The strain tensor

We begin with the definitions of the strain and stress tensors in a solid. The reference configuration is usually taken to be the equilibrium structure of the solid, on which there are no external forces. We define strain as the amount by which a small element of the solid is distorted with respect to the reference configuration. The arbitrary point in the solid at (x, y, z) cartesian coordinates moves to x + u, y + v, z + w) when the solid is strained, as illustrated in Fig. E.l. Each of the displacement fields u,v,w is a function of the position in the solid u x, y, z), v(x, y, z), w(x, y, z). 

There is a different notation which makes the symmetric or antisymmetric nature of the strain and rotation tensors more evident we define the three cartesian axes as Xi, X2, Xs (instead of x, y, z) and the three displacement fields as u, U2, us (instead of u, v, w). Then the strain tensor takes the form 

It is a straightforward geometrical exercise to show that, if the coordinate axes x and y are rotated around the axis z by an angle 0, then the strain components in the new coordinate frame x y z ) are given by 

The last expression is useful, because it can be turned into the following equation  

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

The individual fluid elements of a flowing fluid are not only displaced in terms of their position but are also deformed under the influence of the normal stresses tu and the shear stresses T (i j)- The deformation velocity depends on the relative movement of the individual points of mass to each other. It is only in the case when the points of mass in a fluid element do not move relatively to each other that the fluid element behaves like a rigid solid and will not be deformed. Therefore a relationship between the velocity field and the deformation, and with that also between the velocity field and the stress tensor must exist. This relationship is required if we wish to express the stress tensor in terms of the velocities in Cauchy s equation of motion. 

Normal stresses change the magnitude of a fluid element of given mass. If they are different, for example rn t22, the shape of the fluid element will also be altered. As we can see in Fig. 3.8, a rectangle would be transformed into a prism, a spherical fluid element could be deformed into an ellipsoid. Fig. 3.9 shows the front view of a cube which will be stretched by a normal stress rn. We recognise 

2Auguste-Louis Cauchy (1789-1857) was, as a contemporary of Leonhard Euler and Carl-Friedrich Gauss, one of the most important mathematicians of the first half of the 19th century. His most famous publications are Traite des Fonctions and Mechanique Analytique . As he refused to take the oath to the new regime after the revolution in 1830, his positions as professor at the Ecole Polytechnique and at the College de France were removed and he was dismissed from the Academie Francaise. He spent several years in exile in Switzerland, Turin and Prague. He was permitted to return to France in 1838, and it was there that he was reinstated as a professor at the Sorbonne after the revolution in 1848. 

Corresponding expressions for the volume increase due to normal stresses r22 and t33 can be obtained, so that the total volumetric increase is given by 

---

These are linear equations which give the symmetry of the strain tensor Sij = Sji- In the general case, the strain tensor is nonlinear,... 

In what follows the Kirchhoff-Love model of the shell is used. We identify the mid-surface with the domain in R . However, the curvatures of the shell are assumed to be small but nonzero. For such a configuration, following (Vol mir, 1972), we introduce the components of the strain tensor for the mid-surface,... 

The symbols Sij = SijiyV) stand for the components of the strain tensor of the mid-plane of the plate ... 

Here Sij u) = uij + Uj,i)/2 are the components of the strain tensor. We consider function spaces whose elements are characterized by the conditions... 

Here i —> i is a continuous convex function describing the plastic yield condition. The equations (5.7) provide a decomposition of the strain tensor Sij u) into a sum of an elastic part aijuicru and a plastic part ij, and (5.6) are the equilibrium equations. 

The strain tensor 5 can be written for noncentrosymmetry point group crystals as ... 

It is possible to use directly as a measure of the irrotational part of the deformation, but it is more convenient to use the strain tensor... 

The deformation gradient tensor A is related to the strain tensor n by the equation ... 

Note 2 If the strain tensor is diagonal for all time then the stress tensor is diagonal for all time for isotropic materials. 

Here X is an important parameter which determine whether the time evolution of n is dominated by the strain tensor A or by the vorticity tensor fi (cf. Eq. (42)). If X is larger than unity, the former dominates and n tends to assume a steady-state angle 0 relative to the flow direction which is determined by the equation... 

A little elementary acoustics components of the strain tensor are... 

Starting point is the Taylor expansion of the magnetic exchange parameters with respect to the components of the strain tensor e, leading to the so-called magnetoelastic Hamiltonian. 

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... 

The St. Venant compatibility equations [13,14,15] follow immediately from the strain tensor definition of Equation (2), and are... 

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. 

Following a deformation specified by the strain tensor, E, the unit vector in the deformed state is related to its initial value by... 

The strain tensor AjS describes both a change of form and a change of volume of a body. It is convenient to separate these parts by introducing a strain tensor Ah such that the determinator of matrix Ah = 1. Then we can write... 

The free energy of the deformed isotropic body depends on the strain tensor, it is a function of three invariants of the strain tensor. The volume of the deformed body... 

The strain component S12 is usually the deformation of the body along axis 1, due to a force along axis 2 the strain tensor s is usually symmetrical, = s and thus, of the nine terms of s, at most six are unique. Both P and s can be represented as ellipsoids of stress and strain, respectively, and can be reduced to a diagonal form (e.g., P j along some preferred orthogonal system of axes, oblique to the laboratory frame or to the frame of the crystal, but characteristic for the solid the transformation to this diagonal form is a... 

---