Question $2(45$ points$)$

Two cases for crack propagation relating to the Griffith Energy Balance were shown in lectures:

$(1)$ edge crack with applied tensile stress and $(2)$ double cantilever with constant wedging

displacement. Both cases consist of Mode I failure $($i$.$e$.,$ opening$).$

a$.(10$ points$)$ What is the elastic strain energy $($U$\_($E$))$ during loading for each case for the crack

lengths specified in Figure $1($Hint: Check the course notes $($readings$).$ Case $1$ should be

based on the applied stress, sigma, and Case $2$ on the dimensions of the wedge, h$,$ and the

thickness of the specimen, d $)?$ How does the strain energy relate to the mechanical energy

$($U$\_($M$))$ in each case?

b$.(5$ points$)$ What is the surface energy $($Us$)$ of the crack for each case?

c$.(15$ points$)$ Sketch the total energy diagrams, U$,$ for each case as a function of crack length,

c$,$ including the mechanical energy $($U$\_($M$))$ and surface energy $($Us$).$ Note on the two diagrams

where the equilibrium crack length, c$\_(0),$ is located. The shape of the curves should be correct

$.(10$ points$)$ For each case, derive the equilibrium crack length, c$=$c$0,$ in terms of $($i$)$ the

applied stress for Case $1,$ and $($ii$)$ the wedge dimensions for Case $2.$

e$.(5$ points$)$ For each case, is this equilibrium crack length, c$0,$ stable or unstable? Explain

your answer. d$.(10$ points$)$ For each case, derive the equilibrium crack length, $\backslash ($ c$=$c $0\backslash ),$ in terms of $($i$)$ the applied stress $\backslash (\backslash$sigma $\backslash )$ for Case $1,$ and $($ii$)$ the wedge dimensions for Case $2.$

e$.(5$ points$)$ For each case, is this equilibrium crack length, co$,$ stable or unstable? Explain your answer.