Foundations of Materials Science and Engineering
6th Edition
ISBN: 9781259696558
Author: SMITH
Publisher: MCG
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Chapter 6.13, Problem 8KCP
Distinguish between elastic and plastic deformation (use schematics).
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Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.
Chapter 6 Solutions
Foundations of Materials Science and Engineering
Ch. 6.13 - (a) How are metal alloys made by the casting...Ch. 6.13 - Why are cast metal sheet ingots hot-rolled first...Ch. 6.13 - What type of heat treatment is given to the rolled...Ch. 6.13 - Describe and illustrate the following types of...Ch. 6.13 - Describe the forging process. What is the...Ch. 6.13 - What is the difference between open-die and...Ch. 6.13 - Describe the wire-drawing process. Why is it...Ch. 6.13 - Distinguish between elastic and plastic...Ch. 6.13 - Define (a) engineering stress and strain and (b)...Ch. 6.13 - Define (a) modulus of elasticity, (b) yield...
Ch. 6.13 - (a) Define the hardness of a metal. (b) How is the...Ch. 6.13 - What types of indenters are used in (a) the...Ch. 6.13 - What are slipbands and slip lines? What causes the...Ch. 6.13 - Describe the slip mechanism that enables a metal...Ch. 6.13 - (a) Why does slip in metals usually take place on...Ch. 6.13 - Prob. 16KCPCh. 6.13 - What other types of slip planes are important...Ch. 6.13 - Define the critical resolved shear stress for a...Ch. 6.13 - Describe the deformation twinning process that...Ch. 6.13 - What is the difference between the slip and...Ch. 6.13 - Prob. 21KCPCh. 6.13 - Prob. 22KCPCh. 6.13 - What experimental evidence shows that grain...Ch. 6.13 - (a) Describe the grain shape changes that occur...Ch. 6.13 - How is the ductility of a metal normally affected...Ch. 6.13 - (a) What is solid-solution strengthening? Describe...Ch. 6.13 - What are the three main metallurgical stages that...Ch. 6.13 - Describe the microstructure of a heavily...Ch. 6.13 - Describe what occurs microscopically when a...Ch. 6.13 - When a cold-worked metal is heated into the...Ch. 6.13 - Describe what occurs microscopically when a...Ch. 6.13 - When a cold-worked metal is heated into the...Ch. 6.13 - Prob. 33KCPCh. 6.13 - Prob. 34KCPCh. 6.13 - Prob. 35KCPCh. 6.13 - Prob. 36KCPCh. 6.13 - Prob. 37KCPCh. 6.13 - Why are nanocrystalline materials stronger? Answer...Ch. 6.13 - A 70% Cu30% Zn brass sheet is 0.0955 cm thick and...Ch. 6.13 - A sheet of aluminum alloy is cold-rolled 30% to a...Ch. 6.13 - Calculate the percent cold reduction when an...Ch. 6.13 - Prob. 42AAPCh. 6.13 - What is the relationship between engineering...Ch. 6.13 - A tensile specimen of cartridge brass sheet has a...Ch. 6.13 - A 0.505-in.-diameter rod of an aluminum alloy is...Ch. 6.13 - In Figure 6.23, estimate the toughness of SAE 1340...Ch. 6.13 - The following engineering stress-strain data were...Ch. 6.13 - Prob. 49AAPCh. 6.13 - A 0.505-in.-diameter aluminum alloy test bar is...Ch. 6.13 - A 20-cm-long rod with a diameter of 0.250 cm is...Ch. 6.13 - Prob. 52AAPCh. 6.13 - Prob. 53AAPCh. 6.13 - Prob. 54AAPCh. 6.13 - Prob. 55AAPCh. 6.13 - Prob. 56AAPCh. 6.13 - A specimen of commercially pure titanium has a...Ch. 6.13 - Prob. 58AAPCh. 6.13 - Prob. 59AAPCh. 6.13 - Prob. 60AAPCh. 6.13 - Prob. 61AAPCh. 6.13 - Prob. 62AAPCh. 6.13 - Prob. 63AAPCh. 6.13 - Prob. 64AAPCh. 6.13 - Prob. 65SEPCh. 6.13 - Prob. 66SEPCh. 