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MIT OpenCourseWare http://ocw.mit.edu For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms3.22 Mechanical Properties of MaterialsSpring 2008.������� ������� Problem Set #3 Due: Friday, March 21 by 5:00 PM 1. Because metallic glasses have an amorphous atomic structure, many researchers believe they cannot “strain harden” like their crystalline counterparts (a topic we will cover in depth later). At present, you can understand this to mean that the stress required to continue deformation of the material to strains greater than the yield strain �(σy) increases monotonically, often modeled as a power-law of the form σ = K�np where �p is the plastic strain, n is the material strain hardening exponent, and K is a material constant related to σy. In other words, after yielding the material becomes “harder (i.e. more difficult) to strain.” Regardless of this assumption, you decide to have a UROP in your group conduct some experiments to consider whether strain hardening does occur in a particular metallic glass you have synthesized. You have the UROP compress a cylindrical sample of bulk metallic glass that is 6 mm long and 3 mm in diameter to failure. After the tests, the UROP excitedly presents the data to you (which can be found in 3-1.xls ), claiming to have proven with these data that strain hardening in metallic glasses is possible! As she hyperventilates talking about the Nature article she wants to write, you realize that she is wrong (most likely because she did not have the opportunity to yet take 3.22). (a) Explain why she is wrong. Do this by calculating and discussing the σ − � response(s) and the strain hardening exponent n for the material. Compare your results with what is expected of a material that does strain harden. (b) Determine the elastic and plastic strain at failure. (c) Not one to give up easily, you think that you may be able to get the metallic glass to strain harden by applying a cyclic load (with the minimum load being 0 N) through the plastic region. Show what the resulting stress-strain curve would look like if you applied three cycles before failure. 2. You are given a material and told that it is elastically isotropic and exhibits a yield strength σy of 950 MPa. The material is being used in an application where it experiences a stress state of σij = 0 0 300 300 0 −400 0 0 MPa −800 (a) According to the von Mises and Tresca criteria, respectively, does the material yield? (b) Plot the yield surfaces (on the same graph) of the material based on the von Mises and Tresca criteria. (c) You decide to examine the material more closely by performing your own uniaxial ten-sion and compression tests. The stress-strain data for these tests are found in 3-2.xls. Plot these data on the same graph as you made in (b). Discuss how the von Mises and Tresca criteria predict yielding in the material. Is there a better criterion to use for this material? Explain. 3. Noting that the von Mises and Tresca criteria deviate for specific stress states in the yield surface considered above, determine the following: 3.22 Mechanical Behavior of Materials 1 Prof. K.J. Van Vliet� (a) The stress tensor σij corresponding to the stress state(s) at which there is maximal dif-ference between these two yield criteria, expressed as a matrix and as a representative volume element (RVE) of the material. This will correspond to the biggest gap(s) be-tween these two predictions in the graphical yield surface. (b) The magnitude of this difference, expressed first in terms of an algebraic expression including the σy of the material and second in terms of a percentage basis [%]. (c) Which of these two criteria more accurately predicts the stress states corresponding to material yielding in pure metals and metallic alloys, and why does this tend to be the more accurate predictor? To answer this, you will need to rely on and show/cite experi-mental data for selected materials. 4. The yield strength σy of many materials is poorly described by von Mises and Tresca; these include porous materials, glassy polymers, granular media, and biological materials. For such materials, σy depends in part on the state of hydrostatic stress σii and thus the first invariant of σij , I1. Here, you will explore some of these pressure-sensitive yield criteria. (a) Mohr and Coulomb discussed continuum mechanics and particle interactions, respec-tively. Their ideas enabled modification of the yield criteria to reflect the fact that σy may differ in uniaxial tension vs. compression. This criterion is commonly used for bulk metallic glass and soils, which fail at lower applied stresses in tension than in com-pression. It linearly relates the effect of superposed normal and shear stresses on the stress state required to yield the material, and Mohr’s circle helps us easily understand this. Assume an imposed principal stress state in a material plane; Mohr-Coulomb ex-presses the yield criteria in terms of the stress state at maximal shear stress resulting from σ1 and σ3, and can be expressed as the material yielding when: τ =< σ > sin α (1) where < σ > is the average normal stress corresponding to τ, and (τ, < σ >) is the coordinate on Mohr’s circle defined at the point of max shear stress for defined σ1 > 0 and σ3 > 0, and α is the friction angle. Physically, α is the angle that would naturally arise if you poured the granular material onto a surface due to friction between the particles, and is usually between 15o and 50o. In terms of Mohr’s circle, it is the angle between the vertical line connecting (τ, < σ >) to the τ = 0 horizontal axis, and the tangent of the circle that intersects the point (τ = 0, σ = 0). Draw this Mohr’s circle, and restate the Mohr-Coulomb yield criterion only in terms of these normal principal stresses and the material friction angle. (b) Drucker and Prager developed the creatively named Drucker-Prager yield criterion orig-inally for soil mechanics and it is often applied to composite materials such as cement or concrete. It accounts for normal and shear stresses, and can be stated as: f = J2 − αI1 − k (2) where J2 is the second invariant of the deviatoric stress tensor sij and α and k are mate-rial properties that reflect the friction angle and cohesion,


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