Ring math expression plasticity (#497)

examples/ringPlaneStrain/ringPlaneStrain.m

@@ -152,13 +152,15 @@
 %md The solution is extracted from Hill (The mathematical theroy of plasticity, 1950). 
 %md The yielding pressure $p_0$ is defined as,
 %md```math 
-%md Y = \dfrac{2\sigma_{Y,0}}{\sqrt(3)}  \\
+%md Y = \dfrac{2\sigma_{Y,0}}{\sqrt{3}}  \\
 %md p_0 = \dfrac{Y}{2}\left(1+\dfrac{R_i^2}{R_e^2}\right).
 %md```
 %md The radial displacement of the outer surface of the ring is given by,
-%md```math 
-%md \dfrac{2pR_e}{E\left(\dfrac{R_e^2}{R_i^2-1}\right)}(1-\nu^2)\qquad \textup{if p\leq p_0}
-%md \dfrac{Yc^2}{ER_e}(1-\nu^2)\qquad \textup{if p> p_0}
+%md```math
+%md u_r(R_e) = \text{if}~~ p \leq p_0 \\
+%md \dfrac{2 p R_e}{E \left( \dfrac{R_e^2}{R_i^2-1}\right) }( 1-\nu^2 ) \\
+%md \text{else} \\ 
+%md \dfrac{2pR_e}{E\left(\dfrac{R_e^2}{R_i^2-1}\right)}(1-\nu^2)
 %md```
 %md where $c$ denotes the plastic front surface in the ring and is given by the implicit function,
 %md```math 
@@ -217,14 +219,14 @@
 %
 global Y
 %
-Y = 2*sigmaY0 / sqrt(3) ;
-% p0 = Y/2 * (1-a^2/b^2)  ; % Yielding pressure
-p0 = Y/2 * (1-Ri^2/Re^2)  ; % Yielding pressure
+Y = 2 * sigmaY0 / sqrt(3) 				;
+% p0 = Y/2 * (1-a^2/b^2)
+p0 = Y / 2 * (1 - Ri^2 / Re^2)  		; % Yielding pressure
 %
-pressure_vals = loadFactorsMat(:,3)*p ;
+pressure_vals = loadFactorsMat(:,3) * p ;
 %
-cvals = zeros(length(pressure_vals),1) ;
-ubAna = zeros(length(pressure_vals),1) ;
+cvals = zeros(length(pressure_vals),1) 	;
+ubAna = zeros(length(pressure_vals),1) 	;
 %
 % Plastic front value
 for i = 1:length(cvals)
@@ -277,5 +279,6 @@
 analyticCheckTolerance = 1e-2 ;
 verifBoolean = ( ( numericalRi - analyticValRi ) < analyticCheckTolerance ) && ...
                ( ( numericalRe - analyticValRe ) < analyticCheckTolerance ) && ...
-               ( ( ubNum(end) - ubAna(end)     ) < analyticCheckTolerance )
+               ( ( ubNum(end) - ubAna(end)     ) < analyticCheckTolerance ) ;
+%md