Non-linear analysis of laminated metal matrix composites by an integrated micro/macro-mechanical model

Non-linear Analysis of Laminated Metal Matrix Composites by an Integrated Micro/Macro-Mechanical Model

Antonio Ferreira Ávila

Engenharia Mecânica - UFMG

Tamma Kumar Krishina

University of Minnesota

Abstract

The present paper describes an integrated micro/macro mechanical study of the elastic-viscoplastic behavior of unidirectional metal matrix composites (MMC). The micromechanical analysis of the elastic moduli is based on the Composites Cylinder Assemblage model (CCA) with comparisons also draw with a Representative Unit Cell (RUC) technique. These "homogenization" techniques are later incorporated into the Vanishing Fiber Diameter (VFD) model and a new formulation is proposed. The concept of a smeared element procedure is employed in conjunction with two different versions of the Bodner and Partom elastic-viscoplastic constitutive model for the associated macroscopic analysis. The formulations developed are also compared against experimental and analytical results available in the literature.

Keywords: Metal Matrix Composites, Finite Element Method, Micro/Macro-Mechanics, Elastic-Viscoplastic Materials

Introduction

As structural components, laminated composites are quite common nowadays in many areas of application. They are used in a variety of applications ranging from automotive to aerospace industries. Composite materials are attractive in aerospace uses owing to their inherent advantages over isotropic materials, namely, light weight, yet high strength. Turbine blades, combustor liners, and rocket nozzle are typical examples of aerospace applications of composites. The so-called extreme conditions, i.e., high static and/or dynamic loading, elevated temperature, and corrosive environments are fairly frequent in these applications. One of the most used types of composites for such class of applications is metal matrix composites (MMC). They have considerable potential as structural materials due to their capability of enduring severe environmental conditions. To model such class of materials, it is necessary to characterise the complex interrelationship between the metallic matrix and the fibers.

The importance of characterisation of these materials is evident since the early days of "composites". Considerable efforts have been attempted trying to know more about the mechanical behavior of these materials. One of these efforts pertains to the so called "homogenization techniques". The objective of any homogenization technique is to provide the upper and the lower bounds of the material properties as a function of the phase volume fraction. Considerable work has been done by many researchers attempting to describe the linear (Mori and Tanaka, 1973, Guedes and Kikuchi, 1990, Tandon, 1995, Nemat-Nasser and Hori, 1992) and non-linear (Castañeda and Willis, 1988, Castañeda, 1991, Terada et al. 1996, Fotui and Nemat-Nasser, 1996) behaviour of composites. The Composite Cylinder Assemblage model (CCA), developed by Hashin and Rosen (1964), has been quite well received for unidirectional composites. The Representative Unit Cell (RUC) technique described here is a numerical alternative approach widely used by many researchers for unidirectional laminate (Sun and Vaidya, 1996, Tamma and Ávila, 1999) and for woven fabric composites (Dasgupta et al., 1996, Whitcomb, 1991).

Of the various models, we selected two to predict elastic properties of unidirectional composites. The first is based on variation calculus, and termed as the Composites Cylinder Assemblage (Hashin and Rosen, 1964, Hashin, 1972). The second model estimates the composite properties via an analysis of an RUC corresponding to periodic fiber packing sequence. These models were chosen considering their moduli prediction capabilities and the possibility of computer modelling.

This paper is concerned with the stress analysis of unidirectional laminated high temperature metal matrix composite structures. To pursue this goal, a variation of the Vanishing Fiber Diameter (VFD) model (Dvorak and Bahei-El-Din, 1982) is proposed by incorporating the effective elasticity tensor obtained via micromechanical analysis, and the inelastic strain portion. The fibers are considered linear elastic, and the matrix is assumed to behave in an elastic-viscoplastic manner. The overall composite behavior is considered elastic-viscoplastic. The Bodner and Partom model (1975) is selected as the representative constitutive model. The concept of a smeared element procedure is applied in conjunction with Bodner and Partom's model for the macromechanical analysis. Numerical illustrations using the finite element method are conducted with comparisons also draw with available analytical and/or experimental results.

Composite Cylinder Assemblage

The paper focuses attention to unidirectional laminate composites. We consider a parallel set of cylindrical fibers embedded in a homogeneous matrix. We wish first to determine the effective elastic constants of this material as function of those of the constituents. Christensen (1979) pointed out the idea of equivalent or effective homogeneity. It occurs when the scale of the inhomogeneity is assumed to be orders of magnitude smaller than the characteristic dimension of the problem of interest, so that there exists an intermediate dimension over which the properties averaging can be legitimately performed.

