STRESS ANALYSIS OF AN ISOTROPIC MATERIAL

(MECH 6441)

ByAhmad Abo-Mathkoor 6145531Asmita Dubey 9796924Daniel Modric 6062539Rohit Katarya 6306160

PROBLEM STATEMENT

Point Co-ordinates Before Loading Co-ordinates After Loading

A 0,0,20 0.0001, 0.0002, 20

B 30, 0, 20 30.0001, 0.0, 20.0004

C 30, 10, 20 29.9997, 10.0003, 19.9996

D 0, 10, 20 0.0004, 10.0009, 19.9995

E 0, 0, 0 0, 0, 0.0

F 30, 0, 0 30.0009, 0.0001, 0.00026

G 30,10,0 29.9996, 10.00033, 0

H 0, 10, 0 0.00011, 9.9996, 0.00021

I 0, 0, 10 0.00019, 0.00027, 9.9998

J 30, 5, 20 30.0006, 4.9997, 20.0005

K 15, 10, 20 15.0007, 9.9998, 20.0003

A block made of an isotropic material with dimensions of 30 mm X 20 mm X 10 mm is shown. The coordinates of each corner before and after loading with the addition two extra points (J and K)

The aim of the project

To determine displacements, stresses, strains, principle stresses and strains at the mid-point of each edge of the block.

To determine the change in stress distribution, principle stresses and strains, octahedral stresses at the midpoint of each edge due to temperature change.

To evaluate the most sensitive edge of the block due to temperature change.

Plot and discuss the results with increment of temperature by 5 degree in the range of 0-25 degrees.

Analyse the effect of temperature with increment of 20 degrees on change in octahedral stress of constraints (a) The bottom edge at the front face and (b) the top edge of the block at the rear face.

Property of an Isotropic material

An Isotropic material, has the same properties in every direction. Most material have mechanical properties which are independent of particular coordinate directions, and such material are called the isotropic material. When a solid body or a structure made of isotropic material possesses elastic symmetry that is the symmetric directions exist in the solid body.

Basic definitions and equations used

• Displacements

• strains, , +, +, +• Stresses

• Octahedral Normal and shear stress: [¿𝜎 𝑥

¿𝜎 𝑦

¿𝜎 𝑧

¿𝜏𝑥𝑦

¿𝜏 𝑦𝑧

¿𝜏 𝑥𝑧

]=[𝜆+2𝐺 𝜆 𝜆 0 0 0𝜆 𝜆+2𝐺 𝜆 0 0 0𝜆 𝜆 𝜆+2𝐺 0 0 00 0 0 𝐺 0 00 0 0 0 𝐺 00 0 0 0 0 𝐺

] [¿𝜀𝑥

¿ 𝜀𝑦

¿𝜀𝑧

¿𝜏𝑥𝑦

¿𝜏 𝑦𝑧

¿𝜏𝑥𝑧

]−[¿𝑐 𝛥𝑇¿𝑐 𝛥𝑇¿𝑐 𝛥𝑇

¿0¿0¿0

]

MATLAB programing for finding stress, strains with or without temperature effects

• The programming software MATLAB was used to calculate all of the objectives. various functions that the main program calls upon followed by a flow chart to help the reader understand how the main program works.

RESULTSDisplacementCoefficien

tValue ( * 10-3)

C0 0

C1 0.1167

C2 -0.0029

C3 0.2910

C4 -0.0280

C5 0.0330

C6 -0.0014

C7 -0.0047

C8 -0.0015

C9 0.0010

C10 0.0001Coefficients in the u direction

Coefficient Value ( * 10-3)D0 0

D1 -0.1033

D2 0.0036

D3 -0.2200

D4 0.0180

D5 0.0440

D6 -0.0017

D7 0.0021

D8 -0.0005

D9 0.0055

D10 -0.0002

Coefficients in the v direction

Coefficient Value ( * 10-3)

E0 0

E1 0.1087

E2 -0.0033

E3 0.2210

E4 -0.0200

E5 -0.0400

E6 0.0020

E7 -0.0016

E8 0.0002

E9 -0.0035

E10 0Coefficients in the w direction

STRAINS

Stress

 

Change in Octahedral Stress

AB BC CD DA BF FG GC GH HE EF DH AE0

5

10

15

20

25

Change in Octahedral Stress

AB BC CD DA BF FG GC GH HE EF DH AE0

102030405060708090

100

Change in Equivalent Stresses (TRESCA)

Variation of Temperature (0-25°C) in 5°C Increments

strains

Change in Octahedral Stress

Comparison

Two Constrained Edges

• No Temperature Change• To compare the effect of temperature change, it must first be

calculated without a temperature change. The figures in annex VIII show the principle stresses the principle strains and the octahedral stresses.

• 20°C Temperature Change• The figures in annex IX show the principle stresses the principle

strains and the octahedral stresses after the thermal loading.• Comparison of Octahedral Stress• The following figure shows the change in octahedral stress.