David H. Allen

A thermomechanical constitutive theory for elastic composites with distributed damage—I. Theoretical development

Abstract A continuum mechanics approach is utilized herein to develop a model for predicting the thermomechanical constitution of elastic composites subjected to both monotonic and cyclic fatigue loading. In this model the damage is characterized by a set of second-order tensor valued internal state variables representing locally averaged measures of specific damage states such as matrix cracks, fiber-matrix debonding, interlaminar cracking, or any other damage state. Locally averaged history dependent constitutive equations are posed utilizing constraints imposed from thermodynamics with internal state variables. In Part I the thermodynamics with internal state variables is constructed and it is shown that suitable definitions of the locally averaged field variables will lead to useful thermodynamic constraints on a local scale containing statistically homogeneous damage. Based on this result the Helmholtz free energy is then expanded in a Taylor series in terms of strain, temperature, and the internal state variables to obtain the stress-strain relation for composites with damage. In Part II the three-dimensional tensor equations developed in Part I are simplified using material symmetry constraints and are written in engineering notation. The resulting constitutive model is then cast into laminate equations and an example problem is solved and compared to experimental results. It is concluded that although the model requires further development and extensive experimental verification it may be a useful tool in characterizing the thermomechanical constitutive behavior of continuous fiber composites with damage.

A thermomechanical constitutive theory for elastic composites with distributed damage—II. Application to matrix cracking in laminated composites

Abstract A continuum mechanics approach is utilized herein to develop a model for predicting the thermomechanical constitution of initially elastic composites subjected to both monotonic and cyclic fatigue loading. In this model the damage is characterized by a set of second-order tensor valued internal state variables representing locally averaged measures of specific damage states such as matrix cracks, fiber-matrix debonding, interlaminar cracking, or any other damage state. Locally averaged history dependent constitutive equations are constructed utilizing constraints imposed from thermodynamics with internal state variables. In Part I the thermodynamics with internal state variables was constructed and it was shown that suitable definitions of the locally averaged field variables led to useful thermodynamic constraints on a local scale containing statistically homogeneous damage. Based on this result the Helmholtz free energy was then expanded in a Taylor series in terms of strain, temperature, and the internal state variables to obtain the stress-strain relation for composites with damage. In Part II, the three-dimensional tensor equations from Part I are simplified using symmetry constraints. After introducing engineering notation and expressing the constitutive equations in the standard laminate coordinate system, a specialized constitutive model is developed for the case of matrix cracks only. The potential of the model to predict degradation of effective stiffness components is demonstrated by solving the problem of transverse matrix cracks in the 90° layer of several crossply laminates. To solve the example problems, the undamaged moduli are determined from experimental data. The internal state variable for matrix cracking is then related to the strain energy release rate due to cracking by utilizing linear elastic fracture mechanics. These values are then utilized as input to a modified laminate analysis scheme to predict effective stiffnesses in a variety of crossply laminates. The values of effective (damage degraded) stiffnesses predicted by the constitutive model are in agreement with experimental results. The agreement obtained in these example problems, while limited to transverse matrix cracks only, demonstrates the potential of the constitutive model to predict degraded stiffnesses.

A THREE-DIMENSIONAL FINITE ELEMENT FORMULATION FOR THERMOVISCOELASTIC ORTHOTROPIC MEDIA

SUMMARY This paper is concerned with the development of a numerical algorithm for the solution of the uncoupled, quasistatic initial/boundary value problem involving orthotropic linear viscoelastic media undergoing thermal and/or mechanical deformation. The constitutive equations, expressed in integral form involving the relaxation moduli, are transformed into an incremental algebraic form prior to development of the nite element formulation. This incrementalization is accomplished in closed form and results in a recursive relationship which leads to the need of solving a simple set of linear algebraic equations only for the extraction of the nite element solution. Use is made of a Dirichlet{Prony series representation of the relaxation moduli in order to derive the recursive relationship and thereby eliminate the storage problem that arises when dealing with materials possessing memory. Three illustrative example problems are included to demonstrate the method. ? 1997 by John Wiley & Sons, Ltd.

Internal State Variable Approach for Predicting Stiffness Reductions in Fibrous Laminated Composites with Matrix Cracks

A mathematical model utilizing the Internal State Variable (ISV) concept is proposed for predicting the upper bound of the reduced axial stiffnesses in cross-ply lami nates with matrix cracks. The axial crack opening displacement is explicitly expressed in terms of the observable axial strain and the undamaged material properties. A crack parameter representing the effect of matrix cracks on the observable axial Youngs modulus is calculated for glass/epoxy and graphite/epoxy material systems. The results of the present study show that the matrix crack opening displacement and conse quently the effective Youngs modulus depends not on the crack length but on its ratio to the crack spacing. Comparisons of the present model with experimental data and other models in the litera ture show a good agreement, thus confirming direct applicability of the model to [0 p /90 r ] s type laminates.

