Traction vector

Pmax., Pmin Maximum, minimum values of a parameter t Traction vector... 

In tensor notation the three Cartesian directions x, y, and z are designated by suffixed variables i,j, k, l, etc. (Landau and Lifshitz 1970 Auld 1973). Thus the force acting per unit area on a surface may be described as a traction vector with components rj j = x, y, z. The stress in an infinitesimal cube volume element may then be described by the tractions on three of the faces, giving nine elements of stress cry (i, j = x, y, z), where the first suffix denotes the normal to the plane on which a given traction operates, and the second suffix denotes the direction of a traction component. 

Here, similar to the scalar fields, the velocity is a continuous function therefore there is a unique value of u in every node. Generally, this is not true for the traction vector. However, for a Lyapunov surface, where the normal is continuous, the tractions are also continuous. 

The average force per unit area is Af,-/AS. This quantity attains a limiting nonzero value as AS approaches zero at point P (Cauchy s stress principle). This limiting quantity is called the stress vector, or traction vector T. But T depends on the orientation of the area element, that is, the direction of the surface defined by normal n. Thus it would appear that there are an infinite number of unrelated ways of expressing the state of stress at point P. 

The Cauchy stress principle arises through consideration of the equilibrium of body forces and surface tractions in the special case of the infinitesimal tetrahedral volume shown in fig. 2.7. Three faces of the tetrahedron are perpendicular to the Cartesian axes while the fourth face is characterized by a normal n. The idea is to insist on the equilibrium of this elementary volume, which results in the observation that the traction vector on an arbitrary plane with normal n (such as is shown in the figure) can be determined once the traction vectors on the Cartesian planes are known. In particular, it is found that = crn, where a is known as the stress tensor, and carries the information about the traction vectors associated with the Cartesian planes. The simple outcome of this argument is the claim that... 

Now that we have the notion of the stress tensor in hand, we seek one additional insight into the nature of forces within solids that will be of particular interest to our discussion of plastic flow in solids. As was mentioned in section 2.2.3, plastic deformation is the result of shearing deformations on special planes. Certain models of such deformation posit the existence of a critical stress on these planes such that once this stress is attained, shearing deformations will commence. To compute the resolved shear stress on a plane with normal n and in a direction s we begin by noting that the traction vector on this plane is given by... 

To compute the component of this traction vector in the direction of interest (i.e. the resolved shear stress), it is necessary to project the force along the direction of interest via... 

With respect to this coordinate system, we aim to find the plane (characterized by normal n = niei) within which the shear stress is maximum. The traction vector on the plane with normal n is given by = a n. To find the associated shear stress on this plane, we note that the traction vector may be decomposed into a piece that is normal to the plane of interest and an in-plane part. It is this in-plane contribution that we wish to maximize. Denote the in-plane part of the traction vector by r, which implies t " = n)n - - r. It is most convenient to maximize... 

To make the treatment given above more than just an abstraction, we now consider the case of one-dimensional elasticity. We imagine our one-dimensional elastic medium to have length a and to be characterized by an elastic stiffness k. The traction vector, t x), is directly related to the gradients in displacements by t = kdu/dx. Our weak statement of the equilibrium equations may be written as... 

Rewrite A5i, AS2, AS3 and AV in terms of AS, h and the components of the vector n normal to A 5 and then evaluate the limiting value of the expression given above as h 0. Now, by adopting the convention that the traction vector on the i Cartesian plane can be written as... 

If we now recall that the jump in the displacement field across the slip plane is nothing more than the Burgers vector and we exploit the fact that the traction vector is related to the stress tensor via = a -P n, then the left hand side of eqn (8.28) may be rewritten as... 

To evaluate the resultant electromechanical force, one needs to utilize the fact that the tangential component of the electrical field at the surface of the liquid droplet vanishes, and accordingly, one may express the components of E on a plane with direction cosines n, as , = Utilizing the Cauchy s theorem relating the traction vector with the stress tensor components, one can write... 

Fig. 2.10 Traction vectors represented for undeformed and deformed bodies...

The resolved shear stress can also be calculated for arbitrary stress states a. To calculate, we first calculate the traction vector t on the slip plane with normal vector n... 

Second Piola-Kirchhoff stress tensor and related traction vector Lagrangian strain tensor and displacement vector... 

We are to solve the above system of linear partial differential equations which correspond to the elastic rubber displacement a boundary element method Is applied to the Navler equations. The geometry of Interest Is shown In figure 1. The elastic rubber layer has uniform thickness. The elastic deformation boundary conditions require that either the displacements or the surface tractions be specified on the boundary of the domain. The surface traction vector for the elastic solid Is given as ... 

Is determined provided that either the deformation or surface traction vector Is prescribed on the boundary. The fundamental solutions for the elastic deformation and surface traction are further discussed In Appendix 1. 

Here H and g are determined from the integral of the fundamental solutions. It is appropriate to mention here that both H and G are slngtilar operators. The uniform displacement vector V is mapped to the zero vector by H and the constant pressure surface traction vector t is mapped to zero by G. This is because both t and Hq are homogeneous solutions to the Stokes equations. In the present work the vectors W and are specified on different parts of the boundary. Complete mathematical details of this method and application are to be reported elsewhere. 

The associated surface traction vector depends on the unit outward normal on the surface where the traction is acting. The unit outward normal components are (n, n2) the surface traction vector components are (t ji 2 ... 

Figure 2. Cohesive failure model variation of the normal (a) and tangential (b) components of the cohesive traction vector T with respect to the normal (A ) and shear (A,) crack opening displacements, showing the coupling between tensile and shear failure.

Notice that the stress tensor in each of the examples above is sym metric that is, the rows and columns of the matrix for the components of T can be interchanged without changing T. The components of the traction vectors ti were picked that way intentionsjiy. The symmetry of the stress tensor can be shown by considering the... 

We can visualize the principal stresses in terms of a stress ellipsoid. The surface of this ellipsoid is found by the locus of the end of the traction vector t from P when n takes all possible directions. The three axes of the ellipsoid are the three principal... 

---