The transversely isotropic assumption for unidirectional fiber composites with hexagonal and/or random fiber arrays is feasible according to various researchers (Christensen, 1979, Hashin, 1972). Five independent elastic constants have to be evaluated. A simple unidirectional tensile test in the fiber direction is sufficient to define an axial Young's modulus and the associated axial Poisson's ratio. The axial shear modulus is defined for pure shear in any axial plane. A two-dimensional isotropic state of stress defines the effective plane strain transverse bulk modulus. Once the five constants are known, it is possible to determine from them other convenient elastic constants, e.g. the transverse Young's modulus. The assumption of transversely isotropic composite was also used by Hashin (1972) and is discussed next.

Hashin (1972) considers a certain semi-random fiber reinforced material. He constructs composite circular cylinders each of which is made of a circular cylindrical fiber in a concentric matrix shell. In the n^th composite cylinder, the fiber radius is a_n and the composite cylinder radius is b_n. In all composite cylinders the ratio a_n/b_n are identical and the cylinders are of equal height H. Now a cylindrical specimen of height H and cross section A is progressively filled out with non overlapping composite cylinders. This is done by first placing a number of such composite cylinders into the specimen and then filling out the remaining spaces with smaller and smaller cylinders. At each stage of filling, the volume V consists of V_c, the volume filled out by composite cylinders, and the remaining volume V_i .At the same time, a portion A_c of the cross section is filled out by composite circles and there remains the area A_i. Since all composite cylinders are assumed to be in all small sizes, the remaining volume Vi can be made infinitely smaller by progressive and in limit the volume V consist entirely of non-overlapping composite cylinders. The resulting material is semi-radom and it is termed as the "Composite Cylinder Assemblage" model (CCA), see Fig. 1.

Fig. 1 CCA representation

Expressions and bounds for the five effective elastic moduli of a unidirectional fiber composite, consisting of transversely isotropic fibers and matrix, were derived by Hashin and Rosen (1964) and Hashin (1972), on the basis of analogies between isotropic and transversely isotropic elasticity equations and are summarised for brevity.

The bulk modulus (k*) is given as,

where cm and cf are the correspondent matrix and fiber volume fractions, km and kf are the matrix and fiber bulk moduli, respectively.

(1)

Nomenclature

A = RUC area [mm²]

A_1, A₂ = Bodner's constant [1/s]

b_i = Body forces vector

c_f, c_m = Fiber and matrix volume fraction [%]

D₀ = Bodner's constant [1/s]

D^p₂ = Kinematic variable [1/s]

E*_A = Effective axial Young's modulus [GPa]

E*_T = Effective transverse Young's modulus [GPa]

F_x, F_y, F_z = Constraint normal forces [N]

F_s = Shear loading constraint force [N]

G*_A = Effective in plane shear modulus [GPa]

G*_T = Effective transverse shear modulus [GPa]

J₂ = Second deviatoric stress invariant [MPa]

K_f = Fiber bulk modulus [GPa]

K_m = Matrix bulk modulus [GPa]

K_ij = Stiffness matrix

m₁, m₂ = Bodner's constant [1/MPa]

n = Hardening coefficient dimensionless

N_i = Vector of shape functions

P = Load applied at RUC [N]

Q*_ij = Transformed elasticity tensor

r₁, r₂ = Bodner's constants dimensionless

S_ij = Deviatoric stress components

S*_ij = Compliance matrix

u_i = Displacement [mm]

u_ij, v_ij = Bodner's variables dimensionless

V = RUC volume [mm³]

W_p = Plastic work rate [MPa/s]

Z = Hardening variable [MPa]

Z^I = Isotropic hardening [MPa]

Z^D = Kinematic hardening

d₁ = Prescribed displacement [mm]

g_ij = Shear strain components dimensionless

l = Kinematic hardening variable [MPa]

n*_A = Effective axial Poisson's ratio dimensionless

n*_T = Effective transverse Poisson's ratio dimensionless

s_ij = Stress components [MPa]

eij = Normal strain components dimensionless

The axial Young modulus, (E^*_A) is given by the expression,

(2)

The effective axial Poisson ratio (n*_A) is related to axial Poisson ratio (n_A), as well as the bulk modulus (k) and transverse shear modulus (G_T) of each phase, fiber (f) and matrix (m), by the following expression,

(3)

