Strain tensor infinitesimal

One major discrepancy of the previous model can be attributed to the use of the infinitesimal strain tensor and to derivatives restricted to time changes. Indeed, in the case of large deformations, one has to refer to finite strain tensors, such as the Finger (, t (t ) or Cauchy C t(t ) strain tensors (t being the... 

Here we have used the fact that Fij = 5 + Uij. In addition, we have invoked the summation convention in which all repeated indices (in this case the index k) are summed over. For the case in which all the displacement gradient components satisfy Uij 1, the final term in the expression above may be neglected, resulting in the identification of the small strain (or infinitesimal strain) tensor,... 

Consider a deformation mapping that is nothing more than a rigid rotation about the z-axis. First, explicitly write down the deformation mapping and then compute both the Green strain and the infinitesimal strain tensors associated with this deformation mapping. In general, the infinitesimal strains are nonzero. Why ... 

Eij is the infinitesimal strain tensor is the strain deviator Cij and the volumetric strain e = ea Ui denotes the material displacement in j-th direction g = T - and 0 = 2 - X / are the increments of temperature and liquid content with respect to reference values... 

The first concept is the linear elasticity, that is, the linear relationship between the total stress and infinitesimal strain tensors for the filler and matrix as expressed by the following constitutive equations ... 

The infinitesimal strain tensor is defined in various ways here the notation of Timoshenko and Goodier (8) is adopted in terms of the displacements Ui of a material element and as depicted in Figure la. 

In this limit it is possible to neglect their squares and products. By this approximation (3.21) reduces to the definition of the coordinates Sjd of the infinitesimal strain tensor, again a symmetrical second rank tensor... 

This step is called kinematic or geometrical linearization. For the following development we will employ the coordinates Su of this infinitesimal strain tensor as a measure of the local strain of the body in the immediate vicinity of material point X at position x in the actual configuration. Usually they will simply be referred to as strain coordinates. As a further important consequence of the geometrical linearization the fact should be noted that derivatives with respect to material coordinates Xk may be replaced by the corresponding derivatives with respect to spatial coordinates Xk. Within the limits of applicabiUty of the linear theory of elasticity relations (3.26) and (3.28) are also simplified. The former reduces to... 

The non-diagonal elements of the infinitesimal strain tensor, called shear strains, are one half of the change in angle subtended by two material line elements oriented parallel to the corresponding coordinate axes in the reference configuration. 

It is obvious that the infinitesimal strain tensor Sy is no longer adeqrrate and it is certainly not surprising that the appropriate replacement is the finite strain tensor Vy as defined in (3.20). 

Similarly, the Green-Lagrange strain tensor E j and the infinitesimal strain tensor e-j are related by... 

It is also important to note that the 2nd Piola-Kirchhoff stress tensor is energetically conjugate to the Green-Lagrange strain tensor and the Cauchy stress is energetically conjugate to the infinitesimal strain tensor. In other words, we have... 

Here, sy and ay are the Cartesian components of the infinitesimal strain tensor and Cauchy stress tensor, v is the Poisson s ratio and E is the Young s modulus. Also, Sy are the Cartesian components of the remanent strain tensor. The remanent strain is the irreversible strain and can also be referred to as the plastic strain. In all cases discussed in this paper, the datum for remanent strain is the state of the material as cooled from above the Curie temperature. In such a state, all possible domain orientations are equally likely. 

When the displacement gradient is split into its symmetric and skew symmetric portions, the infinitesimal strain tensor of Eq. (3.18) is identified to be the former, while the latter represents infinitesimal rotations that do not... 

So the infinitesimal strain tensor is established as a symmetric tensor of second order. With provision for the engineering shear-strain measures aside the diagonal, the components can be assigned as given by the left-hand side of Eqs. (3.20). An alternative representation may be gained by resorting the six independent components into a vector as shown on the right-hand side of Eqs. (3.20) ... 

Consider the case of an RVE subject to oiJy small strains and linear elasticity (z.e., sti-ess varies Imearly with the infinitesimal strain tensor for each individual material). We assume that the continuum approximation is valid at all points within the separate materials, that the polymer and matrices are perfectly bonded, and that there is no voiding or cracking. Note that the use of these assumptions creates a somewhat idealistic material model. 

The relaxation processes described above apply to linear viscoelastic behavior. If the deformation is not small or slow, the orientation of the chain segments may be sufficiently large to cause a nonlinear response. We will see that this effect alone can be accounted for in rheological models by simply replacing the infinitesimal strain tensor by one able to describe large deformations no new relaxation mechanism needs to be invoked. Nonlinear effects related to orientation, such as normal stress differences, can be described in this manner. 

The most fundamental deficiency of Eq. 4.4 that prevents it from describing nonlinear behavior is that the product of the strain rate tensor and the time interval dt is not able to describe large strains. In preparation for the introduction of a strain tensor describing finite strain into 4.4 to correct this deficiency, in an ad hoc manner, we rewrite 4.4 in terms of the infinitesimal strain tensor, whose components are This tensor is defined in standard texts [9, p. 31], and it will suffice for our purpose to say that it is totally adequate to describe very small deformations. 

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. 

An equation like 10.5, obtained from the Boltzmann principle by replacing the infinitesimal strain tensor by one that can describe a large deformation, is sometimes called a model of finite linear viscoelasticity . If the memory function in the rubberlike liquid is taken to be the relaxation modulus of a single Maxwell element [G(f) = Gq exp(f/T)], we obtain the special case of the rubber like liquid that we will call Lodge s equation this is shown as Eq. 10.6. 

When applied to particle reinforced polymer composites, micromechanics models usually follow such basic assumptions as (i) linear elasticity of fillers and polymer matrix (ii) the fillers are axisymmetric, identical in shape and size, and can be characterized by parameters such as aspect ratio (iii) well-bonded filler-polymer interface and the ignorance of interfacial slip, filler-polymer debonding or matrix cracking. The first concept is the linear elasticity, that is, the linear relationship between the total stress and infinitesimal strain tensors for the filler and matrix as expressed by the following constitutive equations ... 

---