A Theoretical Model for Estimation of Yield Strength of Fiber Metal Laminate

The paper presents a theoretical model for estimation of yield strength of fiber metal laminate. Principles of elasticity and formulation of residual stress are employed to determine the stress state in metal layer of the laminate that is found to be higher than the stress applied over the laminate resulting in reduced yield strength of the laminate in comparison with that of the metal layer. The model is tested over 4A-3/2 Glare laminate comprising three thin aerospace 2014-T6 aluminum alloy layers alternately bonded adhesively with two prepregs, each prepreg built up of three uni-directional glass fiber layers laid in longitudinal and transverse directions. Laminates with prepregs of E-Glass and S-Glass fibers are investigated separately under uni-axial tension. Yield strengths of both the Glare variants are found to be less than that of aluminum alloy with use of S-Glass fiber resulting in higher laminate yield strength than with the use of E-Glass fiber. Results from finite element analysis and tensile tests conducted over the laminates substantiate the theoretical model.


INTRODUCTION
Fiber metal laminate (FML) is a hybrid material system that is constructed with layers of thin and light metallic sheets which are alternately bonded adhesively with composite prepregs by heat and pressure, each prepreg built up of fiber layers laid in similar or different orientations. FML exhibits excellent fatigue and fracture resistance due to bridging or diversion of load from soft cracked metallic layers towards ultra-strong fibers in prepregs thus making the laminate a better substitute, even at increased cost, for monolithic metals and their alloys in aerospace applications where fatigue and fracture properties assume importance. As the result, research work on FML's has so far been mostly directed at their fatigue and fracture aspects. However results, albeit limited, are also available on strength properties of FML's. In experimental field, Lawcock et al. [1] studied the effect of adhesive bonding between aluminum alloy sheets and carbon fiber based composite prepregs on strength of the laminate. Kawai et al. [2] presented off-axis strength behaviour of glass fiber and aluminum alloy based FML (Glare). Cortes and Cantwell [3] measured tensile properties of laminates constructed with carbon fiber reinforced PEEK and titanium layers. Khalili et al. [4] tested FML samples from various lay ups of glass fiber/epoxy resin with steel and aluminum alloy layers and compared their strengths with each other and with monolithic metals and conventional fiber-resin composites (FRP's). Torshizi  al. [5] examined tensile properties of Glare and kevlar fiber based FML (Arall). Sinmazcelik et al. [6] investigated surface treatment procedures of metals and their bonding techniques with composites that was followed by measurement of tensile strengths of different types of FML's. In numerical field, Rooijen et al. [7] developed a finite element model for understanding the bearing behaviour of Glare. The plasticity in metal layers, failure of fiber layers and frictional effects between the layers were incorporated in the model. Krimbalis et al. [8] presented finite element model for estimation of compressive characteristic dimension (CCD) in various Glare variants. A novel re-definition of conventional CCD governed by yield strength of aluminum alloy was proposed. In theoretical field, Schuerch [9] developed a model with the help of plasticity characteristics to predict uni-axial ultimate compressive strength and failure modes of unidirectional composites. The model was extended to boron fiber and magnesium metal matrix composite that resembled the configuration of a FML. Literature review reveals a limited theoretical work being reported on FML's. As such, there exists the need for theoretical models to assess the mechanical properties of the laminates. Determination of properties in absence of theoretical models calls for extensive numerical analysis or fabrication of large number of test specimens for accumulation of reliable data. Moreover, a theoretical model always proves to be handy in design stage of any structure.
The magnitude of stress that develops or is induced in each metal layer of a FML differs from the stress applied over the laminate in service conditions due to i) Redistribution of stress in dissimilar and elastically un-identical material layers of the laminate to fulfil the requirement of equal strain in all the layers necessitated by the principles of structural mechanics and ii) Inherent presence of variable residual stresses in material layers that are generated during high temperature curing of the laminate due to different coefficients of thermal expansion of the materials. Based on these facts, a theoretical model is presented with the help of the principles of elasticity and formulation of residual stress to obtain the induced stress in metal layer, the magnitude of which is found to be higher than that of applied stress that implies reduced yield strength of the laminate vis-à-vis the metal layer (Yield strength refers to uni-axial value). The model is tested over 4A-3/2 Glare laminate comprising 2014-T6 aerospace aluminum alloy layers and uni-directional glass fiber based prepregs. Laminates with prepregs of E-Glass and S-Glass fibers are investigated separately under uni-axial tension. Yield strengths of both the Glare variants are confirmed to be less than that of aluminum alloy with use of S-Glass fiber resulting in higher laminate yield strength than with the use of E-Glass fiber. The laminates are modelled by finite element method and are fabricated for numerical assessment and experimental measurement of their yield strengths respectively. Theoretical model is validated by finite element and experimental results.

