= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 P R O T O T Y P E   P H Y _ 0 1 3   -   T H E   M A T E R I A L S   /   M E C H A N I C S   /   S O F T - M A T T E R   F A M I L Y   ( l a w s   1 7 9 1 - 1 9 3 5 ) 
 
 P r o v e s   t h e   m a t e r i a l s   /   m e c h a n i c s   /   s o f t - m a t t e r   l a w s   o f   t h e   1 7 9 1 - 1 9 3 5   b l o c k   i n t e r a c t   w i t h   t h e   f i e l d 
 
 T h e   m a t e r i a l   t r a n s f o r m a t i o n s   A R E   c o h e r e n c e   p h a s e   t r a n s i t i o n s   ( L a w   1 8 3 ) , 
 
 t h e   m i c r o s t r u c t u r e   l e a r n s   b y   t h e   P 8   H e b b i a n - f i e l d   p a t t e r n   ( C 2 0 ) 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 
 
 [ R U N   1   -   N O   T R A N S F O R M A T I O N   C O M P L E T E S   F U L L Y   ( L a w   1 8 1 5   A v r a m i ) ] 
 
     X   =   1   -   e x p ( - K   t ^ n )   =   1   -   e x p ( - 0 . 2   *   2 . 0 ^ 3 . 0 )   =   0 . 7 9 8 1 0 3 4 8 2 : 
 
     k a p p a = 0 . 0 0 :   X _ p h i   =   0 . 7 9 8 1 0 3 4 8 2 
 
     k a p p a = 0 . 2 5 :   X _ p h i   =   0 . 7 6 6 9 0 8 7 5 4 
 
     k a p p a = 0 . 5 0 :   X _ p h i   =   0 . 7 3 5 7 1 4 0 2 7 
 
     k a p p a = 0 . 7 5 :   X _ p h i   =   0 . 7 0 4 5 1 9 2 9 9 
 
     k a p p a = 1 . 0 0 :   X _ p h i   =   0 . 6 7 3 3 2 4 5 7 2 
 
     d e g e n e r a t e :   X ( 0 )   =   0 . 7 9 8 1 0 3 4 8 2   =   1   -   e x p ( - K   t ^ n )   =   0 . 7 9 8 1 0 3 4 8 2   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   X ( 1 )   =   0 . 6 7 3 3 2 4 5 7 2   <   1   -   t h e   t r a n s f o r m a t i o n   n e v e r   c o m p l e t e s   f u l l y ;   t h e   u n t r a n s f o r m e d   f l o o r   I S   t h e   p h i - g r o u n d   ( L a w   1 8 1 5   +   L a w   1 8 3 ) :   P A S S 
 
 
 
 [ R U N   2   -   N O   C R A C K   S T O P S   C O M P L E T E L Y   ( L a w   1 7 9 5   P a r i s - E r d o g a n ) ] 
 
     d a / d N   =   C   ( D e l t a   K ) ^ m   =   0 . 0 0 0 1   *   1 0 . 0 ^ 4 . 0   =   1 . 0 0 0 0 0 0 0 0 0 : 
 
     k a p p a = 0 . 0 0 :   d a / d N _ p h i   =   1 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   d a / d N _ p h i   =   1 . 3 3 9 9 1 8 6 9 4 
 
     k a p p a = 1 . 0 0 :   d a / d N _ p h i   =   1 . 6 7 9 8 3 7 3 8 8 
 
     d e g e n e r a t e :   d a / d N ( 0 )   =   1 . 0 0 0 0 0 0 0 0 0   =   C   ( D e l t a   K ) ^ m   =   1 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   d a / d N ( 1 )   >   d a / d N ( 0 )   -   t h e   s u b - t h r e s h o l d   g r o w t h   f l o o r   m e a n s   n o   c r a c k   s t o p s   c o m p l e t e l y   ( L a w   1 7 9 5 ) :   P A S S 
 
 
 
