THEORIES OF FAILURE Yield Criteria for Ductile Materials Under Plane Stress

Fracture Criteria for Brittle Materials Under Plane Stress

1

OBJECTIVES

• To discuss the FOUR THEORIES that are often used in engineering practice to predict the failure of a material subjected to a multiaxial state of stress.

2

HOW DO MATERIALS FAIL?

• Ductile materials – yielding

• Brittle materials – fracture

3

HOW DO MATERIALS FAIL?

These modes of failure are readily defined if the member is subjected to a uniaxial state of stress

4

DUCTILE MATERIALS

5

DUCTILE MATERIALS:

MAXIMUM-SHEAR-STRESS THEORY

The most common type of yielding of a ductile material such as steel is caused by slipping, which occurs along the contact planes of randomly ordered crystals that make up the material.

The edges of the planes of slipping as they appear on the surface of the strip are referred to as Lüder’s lines. These lines clearly indicate the slip planes in the strip, which occur at approximately 45° with the axis of the strip. 6

DUCTILE MATERIALS:

MAXIMUM-SHEAR-STRESS THEORY

• The slipping that occurs is caused by shear stress.

• To illustrate:

7

• This shear stress acts on planes that are 45° from the planes of principal stress, and these planes coincide with the direction of the Lüder lines shown on the specimen, indicating that indeed failure occurs by shear. 8

DUCTILE MATERIALS:

MAXIMUM-SHEAR-STRESS THEORY

• The maximum shear stress is given by:

• Henri Tresca proposed the maximum-shear-stress theory or Tresca yield criterion using the idea that ductile materials fail in shear.

• “Yielding of the material begins when the absolute maximum shear stress in the material reaches the shear stress that causes the same material to yield when it is subjected only to axial tension.”

• It requires that 9

DUCTILE MATERIALS:

MAXIMUM-SHEAR-STRESS THEORY

2max

y

Consider the two cases:

• Two in-plane stresses have the same sign

• Two in-plane stresses have opposite signs

10

DUCTILE MATERIALS:

MAXIMUM-SHEAR-STRESS THEORY

2

maxmax

abs

2

minmaxmax

abs

• Using the equations stated earlier, the limits for the maximum-shear-stress theory are:

11

DUCTILE MATERIALS:

MAXIMUM-SHEAR-STRESS THEORY

y

y

y

21

2

1

• This is based on the concept of strain-energy density (The energy per unit volume).

• An external loading will deform a material, causing it to store energy internally throughout its volume.

• Recall:

(for uniaxial stress)

• The strain-energy density can be considered as the sum of:

• energy needed to cause a volume change (no change in shape)

• energy needed to distort the element 12

DUCTILE MATERIALS:

MAXIMUM-DISTORTION-ENERGY THEORY

13

DUCTILE MATERIALS:

MAXIMUM-DISTORTION-ENERGY THEORY

=

• Experimental evidence has shown that materials do not yield when subjected to a uniform stress, σavg.

• Therefore, M. Huber proposed MDET

• This was also redefined independently by R. von Mises and H. Hencky.

• “Yielding in a ductile material occurs when the distortion energy per unit volume of the material equals or exceeds the distortion energy per unit volume of the same material when it is subjected to yielding in a simple tension test”

14

DUCTILE MATERIALS:

MAXIMUM-DISTORTION-ENERGY THEORY

• Distortion energy per unit volume

• For plane stress

• For uniaxial tension test

15

DUCTILE MATERIALS:

MAXIMUM-DISTORTION-ENERGY THEORY

2

13

2

32

2

216

1

Eud

2

221

2

13

1

Eud

2

3

1YYd

Eu

• Since the maximum-distortion-energy theory requires that

𝑢𝑑 = 𝑢𝑑 𝑌

we have:

16

22

221

2

1 Y

DUCTILE MATERIALS:

MAXIMUM-DISTORTION-ENERGY THEORY

17

DUCTILE MATERIALS:

COMPARISON BETWEEN MSST and MDET

• Same results when σ1= σ2 = σy

or when one of the principal stresses is zero and the other is σy

• Greatest discrepancy in pure shear.

• Actual torsion tests used to develop a condition of pure shear in a ductile specimen show that MDET gives more accurate results

18

DUCTILE MATERIALS:

COMPARISON BETWEEN MSST and MDET

BRITTLE MATERIALS

19

Tension Test: Fracture occurs when normal stresses reaches ultimate stress, 𝜎𝑢𝑙𝑡

20

BRITTLE MATERIALS:

MAXIMUM-NORMAL-STRESS THEORY

Torsion Test: Brittle Fracture occurs due to the maximum tensile stress since the plane of fracture for an element is at 45° to the shear direction.

• Tensile stress to fracture during a torsion test ≈ Tensile stress to fracture in a tension test

• “A brittle material will fail when the maximum principal stress σ1 reaches the ultimate normal stress when subjected to simple tension test.” - W. Rankine

• If the material is subjected to plane stress, we require that

21

BRITTLE MATERIALS:

MAXIMUM-NORMAL-STRESS THEORY

ult

ult

2

1

σ1

σ2

-σult

-σult

σult

σult

• Experimentally, it has been found to be in close agreement with the behavior of brittle materials that have stress–strain diagrams that are similar in both tension and compression.

22

BRITTLE MATERIALS:

MAXIMUM-NORMAL-STRESS THEORY

• When materials have different tension and compression properties

• Developed by Otto Mohr

• To apply it: 3 tests are performed:

1. Uniaxial tensile test (σult)t

2. Uniaxial compressive (σult)c

3. Torsion test (τult)

23

BRITTLE MATERIALS:

MOHR’S FAILURE CRITERION

• From these tests, we can draw three Mohr’s Circle:

• These three circles are contained in a “failure envelope” drawn tangent to all three circles.

• We may also represent this criterion on a graph of principal stresses.

• Failure occurs outside if one of the principal stresses exceeds (σult)t, or (σult)c

24

BRITTLE MATERIALS:

MOHR’S FAILURE CRITERION

𝜎

𝜏