Thermal Strains and Element of the Theory of Plasticity

Thermal Strains• Thermal strain is a special class of Elastic strain that

results from– expansion with increasing temperature, or

– contraction with decreasing temperature

• Increased temperature causes the atoms to vibrate by large amount. In isotropic materials, the effect is the same in all directions.

• Over a limited range of temperatures, the thermal strains at a given temperature T, can be assumed to be proportional to the change, T.

TTT 0(A8-1)

where T0 is the reference temperature ( = 0 at T0). The coefficient of thermal expansion, , is seen to be in units of 1/oC, thus making strain dimensionless.

• Since uniform thermal strains occur in all directions in isotropic material, Hooke’s law for 3-D can be generalized to include thermal effects.

TE zyxx 1

TE zxyy 1

TE yxzz 1

(A8-2)

• The theory of plasticity is concerned with a number of different types of problems. It deals with the behavior of metals at strains where Hooke’s law is no longer valid.

• From the viewpoint of design, plasticity is concerned with predicting the safe limits for use of a material under combined stresses. i.e., the maximum load which can be applied to a body without causing: – Excessive Yielding

– Flow

– Fracture

• Plasticity is also concerned with understanding the mechanism of plastic deformation of metals.

• Plastic deformation is not a reversible process, and depends on the loading path by which the final state is achieved.

• In plastic deformation, there is no easily measured constant relating stress to strain as with Young’s modulus for elastic deformation.

• The phenomena of strain hardening, plastic anisotropy, elastic hysteresis, and Bauschinger effect can not be treated easily without introducing considerable mathematical complexity.

Figure 8-1(a). Typical true stress-strain curves for a ductile metal.

Hooke’s law is followed up to the yield stress 0, and beyond 0, the metal deforms plastically.

Figure 8-1b. Same curve as 8-1a, except that it shows what happensduring unloading and reloading - Hysteresis. The curve will not be exactly linear and parallel to the elastic portion of the curve.

Figure 8-1c. Same curve as 8-1a, but showing Bauschinger effect.

It is found that the yield stress in tension is greater than the yield stress in compression.

Figure 8-2. Idealized flow curves. (a) Rigid ideal plastic material

Figure 8-2b. Ideal plastic material with elastic region

Figure 8-2c. Piecewise linear ( strain-hardening) material.

• A true stress-strain curve is frequently called a flow curve, because it gives the stress required to cause the metal to flow plastically to any given strain.

• The mathematical equation used to describe the stress-strain relationship is a power expression of the form:

where K is the stress at = 1.0 and n, the strain-hardening coefficient, is the slope of a log-log of

Eq. 8-1

That is,

nk (8-1)

logloglog nK (8-2)

FAILURE CRITERIA: FLOW/YIELD and FRACTURE

• The Flow, Yield or Failure criterion must be in terms of stress in such a way that it is valid for all states of stress.

• A given material may fail by either yielding or fracture depending on its properties and the state of stress.

• A number of different failure criteria are available, some of which predict failure by yielding, and others failure by fracture.

• The terms flow criterion, yield criterion and failure criterion have different meanings.– Yield criterion applies mainly to materials that are in

the annealed condition (usually ductile materials).

– Failure criterion applies to both ductile and brittle materials. However, it is mainly used for brittle materials (fracture criterion), in which the limit of elastic deformation coincides with failure.

– Flow criterion applies to materials that have been previously processed via work hardening (usually ductile materials).

• In applying a yielding criterion, the resistance of a material is given by its yield strength.

• In applying a fracture criterion, the ultimate tensile strength is usually used.

• Failure criterion for isotropic materials can be expressed in the following mathematical form:

where failure (yielding or fracture) is predicted to occur when a specific mathematical function f of the principal normal stresses is equal to the failure strength of the material, c, from a uniaxial tension test.

cf 321 ,, (8-3)

• The failure strength is either the yield strength o, or the ultimate strength u, depending on whether yielding or fracture is of interest.

• Let us define an effective stress, , which is a single numerical value that characterizes the state of applied stress. If

where c is a known material property

Failure is not expected if

The safety factor against failure is given as:

That is the applied stress can be increased by a factor of X before failure occurs.

)(_

occursfailurec

)(_

failurenoc

_

cX

Maximum Normal Stress Criterion (Rankine)• Yielding (Plastic flow) takes place when the greatest

principal stress in a complex state of stress reaches the flow stress in a uniaxial tension.

• Since 1 > 2 > 3, Flow occurs when

0 (tension) = 1

Compressive strength is usually greater than tensile strength.

