Write the combined motion in the matrix form

Assignment Help Mechanical Engineering
Reference no: EM131258886

Continuum Mechanics

Instruction: By default, the graduate students are required to answer all the question below. Question 1 part a) and Question 5 are not required for undergraduate students but are bonus problem for undergraduate students
1. Write both the expanded form and matrix form of

a) σij = Cijklεkl with σij = σji (stress), εkl = εlk(strain),Cijkl = Cklij(Stiffness tensor)

(hint: for Cijkl it has 81 components,

σij = σji (this symmetry will reduce from 81 to 54),
εkl = εkl (this symmetry will further reduce from 54 to 36), )
Cijkl = Cklij (this symmetry will further reduce from 54 to 21)

Question a) is not required for undergraduate students but is a bonus problem for undergraduates.

b) TijSij

where (i, j, k, l = 1, 2, 3)

2. In the Cartesian coordinate, the stress-strain relationship can be given as below for isotropic and linear elastic materials, please use the Einstein summation index notation to rewrite the long form of the following equations into one compact form equation.

ε11 = 1/E[σ11 - v(σ22 + σ33)], ε12 = 1+ v/E.σ12

ε22 = 1/E[σ22 - v(σ11 + σ33)], ε23 = 1+ v/E.σ23

ε33 = 1/E[σ33 - v(σ11 + σ22)], ε31 = 1+ v/E.σ31

3. Write the components of b = T.S.a (or b = (TS)a) in both index notation and matrix form, where a and b are vectors, T and S are 2nd order tensors.

4. Prove that

(a) ux( v x w ) = (u.w)v - (u.v)w

(b) u.(v x w) = w.(u x v) = v.(w x u)

(c) δijδjkδki = 3

where u, v, and w are vectors, "." and "x" are scalar products (or inner products or dot products) and cross product, respectively. δij (d) Is u x(v x w ) equal to (u x v )x w , why?

is the Kronecker Delta function.

5. (graduate students only) In the Cartesian coordinates, prove the vector identity ∇x(∇x u) = ∇(∇.u) - ∇2u

6. Three motions are applied to a material in the following sequence

(a) A translational rigid motion moving origin C0 respectively. There is no motion along Z

(b) Then stretching of

λ1 = L/L0, λ2 = HY/HY0, and λ3 = HZ/HZ0 along X, Y, and Z, respectively

(c) rigid rotation Ψ (positive counter clockwise CCW) in the to X, Y about C

Questions:

1) Write the combined motion in the matrix form
2) Write the deformation and displacement gradients in the matrix form
3) Change the sequence of (a)→(b) → (c) to
i) (a)→ (c)→ (b), i.e. translation-rotation-stretching
ii) (c)→(b)→(a), i.e. rotation-stretching-translation

Then, solve the corresponding questions in 1) and 2) and see if there is any difference.

1359_Figure1.jpg

Reference no: EM131258886

Questions Cloud

What is meant by deontological based ethical theories : Give an overview of what is meant by deontological, teleological, and virtue-based ethical theories. Explain which of these types of theories seems to be the most valid way of judging ethical conduct to you.
Solve the maximum flow problem by the labeling algorithm : Specify the residual network before each augmentation. After every augmentation, decompose the flow into flows along directed paths from node s to node t. Finally, specify the minimum cut in the network obtained by the labeling algorithm.
List and discuss android devices location sensors : Q1. List and Discuss Android devices location sensors. Q2. What are Adapters in android, explain different adapter views?
Designing a solution for a programming problem : In your opinion, what are the three biggest challenges in planning and designing a solution for a programming problem? What can you do to overcome these challenges? How would you apply these techniques to the programs in this class?
Write the combined motion in the matrix form : Write the combined motion in the matrix form and write the deformation and displacement gradients in the matrix form - Write the components - In the Cartesian coordinate, the stress-strain relationship can be given as below for isotropic and linear ..
Which activity is part of the operating cycle : Which activity is part of the operating cycle? Public corporations are businesses:
What is the total delay from the time : What is the total delay from the time you click on the webpage until the time that the entire webpage is displayed on your browser? Take into account all messages sent/received by the client host, including DNS, HTTP and TCP-related messages.
Create a presentation on the business you plan to start : Create a 7- to 10-slide Microsoft PowerPoint presentation on the business you plan to start. Provide the A memorable name for your business  in your presentation.
How does the terrorist threat affect emergency management : How does the terrorist threat affect Emergency Management? What issues can arise from dealing with terrorism from an emergency management perspective?

Reviews

Write a Review

 

Mechanical Engineering Questions & Answers

  Allowable bearing capacity of the foundation

a. Determine the allowable bearing capacity of the foundation. Let FS 5 3. b. Plot a graph of the ultimate bearing capacity qu if b is changed from 0 to 6 m.

  Processes for producing a good

A ?rm has two processes for producing a good, X. Each process uses ?ve chemicals: A, B, C, D, and E. The ?rst process, when   operated at unit level, produces 3 tons of X and uses 5, 4, 3, 2, and 1 tons of chemicals A, B, C, D, and E, respectively.

  Determine the minimum standard size bolt required

Three SAE Class 4.8 bolts (Sp = 310 MPa) are used to attach a rigid beam to a rigid support structure. Determine the minimum standard size bolt required from the Table provided

  Fundamental frequency of vibration of the building frame

If the operating speed of the motor is 1500 rpm, design the girder and the columns such that the fundamental frequency of vibration of the building frame is greater than the operating speed of the motor. Use two beam and two bar elements for the i..

  Liquid water at 180oc 2000 kpa is throttled into a flash

liquid water at 180oc 2000 kpa is throttled into a flash evaporator chamber having a pressure of 500 kpa. neglect any

  Determine the thermal efficiency and energy discharged

A power cycle receives 1120 Btu by heat transfer from a reservoir at 1080o F and discharges energy by heat transfer to a reservoir at 280 o F. The thermal efficiency of the cycle is 68% of that for a reversible power cycle operating between the same ..

  The length of the analysis is 15 years compute the simple

the owner of a new home is considering the economics of current and proposed designs. the current design is the

  Solid steel shaft d 20mm is subjected to a torque of 1200nm

solid steel shaft d 20mm is subjected to a torque of 1200nm. what size hollow circular steel shaft could be used to

  The prevailing wind velocity is 50 kmh the atmosphere is in

a 200-m-high 4-m-diameter stack emits 1000 kgs of 100 degree c gases into 5 degree c air. the prevailing wind velocity

  Find the overall efficiency of the pump disregard friction

a storage tank is 40 feet above a river. water is pumped from the river to the storage tank at a rate of 0.095m3s. the

  What is the minimum required length of the rod so that one

a high strength steel drill rod used for boring a hole in the earth has a diameter of 0.5 in. the allowable shear

  Is your pain point likely shared by other people

Recall bugs are the pain points that bother people, e.g., the BZ box Shark Tank video we saw during the lecture. Start from youself, what bothers you in your everyday life. Is your pain point likely shared by other people?

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd