Issue 66

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 66 (2023) 233-260; DOI: 10.3221/IGF-ESIS.66.15

T HE CRACKED VERSUS THE INTACT PLATE UNDER UNIAXIAL COMPRESSION

B

ased on the formulae obtained in previous section, displacements, stresses and SIFs are here calculated approxi mately for a uniaxially compressed square plate, with a short central ‘mathematical’ crack 2 α (Fig.7a). In this con figuration, apart from the limiting case β =0 ο , the ‘initial problem’ provides always unnatural overlapping lips, while the ‘general problem’ introduced in this paper provides always closed cracks with lips under mutual contact stresses. In addition, the intact plate under an identical loading scheme (Fig. 7b) is studied in parallel with the cracked plate in an at tempt to quantify the actual consequences of the presence of the crack on the displacement and stress fields.

 

 

y

y





x

x

b

b

t

t

b

b

(a)

(b)

 

 

Figure 7: (a) The cracked plate uniaxially compressed, and (b) the relevant intact plate under the same loading conditions.

The intact plate under uniaxial compression The complex potentials solving the infinite intact plate under uniaxial pressure, for the xOy system shown in Fig.7b, are:

σ

σ

σ

σ

2i β e z, Φ (z) 



2i β









in φ (z)

z, ψ (z)

, Ψ (z)

e









(49)

in

in

in

4

2

4

2

Substituting from Eqns.(49) in the well-known formulae [27]:

2 Φ (z) z Φ (z) Ψ (z)  

yy,in   

xy,in i τ     in

xx,in σ σ

4 Φ (z), σ

(50)

in

yy,in

in

in

in in 2 μ (u iv ) κφ (z) z Φ (z) ψ (z)     in in in

(51)

stresses and displacements in the plate are obtained as:

σ

σ

σ

  

  



σ

(1 cos2 β ), σ

(1 cos2 β ), τ

sin2 β



(52)

xx,in

yy,in

xy,in

2

2

2

κ 1 

κ 1 

σ

σ

  

  

  

  



x xcos2 β ysin2 β , v  



y ycos2 β xsin2 β  

u





(53)

in

in

4 μ

2

4 μ

2

Comparing the displacement field for a cracked plate versus that for an intact plate The deformed shapes of an intact and a cracked plate, for both unnaturally overlapping lips (of only theoretical interest) and naturally acceptable lips in contact, are obtained in this section. In this context, two square plates ABCD, an intact and a cracked one, of side-length b=0.40 m, are compressed by uniform pressure σ ∞ on their sides AB and CD. In the cracked plate, the crack length is 2 α =0.10 m and forms an angle β =30 o with σ ∞ . The material of both plates is PMMA with E=3.2 GPa and ν =0.36. For clarity of the figures, an extremely large σ ∞ (–1 GPa) is assumed. To draw the deformed configuration of the intact plate use is made of Eqns.(53). In drawing the deformed configuration of the cracked plate for the unnatural overlapping, use is made of Eqns.(6-9) of the unnatural ‘initial problem’, for k=0. The two deformed con figurations are shown in juxtaposition to each other in Fig.8, for plane strain conditions; black color is used for the cracked plate and red color is used for the intact one. The undeformed plate ABCD is also shown with black discontinuous line.

245