Normal stresses, definition

By definition, a brittle material does not fail in shear failure oeeurs when the largest prineipal stress reaehes the ultimate tensile strength, Su. Where the ultimate eompressive strength, Su, and Su of brittle material are approximately the same, the Maximum Normal Stress Theory applies (Edwards and MeKee, 1991 Norton, 1996). The probabilistie failure eriterion is essentially the same as equation 4.55. 

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. 

See Definition 1.2 for the general definition of cr. (7ii, i = 1, 2, 3 are denoted normal stresses. cti2 is called the shear stress. 

MODES OF DEFORMATION AND DEFINITION OF VISCOSITY AND NORMAL STRESS COEFFICIENTS... 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

From Eqs. (15.61) and (15.62) it follows that the shear stress and the first normal stress difference gradually increase from 0 to the steady state value. In this respect sometimes the following definitions are suggested for the transient values of viscosity and first normal stress coefficient... 

We use the definitions of the shear viscosity r , first normal stress coefficient I l, and second normal stress coefficient 1 2 [from Eq. (1-24)] to obtain... 

Fig. 2 Definitions and relationships between stress and strain the vectors of normal force (—) and normal stress (-----) are in one and the same direction, whereas these vec-...

By definition, the mean pressure p = —1/3 5j rkk only includes normal stresses. In order to create a link between mean and thermodynamic pressure we will consider a cubic fluid element at a temperature T and of specific volume v, Fig. Al. We will now assume that the cube is at rest at time t = 0, so that the thermodynamic pressure p prevails inside the element. Now let us assume the mean pressure p is being exerted on the element from outside. When p > p the cube is compressed, should p < p then it expands. So, work — p dy is carried out by the external pressure p. This is equal to the work done during the volume change in the gas — p dy and the dissipated work. It therefore holds that dW = —p dV = —p dy + dWdi88 with the the dissipated work as... 

Performing macro-scale experiments it has been observed that the normal surface tension force induces higher normal stresses in the fluid on the concave side of the interface than on the other fluid on the convex side of the interface. In a micro-scale view we may say that this interfacial tension force is exerted by the interfacial material lying on the convex side of the surface upon the material lying on the concave side. The normal component of the surface force is thus frequently (not always ) defined positive into the mean curvature of the surface, in line with the physical observations. The direction of the normal component of the interface force given by (3.9) is determined by two factors, the interface normal unit vector n/ which we have defined positive into the curvature, and the mean curvature variable which we have chosen to define as an absolute value. That is, the variable used here determining the mean curvature of the surface Hi = ( i + K2)/ 2) is consistent with the definition... 

Similar definitions can be made for p and p1 related to the oscillating secondary normal stress difference. The quantities 0 and p1 are real, whereas tf, 8, and (t are complex. It is customary to write these complex quantities thus12 ... 

The reader should remember that the use of the definition of strain given in equation (2-18) means that this result applies only to small deformations. For example, equation (2-46a) fails to predict that one must apply a normal stress to the plates to constrain the displacements to the x direction. The failure is evident on realizing that development of a normal stress er22 would require a displacement in the x2 direction (i.e., normal to the plates), which is not present. 

As can be seen from (43), the result (45) could also be derived by considering the normal stress at the surface of the cavity, which is by definition dWjdv. This normal stress is purely kinetic and is thus equal to kT times the density of molecular centers at the surface of the cavity pG r). As the cavity becomes very large, the surface of the cavity approaches a plane and the normal stress becomes equal to p. 

The intrinsic permeability k = k(e, volumetric strain or in the case of fractured rock it can be formulated as a function of the effective normal stress using e.g. the following definition of the void aperture... 

This explanation implies the importance of die geometry, flow rate and a characteristic pol)mier time. More detailed explanations are needed to arrive at a quantitative definition for die swell. In this respect, the recoverable strain and the first normal stress difference are critical rheological variables. 

Derivations for almost all analytical models for FRP strengthened flexural members are based on the typical schematic FBDs of Fig. 10.14. This particular case represents a differential segment of an FRP strengthened beam under uniformly distributed load, and the bending stiffness of the FRP laminate is assumed to be much smaller than that of the beam to be strengthened. Forces, moments and stresses acting on these basic FBDs reflect the individual assumptions preset for any analysis. The interfacial adhesive shear and normal stress are denoted by t x) and a(x), respectively. Equation [10.19] is the mathematical representation of the basic definition of shear stress t(x) in the adhesive layer, which is directly related to the difference in longitudinal deformation between the FRP laminate at its interface with the adhesive and the beam s soffit. 

For the pendant drop in figure I, the equations above are subjected to the no-slip boundary conditions at solid surfaces and the kinematic condition on the free surfaces. The kinematic condition implies that there is no liquid crossing the boundary into the gas phase, or in other words forms a definite boundary between the phases. For creeping flows into an atmosphere of gas with minimal velocities there will be no interfacial shear stress tangential to the surface and the normal stress inside the fluid is balanced by the surface tension as described by the famous Young-Laplace equation. 

Normal stresses describe the time rate of the deformation change applied to the fluid element, whereas shear stresses describe the time rate of the volume change applied to the fluid element. Normal stresses are smaller compared to shear stresses and are usually neglected. If we assume the fluid elanent as Newtonian flow, we can write the following definition for normal and shear stresses as... 

By applying the definition for the vector operator gradient of the velocity and normal stresses product for the jc component as... 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

Physical and Mathematical Definition of Normal Stress and Shear Stress Consider a body in equilibrium under the action of external forces Fi as shown in Fig. 2.11(a). If a cutting plane is passed through the body as shown in Fig. 2.11(b), equilibrium is maintained on the remaining portion by internal forces distributed over the surface S. 

Although normal and shear stresses are the same in both tensorial and engineering stress definitions, for strains, the tensorial shear strain components ,7, where i j, represent half of the corresponding engineering strains (6,7 =... 

A generalized interpretation of mode measure has been suggested by Rice [60], and this definition is now widely used. Mode mixity xjr is defined as the ratio of interfacial shear stress to normal stress at a fixed distance I in front of the crack tip. 

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