Displacement Vector and Strain Tensor

Let us consider any point of a solid whose coordinate is x, y, z when in state without stress, and denote r its position vector from the origin. When the solid is deformed by external forces, or internal forces such as those exerted by a gravity field, the position vector becomes r'. The displacement vector u representing the flexure is u = r'-r. (1.118)

where its cartesian components ux(x,y,z), uy(x,y,z) and uz(x,y,z) may be denoted ui,

If dl is the length separating two adjacent points, dl2 = dx2 + dy2 + dz2 = dx2, this length becomes dl' in the deformation, dl'2 = (dxi + dui)2. Using the subscripts i, k, l, the simplifications in the summations lead to dl'2 - dl2 = 2eikdxidxk, (1.119)

where eik is the strain tensor. Following Landau and Lifschitz , one shows that for small deformations the high-order terms can be neglected so the strain tensor becomes symmetric, eik = eki. Hence the strain tensor is defined by

The strain components include the relative axial displacement eii or normal strain in the direction xi, and the associated relative lateral displacements eik and eil or shear strains in the directions xk and xl respectively.

Let us introduce now u, v, w as a quite usual nomenclature for representing the components of the displacement vector; in a cartesian coordinate system we have the equivalent notations u = ux, v = uy, w = uz, where only the u, v, w notation will be used hereafter. From (1.120), the strain components are

2&yz _ dy + dz ' 2£zx _ dz + dx ' 2&xy _ dx + dy '

In a cylindric coordinate system, let us denote by the subscript t a tangential direction which is perpendicular both to the z-axis and a radial direction r so the components of the displacement vector are denoted u = ur, v=ut, and w=uz respectively. The strain components are

£rr ""^T" , £tt =¡"77 + , £zz , dr r dd r dz

2£tz = ~ + d > 2£zr = d + T-, 2£rt = 3---+ " ^ (1.121b)

From a general form strain tensor eik, let £(i) be the three diagonal strains taken in the principal directions at a given point. In a first approximation, the strains or relative elongations of the solid element in these principal directions are, from (1.119),

so the resulting volume when the displacements occur is dV' = (1+ E(1))(1+E(2))(1+E(3))dV ~ (1+ E(1)+E(2)+ E(3))dV. (1.122b)

Since the sum of the principal values e(1)+e(2)+e(3) of a tensor is an invariant, it is also equal to the sum En = E11+e22+e33 of its diagonal terms. Hence, in any coordinate system, the relative volume change is

If En = 0, then the volume element does not change. It can be shown that this corresponds to the case of strains generated by a torsion. Except for substances whose Poisson's ratio is v = 1/2 (such as rubber) and which then conserve a constant volume, a result is the following.

^ Whatever a Poisson's ratio v e [0, 1/2], if the elastic deformation of a small element is with constant volume, then the geometric transformation is a torsion.

1.13.4 The Stress-Strain Linear Relations and Strain Energy

Consider A volume element dV of a perfectly elastic and isotropic substance characterized by E ,v. One defines stress components oik for representing the stress vectors arising at the facets of the element (Fig. 1.49). Since okl = olk, the three axial stress components On and three shear stress components okl are expressed in force per unit surface area. These stresses are functions of the strains such as

°xx = (1+v)E1-2v)[(1 - v) Exx + v ( Eyy + Ezz)], Oyy = (1+vE1-2v) [(1 - v) Eyy + v (Ezz + Exx)], Ozz = (1+v)E1-2v) [(1 - v) Ezz + v (Exx + Eyy)], Oyz = Eyz, Ozx = 1+v Ezx, Oxy = 1+v Exy. (1.123a) Fig. 1.49 Axial and shear stress components acting on a volume element (ok = O»)

The inverse relations are

These so-called stress-strain relations, also known as Hooke's law or Navier's relations, were derived in above general form by Navier  in 1820.13

If both the solid and the deforming loads have a rotational symmetry around the z-axis, the tangential displacement component v = ut = 0, and also du/dd = dw/de = 0; hence, from (1.121b), £tz — £rt — 0. In this case, the so-called thick plate theory involves the stress-strain relation

I Orr\ Ott