Flow stress in uniaxial tension

Maximum normal stress in a complex stress state.

(8-4)

ncompressiotension 010

Where is the flow stress of the material.

• The great weakness of this criterion is that it predicts plastic flow of a material under a hydrostatic state of stress; however, this is impossible, as shown by the example below.

• It is well known that tiny shrimp can live at verygreat depths. The hydrostatic pressure due to water is equivalent to 1 atm (10-5 N/m-2) for every 10 m; at 1000 m below the surface the shrimp would be subjected to a hydrostatic stress of 10-7 N/m-2.

Hence,

0

27321 /10 mNp

• Experiment to determine the yield stress of the shrimp (defined as the stress at which the amplitude of the tail wiggling would have becomes less than a critical value) when crushed between two fingers showed that it occurred at a stress of about 10-5 N/m-2 (14.5 psi).

Hence,

Rankine’s criterion predicts that shrimp failure would occur at

This corresponds to a depth of only 10m. Fortunately for all lovers of crustaceans, this is not the case, and hydrostatic stresses do not contribute to plastic flow.

250 /10 mN

250 /10 mNp

Maximum-Shear-Stress or Tresca Criterion• This yield criterion assumes that yielding occurs when

the maximum shear stress in a complex state of stress equals the maximum shear stress at the onset of flow in uniaxial-tension.

• From Eq,(2.21), the maximum shear stress is given by:

Where is the algebraically largest and is the algebraically smallest principal stress.

231

max (8.5)

1 3

For uniaxial tension, , and the maximumshearing yield stress is given by:

Substituting in Eq. (8.3), we have

Therefore, the maximum-shear-stress criterion is given by:

032,01 0

20

0

220

031

max

031 (8.6)

• This criterion corresponds to taking the differences between 1 and 3 and making it equal to the flow stress in uniaxial tension.

• This criterion does not predict failure under hydrostatic stress, because we would have 1 = 3 = p and no resulting shear stress.

von Mises’ or Distortion-Energy Criterion

• This criterion is usually applied to ductile material • von Mises’ proposed that yielding would occur when

the second invariant of the stress deviator J2 exceeds some critical value.

where

for yielding in uniaxial tension

22 kJ

213

221

2322 6

1 J

0; 3201

220

20 6k

(8.7)

Substituting Eq. 8-8 into 8-7, we obtain the usual form of von Mises’ yield criterion. When the expression

then the material will flow. The above expression above is known as effective stress. It is now accepted that it expresses the critical value of the distortion (or shear) component of the deformation energy of a body.

o 2/12

132

322

21 )()()(2

1(8.9)

k30 (8.8)

Additional Failure Criteria• Octahedral Shear Stress Yield Criteria: This is another

yield criteria often used for ductile metals. It states that yielding occurs when the shear stress on the octahedral planes reaches a critical value.

• Mohr-Coulomb Failure Criterion: This is used for brittle metals, and is a modified Tresca criterion.

• Griffith Failure Criterion: Another criterion used for brittle metals. It simply states that failure will occur when the tensile stress tangential to an ellipsoidal cavity and at the cavity surface reaches a critical level 0.

• McClintock-Walsh Criterion: Another criterion used for brittle metals. It is an extension of the Griffith’s criterion, and considers a frictional component acting on the flaw faces that had to be overcome in order for them to grow.

Example

A region on the surface of a 6061-T4 aluminum alloy component has strain gage attached, which indicate the following stresses:

11 = 70 MPa

22 = 120 MPa

12 = 60 MPa

Determine the yielding for both Tresca’s and von Mises’ criteria, given that 0 = 150 MPa (the yield stress).

SolutionSince we were given the value of 12, we must therefore first establish the principal stresses. Invoke Eq. 4-37.

Hence,

1 = 160 MPa; 2 = 30 MPa; 1 = 0

According to Tresca, max = (160 - 0)/2 = 80 MPa

For yielding in uniaxial tension:

0/2 = 75 MPa

Since the 80 MPa > 75 MPa, Tresca criterion would be unsafe.

2

12

222112211

21 22,

The von Mises criterion can be invoked from Eq. 8-9.

The L.H.S. of the above Eq. gives a value of 175 MPa. This criterion predicts that the material will not fail (flow), unlike the Tresca criterion, which predicts that the material will flow.

Therefore, the Tresca criterion is more conservative than the von Mises’ criterion in predicting failure.

2/1213

232

221 )()()(

21 o