6.13 - A 20-mm-diameter, 350-mm-long rod made of an...Ch. 6.13 - Prob. 68SEPCh. 6.13 - Prob. 69SEPCh. 6.13 - Consider casting a cube and a sphere on the same...Ch. 6.13 - When manufacturing complex shapes using cold...Ch. 6.13 - Prob. 74SEPCh. 6.13 - Draw a generic engineering stress-strain diagram...Ch. 6.13 - (a) Draw a generic engineering stress-strain...Ch. 6.13 - Prob. 77SEPCh. 6.13 - Prob. 78SEPCh. 6.13 - Prob. 79SEPCh. 6.13 - The material for a rod of cross-sectional area...Ch. 6.13 - What do E, G, v, Ur, and toughness tell you about...Ch. 6.13 - A cylindrical component is loaded in tension until...Ch. 6.13 - Referring to Figures 6.20 and 6.21 (read the...Ch. 6.13 - (a) Show, using the definition of the Poissons...Ch. 6.13 - A one-inch cube of tempered stainless steel (alloy...Ch. 6.13 - Prob. 87SEPCh. 6.13 - Prob. 88SEPCh. 6.13 - Prob. 89SEPCh. 6.13 - Prob. 90SEPCh. 6.13 - Prob. 91SEPCh. 6.13 - Prob. 92SEPCh. 6.13 - Prob. 93SEPCh. 6.13 - Prob. 94SEPCh. 6.13 - Starting with a 2-in.-diameter rod of brass, we...Ch. 6.13 - Prob. 96SEPCh. 6.13 - Prob. 97SEPCh. 6.13 - Prob. 98SEPCh. 6.13 - The cupro-nickel substitutional solid solution...Ch. 6.13 - Prob. 100SEP
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- An aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.arrow_forwardProblem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forwardProblem 1 (35 pts). An elastic string of constant line tension1 T is pinned at x = 0 andx = L. A constant distributed vertical force per unit length p (with units N/m) is appliedto the string. Under this force, the string deflects by an amount v(x) from its undeformed(horizontal) state, as shown in the figure below.Force equilibrium in the string requires thatdfdx − p = 0 , (1)where f(x) is the internal vertical force in the string, which is given byf = Tdvdx . (2)(a) [10pts] Write down the BVP (strong form) that the string deflection v(x) must satisfy.(b) [2pts] What order is the governing PDE in the BVP of (a)?(c) [3pts] Identify the type (essential/natural) of each boundary condition in (a).(d) [20pts] Find the analytical solution of the BVP in (a).arrow_forward
- Problem 2 (25 pts, (suggested time 15 mins). An elastic string of line tension T andmass per unit length µ is pinned at x = 0 and x = L. The string is free to vibrate, and itsfirst vibration mode is shown below.In order to find the frequency of the first mode (or fundamental frequency), the string isdiscretized into a certain number of linear elements. The stiffness and mass matrices of thei-th element are, respectivelyESMi =TLi1 −1−1 1 EMMi =Liµ62 11 2 . (2)(a) [5pts] What is the minimum number of linear elements necessary to compute the fundamental frequency of the vibrating string?(b) [20pts] Assemble the global eigenvalue problem and find the fundamental frequency ofvibration of the stringarrow_forwardI need part all parts please in detail (including f)arrow_forwardProblem 3 (10 pts, suggested time 5 mins). In class we considered the mutiphysics problem of thermal stresses in a rod. When using linear shape functions, we found that the stress in the rod is affected by unphysical oscillations like in the following plot E*(ux-a*T) 35000 30000 25000 20000 15000 10000 5000 -5000 -10000 0 Line Graph: E*(ux-a*T) MULT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Arc length (a) [10pts] What is the origin of this issue and how can we fix it?arrow_forward
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