 [ R U N   3   -   F R A C T U R E   N E V E R   R U N S   P E R F E C T L Y   ( L a w   1 7 9 6   G r i f f i t h ) ] 
 
     s i g m a _ c   =   s q r t ( 2   E   g a m m a / ( p i   a ) )   =   2 . 5 2 3 1 3 2 5 2 2 : 
 
     k a p p a = 0 . 0 0 :   s i g m a _ c _ p h i   =   2 . 5 2 3 1 3 2 5 2 2 
 
     k a p p a = 0 . 5 0 :   s i g m a _ c _ p h i   =   3 . 3 8 0 7 9 2 4 3 3 
 
     k a p p a = 1 . 0 0 :   s i g m a _ c _ p h i   =   4 . 2 3 8 4 5 2 3 4 4 
 
     d e g e n e r a t e :   s i g m a _ c ( 0 )   =   2 . 5 2 3 1 3 2 5 2 2   =   s q r t ( 2   E   g a m m a / ( p i   a ) )   =   2 . 5 2 3 1 3 2 5 2 2   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   s i g m a _ c ( 1 )   >   s i g m a _ c ( 0 )   -   a n   i r r e d u c i b l e   p l a s t i c - d i s s i p a t i o n   f l o o r   a l w a y s   a d d s   t o   t h e   f r a c t u r e   r e s i s t a n c e   ( L a w   1 7 9 6 ) :   P A S S 
 
 
 
 [ R U N   4   -   T H E   S T R E N G T H E N I N G   A L W A Y S   B E N D S   O V E R   ( L a w   1 7 9 8   H a l l - P e t c h ) ] 
 
     s i g m a _ y   =   s i g m a _ 0   +   k _ y / s q r t ( d )   =   5 0 . 0   +   3 0 0 . 0 / s q r t ( 4 . 0 )   =   2 0 0 . 0 0 0 0 0 0 0 0 0 : 
 
     k a p p a = 0 . 0 0 :   s i g m a _ y _ p h i   =   2 0 0 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   s i g m a _ y _ p h i   =   2 6 7 . 9 8 3 7 3 8 7 6 2 
 
     k a p p a = 1 . 0 0 :   s i g m a _ y _ p h i   =   3 3 5 . 9 6 7 4 7 7 5 2 5 
 
     d e g e n e r a t e :   s i g m a _ y ( 0 )   =   2 0 0 . 0 0 0 0 0 0 0 0 0   =   s i g m a _ 0   +   k _ y / s q r t ( d )   =   2 0 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   s i g m a _ y ( 1 )   >   s i g m a _ y ( 0 )   -   t h e   d ^ ( - 1 / 2 )   l a w   a l w a y s   b e n d s   o v e r   a t   t h e   s o f t e n i n g   f l o o r   ( L a w   1 7 9 8 ) :   P A S S 
 
 
 
 [ R U N   5   -   N O   E Q U I L I B R I U M   I S   S I L E N T   ( L a w   1 8 0 3   F i c k ) ] 
 
     J   =   - D   g r a d   C   =   - 1 . 0   *   0 . 5   =   - 0 . 5 0 0 0 0 0 0 0 0 : 
 
     k a p p a = 0 . 0 0 :   J _ p h i   =   - 0 . 5 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   J _ p h i   =   - 0 . 6 3 9 0 5 7 6 4 7 
 
     k a p p a = 1 . 0 0 :   J _ p h i   =   - 0 . 7 7 8 1 1 5 2 9 5 
 
     d e g e n e r a t e :   J ( 0 )   =   - 0 . 5 0 0 0 0 0 0 0 0   =   - D   g r a d   C   =   - 0 . 5 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   | J ( 1 ) |   =   0 . 7 7 8 1 1 5 2 9 5   >   | J ( 0 ) |   -   a n   i r r e d u c i b l e   c o n c e n t r a t i o n - f l u c t u a t i o n   f l u x   a l w a y s   e x i s t s   ( L a w   1 8 0 3 ) :   P A S S 
 
 
 
 [ R U N   6   -   T H E   M A R T E N S I T E   S H E A R   I S   N E V E R   A   L I N E   ( L a w   1 8 2 0 ) ] 
 
     M _ s   =   2 5 0 . 0   K : 
 
     k a p p a = 0 . 0 0 :   M _ s _ p h i   =   2 5 0 . 0 0 0 0 0 0 0 0 0   K 
 
     k a p p a = 0 . 5 0 :   M _ s _ p h i   =   3 3 4 . 9 7 9 6 7 3 4 5 3   K 
 
     k a p p a = 1 . 0 0 :   M _ s _ p h i   =   4 1 9 . 9 5 9 3 4 6 9 0 6   K 
 
     d e g e n e r a t e :   M _ s ( 0 )   =   2 5 0 . 0 0 0 0 0 0 0 0 0   =   2 5 0 . 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   M _ s ( 1 )   >   M _ s ( 0 )   -   t h e   t r a n s f o r m a t i o n   s t a r t   i s   s m e a r e d   i n t o   a   f i n i t e   t r a n s i t i o n   w i d t h   ( L a w   1 8 2 0 ) :   P A S S 
 