An experimental and analytical treatment of matrix cracking in cross-ply laminates

The development of damage in cross-ply Hercules AS4/3502 graphite/epoxy laminates has been investigated. Specific endeavors were to identify the mechanisms for initiation and growth of matrix cracks and to determine the effect of matrix cracking on the stiffness loss in cross-ply laminates. Two types of matrix cracks were identified. These include both straight and curved cracks. The experimental study of matrix crack damage revealed that the curved cracks formed after the straight cracks and followed a repeatable pattern of location and orientation relative to the straight cracks. Therefore, it was postulated that the growth mechanism for curved cracks is driven by the stress state resulting from the formation of the straight cracks. This phenomenon was analytically investigated by a finite-element model of straight cracks in a cross-ply laminate. The finite-element results provide supporting evidence for the postulated growth mechanism. The experimental study also revealed that the number of curved cracks increased with the number of consecutive 90-deg plies. Finally, experimental results show as much as 10-percent degradation in axial stiffness due to matrix cracking in cross-ply graphite/epoxy laminates.

A micromechanical model for a viscoelastic cohesive zone

A micromechanical model for a viscoelastic cohesive zone is formulated herein. Care has been taken in the construction of a physically-based continuum mechanics model of the damaged region ahead of the crack tip. The homogenization of the cohesive forces encountered in this region results in a damage dependent traction-displacement law which is both single integral and internal variable-type. An incrementalized form of this traction-displacement law has been integrated numerically and placed within an implicit finite element program designed to predict crack propagation in viscoelastic media. This research concludes with several example problems on the response of this model for various displacement boundary conditions.

Micromechanical analysis of a continuous fiber metal matrix composite including the effects of matrix viscoplasticity and evolving damage

Abstract A thermomechanical analysis of a metal matrix continuous fiber composite is performed herein. The analysis includes the effects of matrix inelasticity and interface cracking. Due to these nonlinearities, the analysis is performed computationally using the finite element method. Matrix inelasticity is modeled with a rate dependent viscoplasticity model. Interface fracture is modeled by the use of a nonlinear interface constitutive model. The problem formulation is summarized, and results are given for a typical SiC-Ti composite at elevated temperature. Preliminary results indicate that rate dependent viscoplasticity can be a significant mechanism for dissipating the energy available for interface fracture, thus contributing to improved macroscopic ductility of the composite.

Formulation of a three-dimensional cohesive zone model for application to a finite element algorithm

A formulation is presented herein for implementing a cohesive zone model to a nonlinear finite element algorithm. The cohesive zone model may be of arbitrary type so long as it can be constructed in an incremental form in time. Problems that can then be solved using this algorithm include a broad array of material types, including: elastic, elastoplastic, viscoplastic, and viscoelastic. Multiple cracks may be modeled by this methodology, and any and all can be evolving simultaneously, with crack interaction included explicitly in the formulation. Example problems are included to demonstrate the efficacy of the algorithm.

Micromechanical Model for Heterogeneous Asphalt Concrete Mixtures Subjected to Fracture Failure

Cracking is a main source of structural distress in asphalt materials and asphaltic pavements. To predict crack-associated fracture damage in asphalt mixtures, this study presents a model using the finite-element method and a cohesive zone fracture model. The approach allows advanced characterization of the microstructural damage evolution in a more realistic length scale, the mixture heterogeneity, the inelastic material behavior, and the interactions among mixture constituents. The model presented herein accounts for (1) actual mixture heterogeneity by using digital image techniques; (2) inelastic material behavior based on elastic-viscoelastic constitutive relations; and (3) microscale fracture damage represented by the cohesive zone fracture model. A computational modeling framework is presented, and the applicability of the model is demonstrated through simulations. Model simulations are further analyzed by comparing numerical predictions to laboratory test results and by conducting parametric analyses of fracture properties. It is expected that the successfully developed computational model can provide better insights into the effect of mixture constituents on overall mixture performance, while minimizing modeling efforts and producing more accurate simulations than traditional approaches, with significant savings in experimental costs and time.

Homogenization techniques for thermoviscoelastic solids containing cracks

Abstract In this paper mathematical techniques are developed for obtaining locally averaged (homogenized) constitutive equations for heterogeneous linear thermoviscoelastic solids. Homogenization principles will be developed for the cases wherein no internal boundaries are present, and also where internal boundaries in the form of sharp cracks are present, thus resulting in damage dependent macroscopic constitutive equations. The microthermomechanics problem will first be formulated, followed by the construction of the locally averaged equations resulting from the homogenization process. It will be shown that homogenized conservation laws and constitutive equations take the same form as do the local equations when locally linear thermoviscoelastic media are considered. However, the resulting homogenized constitutive equations will be nonlinear in the case wherein time dependent damage occurs. In addition, for materials of convolution type at the local scale, the homogenized equations will be shown to contain a term that depends on the time derivative of the strain localization tensor. Example problems will be discussed and the homogenized results will be given for these examples in order to demonstrate the technique.