446 IX- DYNAMICS OF DEFORMABLE BODIES. 



But by 31 this is the condition that the displacement is a lamellar 

 vector and 



' dx y dy' ~ dz 



Then (p is called the strain -potential. Only when the strain is ir- 

 rotational can a strain -potential exist. 



The line integral along any curve AS of the tangential com- 

 ponent of the displacement 



B B 



80} / q cos (q, ds) ds = I (udx + vdy + wdi) 



A A 



is called the circulation along the path, and for irrotational strain is 

 independent of the path, equal to cp B y Aj and vanishes for a closed 

 path. 



Surfaces for which <p is constant are called equipotential surfaces, 

 and the displacement -lines, or lines drawn so that their tangents are 

 everywhere in the direction of the displacements, are normal to the 

 equipotential surfaces. 



The dilatation 



rnx du . dv , cw 



ol) 6 = -K f-^ h -o = ^/<P. 



dx ' dy dz ^ 



Accordingly in a non-dilatational, irrotational strain the displacement 

 is a solenoidal vector and the displacement -potential a harmonic 

 function. Since for a solenoidal vector the magnitude of the vector 

 is inversely proportional to the cross -section of an infinitesimal tube, 

 the displacement cannot vanish except at infinity. By the properties 

 of harmonic functions (p cannot have a maximum or minimum unless 

 <3 is different from zero. 



If (p be a homogeneous quadratic function of the coordinates, 

 the strain is homogeneous throughout and not merely in its smallest 

 parts. The equipotential surfaces are concentric quadric surfaces 

 and since 



the equipotential surfaces, (p = const., are similar to the elongation 

 quadric. 



17O. Stress. When a body is strained a system of forces is 

 generally called into play tending to resist the strain. The system 

 of forces is called a stress. In order to specify the stress at any 

 point in the body, we draw a plane through the point separating 

 the body into two parts. The parts on one side of this plane will 



