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hw6sol - Materials Science and Engineering 765 mechanical...

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1/7 Anderson G4:Users:peterand:Central files:Courses:MSE 765:2005:hw6_sol.doc Materials Science and Engineering 765 mechanical behavior of materials Spring 2005 Homework 6 (w/solutions) (w/corrections 6.12.05) Due: Wednesday, June 8, by 9am. Show all work necessary to arrive at your answers. 1. Conversion between True and Engineering Stress-Strain Response Recall a common relation for uniaxial true stress vs uniaxial true (logarithmic) plastic strain: σ = σ 0 ε p ε 0 n + 1 , where σ 0 , ε 0 are the initial yield strength, initial yield strain and n is the hardening exponent. a. Plot the true stress-true strain response for a material with an initial yield strength of 100 MPa, Youngs modulus E = 100 GPa, and strain hardening exponent n = 0.1, from ε = 0 to 1. You can ignore the contribution of elastic deformation to a change in cross sectional area. According to the constitutive relation, σ = σ 0 when the plastic strain is zero. Therefore, σ 0 in this relation is better described as the limiting stress for elastic behavior. The reference strain ε 0 is approximated by σ 0 /E = 10 -3 . The total true strain is the sum of elastic strain (= σ /E) and true plastic strain ( ε p ). The MathCad program below computes the true stress-true logarithmic strain behavior in uniaxial tension. σ 0 100 E . 100 10 3 ε 0 σ 0 E n 0.1 σ ε p . σ 0 ε p ε 0 n 1 ε ε p ε p σ ε p E ε eng ε p e ε p 1 σ eng ε p . σ ε p e ε p
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2/7 Anderson G4:Users:peterand:Central files:Courses:MSE 765:2005:hw6_sol.doc 0 0.5 1 0 200 400 strain σ ε p σ eng ε p , ε ε p ε eng ε p 0.05 0.06 0.07 0.08 236 238 240 strain σ eng ε p ε p b. Plot the corresponding engineering stress-engineering strain response. Superimpose your result on the same axes as in part (a). Note that engineering quantities can be defined in terms of true quantities. The derivations below assume a deforming section of length l and cross sectional area A , with initial values given by subscript "0". Values associated with plastic deformation are denoted by superscript "p". The MathCad code above implements these equations. ε eng = Δ l l 0 = e ε 1; σ eng = F A 0 = σ A A 0 σ l 0 l p = σ e ε p c. At what plastic strain is localization expected to begin? There are two possible ways to answer this question. The Considére analysis states that necking occurs when the force carried by the specimen reaches a maximum, due to the competition between hardening and area reduction. Graphically, this occurs in the plot when the engineering stress reaches a maximum or when the condition d σ /d ε p = σ is reached on a true stress-true plastic strain curve. The expanded plot below suggests the engineering stress peaks at ε p = 0.06. One might attempt to get an analytic expression using d σ /d ε p =
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