Tensile stress sign convention

The sign associated with the pressure is opposite to that associated with the normal viscous stress. The usual sign convention assumes that a tensile stress is the positive normal stress so that the pressure, which by definition has compressive normal stress, has a negative sign. 

The sign convention for the stress components is important. A positive normal stress is tensile (i.e., tending to expand the control volume) and a negative normal stress is compressive. Thus, for example, referring to Fig. 2.13, a positive xrr points in the positive r direction on the r + dr face while it points in the negative r direction on the r face. 

The stress tensor may be represented as a 3 x 3 matrix, with components Oy, where i and j both go from 1 to 3. The diagonal elements represent normal stresses, whereas the off-diagonal ones represent shear stresses. Positive normal stresses are tensile, while negative ones are compressive (but an opposite sign convention is sometimes used, most notably in the soil mechanics literature). Finally, from the balance of angular momentum (or torque in the static case), it follows that the stress tensor and its matrix representation are symmetric (ay = aji), meaning that only six out of the nine components are in fact independent. 

Sign convention normal stress - Tensile -tve. Compressive -ve. Shear stress - follows coordinates. 

Tu. corresponding lo forces psralld to Ibe y sod z txes, respectively. The sssumption that the stress in the material is homogeneous requites that the stress normal lo opposite foces of the cube be equal and in opposite directions, and the assumption of no body torques requires that > o>. It should be noted that the sign convention used here leads to tensile stresses for positive values of o . 

The negative sign indicates that the liquid is in tension. (The sign convention for pressure is opposite to that for stresses, which are positive when tensile.) Figure 5 shows that when the radius of the tube, o, is very small the pressure is large. Laplace proved [15,16] that there is a pressure difference across a curved liquid-vapor interface equal to... 

Note that we use here the sign convention that tensile stress is positive. Some other texts, particularly Bird et al. (1987a), choose tension as negative. 

For readers not familiar with this notation, a few words of explanation may be useful. The indices on the typical component of the stress tensor have the following meaning. The second index indicates that this component of the stress acts in the Xj direction, while the first index indicates that it acts on a plane normal to the X axis. A component is positive when it acts on a fluid element in the plus Xj direction on the face of that element having the larger value of Xj. Thus, a tensile stress has a positive value, while a compressive stress is negative. Note that this sign convention is not used universally. 

Caution 1 The negative signs in (17.2) and (17.3) arise from the historical conventions of treating both hydrostatic pressure (an inward force on the element) and tensile stress (an outward force on the element) as positive. These conventions are not adhered to universally, however. 

The key points of this example are that the continuity equation helps us find the form of Vr(r, z) and that there are normal stresses for this flow. The values of and T00 come from the extensional deformations generated in this flow. It is noted because of the sign convention, tensile stresses are negative. ... 

The negative sign is a matter of convention a positive pressure is usually understood to be compressive (i.e., directed inward), whereas a positive normal stress is taken to be tensile (i.e., directed outward). Hence the need for the negative sign. 

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