That is stress in three dimensions. Biaxial stress is in two dimensions.
Triaxial stress has normal and shear stresses along each of three planes. Most beams can be analyzed with uniaxial stress; plates with biaxial; solids with triaxial
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Triaxial testing will yield static strength properties of the material while ultrasonic measurements will give dynamic strength properties. The two can be related to each other by using various relationships between Young's Modulus, Bulk Modulus, Shear Modulus and Poisson's ratio. Static results should be lower than the dynamic results.
Strictly speaking, in 2D it is a yield line, not a surface (you would have to do a £D plot for that). In order to construct a deviatoric yield surface plot you require the triaxial data for a sample for at least three differing confining pressures. You will then essentially be creating a series of mohr circle plots (shear stress vs Sigma1 - Sigma3) of the deviatoric stress circles at the failure stress of the samples. A straight line is then drawn that touches the upper bounds of the three circles without passing through them. This line is the 2D deviatoric yield plot. There is free software available that will do this for you. Please see the related links.
Normal stress and shear stress
In a consolidated drained test the sample is consolidated and sheared in compression with drainage. The rate of axial deformation is kept constant, i.e. is strain controlled. The idea is that the test allows the sample and the pore pressures to fully consolidate (i.e. adjust) to the surrounding stresses. The test may take a long time to allow the sample to adjust, in particular low permeability samples need a long time to drain and adjust strain to stress level
A factor of safety against yield is applied to design stress Yield Stress/ Design Stress = Factor of safety The factor of safety varies for different industries; 1.5 is used in structural steel design for buildings; 1.25 or even 1.1 for aircraft/space systems