G. F. Becker — Finite Elastic Stress-Strain Function. 351 



where the second member expresses the moments of the ex- 

 ternal forces, which are as usual measured per unit area, and 

 x l y l are the coordinates of any point the mass of which is m. 



Reduction of equation (7). — Let x and y represent the posi- 

 tion of the corner of the strained cube ; then the abscissa of 

 the center of inertia of the surface on which the stress Y acts 

 is x/2, and since Y is uniform, Ix x Y — xY/2. Similarly 

 Iy^X= yX/2. Now xY and yX may also be regarded as 

 the loads or initial stresses acting on the two surfaces of the 

 mass parallel respectively to ox and oy, and in a shear these 

 two loads are equal and opposite. Hence the second member 

 of (7) reduces to x Y It has been shown above that, if Q is 

 an initial tractive load, Q/S is the common value of the two 

 equal and opposite loads producing one shear. But to obtain 

 comparable results for shear dilation and extension, Q/S must 

 be measured in appropriate units of resistance. Since M is 

 the unit of resistance appropriate to extension, the separate 

 parts of the force must be multiplied by M and divided by 

 resistances characteristic of the elementary strains. Now 



MQ.MQMQ 



2n '3 2n ' 3 Sk ' 3 V ' 



and it is evident that 2/i/M is the unit in which Q/S should 

 be measured for the single shear.* Thus the second member 

 of (7) becomes MQ/6n. 



This, then, is the value which the moment of the external 

 forces assumes when these hold the strained unit cube in equi- 

 librium. This unit cube forms an eighth part of the cube 

 circumscribed about the sphere of unit radius. When the 

 entire mass is considered, the sum of all the moments of the 

 external forces is zero ; since they are equal and opposite by 

 pairs. If the entire mass thus strained is suddenly released 

 and allowed to perform free vibrations, the sum of all the 

 moments of momenta will of course remain zero. On the 

 other hand the quantity MQ/Qn will remain constant. For 

 this load determines the limiting value of the strain during 

 vibration and is independent of the particular phase of vibra- 

 tion, or of the time counted from the instant of release. It 

 may be considered as the moment of the forces which the other 

 parts of the entire material system exert upon the unit cube. 



* In this paper changes of temperature are expressly neglected. The changes 

 of temperature produced by varying stress in a body performing vibrations of 

 small amplitude can be allowed for by employing ''kinetic" moduluses, which 

 are a little greater than the ordinary '■ static " moduluses. Thomson and Tait, 

 Xat. Phil , section 687. 



