Download - Chap-5. Plasticity Theory
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Plasticity
theory
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Introduction
• The study of yielding wasmotivated by the wish to
predict mechanical failure ofmaterials
• Yielding is considered as thebeginning of a process whichwill eventually lead to fracture
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STRESS FRACTURE
CRITERION• the simplest failure criterion is thatfailure is expected when the largestprincipal normal
stress reaches the uniaxial strength ofthe material.
• This approach is successful inpredicting fracture of brittle materialsunder tension-dominated loading
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MAXIMUM NORMAL
STRESS FRACTURE
CRITERION
the effective stress
the safety factor against fracture
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STRESS FRACTURE
CRITERION
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Yield Criteria
• A yield criterion is amathematical expression of thestress states that will causeyielding or Fracture.
• The most general form of ayield criterion is f!σx, σy, σz, τyz, τzx, τxy) = C
• For an isotropic materialf!"#$ "%$ "&' ( ) ( f (σ1, σ2, σ3)
e*ective stress is
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Yield Criteria
• Failure is not expected if is lessthan σ c:
< σ c
σ c is the materials property• Also$ the safety factor against failure is
+(
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Yield criterion for ductilematerial
• Instead of the rupture of atomicbonds$ the material is separated bysliding of atoms as shown
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Tresca criterion
• failure in ductile materials iscaused by shear deformations.
,o$ it is logical to establish ayield criterion in terms of theamount of shear stress a
material is able to sustain. This is the principle of the
Tresca theory
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Tresca (maximum sear
stress criterion!
• The simplest yield criterionis one rst proposed by
Tresca. It states thatyielding will occur when
the largest shear stressreaches a critical value.
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Tresca criterion
This yield criterion can be stated asfollows
The yield stress inshear
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,ample Problem
• A Plane ,tress element in a part made ofthe /#-T is found to have the followingstress
σx ( 0. 1si2 σy ( 3.3 1si$ and τxy ( 0./ 1si. The Axial Yield ,trength$ " Y$ of /#-T
aluminum is &0 1si$ and its ,hear Yield,tress$ τ Y$ is %/ 1si.
• Find Factor of ,afety using Tresca)riterion
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4on 5ises )riterion
Another yield criterion often used for ductile metals is theprediction that yielding occurs when the shear stress on the
octahedral planes reaches the critical value
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4on 5ises )riterion
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,ample Problem
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,ample Problem