The effective in-plane shear modulus (G*_A) can be calculated using the expression,

(4)

The effective transverse shear modulus (G*_T) is bonded by the following expression,

(5)

where, for

(6)

(7)

(8)

and, for

(9)

(10)

(11)

The remaining other variables are defined as:

(12)

(13)

(14)

(15)

The lower bounds for k*, E*_A, n*_A, and G*_A are given by Eqs. (1-16). The corresponding upper bounds are obtained by interchanging the phase indices m and f in Eqs. (1-16) provide that the phase f moduli appearing in the bounds (but not necessarily Poisson's ratios) are larger than phase m moduli . The general lower bound on G*_T is given by Eq. (7) and the upper bound is obtained by the interchange of m with f . Again, phase f moduli must be larger than phase m moduli.

Bounds for E*_T are give in terms of general bounds, by

(16)

(17)

where

(18)

Bounds for v*_T are given as,

(19)

The plus and minus signals are representative of quantity upper bound and lower bound, respectively. It is important to point out that these equations derived above are valid only when "perfect bonding" is considered. The term bonding as used here does not imply a physical attraction between materials. Rather perfect bonding is synonymous with perfect contact.

Representative Unit Cell

The actual fiber distribution in a composite lamina is random across the cross-section. For simplicity, most micromechanical models assume a periodic arrangement of fiber for which a representative unit cell can be isolated. One of the most important issues when modelling composites using a RUC is to understand how it deforms when a uniform tensile or shear load is applied at the boundary of the composite (Sun and Vaidya, 1996, Avila et al., 1997). It is a well known fact that in a homogeneous material a uniform stress and strain state will exist under uniform loading, but it is not the case in composites. Since by definition all RUC's are identical, they should exhibit the same stress and strain fields. The stress and strain fields will be periodic in nature, except in a narrow boundary layer where the external load is applied. These periodicity constraints are used to determine the appropriate displacement constrains at the boundary of the RUC.

By isolating a RUC and the imposition of specific boundary conditions, the elastic moduli can be predicted using the principle of superposition, which is the focus of the following paragraphs. Assuming the usual Cartesian system of coordinates where the x-direction is the longitudinal axis, and considering the composite as an overall transversely isotropic material, a typical RUC can be represented by Fig. 2. A displacement (u =u₀) is imposed on x = 2a and the other constrained displacements are set to zero. The normal constrain forces are calculated and termed F¹_x, F_y¹_y, F¹_z, where the superscript indicates that these forces are from loading case1. Next, a displacement ( v= v₀) is imposed on y = 2b and the other constrained displacements are set to zero. The corresponding constraint forces F_x², F_y², F_z², are then calculated. Finally, a displacement (w = w₀) is imposed on z = 2c and the other constrained are set to zero. The constraint forces F_x³, F_y³, F_z³ are calculated next. These nine constraint forces are used in the following equations

(20)

(21)

(22)

Fig. 2 RUC representation

The unknowns are the load in the x-direction, P, and the scaling coefficients x and h. Using algebraic manipulations, the coefficients x and h can be calculated from Eqs. (21-22).

(23)

(24)

The average normal strain and the average Poisson ratio are defined as,

(25)

(26)

(27)

(28)

(29)

Finally, using the energy balance equation, the axial Young's modulus can be computed as,where V is the RUC volume.

(30)

The effective axial shear modulus can be calculated using the same methodology described earlier. The boundary conditions are such that a state is imposed. Once the constraint forces are evaluated, the shear strain energy balance is applied.

(31)

where, d₁ is the prescribed is displacement, F_s¹, F_s² are the shear loading cases constraint forces, g is the shear strain, and A = 4bc is the RUC cross section area.

The effective transverse Young's modulus, the transverse Poisson ratio, and the effective transverse shear modulus can be calculated using the same methodology described earlier.

Bodner and Partom's Viscoplastic Modulus

Unified viscoplastic constitutive models have evolved over the last three decades to provide a means for analytically representing a material response from the elastic through the plastic range, including strain-rate dependent plastic flow, creep and stress relaxation. The theories are guided by physical considerations, which include the theory of dislocations and are based on the principles of continuum mechanics. For years, many models have developed and among the various efforts one of the most common is the model developed by Bodner and Parton (1975). Many finite element researchers have implemented Bodner and Parton's model for isotropic materials (Thornton et al., 1990, Rowley and Thornton, 1996) and composites (Robertson and Mall, 1994, Allen et. al., 1994, Lee et al., 1991, Aboudi, 1989) with good results when compared against experiments.

