7.2.1 Compression

When loaded in pure compression, a joint is less likely to fail than when loaded in any other manner, but compression-loaded joints are limited in application.

7.2.2 Shear

This type of loading imposes an even stress across the whole bonded area, utilizing the joint area to the best advantage and providing an economical joint that is most resistant to joint failure. Whenever possible, most of the load should be transmitted through the joint as a shear load [4].

7.2.3 Tension

The strengths of joints loaded in tension or shear are comparable. As in shear, the stress is evenly distributed over the joint area, but it is not always possible to be sure that other stresses are not present. If the applied load is offset to any degree, the advantage of an evenly distributed stress is lost and the joint is more likely to undergo failure. The adherends should be thick with this type of joint and not likely to be deflected to any appreciable degree under the applied load. Such a situation will result in nonuniform stress [4]. Tensile stress develops when forces acting perpendicularly to the plane of the joint are distributed uniformly over the entire area of the bond. The types of stress likely to result when other than completely axial loads are applied are cleavage and peel. As adhesives generally have poor resistance to cleavage and peel, joints designed to load the adhesive in tension should have physical restraints to ensure axial loading [6].

7.2.4 Peel

One or both of the adherends must be flexible in this type of loading. A very high stress is applied to the boundary line of the joint, and unless the joint is wide or the load is small, failure of the bond will occur. This type of loading is to be avoided if possible [4].

7.2.5 Cleavage

Cleavage is somewhat similar to peel and occurs when forces at one end of a rigid bonded assembly act to split the adherends apart [4]. It may be considered as a situation in which an offset tensile force or a moment has been applied. The stress is not evenly distributed (as is the case with tension) but is concentrated on one side of the joint. A sufficiently large area is needed to accommodate this stress, resulting in a more costly joint [4].

7.10.1 Flexible Materials

Thin or flexible polymeric substrates may be joined using a simple or a modified lap joint. The double strap joint shown in Figure 7.3 is best but time-consuming to form. The strap material should be fabricated from the same material as for the parts to be joined. If this is not possible, it should have approximately equivalent strength, flexibility, and thickness. The adhesive should have the same degree of flexibility as the adherends. If the sections to be bonded are relatively thick, a scarf joint, also shown in Figure 7.3, is acceptable. The length of the scarf should be at least four times the thickness, as shown in Figure 7.15 [6].

Figure 7.16 shows several types of joints for rubber under tension. The horizontal white lines are equidistant when the joints are unstressed. It is obvious that the scarf joint is least subject to stress concentration with materials of equal modulus, and the double scarf joint is the best for materials of unequal modulus. These designs offer the best resistance to peel and, all other factors being equal, represent the best choices [11].

When bonding elastic material, forces on the elastomer during cure should be carefully controlled, as too much pressure will cause residual stresses at the bond interface. Stress concentrations may also be minimized in rubber-to-metal joints by elimination of sharp corners and by the use of metal adherends sufficiently thick to prevent peel stresses that may arise with thinner-gauge metals. As with all joint designs, polymeric joints should avoid peel stresses. Figure 7.17 illustrates methods of bonding flexible substrates so that the adhesive will be stressed in its strongest direction [11].

7.10.2 Rigid Plastics

In the case of rigid plastics, the greatest problems is in reinforced plastics, which are often anisotropic, having directional strength properties. Joints made from such substrates should be designed to stress both the adhesive and the adherend in the direction of greatest strength. Laminates should be stressed parallel to the laminations to prevent delamination of the substrate [6].

Single and joggle lap joints (Figure 7.3) are more likely to cause delamination than scarf or beveled lap joints. Strap joints may be used to support bending loads [6].

7.11.1 Theoretical Analysis of Stresses and Strains

Most theoretical analyses have been carried out on single or double lap joints, which are the primary types of joints used for determining the strength of adhesives. Properly designed joints stress the adhesive in shear. Adhesives are especially weak in peel, and are also weak under tensile loads applied normal to the plane of the joint. The earliest theoretical lap joint work involved simplifying assumptions that: (1) the joint was a simple overlap type; (2) both adherends were made of the same metal and had the same geometry; (3) the adherends and adhesive behaved elastically; (4) bending or peeling stresses were not involved; (5) thermal expansion or residual stresses were ignored; and (6) the deflections were small [12].

