[Deformation Theory Of Plasticity By Robert Millard Jones Pdf Download
Everardo Laboy
In continuum mechanics, the maximum distortion energy criterion (also von Mises yield criterion[1]) states that yielding of a ductile material begins when the second invariant of deviatoric stress J 2 \displaystyle J_2 reaches a critical value.[2] It is a part of plasticity theory that mostly applies to ductile materials, such as some metals. Prior to yield, material response can be assumed to be of a nonlinear elastic, viscoelastic, or linear elastic behavior.
In materials science and engineering, the von Mises yield criterion is also formulated in terms of the von Mises stress or equivalent tensile stress, σ v \displaystyle \sigma _\textv . This is a scalar value of stress that can be computed from the Cauchy stress tensor. In this case, a material is said to start yielding when the von Mises stress reaches a value known as yield strength, σ y \displaystyle \sigma _\texty . The von Mises stress is used to predict yielding of materials under complex loading from the results of uniaxial tensile tests. The von Mises stress satisfies the property where two stress states with equal distortion energy have an equal von Mises stress.
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Because the von Mises yield criterion is independent of the first stress invariant, I 1 \displaystyle I_1 , it is applicable for the analysis of plastic deformation for ductile materials such as metals, as onset of yield for these materials does not depend on the hydrostatic component of the stress tensor.
where σ y \displaystyle \sigma _y is tensile yield strength of the material. If we set the von Mises stress equal to the yield strength and combine the above equations, the von Mises yield criterion is written as:
An equivalent tensile stress or equivalent von-Mises stress, σ v \displaystyle \sigma _\textv is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define
In this case, yielding occurs when the equivalent stress, σ v \displaystyle \sigma _\textv , reaches the yield strength of the material in simple tension, σ y \displaystyle \sigma _\texty . As an example, the stress state of a steel beam in compression differs from the stress state of a steel axle under torsion, even if both specimens are of the same material. In view of the stress tensor, which fully describes the stress state, this difference manifests in six degrees of freedom, because the stress tensor has six independent components. Therefore, it is difficult to tell which of the two specimens is closer to the yield point or has even reached it. However, by means of the von Mises yield criterion, which depends solely on the value of the scalar von Mises stress, i.e., one degree of freedom, this comparison is straightforward: A larger von Mises value implies that the material is closer to the yield point.
This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is 3 \displaystyle \sqrt 3 times lower than the yield stress in the case of simple tension. The von Mises yield criterion for pure shear stress, expressed in principal stresses, is
Hencky (1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value.[6] For this reason, the von Mises criterion is also known as the maximum distortion strain energy criterion. This comes from the relation between J 2 \displaystyle J_2 and the elastic strain energy of distortion W D \displaystyle W_\textD :
In 1937 [9] Arpad L. Nadai suggested that yielding begins when the octahedral shear stress reaches a critical value, i.e. the octahedral shear stress of the material at yield in simple tension. In this case, the von Mises yield criterion is also known as the maximum octahedral shear stress criterion in view of the direct proportionality that exists between J 2 \displaystyle J_2 and the octahedral shear stress, τ oct \displaystyle \tau _\textoct , which by definition is
As shown in the equations above, the use of the von Mises criterion as a yield criterion is only exactly applicable when the following material properties are isotropic, and the ratio of the shear yield strength to the tensile yield strength has the following value:[10]
Since no material will have this ratio precisely, in practice it is necessary to use engineering judgement to decide what failure theory is appropriate for a given material. Alternately, for use of the Tresca theory, the same ratio is defined as 1/2.
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