The Bodner and Partom constitutive model is of the internal state variable type that is based on phenomenological observations and its basis lies on the dynamics of the theory of dislocations. Since its first publication, the models has gone through several modifications and was extended for anisotropic work hardening materials (Bodner, 1987). It was assumed that the deformations are small and therefore the total strain can be divided into the elastic e^e_ij and the inelastic e^p_ij components.

(32)

(33)

For the inelastic strain rate component, the isotropic form of the Prandtl-Reuss law is assumed

(34)

(35)

where Sij are the deviatoric stress components given by

and l is a scalar representation of kinematic hardening defined by the equation

(36)

It should be noted that Eqs. (33-34) denote plastic incompressibility.

Bodner relates the second invariant of the inelastic strain rate, D^p₂, to the invariant of the deviatoric stress, J₂, through an exponential function. The kinematic equation is defined as,

(37)

where D₀ and n are material parameters, and Z is the internal state variable that represents resistance to plastic flow.

The inelastic strain rate term of Eqs. (33-34) obeys a single flow designed to account simultaneously for plasticity and creep. Various models based on the original Bodner and Partom's model have been proposed by many researchers (Bass, 1985, Bass and Oden, 1987, Chan et al., 1989, Ramaswamy et al.,1990, Kroupa et al., 1996). In this study, we focus our attention on the original isotropic hardening Bodner and Partom's model (1975), and the version proposed by Chan et al. (1989) which also includes directional hardening.

The original Bodner and Partom's model

The inelastic strain rate flow developed by Bodner and Partom (1975) is represented by,

(38)

The evolution of the hardening parameter Z is described by the following expression:

(39)

(40)

where m, Z₀,and Z₁ are parameters that characterise the material behaviour in the plastic regime, and W_p is the work rate defined by,

Robertson and Mall (1994) observed that although this model requires fewer experimental parameters, it only considers the isotropic portion of hardening, and hence it has limited application in cyclic loading environments.

Bodner and Partom's model with isotropic and kinematic hardening

The flow rule for the Bodner and Partom's model which considers isotropic and kinematic hardening is expressed by,

(41)

The anisotropic form of Bodner and Partom's model (1975) decomposes this internal state variable into components that correspond to isotropic and kinematic hardening as,

(42)

The internal state variable which represents the hardening effects are described by a set of evolutionary equations. Bodner and Patron used the Orowan-Bailey hardening/recovery format to cast the isotropic and kinematic hardening evolutionary equations. The isotropic hardening parameter is provided by

(43)

where m₁, Z₁, Z₂, A1, and r₁ are material parameters.

The initial condition on Z^I(t) is

(44)

where Z₀ represents the initial value of the isotropic hardening. In the same manner, the kinematic hardening can be calculated as,

(45)

where b_ij (t) has the general form as for isotropic hardening

(46)

m₂ is the hardening rate, A₂ and r₂ are temperature dependent material constants, u_ij and v_ij are defined by

(47)

(48)

The initial boundary condition for the kinematic hardening is expressed by,

(49)

It should be mentioned that the expressions shown above describer the material's elastic-viscoplastic behavior based on dislocation theory, and yield surface is considered.

Integrated Micro/Macro Finite Element Formulation

A finite element analysis is performed at the macroscopic level, however the composite material properties are first computed at the microscopic level. The coupling between the micro/macro approaches is attempted by a new formulation. It is assumed that the total strain is a linear combination of the elastic strain and the inelastic strain. Bodner (1975) pointed out that this assumption is valid for small deformations. The fibers and matrix are considered perfectly bonded. The fibers are linear elastic and the matrix is elastic-viscoplastic. From the VFD theory (Dvorak and Bahei-El-Din, 1982), we have the following strain field:

(50)

(51)

where the superscript c stands for composite, f for fiber, m for matrix, c_f is the fibers volume fraction and c_m is the matrix volume fraction.

The decomposition of the total strain into the elastic and inelastic portions leads to

(52)

(53)

With the assumption of fibers as linear and the matrix as of elastic-viscoplastic behavior, Eqs. (52-53) can be rewritten as,

(54)

(55)

(56)

It is a well known fact that elastic strain components are related to the stresses by the following expression

where S*_ij is the effective compliance tensor, defined for a transversely isotropic material as

(57)

Substituting Eq. (57 ) into Eq. ( 56 ) and the results into Eqs. (54-55) leads to

(58)

where D^*_ij=(S^*_ij )^-1

A generalization of Eq. (58) can be obtained adding the overall composite initial strain, e^o_ij , and the initial stress, s^o_ij .

