VonMisesCriterion - 0.1. Failure Theories In the previous...

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3 0.1. Failure Theories In the previous section, we introduced the concept of stress, strain and the relationship between stresses and strains. We also discussed failure of materials under uniaxial state of stress. Failure of engineering materials can be broadly classified into ductile and brittle failure. Most metals are ductile and fail due to yielding. Hence, the yield strength characterizes their failure. Ceramics and some polymers are brittle and rupture or fracture when the stress exceeds certain maximum value. Their stress–strain behavior is linear up to the point of failure and they fail abruptly. The stress required to break the atomic bond and separate the atoms is called the theoretical strength of the material. It can be shown that the theoretical strength is approximately equal to E /3 where, E is Young’s modulus. 1 However, most materials fail at a stress about one–hundredth or even one–thousandth of the theoretical strength. For example, the theoretical strength of aluminum is about 22 GPa. However, the yield strength of aluminum is in the order of 100 MPa, which is 1/220 th of the theoretical strength. This enormous discrepancy could be explained as follows. In ductile material yielding occurs not due to separation of atoms but due to sliding of atoms (movement of dislocations) as depicted in Figure 1.1. Thus, the stress or energy required for yielding is much less than that required for separating the atomic planes. Hence, in a ductile material the maximum shear stress causes yielding of the material. In brittle materials, the failure or rupture still occurs due to separation of atomic planes. However, the high value of stress required is provided locally by stress concentration caused by small pre-existing cracks or flaws in the material. The stress concentration factors can be in the order of 100 to 1,000. That is, the applied stress is amplified by enormous amount due to the presence of cracks and it is sufficient to separate the atoms. When this process becomes unstable, the material separates over a large area causing brittle failure of the material. Figure 1.1: Material failure due to relative sliding of atomic planes Although research is underway not only to explain but also quantify the strength of materials in terms of its atomic structure and properties, it is still not practical to design machines and structures based on such atomistic models. Hence, we resort to phenomenological failure theories, which are based on observations and testing over a period of time. The purpose of failure theories is to extend the strength values obtained 1 T.L. Anderson, Fracture Mechanics – Fundamentals and Applications , Third Edition, CRC Press, Boca Raton, FL, 2006.
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4 Finite Element Analysis and Design from uniaxial tests to multi-axial states of stress that exists in practical structures. It is not practical to test a material under all possible combinations of stress states. In the following, we describe some well-established phenomenological failure theories for both ductile and brittle materials.
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VonMisesCriterion - 0.1. Failure Theories In the previous...

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