S.E. Groves

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.

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 cumulative damage model for continuous fiber composite laminates with matrix cracking and interply delaminations

Experimental evidence has shown that significant stiffness loss occurs in graphite/ epoxy laminates when matrix cracking and interply delaminations exist. Therefore, a cumulative damage model for predicting stiffness lossin graphite/epoxy laminates is proposed herein by applying a thermomechanical constitutive theory for elastic composites with distributed damage. The model proceeds from a continuum mechanics and thermodynamics approach wherein the distributed damage is characterized by a set of second-order tensor-valued internal state variables. The internal state variables represent locally averaged measures of matrix cracking and interply delaminations. The model formulation provides a set of damage dependent laminated plate equations. These are developed by modifying the classical Kirchhoff plate theory. The effect of the matrix cracking enters the formulation through alteration in the individual lamina constitution. The effect of interply delamination enters the formulation through modifications of the Kirchhoff displacements. The corresponding internal state variables are defined utilizing the kinematics of the interply delaminated region and the divergence theorem. These internal state variables depend on the components of the displacements created by the delamination.

The Effects of Through-thickness Compression on the Interlaminar Shear Response of Laminated Fiber Composites

The effects of through-thickness compression on the interlaminar shear response of laminated fiber composites were studied. The combined stresses were generated using a hollow cylindrical specimen that was subjected to axial compression and torsion. For both glass- and carbon-fiber composites, through-thickness compression resulted in a significant enhancement in the interlaminar shear stress and strain at failure. Under moderate compression levels, the failure mode changed from elastic to plastic. An attempt was made to predict the observed increase in shear strength for carbon fiber epoxy laminates using three-dimensional lamina failure criteria. Although all the failure theories correctly predicted the trend of increasing shear strength with compression, none were able to predict the full extent of the observed strength increase. These results indicate that improved models are needed for determining failure under a combined state of interlaminar stress. The experimental results demonstrate that there are significant gains to be made in improving interlaminar strengths of composite structures by applying through-thickness compression. This effect could be exploited for improved strength and possibly improved fatigue life of composite joints and other regions in structures where interlaminar stress states are critical.

Stress analysis of a matrix-cracked viscoelastic laminate

Two distinctly different approaches to viscoelastic stress analysis are employed herein for the purpose of predicting the response of a matrix-cracked viscoelastic laminate to a given loading history. A viscoelastic correspondence principle is developed to provide an analytical solution and a finite element formulation is developed to provide a numerical solution. The two methods are demonstrated through the solution of a simple illustrative example problem. Results from the two methods of analysis are compared.

High Strain Rate Effects for Composite Materials

We have been developing the capability to characterize the high strain rate response of continuous fiber polymer composites. The data presented covers strain rates from 0/sec to 3000/sec. A combination of test machines and specimen geometries was investigated. Strain rates from 0--100/sec were generated using conventional and high speed hydraulic test machines. Strain rates from 10--1000/sec were generated using a high energy drop tower, and rates from 1000--3000/sec were generated using a split Hopkinson bar. Strain rates above 100/sec have only been generated for uniaxial compression. Our efforts have primarily focused on developing the high energy drop tower for these purposes. Specimen geometries for compression include tapered cubes, one inch tubes, and solid rods. For tension a smaller 0.5 in. diameter version of our 2.0 in. diameter multiaxial test specimen was developed and has been successfully used at strain rates up to 100 per second. Fixtures were also developed for performing high strain rate shear testing and through thickness penetration studies of composite plates. The objective of these experiments is to develop dynamic material models for use in finite element design tools. This presentation will focus on the methods and results obtained from this study.

Evaluation of cylindrical shear joints for composite materials

The objective of this work was to evaluate the strength of four candidate cylindrical shear joints for composite tubes. The basic design for the joint entails an axial length of one inch with an external 15° tapered cone. The purpose of the joint is to transfer axial load from a cylinder through a steel shear attachment with a matching internal coni cal seat. The design candidates consisted of a bonded wedge cone, pinned wedge cone, and a bonded and pinned wedge cone attached to a two-inch diameter composite tube and a wedge cone integrally wound into the tube. The actual joint strengths achieved were higher than expected for some designs and were found to be dependent on the amount of hydrostatic or radial compression applied to the joint. The bonded wedge ring and the inte gral wedge ring both achieved over 96 MPa (14 ksi) of shear strength without failure. Due to the unexpectedly high shear strengths of the joints most of the failures were located in other parts of the test specimen. The bonded wedge cone was probably close to its maxi mum load carrying capacity while the integral wedge design showed potential for even higher shear strengths. The bonded and pinned joint reached a peak shear strength of 78.9 MPa (11.5 ksi) and the pinned only configuration achieved 70.6 MPa (10.3 ksi). When loaded without any hydrostatic compression the joint strengths were less than 34.3 MPa (5 ksi); however, the failure mode was hoop compression buckling of the tube itself as op posed to a joint shear failure.

Evaluation of First Ply Failure in a Three-Dimensional Load Space

Numerically generated failure envelopes for several three-dimensional failure criteria are presented and compared to experimental data. These envelopes are developed through finite element analysis and are based on first ply failure (FPF). The experimental data are based on ultimate failure of filament wound tubes constructed from Toray 1000/DER332-T403 and loaded in various combinations of axial traction (both tension and compression), internal pressure, and torsion. All tubes have a layup of [± 1.5, ± 45, ± 89] T . Five three-dimensional failure criteria are considered: max-stress, max-strain, one proposed by Tsai, and two recently proposed by Feng. In addition to the three-dimensional failure criteria evaluated, the finite element results are used in conjunction with classical lamination theory to test the predictive capability of some of the more common two-dimensional failure criteria (max-strain, Tsai-Hill, and Tsai-Wu). None of the predicted envelopes compares well with the experimental data; thereby illustrating the need for progressive failure analysis in structures subjected to complex stress states. It is shown that it is possible to improve the accuracy of at least one of the three-dimensional failure criteria through a very minor modification of the theory.