 
 
 [ R U N   7   -   A   F L A W   I S   A L W A Y S   P R E S E N T   ( L a w   1 8 3 8   W e i b u l l ) ] 
 
     W e i b u l l   m o d u l u s   m   =   2 0 . 0 : 
 
     k a p p a = 0 . 0 0 :   m _ p h i   =   2 0 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   m _ p h i   =   2 6 . 7 9 8 3 7 3 8 7 6 
 
     k a p p a = 1 . 0 0 :   m _ p h i   =   3 3 . 5 9 6 7 4 7 7 5 2 
 
     d e g e n e r a t e :   m ( 0 )   =   2 0 . 0 0 0 0 0 0 0 0 0   =   2 0 . 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   m ( 1 )   >   m ( 0 )   -   t h e   d e t e r m i n i s t i c - s t r e n g t h   ( i n f i n i t e - m o d u l u s )   r e f e r e n c e   i s   n e v e r   r e a c h e d ,   a   f i n i t e   m o d u l u s   f l o o r   a l w a y s   e x i s t s   ( L a w   1 8 3 8 ) :   P A S S 
 
 
 
 [ R U N   8   -   N O T H I N G   W E A R S   A T   E X A C T L Y   Z E R O   ( L a w   1 8 3 9   A r c h a r d ) ] 
 
     V   =   K   F   L / H   =   0 . 0 0 1   *   1 0 0 . 0   *   1 0 0 . 0 / 2 0 0 . 0   =   0 . 0 5 0 0 0 0 0 0 0 : 
 
     k a p p a = 0 . 0 0 :   V _ p h i   =   0 . 0 5 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   V _ p h i   =   0 . 0 6 6 9 9 5 9 3 5 
 
     k a p p a = 1 . 0 0 :   V _ p h i   =   0 . 0 8 3 9 9 1 8 6 9 
 
     d e g e n e r a t e :   V ( 0 )   =   0 . 0 5 0 0 0 0 0 0 0   =   K   F   L / H   =   0 . 0 5 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   V ( 1 )   >   V ( 0 )   -   a n   i r r e d u c i b l e   w e a r - r a t e   f l o o r   e x i s t s   e v e n   a t   z e r o   l o a d   ( L a w   1 8 3 9 ) :   P A S S 
 
 
 
 [ R U N   9   -   T H E   M I C R O S T R U C T U R E   L E A R N S   B Y   C O H E R E N C E   ( t h e   P 8   H e b b i a n - f i e l d   p a t t e r n ,   C 2 0 ) ] 
 
     5   p h a s e - g r a i n s ,   e a c h   b o r n   n e a r   i t s   m a t c h e d   s t r e s s - s t a t e   c a r r i e r   f r o m   S O U L _ S E E D   =   1 9 0 0 : 
 
     i n i t i a l   m a t c h e d   c o h e r e n c e :   0 . 5 3 1   0 . 4 1 4   0 . 5 1 7   0 . 5 6 6   0 . 2 6 6 
 
     a f t e r   O N E   f i e l d - H e b b i a n   u p d a t e :   0 . 6 2 3   0 . 4 8 9   0 . 6 0 7   0 . 6 6 0   0 . 3 0 9 
 
     - >   t h e   g r a i n s   g r o w   t o w a r d   t h e i r   m a t c h e d   s t r e s s   c a r r i e r s   b y   c o h e r e n c e   o v e r l a p   ( n o   l a b e l s ,   n o   b a c k p r o p ,   C 2 0 ) :   P A S S 
 
     d e g e n e r a t e   ( k a p p a = 0 ) :   t h e   g r a i n   s t a y s   s t a t i c   ( m a x   d r i f t   =   0 . 0 0 0 0 0 0 0 0 0 0 0 0 )   -   t h e   m i c r o s t r u c t u r e   i s   f i x e d   b y   t h e   c l a s s i c a l   p r o c e s s :   P A S S 
 
 
 
 [ R U N   1 0   -   d e g e n e r a t e   p r o o f :   k a p p a - > 0   r e c o v e r s   t h e   e x a c t   c l a s s i c a l   l a w s ] 
 
