The von Mises stress is always positive, while the hydrostatic stress can be positive or negative. Under simple uniaxial tension or compression, the von Mises stress is equal to the applied stress, while the hydrostatic stress is equal to one third of it. The hydrostatic stress, which is also a scalar, can be written The von Mises stress, which is a scalar quantity, is related to the principal stresses by: One consequence of this is that the material response (stress-strain relationship) is the same in tension and compression (provided that true stresses and strains are being used). This is follows from the fact that plastic deformation (of metals) occurs at constant volume. Plastic deformation of metals is stimulated solely by the deviatoric (shape-changing) component of the stress state, often termed the von Mises stress, and is unaffected by the hydrostatic (volume-changing) component. While the loading configurations described above are simple, with just a single stress being generated in the sample, more complex (multi-axial) stress states are common. Deviatoric (von Mises) and Hydrostatic Stresses and Strains Furthermore, a general state of stress can always be represented just by three principal stresses, σ1, σ2 and σ3, which are normal stresses acting in three orthogonal directions (the principal directions). A general stress state (and a strain state) should therefore be expressed as an array of 9 numbers, although this is reduced to 6 when it is recognised that, in order to avoid rotation, σ ij = σ ji (ie the tensor is symmetric). (A scalar is a tensor of zeroth rank, with no subscript, while a vector is a tensor of first rank, with one subscript.) The stress σ ij is a normal stress if i = j and a shear stress otherwise. A stress should thus be expressed as σ ij, which is a tensor of second rank. The values of i and j can be 1, 2 or 3 (the three orthogonal reference directions). Stresses and Strains as Second Rank TensorsĪ stress, or a strain, therefore has two directions associated with it – the direction of the applied force ( i) and the direction of the normal of the plane on which the force acts ( j). This is illustrated in Fig.2.įig.2: A cuboid under load by application of forces to two pairs of opposing faces, with the forces acting parallel to the faces, to generate a shear stress, Tau. Shear forces must be applied in pairs, if the body is not to rotate. It’s also possible to apply forces parallel to two opposite faces of a body, so as to generate shear.

In the same way as for stresses, a distinction can be drawn between this nominal or engineering strain and the true strain (defined as the change in length divided by the current length). The normal strain in a particular direction, usually termed ε (epsilon), is the change in length along that direction, divided by the original length. The response of the material to the applied stress is characterised by the strain. Therefore, at relatively large strains, the stress defined as F/ A0, termed the nominal stress or the engineering stress, differs significantly from the true stress (defined as the force divided by the current area). The area being loaded will reduce as F increases and the material progressively deforms. This may be tensile (positive) or compressive (negative).įig.1: A cuboid under load by application of forces to a pair of opposing faces, to generate a normal stress, Sigma. If these forces act perpendicular to a face (ie parallel to its face normal), as shown, the stress σ (sigma) is termed a normal stress. Consider a body subjected to equal and opposite forces on two faces (Fig.1). Stress is a measure of the local concentration of mechanical force.

Sigma 1 sigma 2 sigma 3 stress