Roman Josi

Roman Josi is a Swiss professional ice hockey defenceman who serves as captain of the Nashville Predators of the National Hockey League. Josi began playing professionally with SC Bern in 2006, he was selected by the Nashville Predators in the second round, 38th overall, in the 2008 NHL Entry Draft. He was considered one of the top Swiss prospects in the draft. Josi became a full-time member of Bern during the 2007–08 season after spending the previous two seasons with the club's NLB team. Josi won a National League A championship in 2010. Josi joined the Predators' organization during the 2010–11 season, spending the entire year with the Milwaukee Admirals of the AHL. Josi scored his first NHL goal on 10 December 2011, against Dan Ellis of the Anaheim Ducks, he began the 2012 -- SC Bern, because of the NHL lockout. He played alongside NHL stars John Mark Streit. With the lockout resolved, Josi returned to play with the Predators and on 25 February 2013, Josi scored a career high four points in an overtime home win against the Dallas Stars.

Josi had a career high year during the 2014–15 season, ranking fifth among defensemen in scoring and finishing with a then-career high 55 points. At the end of the season Josi finished in the top five for the Norris Trophy vote. During the 2015–16 season, Josi was selected to play in the 2016 NHL All-Star Game along with teammates Pekka Rinne, Shea Weber, James Neal, he finished in the top five again for the Norris Trophy vote. During the 2016–17 season, Josi was selected as an alternate captain along with James Neal. Josi's defensive skill played a crucial role in the Predators 2017 Stanley Cup playoffs run. Despite being placed on long term injury reserve in January, he ended the season with 12 goals and 49 points. Before the 2017–18 season Josi was named the seventh captain in Predators history on 19 September 2017. On October 29, 2019, Josi signed an eight-year, $72.472 million contract extension with the Predators. Josi was selected to play for Switzerland at the 2010 Winter Olympics, he represented Switzerland at the 2007 and 2008 IIHF World U18 Championships, the 2007, 2008 and 2009 IIHF World U20 Championship and the 2009 Ice Hockey World Championship.

Josi was selected as the tournament MVP at the 2013 IIHF World Championship in Sweden. He was selected to play for Team Europe at the 2016 World Cup. Josi won a second silver medal with Switzerland at the 2018 IIHF World Championship in Denmark, falling to Sweden in the shootout. All statistics taken from 2010 National League A championship 2013 National League A championship 2013 IIHF World Championship MVP 2013 IIHF World Championship Silver medal 2013 IIHF World Championship Media All-Star Team 2013 IIHF World Championship Best Defenceman Clarence S. Campbell Bowl Presidents' Trophy 3x NHL All-Star Game Biographical information and career statistics from, or, or, or, or The Internet Hockey Database

Stress functions

In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces on the boundary are the equilibrium equation: σ i j, i = 0 where σ is the stress tensor, the Beltrami-Michell compatibility equations: σ i j, k k + 1 1 + ν σ k k, i j = 0 A general solution of these equations may be expressed in terms the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations, it can be shown that a complete solution to the equilibrium equations may be written as σ = ∇ × Φ × ∇ Using index notation: σ i j = ε i k m ε j l n Φ k l, m n where Φ m n is an arbitrary second-rank tensor field, continuously differentiable at least four times, is known as the Beltrami stress tensor. Its components are known as Beltrami stress functions. Ε is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated.

For a set of non-repeating indices the component value will be +1 for permutations of the indices, -1 for odd permutations. And ∇ is the Nabla operator; the Maxwell stress functions are defined by assuming that the Beltrami stress tensor Φ m n is restricted to be of the form. Φ i j = The stress tensor which automatically obeys the equilibrium equation may now be written as: The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami–Michell compatibility equations for stress. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions: ∇ 4 A + ∇ 4 B + ∇ 4 C = 3 /, These must yield a stress tensor which obeys the specified boundary conditions; the Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only. This stress function can therefore be used only for two-dimensional problems.

In the elasticity literature, the stress function C is represented by φ and the stresses are expressed as σ x = ∂ 2 φ ∂ y 2. In polar coordinates the expressions are: σ r r = 1 r ∂ φ ∂ r + 1 r 2 ∂ 2 φ ∂ θ