Continuum mechanics studies the foundations of deformable body mechanics from a mathematical perspective. It also acts as a base upon which other applied areas such as solid mechanics and fluid mechanics are developed. This book discusses some important topics, which have come into prominence in the latter half of the twentieth century, such as material symmetry, frame-indifference and thermomechanics. The study begins with the necessary mathematical background in the form of an introduction to tensor analysis followed by a discussion on kinematics, which deals with purely geometrical notions such as strain and rate of deformation. Moving on to derivation of the governing equations, the book also presents applications in the areas of linear and nonlinear elasticity. In addition, the volume also provides a mathematical explanation to the axioms and laws of deformable body mechanics, and its various applications in the field of solid mechanics.

Continuum Mechanics : Foundations and Applications of Mechanics

Studies into the stress-strain state of rock mass are reviewed. The solutions of a wide range of applied problems are analyzed and methods used to solve them are generalized. Important areas of further research in rock mechanics are pointed out.

Rock deformation problems

The paper describes a numerical method for determining the stress distribution in the interior of a three-dimensional body using experimentally determined surface stresses, and the interior displacements from surface displacements. The normal and shear stresses inside the body are obtained by solving Laplace's equation in terms of sum of normal stresses together with the three-dimensional compatibility equations in terms of stresses, using the finite difference technique, when the stresses on the surface of the body are known. On the other hand if surface displacements are known (from which strain components could be determined) then displacement components in the interior of a body can be determined by solving Laplace's equation in terms of sum of normal strains together with the three-dimensional equilibrium equations in terms of displacements. It is shown that axi-symmetric problems can also be solved in an identical way by transforming the equations into cylindrical co-ordinates. The application of the method has been illustrated through several examples.

An experimental–numerical hybrid technique for three dimensional stress analysis

The stress distribution within a prismatic specimen loaded uniaxially was determined by the three-dimensional finite element method. The stresses are nonuniform and vary from point to point. Prismatic specimens of Tennessee marble were loaded uniaxially. Pictures of the two adjacent side surfaces were taken simultaneously at various deformation levels. Crack patterns in these pictures were then traced in the transparency and compared for crack growth study. Grain boundary cracks were identified as Griffith cracks. A model consisting of a crack subjected to the combination of modes 1 and 2 of basic crack surface displacement was developed. Qualitatively, it was found applicable for describing the fracture processes of Tennessee marble specimens.

Crack propagation and fracture of rock specimens loaded in compression

Transverse shear force effects on the stress and strain distributions of post-tensioned, rectangular, concrete anchorage blocks were studied, for both concentrically and eccentrically prestressed anchorage zones. From the three-dimensional, isoparametric finite element model used, it was found that the peak lateral bursting stress in the beam width direction is always higher and nearer to the loaded face of the concrete than the peak transverse bursting stress in the direction of the beam height. Experimental verification involved the use of three-dimensional, embeddable, strain gage tripods, fabricated to measure the interior concrete strains in the anchorage blocks of fifteen beams subjected to transverse shear loadings up to failure. The measured strains were compared with the analytical strain curves and strain contours given by the three-dimensional finite element model, and found favorable. It was also found that the transverse shear load reduces the peak transverse bursting strain, but has little effect on the peak lateral bursting strain. The test ultimate transverse loads compared well with the respective theoretical loads, with a mean value of 1.10.

Anchorage zone stresses of beams subjected to shear forces

Fur ein Rechteckprisma, an dessen Enden vorgegebene Krafte wirken, wird auf der Grundlage der die dreidimensionale Elastizitatstheorie beherrschenden Gleichungen das Problem des dreidimensionalen elastischen Verhaltens gelost. Die Losung hat die Form eines Galerkin-Vektors, dessen Komponenten in Doppel-Fourier-Reihen entwickelt sind, die der zugrunde liegenden biharmonischen Gleichung genugen und so gewahlt sind, das sie alle Randbedingungen befriedigen. Es werden Rechteckprismen mit symmetrischen oder exzentrischen Normalkraften an den Enden untersucht und fur typische Falle numerische Ergebnisse angegeben. Die gewonnenen Ergebnisse werden mit denen der zweidimensionalen Theorie verglichen.

Raja Iyengar

Papers

A three dimensional elasticity solution for rectangular prism under end loads