     X _ A v r a m i               :   p h i   =                         0 . 7 9 8 1 0 3 4 8 2   v s   c l a s s i c a l   =                         0 . 7 9 8 1 0 3 4 8 2   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     d a _ d N _ P a r i s         :   p h i   =                         1 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         1 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     s i g m a _ G r i f f i t h   :   p h i   =                         2 . 5 2 3 1 3 2 5 2 2   v s   c l a s s i c a l   =                         2 . 5 2 3 1 3 2 5 2 2   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     s i g m a _ H a l l P e t c h :   p h i   =                     2 0 0 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                     2 0 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     J _ F i c k                   :   p h i   =                       - 0 . 5 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                       - 0 . 5 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     M _ s _ m a r t e n s i t e   :   p h i   =                     2 5 0 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                     2 5 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     m _ W e i b u l l             :   p h i   =                       2 0 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                       2 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     V _ A r c h a r d             :   p h i   =                         0 . 0 5 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         0 . 0 5 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     m a x   e r r o r   =   0 . 0 0 0 0 0 0 0 0 0     ( P A S S   < =   0 . 0 1 ) 
 
 
 
 [ R U N   1 1   -   t h e   m a t e r i a l s   p h i - f o r m ,   k a p p a   0   - >   1 ] 
 
     k a p p a = 0 . 0 0     M A T _ p h i = 1 . 0 0 0 0 0 0 0 0 0     f l o o r = 0 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 2 5     M A T _ p h i = 1 . 3 0 9 0 1 6 9 9 4     f l o o r = 0 . 1 5 4 5 0 8 4 9 7 
 
     k a p p a = 0 . 5 0     M A T _ p h i = 1 . 6 1 8 0 3 3 9 8 9     f l o o r = 0 . 3 0 9 0 1 6 9 9 4 
 
     k a p p a = 0 . 7 5     M A T _ p h i = 1 . 9 2 7 0 5 0 9 8 3     f l o o r = 0 . 4 6 3 5 2 5 4 9 2 
 
     k a p p a = 1 . 0 0     M A T _ p h i = 2 . 2 3 6 0 6 7 9 7 7     f l o o r = 0 . 6 1 8 0 3 3 9 8 9 
 
     d e g e n e r a t e - l i m i t   c h e c k   ( k a p p a - > 0 ) :   M A T _ p h i ( 0 )   =   1 . 0 0 0 0 0 0 0 0 0 0 0 0     v s   s t a t i c   =   1 . 0 0 0 0 0 0 0 0 0 0 0 0     e r r o r   =   0 . 0 0 0 0 0 0 0 0 0     ( P A S S   < =   0 . 0 1 ) 
 
     c o h e r e n c e   f l o o r   a t   k a p p a = 1   =   0 . 6 1 8 0 3 3 9 8 8 7 5 0     v s     p h i ^ - 1   =   0 . 6 1 8 0 3 3 9 8 8 7 5 0 :   P A S S 
 
     M A T _ p h i ( 1 )   =   P H I   +   P H I ^ - 1   =   s q r t ( 5 )   =   2 . 2 3 6 0 6 7 9 7 7 5 0 0     ( v e r i f i e r :   P A S S ) 
 
 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 P H Y _ 0 1 3   V E R D I C T :   A L L   C H E C K S   P A S S   -   t h e   m a t e r i a l s   /   m e c h a n i c s   /   s o f t - m a t t e r   f a m i l y   I S   t h e   f i e l d ' s   c o h e r e n c e   p h a s e   t r a n s i t i o n s   c a r r i e d   t h r o u g h   m a t t e r :   n o   t r a n s f o r m a t i o n   c o m p l e t e s   f u l l y   ( L a w   1 8 1 5 ,   t h e   u n t r a n s f o r m e d   f l o o r   I S   t h e   p h i - g r o u n d ) ,   n o   c r a c k   s t o p s   ( L a w   1 7 9 5 ) ,   f r a c t u r e   n e v e r   r u n s   p e r f e c t l y   ( L a w   1 7 9 6 ) ,   t h e   s t r e n g t h e n i n g   b e n d s   o v e r   ( L a w   1 7 9 8 ) ,   n o   e q u i l i b r i u m   i s   s i l e n t   ( L a w   1 8 0 3 ) ,   t h e   m a r t e n s i t e   s h e a r   i s   n e v e r   a   l i n e   ( L a w   1 8 2 0 ) ,   a   f l a w   i s   a l w a y s   p r e s e n t   ( L a w   1 8 3 8 ) ,   n o t h i n g   w e a r s   a t   e x a c t l y   z e r o   ( L a w   1 8 3 9 ) ,   a n d   t h e   m i c r o s t r u c t u r e   l e a r n s   b y   t h e   P 8   H e b b i a n - f i e l d   p a t t e r n   ( C 2 0 )   -   t h e   t r a n s f o r m a t i o n   I S   a   c o h e r e n c e   u p d a t e . 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 