Net local stress

The net local stress is the network stress discussed earlier. 

In this section, we assume that the green body is already dry and the stress is caused by the thermal expansion of the ceramic particles that make up the porous ceramic due to the temperature profile in the green body in either heating or cooling. For an infinite plate of thickness 2xq> the normal stress ofx) at a position x in the green body depends on the temperature difference between that point, T, and the average temperature, T. This gives the strain at that point and fixes the net local stress at [28]... 

A particularly interesting feature of solutions like that given above is their interpretation as providing a source of screening for the crack tip. The use of the word screening in this context refers to the fact that the local stress intensity factor is modified by the presence of the dislocation and can result in a net reduction in the local stresses in the vicinity of the crack tip. In particular, the total stress intensity factor is a sum of the form... 

If we think about the physics of establishing the immobile cap at the rear of the bubble, it is necessary that the local Marangoni stress balance the net shear stress on the interface. In effect, this means that the condition (7-258) must be satisfied at all points on the surface where the cap exists,... 

Consider a fluid element of constant mass pAxAyAz moving along with the local fluid velocity v. The x component of momentum of this fluid element is pvxAxAyAz. The momentum of the fluid element as it moves along with the local fluid velocity is a function of both space and time. The total derivative of the momentum of the fluid element with respect to time is then pAxAyAz Dvx/Dt). According to Newton s second law this quantity is to be equated to the forces acting on the element of mass the net force in the x direction due to the difference in pressure on faces a and b, which is [p x)AyAz — p(x + Ax)AyAz], the net force in the x direction due to the difference in the viscous stresses,2 which is... 

Since the stress field varies spatially, there are differential forces across the differential element. These net forces serve to accelerate a fluid packet. Determining the net forces on an element of fluid requires understanding how the stresses vary from one face of a differential element to another. Assuming that the stress field is smooth and differentiable, local variations can be expressed in terms of Taylor-series expansions. 

In this process, the net flux of substitutional atoms across the interface plane results in local volume changes (i.e., as a crystal plane is removed by climb, the crystal contracts in a direction normal to the plane). However, free expansion in directions parallel to the interface plane is constrained by the specimen ends, where significant diffusion has not occurred, and by the coherence of the interface between the expanding and contracting regions. Therefore, dimensional changes parallel to the interface (i.e., normal to the diffusion direction) are restricted, and in-plane compatibility stresses are generated. No out-of-plane compatibility stresses develop because the diffusion couple can expand freely in the diffusion direction. 

The relation between the frequency v of local jumps, shear stress r, temperature T and macroscopic rate of creep e was well established by Eyring s reaction rate theory [41]. Let us consider that a number of vl0 thermally activated structural units attempt per unit time to cross a potential barrier Ur, the net flow v, of units that will succeed is then given by ... 

The left-hand side is just the time rate of change of linear momentum of all the fluid within the specified material control volume. The first term on the right-hand side is the net body force that is due to gravity (other types of body forces are not considered in this book). The second term is the net surface force, with the local surface force per unit area being symbolically represented by the vector t. We call t the stress vector. It is the vector sum of all surface-force contributions per unit area acting at a point on the surface of Vm(t). 

In the steady, unidirectional flow problems considered in this section, the acceleration of a fluid element is identically equal to zero. Both the time derivative du/dt and the nonlinear inertial terms are zero so that Du/Dt = 0. This means that the equation of motion reduces locally to a simple balance between forces associated with the pressure gradient and viscous forces due to the velocity gradient. Because this simple force balance holds at every point in the fluid, it must also hold for the fluid system as a whole. To illustrate this, we use the Poiseuille flow solution. Let us consider the forces acting on a body of fluid in an arbitrary section of the tube, between z = 0, say, and a downstream point z = L, as illustrated in Fig. 3-4. At the walls of the tube, the only nonzero shear-stress component is xrz. The normal-stress components at the walls are all just equal to the pressure and produce no net contribution to the overall forces that act on the body of fluid that we consider here. The viscous shear stress at the walls is evaluated by use of (3 44),... 

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