Strains tensor shear strains

The off-diagonal strain elements of the expanded matrix of eq. (3.7), which are referred to as tensor shear strains, represent rotation-free distortions of the body, complementing the dilatations of eq. (3.9). 

For compliance constants the substitution is based on the above conversion for stiffness constants, but additional rules apply because of the factor 2 difference between the definition of engineering and tensor shear strains ... 

With these definitions it is clear that, if b, a -b the solid has undergone a rigid rotation without deformation. The tensor shear strains t, are, therefore, defined as the symmetric part of so that they are insensitive to rigid rotations ... 

It is conventional to replace the tensor shear strains by the angles of shear y, which are twice the corresponding shear component in the strain tensor. There is also a corresponding contraction in c,yw which can then be written... 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

Note that Y represents engineering shear strain whereas (l ) represents tensor shear strain. 

Figure 2-3 Engineering Shear Strain versus Tensor Shear Strain...

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

We now use this to calculate the stress in the melt after the retraction has occurred. The deformation is described by the tensor E defined so that an arbitrary vector V in the material is deformed affinely into the vector E.v. For example, in simple shear of shear strain 7, and in uniaxial extension of strain e, the tensor E takes the forms... 

Note 5 The Finger strain tensor for simple shear flow is... 

Fig. F.2. Shear stress and shear strain, (a) The shear force per unit area is a component of the stress tensor, (b) The shear stress causes a shear strain.

There is a corresponding strain tensor (which is not presented here due to its complexity), and for each stress component there is a corresponding strain component. Hence, there are three normal strains, and e, and six shear strains, which... 

Sect. 6 a Aa T Do Fsh T X A M R De final radius of cylindrical gel after swelling displacement relaxation time for swelling collective diffusion constant shear energy trace of the strain tensor u,k swelling rate ratio total change of the radius of the gel longitudinal modulus ratio of the shear modulus to the longitudinal modulus effective collective diffusion constant... 

As discussed in Appendix A, symmetric tensors have properties that are important to the subsequent derivation of conservation laws. As illustrated in Fig. 2.9, there is always some orientation for the differential element in which all the shear strain rates vanish, leaving only dilatational strain rates. This behavior follows from the transformation laws... 

Seebeck sclerometer, 48 Shear modulus, 9 Shore sclerometer, 118 Sonic Mill method, 39 Strain tensor, 11 Stress tensor, 11... 

Here, pa,- is the bead momentum vector and u(rm. f) = iyrV is the linear streaming velocity profile, where y = dux/dy is the shear strain rate. Doll s method has now been replaced by the SLLOD algorithm (Evans and Morriss, 1984), where the Cartesian components that couple to the strain rate tensor are transposed (Equation (11)). 

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

The Invariants of the Rate of Strain Tensor in Simple Shear and Simple Elogational Flows Calculate the invariants of a simple shear flow and elonga-tional flow. 

GNF-based constitutive equations differ in the specific form that the shear rate dependence of the viscosity, t](y), is expressed, but they all share the requirement that the non-Newtonian viscosity t](y) be a function of only the three scalar invariants of the rate of strain tensor. Since polymer melts are essentially incompressible, the first invariant, Iy = 0, and for steady shear flows since v = /(x2), and v2 V j 0 the third invariant,... 

The diagonal elements of Eq. 10.3 are the stretches or tensile strains. The nondiagonal elements are the shear strains. The variation of the displacement vector, u, with the position vector, d, for a point in the solid is used to define the nine tensor components in Eq. 10.3, as follows ... 

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 ... 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

Rheological properties of food materials over a wide range of phase behavior can be expressed in terms of viscous (viscometric), elastic and viscoelastic functions which relate some components of flie stress tensor to specific components of the strain or shear rate response. In terms of fluid and solid phases, viscometric... 

Doi and Ohta (1991) derived very similar expressions for cri2, A/i, N2 for the elastic stresses of emulsions in step shearing strains. Larson (1997) has shown that Eqs. (9-57a) and (9-57b) can be derived as an approximation from a general phenomenological film model for affinely stretching, constant-tension interfaces thus cti2, N, and N2 can all be represented by the simple tensor expression... 

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