Page 298 - Mechanical Behavior of Materials

Section 7.6 Discussion of the Basic Failure Criteria 299

where the X-Y-Z axes are aligned with the planes of material symmetry, and H, F, G, N, L, and M
are empirical constants for the material. Let σ oX , σ oY , and σ oZ be the uniaxial yield strengths in the
three directions, and let τ oXY , τ oY Z , and τ oZ X be shear yield strengths on the respective orthogonal
planes. The empirical constants can be evaluated from the various yield strengths as follows:

1 1 1
H + G = 2 , H + F = 2 , F + G = 2
σ oX σ oY σ oZ
(7.40)
1 1 1
2N = 2 , 2L = 2 , 2M = 2
τ τ τ
oXY oY Z oZ X
The Hill criterion as just described can also be used with reasonable success as a fracture criterion
for orthotropic composite materials. The equations are the same, except that the various yield
strengths are replaced by the corresponding ultimate strengths. However, different values of the
constants are generally needed for tension versus compression, and other complexities exist for
composite materials that may not be fully predicted by this criterion.
If a material has different yield strengths in tension and compression, this suggests that a
dependence on hydrostatic stress needs to be added. One proposed yield criterion for this situation is

2 2 2
(σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) + 2 (|σ oc | − σ ot )(σ 1 + σ 2 + σ 3 ) = 2 |σ oc | σ ot (7.41)

where σ ot and σ oc are the yield strengths in tension and compression, respectively, with the negative
sign on σ oc being removed by use of the absolute value.
Polymers often have somewhat higher yield strengths in compression than in tension, with
the ratio |σ oc |/σ ot often being in the range from 1.2 to 1.35. This is illustrated by biaxial test
results for three such materials in Fig. 7.12. The behavior expected from Eq. 7.41 with a typical
value of |σ oc |/σ ot = 1.3 is plotted. The resulting off-center ellipse is in reasonable agreement with
the data.
In some exceptional cases, the yield strength of ductile metals has been observed to be
decreased by hydrostatic compression. See the review of Lewandowski (1998) for details and a
discussion of the physical mechanism involved, which is associated with upper/lower yield point
behavior.

7.6.5 Fracture in Brittle Materials
The maximum normal stress criterion gives reasonably accurate predictions of fracture in brittle
materials, as long as the normal stress having the largest absolute value is tensile. However,
deviations from this criterion occur if the normal stress having the largest absolute value is
compressive. Data illustrating this trend for gray cast iron are shown in Fig. 7.13. A prominent
feature of the deviation is that the ultimate strength in compression is higher than that in tension by
more than a factor of three.
Recall from Chapters 2 and 3 that brittle materials, such as ceramics and glasses and some
cast metals, commonly contain large numbers of randomly oriented microscopic cracks or other
planar interfaces that cannot support significant tensile stress. For example, the numerous flaws in