recovered – Where has it gone?

Stretching a rubber band Is the area greater under the loading or unloading curve? What does this tell you about the work done and energy released? How can we work out and account for the difference?

Practice of stress and strain: SQs p.169 Practice of curves: SQs p.171 Now make sure you can do the exam-style questions on pages After that you can try Materials Revision 1