Joining And Repair Of Composite Structures (ASTM Special Technical Publication, 1455)
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In addition, the same order polynomials are applied. When the material properties are updated, the property as a function of local strain is computed for a series of points. The Legendre polynomial is then computed using a least-squares method. Because the polynomial order is variable, the energy integration scheme must also be flexible.
A heuristic relation between the number of Gaussian points and the polynomial order has been established to assure the energy integrals are accurate. Exact integration is theoretically possible, but the Gaussian method was found to be faster and equally precise. The method for handling material nonlinearity is not theoretically restricted to the transverse shear modulus, and could be applied to all the properties of an orthotropic material. However, the current implementation of method only updates the transverse shear modulus based on the effective layer shear strain y.
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The nonlinear version of S U B L A M has the capability of accepting the definition of a material shear stress-strain curve in terms of either a Ramberg-Osgood fit, an elastic-perfectly-plastic model, or direct table lookup. Iteration The solution also requires an iteration scheme to achieve equilibrium. A secant modulus approach was chosen because it was relatively easy to implement, and it converges reliably even for highly nonlinear materials.
Figure 4 illustrates the scheme. The disadvantage of this approach is that a large number of iterations may be required to achieve a specified degree of convergence. Fracture Calculation For stress analysis of laminated composites, laminated plate theories of various orders and sublaminate assemblage models are often employed. Obviously, the delaminations considered are between two adjacent laminae. It is known  that the displacements u, v, w in x, y, z directions, z being the thickness coordinate have to be continuous across the delamination front or periphery but their spatial derivatives with respect to n, n being measured in the direction normal to the front in x-y plane are usually discontinuous.
As a result, the gradients of the displacement discontinuities across the delamination surfaces with respect to n do not have the inverse square root singularity at the delamination front as in the case of elasticity solution for homogeneous materials , but they have finite values. It will be illustrated later with a simple example that the discontinuities in the gradients of the displacements at the delamination front yield singularities in stress fields in the form of interactive concentrated line forces at the front between each of the sublaminates used for stress analysis.
This is illustrated in Figs.
In this case, the direction n normal to delamination front coincides with y-axis. In using Irwin's virtual crack closure technique  one needs to consider a virtual self similar extension of the delamination by an amount 5a as shown in Fig. In this case, it is in the negative y direction. Since we are considering infinitesimal extension, the calculated displacements shown in Fig. Only the origin or the tip is shifted to the left by an infinitesimal amount 5a and the displacements and tractions for the extended delamination should now be considered as functions ofyl instead of y measured from the shifted tip.
The model being considered is shown in Fig. The adherend layers are aluminum.
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The adhesive is 0. For the linear examples, the adhesive is treated as an isotropic material with a shear modulus of 4. One strength of the sublaminate method is the ability to accurately represent the interracial tractions between sublaminates. The shearing interfacial traction is a measure that could be used to predict adhesive failure. However, some care must be used in interpreting these tractions. The first problem is that the adhesive layer has two interfaces, and the tractions may not be identical. Figure 9 shows the interracial tractions in the boundary region at the left side of the joint.
Also shown on the graph is the average adhesive layer shear. This is obtained by evaluating the layer natural boundary conditions to determine a net shear force. Dividing by layer thickness gives an average shear stress. As expected, the average shear stress falls between the bounds of the interface values.
The average value approaches zero as required by the stress boundary conditions. The average does not exactly equal zero because of the shared boundary conditions with the adjacent adherend layers. The nodal degrees-of-freedom do not represent sufficient boundary conditions to independently satisfy the free-edge conditions for each layer.
Figure 9 has been truncated to show only positive shears. The sudden traction reversal near the transition can be viewed as the plate theory response to the singularities that would be present in an elasticity solution. In other problems, such as the free-edge stresses in a laminated composite, these extremely large gradients correlate well with elasticity results.
