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By David Roylance

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A measure of strain often used in conjunction with the true stress takes the increment of strain to be the incremental increase in displacement dL divided by the current length L: d t = dL → l L t = l0 L 1 dL = ln L L0 (4) This is called the “true” or “logarithmic” strain. During yield and the plastic-flow regime following yield, the material flows with negligible change in volume; increases in length are offset by decreases in cross-sectional area. Prior to necking, when the strain is still uniform along the specimen length, this volume constraint can be written: dV = 0 → AL = A0 L0 → L A = L0 A0 (5) The ratio L/L0 is the extension ratio, denoted as λ.

Replot the the results of the previous problem using log-log axes as in Fig. 9 to determine the parameters A and n in Eqn. 8 for aluminum. 5. Using Eqn. 4. Does the material neck? Explain why the curve is or is not valid at strains beyond necking. 6. 221. 7. Use a Consid`ere construction (plot σt vs. λ, as in Fig. 10 ) to verify the result of the previous problem. 8. Elastomers (rubber) have stress-strain relations of the form σe = E 1 λ− 2 , 3 λ where E is the initial modulus. Use the Consid`ere construction to show whether this material will neck, or draw.

Strain energy The area under the σe − e curve up to a given value of strain is the total mechanical energy per unit volume consumed by the material in straining it to that value. This is easily shown as follows: U∗ = 1 V L P dL = 0 P dL = A0 L0 0 σd (9) In the absence of molecular slip and other mechanisms for energy dissipation, this mechanical energy is stored reversibly within the material as strain energy. When the stresses are low enough that the material remains in the elastic range, the strain energy is just the triangular area in Fig.

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