Cauchy stress tensor

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

Therein, T" = (T )r are the symmetric partial Cauchy stress tensors, // b represent the body forces and p" are the momentum productions, where ps + pF = o must hold due the overall conservation of momentum. 

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

The constitutive form of the Cauchy stress tensors for the solid and liquid phases are [5]... 

Therein, A denotes the interface pressure (whole pore pressure), p l the realistic pressure, T denotes the partial effective Cauchy stress tensor and I is the unit tensor. With the expression A = ns A + nL pLR + nG pGR (Dalton s law), the constrain... 

In the simplified case of a long cylinder (plane strain), as shown in Figure 4, the Cauchy stress tensor at point M can be written as ... 

The starting point of the investigation is the introduction of a scalar microstruc-tural parameter k which contributes to the total energy E of the body under study as pointed out in Refs. [38] and 39]. In Eq. (1) p, s and x are the mass density, the specific internal energy density and the velocity, respectively. The parameter k in the product pk describes microstructural properties and transfers the square of the rate of k to the dimensions of a specific energy density. In addition, the energy supply Ri and the energy flux R2 are also modified in the form of Eqs. (2) and (3), wherein pb is the body force density, pg is the supply of K, and p r is the heat supply. Further quantities are the stress vector t = T n associated to the Cauchy stress tensor T and to the outer normal n, the microstructural flux s = S n and the heat flux qi = —q n. 

In these equations, p denotes the mass density and p the production term due to a spray u is the velocity, t is the Cauchy stress tensor, g is the constant of gravity, and M" is the spray momentum production e denotes the specific internal energy, q is the mass specific heat flux vector, and Q is the energy contribution due to the spray. 

In the specific conservation equations for mass, momentum, and energy given in (19.14)-(19.16), there are material-dependent expressions such as the Cauchy stress tensor t, the specific heat flux vector q, and the specific internal energy e. The expressions for these quantities are called the constitutive equations. 

The most commonly encountered fluids are Newtonian fluids, that is, fluids whose stress tensor depends linearly on its rate-of-strain tensor (at a fixed temperature). For isotropic Newtonian fluids, the Cauchy stress tensor is given by... 

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. 

Fundamentally, it is possible to characterize a class of fluids known as simple fluids [1], by postulating a functional relationship between the stress tensor at time t and the strain history of the material point with respect to its current configuration. For such fluids, the Cauchy stress tensor, Tij, can be expressed as... 

The Cauchy stress tensor , a, at any point of a body (assumed to behave as a continuum) is completely defined by nine component stresses-three orthogonal, normal stresses and six orthogonal, shear stresses. It is used for the stress analysis of materials undergoing small deformations, in which the differences in stress distribution, in most cases, can be neglected. 

The tension T across a normal cross-section of a fiber undergoing an axially symmetric extension equals the integral over the cross-section of the S component of the Cauchy stress tensor ... 

Since rik are the coordinates of the unit normal vector n and ti the coordinates of the stress vector t we conclude that Ty must be the coordinates of a second rank tensor. It is called stress tensor or Cauchy stress tensor. In general it is position and time dependent. 

Similar conditions are obtained immediately for the other two directions and we come to the conclusion that the Cauchy stress tensor is symmetric... 

Keeping in mind that our aim is to describe the mechanical behaviour of a piezoelectric element we start by using the model of an ideal elastic material. The basic property of this model is that the Cauchy stress tensor at an arbitrary material poirrt at a certain moment depends only on the deformation gradierrt at this same poirrt at the same moment. This implies that a rigid body motion cannot produce ary stresses. It is also presupposed that the elastic properties of the rtraterial are irrde-pendent of time. Therefore, the stresses are independent of the strairring rate as well as of previous treatment, in short of the history of the rrraterial. 

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. 

During the motion of the body, its volume, surface area, density, stresses, and strains change continuously. The stress measure that we shall use is the 2nd Piola-Kirchhofif stress tensor. The components of the 2nd Piola-Kirchhoff stress tensor in Cj will be denoted by To see the meaning of the 2nd Piola-KirchhofiF stress tensor, consider the force dF on surface dS in C2. The Cauchy stress tensor t is defined by... 

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