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In the case of a bonded joint, values that change rapidly in a dimension comparable to the bondline thickness are probably best treated as artifacts of exact solution method. For the purposes of joint evaluation, the peak stress shown in Fig. This value can be shown to be stable with respect to modeling details, and additional throughthickness discretization. The peak stress exactly at the free-edge is sensitive to modeling details. If the interface is to be evaluated, then fracture methods are recommended.
The tractions corresponding to peel stress are shown in Fig. Again, the plot has been truncated because the edge values are very large. This shows the power of the exact solution method to extract rapidly changing stress. For the purpose of failure prediction, fracture mechanics is probably more meaningful. For linear material properties and uniform thickness elements, the exact method is available. The P-method can be applied to the same joint for the purpose of determining approximate solution accuracy. Although the P-elements may be large, experience shows that it is impractical to model an entire joint with a single element.
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The approximation functions will attempt to follow the large edge tractions discussed above and therefore generate large errors elsewhere. A better modeling approach is to provide small "sacrificial" elements at the joint ends, as shown in Fig. The joint is also subdivided in the center to further increase accuracy.
In this model, the extensions to the adherends beyond the joint region are exact elements. The number of terms in the Legendre polynomial approximation function, referred to as the P-Order determines the convergence of the method. The element boundary conditions do not enforce continuity of the interface tractions. Figure 13 shows how the magnitude of the discontinuity decreases with the P order. The plot also includes a distribution for a classical Volkerson-type solution which assumes that all axial load is carried by the adherends, and all shear deformation is in the adhesive.
Peel stress is given in Fig. Discontinuity 3i 2. Exact P-Method Order10 5 2. The same joint configuration was used in conjunction with an elastic-perfectly-plastic adhesive model. The plastic stress was set to 21 MPa.
The elastic-plastic material model is challenging for the method because the sudden change in slope cannot be represented exactly with the continuous approximation functions. Getting acceptable results required further subdividing the model into a total of 6 elements, counting the 2 small sacrificial elements at the ends. Figure 16 shows that the solution tends to overshoot and oscillate at the points where the stress reaches the plastic value.
The figure shows the ability of method to follow the transition between elastic and plastic regions, independent of the location of the element boundaries. The first is a straight overlap. The second uses a joggle at each end of the joint so that the load line runs along the center of the adhesive layer. The third joint applies a single joggle such that the two adherends are parallel. An exact, linear solution will be used.
When dealing with eccentric joints, it must be emphasized that the existing code does not include large-deflection, geometric nonlinearity. Figure 18 shows the deformed shape for a finite element model of the symmetric overlap joint. The finite element solution uses a commercial P-element approach. The contours show variations in axial stress. The boundary conditions and overall dimensions for one of the joints is shown in Fig.
Each of the models was constructed to have the same overall dimensions. The loading introduction is by a uniform displacement at the ends, with the magnitude of the displacement selected such that the integrated load is unity. The adhesive layer is 0. Deformations are to the same magnification scale. A surprising result is that the peak stresses are actually lower for the straight overlap than for either of the joggled configurations. There is a constant shear load in the straight overlap so that the shear stress distribution never goes to zero at the center of the joint.
This constant shear comes from the particular choice of end boundary conditions, in particular vertical restraints at each end, and serves to transfer a portion of the load. Also shown on the plot is the shear stress distribution from the finite element FE model shown in Figure 18 for the symmetric joggle. The FE results track reasonably well, but there is an inherent difference in the solutions. This is believed to be related to the difference in bending stiffness between the models. The peel stress distribution Fig. However, the average peel stress integrated over an arbitrary distance of two adhesive thicknesses for the straight joint is less than either of the joggled joints by a substantial factor.
The single joggle joint has an average peel stress at the edge that is twice the value for the straight joint. The FE result is also included for comparison. Straight Overlap Symmetric Joggle ,,, Single Joggle Finite Element Sym J, '. A more meaningful measure of peel is to look at the strain-energy-release-rates G for joints with pre-existing cracks. Each of the models was modified to include a 2h debond between the adhesive layer and the adjacent adherent.