Cauchy deformation tensor

The angle Xo can be compared on the one hand to the extinction angle of birefringence, and on the other hand to the orientation of the principal directions of the Cauchy deformation tensor, which would correspond to a molecular deformation purely affrne with the macroscopic deformation shear strain. For a simple shear deformation y, is given by ... 

The molecular deformation ratio K in the directions I and II can be estimated in the following way the difference Aa between the principal stresses in the x-y plane can be readily calculated from the birefringence A (measured parallel and perpendicular to the direction of extinction) and the stress-optical coefficient C for molten polystyrene (C= 4.8xlO Pa , see Chapter III.l). According to the classical network theory, the stress tensor is proportional to the Cauchy deformation tensor which means that the network deformation along the principal directions of the stress tensor are X and 1/ where ... 

For the deformations illustrated in Figure 1.4.2 and Example 1.4.1, evaluate the components of the Finger and the Cauchy deformation tensors. 

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. 

Rivlin-Ericksen tensor of order n, for a viscoelastic liquid or solid in homogeneous deformation, is the nth time derivative of the Cauchy strain tensor at reference time, t. Note 1 For an inhomogeneous deformation the material derivatives have to be used. 

Customarily, the following three invariants of Cauchy-Green deformation tensor,... 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

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]... 

Therein, Cs = FsFj and Jl,g = detFL, g are the right Cauchy-Green deformation tensor of the solid and the Jacobian of the gas and liquid phases respectively, where Fa denotes the deformation gradient of free energies and the specific entropies of the constituents [Pg.333]

Hooke himself interpreted that dependence as follows if a tensile stress is applied to the ends of a thin rod, then the increment in the rod length Al will be proportional to the force applied. The present-day formulation of Hooke s law was given as early as the 19th century by Cauchy and Poisson and is read as follows if a small deformation occurs in an isotropic body, the stress tensor r is a linear function of the deformation tensor U (and vice versa). 

The Cauchy tensor is a device for describing the change in length at any point of the material body. Equation (5.29) indicates that in performing such a function the Cauchy tensor operates on the unit vector u at some time t in the past. However, the stress tensor is always measured with respect to the present form of the material body or the deformed state. Thus, there is the need to find a deformation tensor that operates on the unit vector at the present time. This can be done by using the inverse of E, E to express dX in terms of dX, namely. 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

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. 

Corresponding to U and V, two new tensors can be defined, which are used to calculate U and V. We have the right Cauchy-Green deformation tensor C and the left Cauchy-Green deformation tensor B ... 

To hold the incompressibihty assumption the volume preserving part of the apphed deformation gradient needs to be utilized in the trial elastic part, with the left and right Cauchy-Green tensors given by [14]... 

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 K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

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. 

C is referred to as the right Cauchy-Green tensor and B the left Cauchy-Green tensor. Then the deformation measure can be written as... 

Therefore, using Eq. (23) the following relation for the Cauchy stress tensor holds for each deformation gradient F and for all Q Ortha ... 

Exactly the same technique can be used to analyse the Cauchy-Green tensor C. When a deformation gradient F includes rigid-body rotation, it is necessary to first form the Cauchy-Green measure C and then find its principal components and directions using the methods outlined above for V. The principal directions of C are the same as those of the pure deformation V that underlies F (F = VR). Writing the analogue of Equation (3.12) forC... 

Fig. 7.9. Notions used in the definition of the Cauchy strain tensor The material point at r in the deformed body with its neighborhood shifts on unloading to the position r. The orthogonal infinitesimal distance vectors dri and drs in the deformed state transform into the oblique pair of distance vectors dri and dr. Orthogonality is preserved for the distance vectors dra, drc oriented along the principal axes...

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