Structural Mechanics (Wierzbicki)

Strain is a fundamental concept in continuum and structural mechanics. Displacement fields and strains can be directly measured using gauge clips or the Digital Image Correlation (DIC) method. Deformation patterns for solids and deflection shapes of structures can be easily visualized and are also predictable with some experience. By contrast, the stresses can only be determined indirectly from the measured forces or by the inverse engineering method through a detailed numerical simulation. Furthermore, a precise determination of strain serves to define a corresponding stress through the work conjugacy principle. Finally the equilibrium equation can be derived by considering compatible fields of strain and displacement increments, as explained in Chapter 2. The present author sees the engineering world through the magnitude and shape of the deforming bodies. This point of view will dominate the formulation and derivation throughout the present lecture note. Chapter 1 starts with the definition of one dimensional strain. Then the concept of the three dimensional (3-D) strain tensor is introduced and several limiting cases are discussed. This is followed by the analysis of strains-displacement relations in beams (1-D) and plates (2- D). The case of the so called moderately large deflection calls for considering the geometric non-linearities arising from rotation of structural elements. Finally, the components of the strain tensor will be re-defined in the polar and cylindrical coordinate system.

In this section the strain-displacement relations will be derived in the cylindrical coordinate system . The polar coordinate system is a special case with . The components of the displacement vector are . There are two ways of deriving the kinematic equations. Since strain is a tensor, one can apply the transformation rule from one coordinate to the other. This approach is followed for example on pages 125-128 of the book on “A First Course in Continuum Mechanics” by Y.C. Fung. Or, the expression for each component of the strain tensor can be derived from the geometry. The latter approach is adopted here. The diagonal (normal) components , , and represent the change of length of an infinitesimal element. The non-diagonal (shear) components describe the change of angles.

The word “kinematics” is derived from the Greek word “kinema”, which means movements, motion. Any motion of a body involves displacements , their increments and velocities . If the rigid body translations and rotations are excluded, strains develop. We often say “Kinematic assumption” or “Kinematic boundary conditions” or “Kinematic quantities” etc. All it means that statements are made about the displacements and strains and/or their rates. By contrast, the word “static” is reserved for describing stresses and/or forces, even though a body could move. The point is that for statically determined structures, one could determine stresses and forces without invoking motion. Such expressions as “static formulation”, “static boundary conditions”, “static quantities” always refer to stresses and forces.