Recent theoretical studies have become much more complex. New computer-assisted techniques permit the use of finite-element matrix-theory type approaches. The effects of important variables are being determined by parametric studies. More complex joints are also being studied. New adherend materials, including advanced filamentary composites, are also being evaluated. The elastic, low-deflection, constant temperature behavior of scarf and stepped-lap joints has been replaced by elastic-plastic, large-deflection behavior, combined with thermal expansion differences, or curing shrinkage-induced residual stresses.

7.11.2 Experimental Analyses

Typically, the yardstick for qualitatively measuring the internal resistance of an adhesive bond to an external load has been the determination of the strain distribution in the adhesive and adherends. This is a difficult task. Even in simple lap joints, the actual stress–strain distributions under load are extremely complex combinations of shear and tensile stresses, and are very prone to disturbance by nonuniform material characteristics, stress concentrations or localized partial failures, creep and plastic yielding, etc. It is extremely difficult to accurately measure the strains in adhesive joints with such small glue line thicknesses and such relatively inaccessible adhesive. Extensometers, strain gauges, and photoelasticity are being used with limited success [12].

The stress distribution on the adhesive affects the ability of the joint to accommodate loads. Joint design should strive to distribute the stresses equally over the bond area in order to create uniform stress on the adhesive. Adhesive bonds subjected to tensile, compressive, or shear stress during loading experience a more uniform stress distribution than bonds exposed to cleavage and peel stress. Tensile and compressive stresses are evenly distributed throughout the bond area; stress is represented by a straight line in Figure 7.18. The compressive strength of most adhesive films is greater than their tensile strength; optimal joint design should maximize compression and minimize tensile stresses. The stress distribution of cleavage and peel stress is concentrated at one end of the joint (Figure 7.19). Peel strength of any adhesive may be as low as 1% of its shear strength; low-modulus elastic adhesives usually have higher peel strengths. Peel stress can be reduced through symmetrical joints, such as double lap. Joint design should ensure that peel and cleavage stress are minimized, and shear stress is maximized [14,15].

In shear stress, the ends of the bond resist a greater amount of stress than the middle of the bond (Figure 7.20). The maximum stress experienced by the ends of the joint is greater than the average stress (joint load divided by bond area); lower than average stress occurs in the middle of the joint. This stress distribution is due to the flexibility of plastic materials, which tend to bend when a load is applied, increasing stress concentrations at the joint ends. Stress ratios (highest stress/average stress) of plastics (15), with relatively low elastic moduli (2068 MPa unfilled plastic), are much greater than stress ratios for steel (1.7), with a high elastic modulus (212,000 MPa). Stress concentrations can lead to joint failure at relatively low loads; however, stress concentrations can be reduced by a joint design that takes into account the elastic modulus of the adhesive, the joint overlap length, and the bond line thickness [14,16].

The elastic modulus of the adhesive influences the stress distribution of the joint. The shear stress distribution of a more brittle, higher modulus adhesive, with a stress–strain curve as shown in Figure 7.20(a), shows a linear increase in stress from the center to the ends of the joint (Figure 7.20(b)). A more elastic adhesive with a higher elongation and stress–strain behavior as shown in Figure 7.20(d) exhibits a nonlinear stress distribution (Figure 7.20(c)). The flexibility of the more elastic adhesive allows the joint to more readily accommodate motion of the adherends; stress is then distributed over a larger area, and the stress ratio (highest stress/average stress) is reduced. Substitution of a lower modulus (1.4 MPa) adhesive for a higher modulus (2068 MPa) adhesive can reduce the stress ratio from 22.4 to 1.2. Although a lower modulus adhesive may, for some applications, produce a stronger joint, it may not be able to accommodate structural loads without excessive deformation. Due to the greater area under the stress distribution curve, the more elastic adhesive experiences a higher average stress than a brittle adhesive of the same strength. For two adhesives of the same strength and elongation, the higher modulus, brittle adhesive would tolerate higher loads. Brittle adhesives, however, are more sensitive to crack propagation and generally have lower fatigue life than more elastic adhesives [14,16].

