Mechanical Properties of Materials
𝐴
𝐴
Is 𝑘 a good measure of the
material property of the rod?
Tensile Test
𝑘 =?
45°
𝐶
𝐶
𝐵
𝜃
𝑎
𝑘 =?
𝐵
𝐹
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台大機械系材料力學 一盧中仁
Rod as a Linear Spring
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𝑘 vs Dimension
𝑃
𝑃
𝐿

𝑘
1
𝛿

𝛿
𝑃
Is 𝑘 a good measure of the
material property of the rod?
台大機械系材料力學 一盧中仁


𝛿
𝛿
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

𝐿
𝑃
2
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Normal Strain and Normal Stress
Stress-Strain Diagram
𝑃
𝑃/𝐴

𝐿
 




Normal Strain (Average Strain):
𝑃
𝛿/𝐿
𝛿
𝜀 = 𝛿/𝐿
𝑃
Strain is dimensionless.


𝛿
𝑃
𝑃
𝑃 𝑃
𝑃
Normal Stress (Average Stress):
𝜎 = 𝑃/𝐴
𝑃
has the unit of pressure. (Pa: N/m2)
𝜎
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Test Machine
Stress-Strain Diagram for Material Properties
http://www.czhtest.com/pddetaildate/image/20131206_19052720.html
𝜎
O
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𝜀
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Extensometer
Test Specimen
https://knowledge.ulprospector.com/1420/pe-extensometer/
https://knowledge.ulprospector.
com/1420/pe-extensometer/
https://www.researchgate.net/profile/Sudipto_Mondol/publication/305806934/figure
/fig2/AS:391294028402690@1470303174382/Figure-2-UTM-Test-SpecimenNomenclature-Tensile-testing-by-Joseph-R-Davis-p-50.ppm
http://hill-engineering.com/general-interest/tensile-test/
試片的形狀、尺寸請參考各種標準，例如
Combination of fully automatic feeler arm extensometers, with dual
averaging axial and transverse strain extensometer.
ASTM International, originally known as the American Society for Testing and
Materials (ASTM)
https://www.epsilontech.com/wpcontent/uploads/2016/04/3442_min
iature_extensometerside_view_overview.jpg
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Experiment
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台大機械系材料力學 一盧中仁
Ductile Material
𝜎 >𝜎 ?
\s
B&J
necking
much less than 1
https://www.youtube.com/watch?v=D8U4G5kcpcM
https://i.ytimg.com/vi/D8U4G5kcpcM/maxresdefault.jpg
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True Stress and Strain
True Stress-True Strain Diagram
𝐴
𝐿
𝐿0
https://i.ytimg.com/vi/D8U4G5kcpcM/maxresdefault.jpg
A
𝐴0
Nominal (engineering) strain: Δ𝐿/𝐿0
For most engineering purposes
Nominal (engineering) stress: 𝐹/𝐴0
(small deformations), nominal
True strain: ∑
strain and stress are adequate.
=∫
= ln( )
True stress: 𝐹/𝐴
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By Nicoguaro - Own work, CC BY 4.0,
https://commons.wikimedia.org/w/index.php?curid=89891679
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Brittle Material
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Concrete
All from Hibbeler
From B&J
• Very small tensile strain
• Much stronger in compression
than in tension
Brittle materials do not have a well-defined tensile fracture stress, since appearance of initial
cracks in a specimen is quite random.
Instead, the average fracture stress from a set of observed tests is generally reported.
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台大機械系材料力學 一盧中仁
https://en.Wikipedia.org/wiki/Hoover_Dam#/media/File:Hoover_Dam,_Colorado_River.JPG
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Ductile vs Brittle
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https://www.pinterest.com/pin/522206519277323595/
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Ductile and Brittle – C1
http://www.laserextensometer.de/en/matkwmetall.html
注意不同材料伸長量和斷裂面的形狀不同
http://wiki.dtonline.org/images/1/1b/StressStrainGraph2.png
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Elastic vs Plastic Behavior
Ductile and Brittle – C2
All from Gere
• If the strain disappears when the stress is removed, the material is said to behave elastically.
web.vtc.edu/mt/114SUM01/TTsum01/Spec19/CupandCone.jpg
• The largest stress for which this occurs is called the elastic limit.
“The structure and properties of materials” Vol. 3, by Hayden,
Moffatt and Wulff
• When the strain does not return to zero after the stress is removed, the material is said to
behave plastically.
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Hooke’s Law
• Below the yield stress
𝜎 = 𝐸𝜀 : Hooke′s Law
𝐸 = Young′s Modulus or
Modulus of Elasticity
• Strength is affected by alloying, heat
treating, and manufacturing process but
stiffness (modulus of elasticity) is not.
• 鋼的楊氏係數約為 200 GPa
from Hibbeler
Effect of Temperature on Strength and Elastic Modulus of High-Strength Steel
From B&J
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174 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2013
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