(59)

For a laminate the concept of "smeared" element is employed. The effective elasticity tensor D*_ij is substituted by the transformed elasticity tensor Q*_ij,

(60)

where h is the laminated thickness, z is a coordinate measured from the lamina semi-thickness, and Q_ij is the lamina transformed elasticity tensor. For more details see Ávila (1996), Tamma and Ávila (1999). In the study thermal strains are not considered.

(61)

The finite element approach (Zienkiewicz and Taylor, 1989, Bass and Oden, 1987) approximates the displacement rates within an element as

where Niare the interpolation functions, and d'_i represent it the nodal displacement rates (Thomton etal., 1990, Bass and Oden , 1987). Using the strain-displacement equations in the rate form, the element strain rate is given by

(62)

where B_ij is the strain-displacement matrix as defined by Zienkiewicz and Taylor (1989). The usual finite element procedures are used, and for a representative element, the finite element equations are obtained as

(63)

where K_ij is the stiffness matrix, and the right hand side is composed of the element load vectors due to the rate of inelastic strain, surface tractions, and body forces, respectively. These matrices are defined by

(64)

(65)

(66)

(67)

The strategy employed to solve the finite element equations is the following: With the initial stress distribution and internal variables pre-determined the equilibrium equations, described by Eq. (63), are used to calculate the nodal displacement rates. We then integrate the constitutive equations forward in time at the element Gauss points. With the update values of stress and the internal variables at a new time step, the equilibrium equations are solved again. The procedure is repeated until the desired total deformation or total time is achieved. The algorithm the following steps:

if n < TIME

then

Calculate the effective elastic moduli using CCA or RUC

for each element do

end do loop

Assemble and on the global basis

Solve = for on the global basis

for each element do

end do loop

Assemble and on the global basis

Updaten = n + 1

else

Print the step solution

end if

stop

The stress components and the internal state variables are advanced in time with the conditionally stable Euler algorithm. A small time step may be required because of the stiff character of the ordinary differential equations describing the internal state variables.

Numerical Applications

In this section, the overall developments are analyzed, and the present results are compared against other, analytical/numerical and experimental results available in the literature. Four test problems are presented. The homogenized material properties are obtained considering the CCA equations, and the RUC approach which are implemented in a finite element code. A typical mesh for the RUC is shown in Fig. 3. It employed 1300 hexahedron elements and its degenerated form represents a wedge.

Fig. 3 Typical RUC FEM mesh

Test case 1:

The first test problem is a SiC/Al metal matrix composite with 30% fiber volume fraction. The objective is to compare the elastic moduli predicted using the RUC approach against analytical solutions available in the literature. The material properties are shown in Table 1.

Table 1 SiC/Al material properties

In general, good agreement is observed between the finite element predictions and the analytical results, see Table 2 (UB and LB stand for upper and lower bounds, respectively). A slightly greater variation, ~5%, in the shear modulus and the Poisson ratio in the transverse direction could be due the application of boundary conditions on each face of the RUC's.

Table 2 Effective elastic moduli for SiC/Al

Test case 2:

The second test problem is a B/Al-6061-T0 metal matrix composite laminate [0₈] with 47%fiber volume fraction. The numerical simulation is performed considering all phases at room temperature (25 ºC). The finite element predictions of the elastic moduli are compared with analytical solutions, CCA (based on variational method), and experimental available literature.

The material properties of each phase are given in Table 3. Bodner and Partom's constants are given from Aboudi (1983), and are summarized in Table 4. It is worthy to mention that the Al-6061-T0 exhibits only isotropic hardening at room temperature, hence the "original" isotropic version of Bodner and Partom's model is employed.

Table 3 B/Al material properties

Table 4 Bodner's constants for B/Al

The results for the elastic moduli are compared with experimental (Kenaga et al., 1987) and analytical results in Table 5. It should be mentioned that only the square packing array is considered for comparison. Again the results are good agreement with analytical and experimental data available.

