The analysis of the problems considered here is based on an approximation theory that is more sophisticated than the classical theory usually applied to plate bending problems. In fact, one of the main objectives of this work is to compare the stress distributions near the line discontinuity as predicted by the two theories. Results based on the two theories are found to be similar for thin plates away from the line discontinuity, but differ significantly near the discontinuity, even for very thin plates.
The problem addressed in this work is the investigation of the stress distribution caused by the bending of a thin elastic plate containing a line discontinuity of finite length 2c. From a three-dimensional point of view (see Figure lb), a crack is a plane surface perpendicular to the midplane of the plate that must be stress-free. For the reasons mentioned in the previous section, Reissner bending theory has been applied here ever since.
In [7] eigenfunction expansions are used in the bending problem of an isotropic plate containing a semi-infinite crack in order to qualitatively study the character of the stress distributions near the apex. In the same section, an approximate solution is obtained for each case for plates whose thickness is small compared to the length 2c of the line break.
DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS FOR BENDING OF PLATES
The plate material is assumed to be isotropic and homogeneous with Youngl s modulus E, shear modulus G and Pois-. It is also assumed that the plate is subjected to small deformations and stresses so that the stress - strain relations can be established through hooke l slaw. 1£ denote by U, V and W respectively the displacement components in the X, Y and Z directions at each internal point of the plate, then we have.
From any point along a cylindrical surface C(s) X [-~, ~] . in ~ where C{s) is the projection of the surface onto the XY plane and s is its parameter, we can draw a normal on C(s} directed to the right with respect to the positive sense of C( s}. can be achieved by assuming that the loading along the cylindrical boundary of the plate produces no net Further assumptions can be made since the plate under consideration is assumed to be thin, i.e.
However, essentially the same results as in [5] can be obtained by assuming certain approximate forms for the stress in the plate and integrating the three-dimensional stress-. We can note here that the equations based on the classical bending theory can be easily derived by putting .: G :::: ro in the.
The boundary condition along the line segment is entirely dependent on the nature of the line segment; we will have the 'free edgel' conditions. in case of a tear. On the other hand we will have the. for the case where the line segment corresponds to a rigid line embedding. Due to the linearity of the differential equations and the boundary conditions, it can easily be shown that the pair r {w, x}.
Since the pair {;;:', , 24) is known. It is known that crack problems lead to infinities in the stress distribution at the crack tips. Then, with respect to the boundary condition (3.1,26), we group our problems into two cases: the case of a crack and the case of a rigid inclusion.
Instead of reducing the problem to double integral equations, as in the previous section, one can proceed in an alternative way. All physical quantities can be expressed in terms of v(s) and w(s) as follows. We further assume that dX and dx exist and are H!:)lder continuous for all x in the open interval (-l, 1). The reduction to singular integral equations for the rigid line illusion problem is essentially the same.
In the above subsections, we reduced our problems to problems of solving systems of singular integral equations.
SOLUTION OF THE INTEGRAL EQUATIONS IN DIFFERENT CASES
Of the various classes of functions listed in that reference, we look for our solution in the class of functions which are HBlder continuous for all x in the closed interval [-1,1]. 22' we find that € depends linearly on the ratio of the plate thickness h to the length 2c of the crack or solid inclusion. We seek the solution of (4.2,3) in the same class of functions accepted in the previous section.
To investigate the stress inside the plate for E small, we divide the plate into three regions: the region away from the crack, the regions near the vertices x :z: :: 1, y ::: ° and the region near the just crack away from the vertices as in the next sub-. In the first and third of these integrals, R is bounded away from zero so that. Let's again make a dimensionless coordinate transformation X :=: cx, Y :0: cy in the above equations and we will use (x, y) hereafter.
These are exactly the same as the limit values as E to (4.2,16) of the pairs and resultants calculated according to Reissner's theory, provided we stay away from the crack. Using the same method as in the previous section, we find for points near the end x == 1, Y x 0 that, as ·r - 0, the asymptotic expressions for the voltage pairs are the same. Furthermore, the requirement in Section 3.3 that d~~) and . dw(x) exist and are HlHder continuous' for all x in open inter-dx.
For the considered case, we will find the solution w (2)(x, y) according to the classical theory in the form c. In the three cases we have considered so far, the stress fields away from the crack or from the rigid line inclusion are the same. for the Reissner theory and the classical theory for thin plates. However, significant differences occur near the vertices of the line segment y = 0, Ixl ~ 1 and near the line segment but away from the points.
Comparing (4.10,1) with (4.10,2) shows that the angular distribution of the stress based on the two theories is different. Therefore, Reissner's theory predicts that along crack extension near the tip the state of stress is uniform hydrostatic tension or compression, whereas in the classical theory M(l) and M(l) apply. So, to conserve a finite amount of energy near the vertex, the classical theory would have to have a transverse shear modulus of G = 00.
Moreover, there is a significant difference in the maximum shear stress behaviors calculated according to the two theories. It is established that the maximum shear stress according to the classical theory possesses il.
97- in the plate is
Hereafter we will consider only the reduced problem, i.e. the problem associated with the boundary conditions. We digress for a moment to note that the uniqueness of the solutions can be easily established using (S.l, 4). For the case of cracking, we will use the theorem of minimum potential energy as our guide to derive a variational method applicable to our problems.
Then, using a technique similar to that discussed by Noble [17], we can show that the double integral equations (3.2,39) and (3.2,40) corresponding to Case I again arise from the variational principle. The first integral in (5.2,8) is apparently a Hermitian form in A{a), B(a), C(a) and therefore (5.2,8) is a suitable form to which we will again apply the variation principle. It is this form that we will use to obtain an approximate solution for f) (l)(x,O).
In the usual way (see [17]), we will assume that our solution can be approximated by a finite linear combination of appropriately chosen functions. The choice of the minimizing sequence of functions generally depends on the concept of the class of the solution. For the present problem, we further require that the ~ourier transform of each term of the sequence can be evaluated explicitly, so that our subsequent computation will be greatly simplified.
It is worth noting here that the problem or approximate solution tends towards the true solution if n -- 00 is also of interest. Again, using perturbation methods, it can be directly shown that the solution of the integral equation (4.1,1) for E - CD agrees. We finally conclude that for this special case (flc=oMO=oconst.) the approximate one-term solution, based on the variational principle, corresponds to the approximate solution of the integral equation (4.1,1) for small € and for large ones.
However, large € means physically a plate that sees its thickness is large compared to the length of the crack. In such a case, the differential equations may not be accurate near the crack. Thus, while the physical validity of the approximate solution for large € is questionable, it is still useful to note that the variational approximation agrees for large € with a limiting solution obtained directly from the integral equation by perturbation methods.
Following the same procedures used for case I, we will apply the variational principle to case II to find an approximate solution. It can be easily shown that the integrand on the left side of (C,6) is positive definite for almost all a since then.