THEORETICAL MODEL
Refer Fig. 1 that illustrates the stacking sequence of material layers in a FML. Elastic stress-strain, where τ σ and are normal and shear stresses whereas γ ε and are normal and shear strains respectively. Since the material layers in lam, under tension in service condition is considered to yield and therefore fail when its soft metal layers yield with prepregs remaining intact due to the presence of ultra-strong fibers in them, attention is focused on elastic stress state in metal layers whose magnitude is decided by overall stiffness of the laminate. Generalized form of elastic stiffness matrix of the laminate, where α is coefficient of thermal expansion and curing T and ambient T are laminate curing and ambient temperatures. Residual strain in z direction is neglected due to small thickness of the layer. Strain developing in metal layer upon application of tensile stress, applied σ , over the laminate in service conditions is equal in all the material layers and is written as Total strain in metal layer is found by superimposing residual and applied strains as, , following which the induced stress developing in each metal layer is determined from   Eq. (4) indicates induced stress in metal layer to be higher than the applied stress. Since the possibility of buckling and separation of material layers of the laminate are minimal in case of tensile loading, design stress in metal layer that is the function of induced stress is justifiably presumed to be equal to yield strength of metal at the juncture when the laminate yields.
Corresponding stress applied over the laminate represents the uni-axial yield strength, { } lam YS , of the laminate. Application of Eq. (4) at yield or critical condition (*) of the laminate results in

Solution
A numerical iterative procedure is adopted to solve Eq. (6). Suitable value of { } lam YS is assumed at LHS in each iteration to determine the unknown induced stress state in metal layer in x and y directions at RHS with shear stress component equal to zero. Using distortional energy theory as yield criterion of ductile metal layer, the magnitude of von-Mises stress, in case of principal stress state existing in x-y plane of the layer, given by , is obtained in every iteration.   Also, S-Glass fiber is found to result in increased yield strength of Glare because of its higher modulus of elasticity than that of E-Glass fiber.

MODEL VALIDATION
The theoretical model is validated with the help of finite element analysis and experimental work that are discussed as follows:- Fig. 3. 3D finite element models of investigated Glare with prepregs of E-Glass and S-Glass fibers were constructed separately in Ansys software. Solid 185 elements were used to discretize aluminum alloy layers. Since solid elements tend to lock in thin applications, shell 190 elements were chosen to mesh thin fiber and resin layers. Known elastic-plastic stress strain data of aluminum alloy, linear elastic data of fiber and non-linear elastic data of resin were substituted in the material models. Uniform residual stresses in x and y directions in aluminum alloy, fiber and resin layers, obtained theoretically from the expressions,

Finite element analysis Refer
respectively, as explained in Section 2, were externally introduced as initial stress at all respective nodes in material layers with the help of a pre-processor code. The residual stresses were different in laminates with E-Glass and S-Glass fibers. Their magnitudes are presented in Table 2. The nodes at material interfaces were merged and their connectivity was checked. The model was constrained in all degrees of freedom at one end whereas tensile stress, Error of the order of 7.4 % was recorded in both the types of laminates that is acceptable in numerical solutions. Yield strength of laminate with S-Glass fiber was again found to be higher than that using E-Glass fiber.

Experimental work
Glare laminates with prepregs of E-Glass fibers were fabricated in accordance with the procedure specified in Section 2.1. Aluminum alloy sheets, cold rolled from thickness of 2 mm to 0.4 mm, were re-heat treated to achieve T6 state. Prepregs were prepared and stored in cold environment prior to use. Post laminate fabrication, residual stresses were measured in y direction at various arbitrary locations on external aluminum alloy layers of the laminates by X-ray diffraction technique. Following parameters were used in the measurement system:-  As expected, the residual stresses were mostly found to be tensile in nature due to higher coefficient of thermal expansion of aluminum alloy than that of the laminate. The magnitudes of residual stresses (MPa) were obtained as follows :-+22.2, +36.9, +12.6, -0.5, +9.5 in Test laminate 1 and +14.3,+18.9, +24.9, -0.2, +17.3 in Test laminate 2. Average residual stress was equal to +15.59 MPa (tensile). The value was lower than theoretically estimated value of +48.36 MPa (tensile). The experimental findings also revealed non-uniform residual stress patterns. Glare with S-Glass fiber Full yielding across width Refer Fig. 5. Several Glare laminates were subjected to uni-axial tension in y direction, as per the tension test (ASTM D3039), at test speed of 1-2 mm/min [10]. The load was gradually increased until the fibers split upon breaking leading to the fracture of the laminate. Since aluminum alloy and fiber layers did not separate from each other in each laminate during the test, a strong adhesive bond between the layers was confirmed that in turn supported the correctness of adopted volume fractions of fiber and resin in prepregs and the bonding parameters employed to join prepregs with aluminum alloy stress in aluminum alloy layers. Lower residual stress necessitated increase in applied tension over test laminates for aluminum alloy layers to yield thereby resulting in higher laminate yield strength in comparison with theoretical result. Reduced yield strength of Glare vis-à-vis aluminum alloy was once again proved.

CONCLUSIONS
A theoretical model for estimation of yield strength of fiber metal laminate is presented. The model is tested over 4A-3/2 Glare laminate that comprises three thin 2014-T6 aerospace aluminum alloy layers alternately bonded adhesively with two prepregs, each prepreg built up of three uni-directional glass fiber layers laid in longitudinal and transverse directions. Laminates made with prepregs of E-Glass and S-Glass fibers are investigated separately under uni-axial tension. The model is validated by finite element analysis and experiments. The following conclusions are drawn based on the results reported in the paper:i) Yield strength of Glare is less than that of 2014-T6 aluminum alloy with use of S-Glass fiber resulting in higher laminate yield strength than with the use of E-Glass fiber. Since light weight composite prepregs are employed in the construction of Glare, reduced yield strength of Glare vis-à-vis monolithic aluminum alloy member of identical dimensions as that of the laminate is not expected to result in substantial difference between the specific strengths of the two. ii) Despite minor variations in patterns and magnitudes of theoretically predicted and experimentally measured residual stresses in aluminum alloy layers of the laminates, the results from finite element analysis and tensile tests conducted over the laminates are in good agreement with theoretical estimations that supports the viability and usefulness of the model. iii) Since the model is developed from basic principles, it is versatile and is applicable to any type of fiber metal laminate involving different material combinations, varying number and dimensions of material layers. ,