Although bonds with larger areas generally have higher strength, bond width is a more important design parameter than bond length or overlap. Bond strength increases slightly with overlap length (Figure 7.21) up to a point, and then remains constant. Due to the shear stress concentration at the ends of the joint, however, shear strength is directly proportional to bond width (Figure 7.21). A 2-cm-wide joint is twice as strong as a 1-cm (0.4 in)-wide bond, but a 2-cm-long joint is not twice as strong as a 1-cm joint. A short, wide bond area (Figure 7.22(b)) is stronger than a long, narrow bond area (Figure 7.22(a)). Bending and differential shear stress concentrations are reduced with shorter bond overlaps; decreases in overlap length from 2.54 to 0.32 cm can result in reduction of the stress ratio (highest stress/average stress) from 22.5 to 3.78 [14,16,17].

A thicker bond line can reduce shear stress concentration by spreading the strain over a larger dimension, resulting in less strain on the adhesive. An increase in bond line gap from 0.0025 to 0.10 cm can decrease the stress ratio from 18.2 to 3.06 [14,16].

The most common method for reducing stress concentration in lap joints is tapering the adherends, in a tapered lap joint (Figure 7.23(a)). Stress at the joint ends is reduced, allowing for a more uniform stress distribution. Modeling studies indicate that both adhesive peel and shear stresses decrease with a decrease in taper angle, with the optimum angle being the smallest angle that can be economically machined and assembled. A step lap joint (Figure 7.23(b)) can be used to avoid a large change in stress concentration from the ends of the joint to the center when long overlap lengths are necessary [14,16].

7.11.3 Failure Analyses

The function of a structural adhesive joint is to transmit an external load to the structural member. If the joint fails to function as it is intended, it will undergo damage or failure. The damage could be actual fracture of the structure, excessive elastic deformation, or excessive inelastic flow. The criteria for what constitutes structural failure depend on the performance requirements of the joint. The fundamental problem in the mechanics of adhesives and joints is to obtain some relationship between the loads applied to the joint and a parameter that will adequately describe the criteria for structural failure. The most common criterion for such failure of lap-type joints is actual fracture of the joint. For a given combination of adherend and adhesive, the stress analyst must decide what the mode or theory of failure would be if the applied loads become large enough to cause failure. The decision as to which theory would realistically determine the mode of failure is usually based on past experience, or upon some form of experimental evidence [12].

The next step is to determine a relationship between the applied load and a parameter that will describe the failure of the joint. Such parameters might be stress, strain, strain energy, etc. Finally, when maximum tolerable stresses have been obtained, the allowable stress values or factors of safety are decided upon to allow for factors such as long- and short-term loading, fatigue loading, special environmental conditions, and other special considerations. This step is ordinarily based on experience, engineering judgment, and legal, government, or commercial specifications [12].

7.11.4 Methods of Stress Analysis

Theory of Volkersen: In 1938, Volkersen analyzed the distribution of shearing stresses in the adhesive layers of a lap joint. Volkersen’s model is useful only with very stiff adhesives, which do not bend on loading the joint. A dimensionless stress concentration factor is found to depend on the geometry and the physical parameters of the joint. By introducing further simplifications, certain reasonable geometric conditions, and identical adherends, a simple formula is obtained [14]:

$\mathrm{\Delta }=G{L}^2/Etd$

where G is the shear modulus of the adhesive, E is the Young’s modulus of the adherend, d is the thickness of the adherend, L is the length of the overlap, and t is the thickness of the adherend.

DeBruyne has suggested that, when all other variables are kept constant, the quantity √t/L with dimension (length)^−1/2, the “joint factor” derived from the above equation, is useful in correlating joint strengths [14].

Volkersen’s theory predicts that the shear stresses in the adhesive layer reach a maximum at each end of the overlap, when the bonded plates are in pure tension. Photoelastic analyses of these composite structures show that stresses are uniform in the central part of the model adhesive, but high near the edges of the steel plate used in the analysis (Figure 7.3). Stress distributions at the end were found to be independent of the length of the overlap, when its length was at least three times the thickness of the adhesive layer [14].

Theory of Goland and Reissner: In Volkersen’s theory, the so-called tearing or peeling stresses were ignored. Goland and Reissner [16] took the bending deformation of the adherends into account, as well as the transverse strains in the adhesive and the associated tearing stresses. These researchers showed that the maximum tearing and shear stresses reach asymptotic values for large overlap lengths. Provided the system remains linearly elastic, the joint strength reaches a limiting value with increasing overlap length. In actual practice, however, a limiting strength is obtained because the adherends are loaded to their ultimate strength [15].