Consider a circular disc of homogeneous material (centre O, radius R, thickness e) to which is attached a laboratory Cartesian coordinate system (O; xj), containing a cylindrical hole (diameter 2r, axis x3), displaced vertically by h from the origin O; OO’ = h < R and r
The crack is moving under mode I loading with the tension Poisson’s stress νAσa , perpendicular to the plane of the moving crack that is parallel to the compression direction. The energy release rate G is given in Anongba (2020), see below. No reference is made to Rayleigh wave velocity. Hence, we have in the subsonic velocity regime (as example) at crack front position P (x1= 0, x2= a, x3) (use Figure in attached file)
G (P) = bK2/ 4πC
where C= [ µb(2-v2)/πv2] (1/Pt – 1/Pl ), v= v0/ ct, µ is the shear modulus, b Burgers vector; Pt , Pl , ct, cl are defined in Anongba (2020).
K2 = (νAσa)2 π|a-h|,
|a-h| = r + v0t (absolute value), v0 is the constant velocity of the crack-front, t is the time. In absence of Poisson's stress, G= 0; consequently, no motion of the crack is possible.
P.N.B. ANONGBA, Planar cracks in uniform motion under mode I and II loadings, Rev. Ivoir. Sci. Technol., 35 (2020) 1 - 22
I don't have a clear understanding of your problem but suspect that its stress-strain state is 3-dimensional. In such a condition principal strain in one dimension can be tensile while all other principal stress-strain components are compressive. Refer for example to DOI:
- 10.13140/RG.2.2.32291.26407 and its references.
In the present problem, there is only one externally applied compression (-σa) along x2. As a result, induced tensile Poisson's stresses are along both x1 and x3. Along x3, there is no relative displacement of the faces of the observed moving crack; hence this contributes zero value to G the crack extension force per unit length of the crack front. The opening of the faces of the crack is along x1 only and due to Poisson's stress νAσa . Again "the crack is moving under mode I loading with the tensile Poisson's stress νAσa " so you may follow my first answer (july 28) above.
I understand now. Thank you. Your explanation confirms my suspicion that it is a 3-dimensional problem. In my experience there is a tendency to try to apply 2-dimensional solutions.
Thank you very much for your contribution in clarifying this rather difficult subject of 3-d propagating fracture. The dislocation concept (Friedel, Bilby, Cottrell, Eshelby, Stroh ...) is really helpful on the road. Sincerely, Patrick A.
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