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Complex Constitutive Adhesive ModelsErol Sancaktar
Abstract A complete approach to modeling adhesives and adhesive joints needs to include considerations for: deformation theories, viscoelasticity, singularity methods, bulk adhesive as composite material, adhesively bonded joint as composite and the concept of the interphase, damage models, and the effects of cure and processing conditions on the mechanical behavior. The adherend surfaces have distinct topographies, which result in a collection of miniature joints in micron, and even nano scale when bonded adhesively. The methods of continuum mechanics can be applied to this collection of miniature joints by assuming continuous, or a combination of continuous/discontinuous interphase zones.
4.1 IntroductionAdhesively bonded joints are complex composite structures with at least one of the constituents, namely the adhesive, most often, being a composite material itself due to the presence of secondary phases such as llers, carriers, etc. The joint structure possesses a complex state of stress with high stress concentrations, and often, singularities due to the terminating adhesive layer where the substrates may possess sharp corners. Consequently, accurate analysis and modeling of adhesive materials and bonded joints require the use of the methods of composite materials and composite mechanics. The inclusion of the interphase region is necessary in this analysis as a distinct continuum. The presence of geometric discontinuities creates stress concentrations and, possibly, singularities adding additional complexity to the topic of adhesively bonded joints. This problem, however, can be alleviated, at least partially, by making the proper changes in the geometry of the bonded joint. Furthermore, since most adhesive materials are polymer-based, their natural viscoelasticErol Sancaktar Polymer Engineering, Adjunct Professor, Mechanical Engineering, The University of Akron, Akron, OH 44325-0301, e-mail: firstname.lastname@example.org
L.F.M. da Silva, A. Ochsner (eds.), Modeling of Adhesively Bonded Joints, c Springer-Verlag Berlin Heidelberg 2008
behavior usually serves to reduce localized stress concentrations. In those cases where brittle material behavior prevails or, in general, when inherent material aws such as cracks, voids, disbonds exist, then the use of the methods of fracture mechanics are called for. For continuum behavior, however, the use of damage models is considered appropriate in order to be able to model the progression of localized and non-catastrophic failures. Obviously, a technical person involved in adhesive development and/or applications should keep the above mentioned issues in mind, along with insight into typical joint stress distributions for adhesive joints as well. Stress distributions are relevant from the mechanical adhesion point of view also, since they depend on surface topography, which can be considered a collection of many geometrical forms. Therefore, mechanical adhesion depends on the stress states of different adhesive joint geometries on the scale of the surface topography, which may include many lap, butt and scarf joints in the interphase region. A complete approach to modeling adhesives and adhesive joints, therefore, needs to include considerations for: deformation theories, viscoelasticity, singularity methods, bulk adhesive as composite material, adhesively bonded joint as composite the concept of the interphase, damage models, and the effects of cure and processing conditions on the mechanical behavior.
4.2 Deformation TheoriesThe deformation theory was rst introduced by Hencky (1924) as reported by Hill (1956) and Kachanov (1971) in the form:
i j = (kk /9 K)i j + Si j
where, Si j is the deviatoric stress tensor. Equation (4.1) reduces to the elastic stressstrain relations when = 1/2G, where G is the elastic shear modulus. If the scalar function is dened as = (1/2 G) + (4.2) then Eq. (4.1) can be interpreted in the form
i j = i j E + i j V + i j Pwhere
(4.3) (4.4) (4.5)
i j = (kk /9 K)i j + (Si j /2G) + Si jand,
i j P = Si j
with being a scalar function of the invariants of the stress tensor, and the superscripts E, V and P representing elastic, viscoelastic and plastic behaviors, respectively.
4 Complex Constitutive Adhesive Models
Consequently, the relation
= ( /E) +
is obtained for uniaxial tension on the basis of Eqs. (4.4) and (4.5) with = 2/3. Ramberg and Osgood (1943) used a special form of Eq. (4.6) with = K n1 to result in: = ( /E) + K n (4.7) where, K and r are material constants. They reported that Eq. (4.7) could be used successfully to describe uniaxial tension and compression behavior of various metal alloys. Equation (4.7) was later modied by McLellan (1966, 1969) to accommodate strain rate effects. McLellan interpreted the terms E, K and n of Eq. (4.7) as material functions with the function E representing viscoelastic behavior and functions K and n representing workhardening characteristics. The terms E, K and n were all described as functions of the strain rate (d /dt) so that rigidity, stress and plastic ow respectively were all affected by variations in the strain rate. Renieri et al. (1976) used a bilinear form of rate dependent Ramberg-Osgood equation to describe the stress-strain behavior of a thermosetting adhesive in the bulk tensile form. The bilinear behavior was obtained when log p was plotted against log , where p represents the second term on the right-hand side of Eq. (4.7). The model adhesive they used was an elastomer modied epoxy adhesive with and without carrier cloth. They made several modications on the form of the equation previously used by McLellan. First, the plastic strain p was assumed to be a function of the over-stress above the elastic limit stress (the development of over-stress approach will be presented subsequently) and second, the stress levels dening the intersection point for the bilinear behavior were found to occur slightly below the stress whitening stress values. The equations they developed in this fashion are given as:
= /E, = ( /E) + K1 [ ]n1 = ( /E) + K2 [ ]n2