Infinitesimal deformation

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

Viscoelastic behavior is classified as linear or non-linear according to the manner by which the stress depends upon the imposed deformation history (SO). Insteady shear flows, for example, the shear rate dependence of viscosity and the normal stress functions are non-linear properties. Linear viscoelastic behavior is obtained for simple fluids if the deformation is sufficiently small for all past times (infinitesimal deformations) or if it is imposed sufficiently slowly (infinitesimal rate of deformation) (80,83). In shear flow under these circumstances, the normal stress differences are small compared to the shear stress, and the expression for the shear stress reduces to a statement of the Boltzmann superposition principle (15,81) ... 

A morphism f X — S can always be considered as a family of schemes if we view it as the totality of its fibres, if f is flat we call it a fiat family of schemes parametrized bv 5 X is called the total space of the family f If 5 is connected and s e 5 is a k-rational point, f is also called a flat family of deformations of the fibre X(s). When S=Spec(A) with A a local k-algebra with residue field k, the flat morphism f is called a local family of deformations of %(q), the ribre over the closed point 5 if moreover the local k-algebra A is artinian, then f is called an infinitesimal family of deformations, or an infinitesimal deformation, of X(o). 

Infinitesimal deformations of a given scheme X or of a closed subscheme X of a scheme Y are easier to study than general deformations and contain a great deal of information. In this section we will discuss the case of affine schemes, the results we obtain here easily yield corresponding results in the projective case, as we will see in section 8. 

We will study infinitesimal deformations of X in Y. namely flat families of subschemes of Y of the form ... 

This is a prorepresentable functor of Artin rings, because the functor Hilbrp(t) is representable Hilbx is prorepresented by the completion oz. The study of Hilbx gives informations on oz and consequently on the properties of oz which are preserved under completion. For example the Zariski tangent space Tz of oz is naturally identified with Hilbx(k(c]), the set of first order deformations of X in Pr, that is with the set of infinitesimal deformations of X in Pr parametrized by D = SpecOtW)... 

Suppose we have homogeneous polynomials. .., fs k[x0,. .., xr] of degrees d1(.., d5 respectively defining a closed subscheme X c Pr Then, in this section, we will refine some of the results of section 8 by investigating those infinitesimal deformations of X in Pr, parametrized by a k-algebra A in a, which can be defined by s homogeneous polynomials. .., Fs t A[x] whose reductions mod m.A are fi, respectively... 

For infinitesimal deformations, we assume that the relation between strain and stress is expressed by Hooke s law the deformation is proportional to the applied force For isotropic bodies, this linear relation... 

We analyze, within a linearized second gradient theory, the static infinitesimal deformations of an annular porous cylinder filled with an inviscid fluid and with the inner and the outer surfaces subjected to uniform external pressures pj xt and iff1 respectively. We assume that surface tractions on the inner and the outer surfaces of the cylinder, in the reference configuration, equal -po and postulate that... 

A schematic representation of the domain contributions to the tensile deformation of a fibre is depicted in Fig. 13.97 and shows that the total deformation consists of chain stretching and shear deformation resulting in rotation of the chain axis. For an infinitesimal deformation and well-oriented fibres Northolt and Baltussen found for the initial modulus, El, and for the initial compliance,S[n, of the fibre... 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

Let us consider an elastic solid of initial length Iq under a uniaxial tensile force / that causes an infinitesimal deformation dl. The work done on the solid is... 

A variety of experiments show that for a solid under an infinitesimal deformation, the stress tensor is a linear function of the strain tensor,... 

To carry out the second step, one applies Kodaira-Spencer-Grothendieck deformation theory to calculate the vector space of infinitesimal deformations of 7r X -> C — S and of s C — S - X. More precisely, one looks at deformations of X such that the map 7r X - C extends to this deformation and all singular fibres remain concentrated in 7r 1(5). It turns out that ... 

The two definitions introduced above are obviously equivalent at infinitesimal deformation, but their difference increases rapidly with strain. For = 20%, the relative discrepancy is already as large as 9%. Consequently, Hencky definitions should always be employed for describing the stretching properties of polymers. 

We neglect higher order terms in the expansion as we are concerned only with infinitesimal deformations. Since these deformations relate to changes in the orientation of the director, may appropriately be called the curvature strain tensor (Frank ). In order to define the components of this tensor, let us choose a local right-handed system of cartesian coordinates with n parallel to z at the origin. The components of strain are then... 

The linear Young s modulus E is defined by the Young s modulus for an infinitesimal deformation. Fixing at X = 1 in the above equation, we find... 

Inverting the procedure that led to the function Equation 4.83, we may assert that the linear superposition of wave functions (Equation 4.78) may be generated by the following infinitesimal deformation-dependent relabeling of the trajectories, a symmetry of the law of motion (Equation 4.11) ... 

Equations 9.8 and 9.10b are written for rectilinear flows and infinitesimal deformations. We need equations that apply to finite, three-dimensional deformations. Intuitively, one might expect simply to replace the strain rate dy/dt by the components of the symmetric deformation rate tensor (dv /dy -y dvy/dx, etc.) to obtain a three-dimensional formulation, as in Section 2.2.3, and dr/dt by the substantial derivative D/Dt of the appropriate stress components. The first substitution is correct, but intuition would lead us badly astray regarding the second. Constitutive equations must be properly invariant to changes in the frame of reference (they must satisfy the principle of material frame indifference), and the substantial derivative of a stress or deformation-rate tensor is not properly invariant. The properly invariant... 

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