Thermodynamic considerations
In thermodynamic terminology, a state of purely elastic material response corresponds to an equilibrium state, and a process during which there is purely elastic response corresponds to a sequence of…
Thermal strains
Temperature change can also cause strain. In an isotropic material the thermally induced extensional strains are equal in all directions, and there are no shear strains. In the simplest cases,…
Stress-strain relations
Linear elastic isotropic solid The simplest type of stress-strain relation is that of the linear elastic solid, considered in circumstances for which |∂ui/∂Xj|<< 1 and for isotropic materials, whose mechanical…
Finite deformation and strain tensors
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…
Small-strain tensor
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…
Geometry of deformation
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…
Equations of motion
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…
Stress
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…
Linear and angular momentum principles: stress and equations of motion
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…
Basic Principles
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…
