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In the theory of finite deformations, extension and rotations of line elements are unrestricted as to size. For an infinitesimal fibre that deforms from an initial point given by the vector dX to the vector dx in the time t, the deformation gradient is defined by Fij = ∂xi(X, t)/∂Xj; the 3 × 3 matrix [F], with components Fij, may be represented as a pure deformation,…

The small strains, or infinitesimal strains, εij are appropriate for situations with |∂uk/∂Xl|<< 1 for all k and l. Two infinitesimal material fibres, one initially in the 1 direction and the other in the 2 direction, are shown in Figure 6 as dashed lines in the reference configuration and as solid lines in the deformed configuration. To first-order accuracy in components of…

Strain and strain-displacement relations The shape of a solid or structure changes with time during a deformation process. To characterize deformation, or strain, a certain reference configuration is adopted and called undeformed. Often, that reference configuration is chosen as an unstressed state, but such is neither necessary nor always convenient. If time is measured from zero at a moment when the…

Symmetry of the stress tensor has the important consequence that, at each point x, there exist three mutually perpendicular directions along which there are no shear stresses. These directions are called the principal stress directions, and the corresponding normal stresses are called the principal stresses. If the principal stresses are ordered algebraically as σI, σII, and σIII (Figure 4), then the…

Now the linear momentum principle may be applied to an arbitrary finite body. Using the expression for Tj above and the divergence theorem of multivariable calculus, which states that integrals over the area of a closed surface S, with integrand ni f (x), may be rewritten as integrals over the volume V enclosed by S, with integrand ∂f (x)/∂xi; when f (x) is a differentiable function, one may derive thatat least…

Assume that F and M derive from two types of forces, namely, body forces f, such as gravitational attractions—defined such that force fdV acts on volume element dV (see Figure 1)—and surface forces, which represent the mechanical effect of matter immediately adjoining that along the surface S of the volume V being considered. Cauchy formalized in 1822 a basic assumption of continuum mechanics that such surface forces could be represented as a stress…

Let x denote the position vector of a point in space as measured relative to the origin of a Newtonian reference frame; x has the components (x1, x2, x3) relative to a Cartesian set of axes, which is fixed in the reference frame and denoted as the 1, 2, and 3 axes in Figure 1. Suppose that a material occupies the part of space considered, and…

In addressing any problem in continuum or solid mechanics, three factors must be considered: (1) the Newtonian equations of motion, in the more general form recognized by Euler, expressing conservation of linear and angular momentum for finite bodies (rather than just for point particles), and the related concept of stress, as formalized by Cauchy, (2) the geometry of deformation and thus the expression…

The digital computer revolutionized the practice of many areas of engineering and science, and solid mechanics was among the first fields to benefit from its impact. Many computational techniques have been used in this field, but the one that emerged by the end of 1970s as, by far, the most widely adopted is the finite-element method. This method was outlined by…

The German physicist Wilhelm Weber noticed in 1835 that a load applied to a silk thread produced not only an immediate extension but also a continuing elongation of the thread with time. This type of viscoelastic response is especially notable in polymeric solids but is present to some extent in all types of solids and…