Stress vectors with associated components

Transitioning from the stress state of a particle to the stress field of the continuum, the interaction of the Cauchy stress tensor components of neighboring points needs to be investigated. They have to satisfy the conditions of local equilibrium to be established with the aid of an arbitrary infinitesimal volume element. Such an element with faces in parallel to the planes of the Cartesian coordinate system is subjected to the volume force and on the faces to the components of the Cauchy stress tensor with additional increments in the form of the first element of Taylor expansions on one of the respective opposing faces. The balance of moments proves the symmetry of the stress tensor, 

Here the continuum is denoted by the domain A and the respective boundary dA is subdivided to consider two types of boundary conditions. The area dA is subjected to the prescribed loads of the physical boundary conditions in equilibrium with the boundary stresses expressed by application of the Cauchy theorem of Eq. (3.13) (Neumann boundary conditions)  

The prescribed displacements usa of the geometric boundary conditions are imposed on the area dAu (Dirichlet boundary conditions)  

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Fig. 3.1. Stress vectors with associated components by means of an infinitesimal...

Here r is the stress vector, with components that are typically taken to align with the coordinate directions. Recognize that both normal stress and shear stress contribute to work. That is, work is associated with both dilatation and deformation. It is important to note that there is not a -iid A construct in the work-rate integral, for example, as is the... 

Any or all of these forces may result in local stresses within the fluid. Stress can be thought of as a (local) concentration of force, or the force per unit area that bounds an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, which are the characteristics of a second-order tensor or dyad. If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (normal) of the local area element upon which it acts is designated by subscript i, then the corresponding stress component (ay) is given by... 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

Equations (5.206) and (5.207) describe the vector phenomena of heat conductivity, diffusion, and cross-effects. Coefficients Lqq, Liq and La are scalars. Equations (5.208) relate components of the stress tensor to components of a symmetric tensor. Equations (5.209) and (5.210) describe the scalar processes of the chemical character associated with the phenomena of volume viscosity and the cross-phenomena. 

Because of chain inextensibility, the shear rate of any slip system is not dependent on the normal-stress component in the chain direction (Parks and Ahzi 1990). This renders the crystalline lamellae rigid in the chain direction. To cope with this problem operationally, and to prevent global locking-up of deformation, a special modification is introduced to truncate the stress tensor in the chain direction c. Thus, we denote by S° this modification of the deviatoric Cauchy stress tensor S in the crystalline lamella to have a zero normal component in the chain direction, i.e., by requiring that 5 c,c = 0, where c,- and c,- are components of the c vector (Lee et al. 1993a). The resolved shear stress in the slip system a can then be expressed as r = where R is the symmetrical traceless Schmid tensor of stress resolution associated with the slip system a. The components of the symmetrical part of the Schmid tensor / , can be defined as = Ksfw" + fs ), where if and nj are the unit-vector components of the slip direction and the slip-plane normal of the given slip system a, respectively. 

The remaining stress components are associated with the screw component of the Burgers vector, and the fields are... 

Previous formulations of ruhber elasticity theory have been based on the assumption that either the defonmtion , or force field associated with crosslink deformations is the same as in a continuum. There is no apriori justification for such an assumption. In the present theory, consideration is given to a completely arbitrary distribution of crosslink deformations. Generalized equations for the six components of the macroscopic stress tensor and for the stored strain energy function are formulated. The unknown deformation functions of the radius vectors between crosslinks are then found by postulating that the deformation functions are such as to minimize the free energy of the network, subject to the constraints imposed by the macroscopic stress components applied to the body. 

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