Table 5 Effective elastic moduli of B/Al

The macroscopic elastic-viscoplastic behavior is next analyzed. The finite element solutions for the transverse and shear loading are shown and compared with some of the available results in the literature. The transverse loading test case is performed considering a rectangular shape [0₈] specimen and dimensions .0.18 x 1.91 x 15.25 cm, and strain rate of 0.01 1/s. For the in-plane shear loading test a 10 deg off-axis tensile test of a [0₈] specimen with rectangular shape and nominal dimensions 0.15 x 1.27 x 17.8 cm following Chamis and Sinclair (1992) is employed. The same strain rate used in the transverse loading test case is also used in the in-plane shear loading. As shown in Fig. 4 and 5 the numerical results are in qualitative agreement and seem to produce a slightly stiffer response than the available models. It could be due to the assumption of perfect bonding and/or the homogenization procedure itself. It is interesting to mention that the values for the axial Young's modulus in Table 5 are close to the rule of mixtures. For more details about the rule of mixtures see Jones (1999).

Fig. 4 B/Al response under transverse normal loading

Fig. 5 B/Al in-plane shear response

Test case 3:

The third test problem is a SiC/Ti metal matrix composite, SCS-6/Ti-15-3, with a 29% fiber volume fraction. It employed a [0₈] laminated rectangular specimen with nominal dimensions of 0.18 x 1.91 x 15.25 cm, and constant strain rate of 0.01 1/s. The phases elastic moduli are from Rogacki and Tuttle (1992) for a temperature of 649 ^oC, and are listed in Table 6. The elastic-viscoplastic model constants are given by Jeong et al. (1993) are summarized in Tables 7 and 8. The object here is not only to investigate how the second version of Bodner and Partom's model, which includes isotropic and directional hardening, predicts the material behavior, but also the comparison against the experimental stress- strain relation obtained by Rogacki and Tuttle (1992). The MMC exhibits isotropic hardening and this fact is represented by Bodner and Partom's model by setting the variable m₂ equals to zero. It is important to point out that although the numerical simulations and the experiments made by Rogacki and Turtlle (1992) are accomplished in high temperature (649 ^oC) no thermal stresses were considered due to the isothermal condition. The temperature effects are represented only by lower values for the elastic moduli.

Table 6 SCS-6/Ti-15-3 material properties at 649 ^oC

Table 7 Bodner's temperature independent constants for SCS-6/Ti-15-3

Table 8 Bodner's temperature dependent constants for SCS-6/Ti-15-3

The elastic moduli obtained using the CCA upper and lower bounds, and the RUC approach are listed in Table 9. The numerical stress-strain results are compared against experimental data (Rogacki and Tuttle, 1992) in Fig. 6. Again the results of the integrated micro /macro formulation are in good agreement. An even closer agreement may likely be obtained considering the inclusion of some form of damage mechanism model in the present formulation.

Table 9 Effective elastic moduli for SCS-6/Ti-15-3

Fig. 6 0

Concluding Comments

An micro/macro mechanics model formulation and approach for the stress analysis of unidirectional laminated metal matrix composites has been described and a new constitutive equation is proposed.

(68)

After applying the new constitutive equation to B/Al and SiC/Ti metal matrix composites the results indicate that:

• In the micromechanical analysis, in which the elastic moduli have been predicted using the RUC and the CCA techniques, the results are very encouraging. Both the RUC and CCA seem to be adequate for predicting the elastic moduli of unidirectional metal matrix composites.

• In the macromechanical analysis, the present elastic-viscoplastic analysis and the integration of the micro/macro behavior closely matched with the analytical and experimental data available in the literature. Two different versions of Bodner and Partom's elastic-viscoplastic model were employed during the macromechanical analysis. The isotropic hardening version can be applied where loading conditions are monotonic and the boundary conditions, e.g, strain rate and test temperature, are such that no kinematic hardening is present . For more complex situations, a second version of Bodner and Partom's model was included in the integrated micro/macro model proposed. The inclusion of kinematic hardening besides the isotropic hardening further extended the applicability of the proposed model. To be able to analyze laminated composites the concept of smeared element was employed with good results.

The integrated micro/macro model seems to be a promising technique. The inclusion of interface model via integrated micro/macro model is under development. Experiments and numerical simulations must conducted to further validate the inclusion of debonding effects via proposed formulations.

Acknowledgments

The authors would like to acknowledge the financial support provides by the Universidade Federal de Minas Gerais, the CAPES Foundation. The Minnesota Supercomputer Institute furnished additional support and computer grants at the University of Minnesota.

Manuscript received: December 1997. Technical Editor: Agenor de Toledo Fleury

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Publication Dates

• Publication in this collection
  11 Oct 2001
• Date of issue
  Dec 1999

History

• Received
  Dec 1997