Cauchy stress introduced

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. 

Please note the identity between the velocity v and the time derivative of u. Furthermore the quantities f, a, q and r stand for the sum of externally applied body forces, the transposed Cauchy stress tensor, the heat flux, and an arbitrary energy production term (e.g. due to latent heat during phase transitions). Eqs. (1-3) are equations of motion for the five unknown fields p, u and e. They are universal, namely material-independent. To solve these equations, the constitutive quantities, viz. heat flux and stress tensor, must be replaced by constitutive equations (cf. subsequent paragraph) q = (T, VT,...) and = (T,u,...). Moreover, up to now no temperature T occur in the balances (1-3). For this reason a caloric state equation, e = e T), must be introduced, which allows for replacing the internal energy e by temperature T. 

We introduce a relationship between the Cauchy stress a defined in the deformed body with its basis e, and the first Piola-Kirchhoff stress II defined in the undeformed body with its basis Ej as follows ... 

The governing equations that control material responses are given by the mass conservation law (2.97) and the equation of motion (2.104) if no energy conservation is considered. Note that the Cauchy stress is symmetric under the conservation law of moment of linear momentum. Furthermore, if the change of mass density is small (or it may be constant), the equation to be solved is given by (2.104). The unknowns in this equation are the velocity v (or displacement u in the small strain theory) and the stress a, i.e. giving a total of nine, that is, three for v (or u) and six for three components, therefore it cannot be solved, suggesting that we must introduce a relationship between v (or u) and [Pg.40]

Let Cauchy stress and pressure (compression positive), respectively. We first introduce the Kirchhoff energy stress o- (at, t) and the corresponding pressure p (x,t) by... 

The stress tensor has been introduced in Chapter 2. In small strain elasticity theory, the components of stress are defined by considering the equilibrium of an elemental cube within the body. When the strains are small, the dimensions of the body, and therefore the areas of the cube faces, are to a first approximation unaffected by the strain. It is then of no consequence whether the components of stress are defined with respect to the cube before deformation or the cube after deformation. For finite strains, however, this is not true and there are alternative definitions of stress depending on whether the deformed or undeformed state is chosen as a reference. We will choose to adopt the stress associated with the deformed state - the true stress or Cauchy stress - throughout this work. In our present axis notation, we can express this stress tensor X as... 

A.8 confers consistency to A.I with A.6 and allows us to introduce B by means of which we can determine the evolution of (P,a,Z) when the history of the motion is known. As a mater of fact, the knowledge of Y requires tlie knowledge of the ir-history which depends on P and consequently, in the case of large elastic deformation we can not determine the Ti-history even when the Cauchy stress liistory is given. 

Summary. The first three balance equations are formulated in this section. The balances are necessary conditions to be fulfilled not only in thermodynamics but generally (in continuum mechanics). The balance of mass was formulated locally in several alternatives—(3.62), (3.63), or (3.65). The most important consequence of the balance of momentum is the Cauchy theorem (3.72), which introduces the stress tensor. The local form of this balance is then expressed by (3.76) or (3.77). The most relevant outcome of the balance of moment of momentum is the symmetry of the stress tensor (3.93). Note that in this section also an important class of quantities— the specific quantities—was introduced by (3.66) note particularly their derivative properties (3.67) and (3.68). 

We could summarize Eqs. 1-1 through 1-3 by sa3dng that they introduced the concepts of kinematics and stress. More than half a century would elapse before the concept of stress would be presented in a modem framework by Cauchy, and it would require a slightly longer period of time before a constitutive equation would be developed leading to the Navler-Stokes equations. In the century between Euler and Stokes, the basic ideas associated with kinematics, stress and constitutive relations were formulated. Two centuries later, these same concepts represent the building blocks of fluid mechanics. Before we comment on the development of these concepts, we need to examine how Eqs. 1-1 through 1-3 compare with Newton s three laws of mechanics. 

At this point, we have seen that 100 years was required for the concepts of kinematics, stress, and constitutive relations to be developed. It was Euler who provided a precise statement of the non-relativistic laws of mechanics, introduced the cut principle and the concept of stress, and developed important results concerning the kinematics of continua. This was done primarily in the period between 1750 and 1766. In 1822, Cauchy placed the concept of stress on a modem basis and in... 

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