Guided Lamb waves for identification of damage in composite structures: A review

The problem of impact damage in laminated composite structures, and the consequent reduction in residual strength, has been a topic of continual research for over two decades. The number of journal papers on the subject now runs into four figures and most have been conscientiously reviewed by Abrate(1991, 1994, 1998). This review is not intended to be in the academic tradition, with emphasis on acknowledging the authorship of all the various research initiatives. Instead we present our opinions so that the reader can appreciate our current understanding of the problem, our capability of predicting by analysis, and the scope of the design tools for avoiding structural damage, or at least designing damage tolerant aerospace structures.

Impact on composite structures

Impact duration strongly influences the impact response of plates. Long impacts cause a quasi-static response influenced by the plate size and boundary conditions. Short impacts cause a response governed by wave propagation unaffected of plate size and boundary conditions. This paper shows that the response type is governed by the impactor–plate mass ratio and not by impact velocity and derives a criterion for small-mass (wave controlled) impact response of orthotropic plates. Published criteria for large-mass (quasi-static) impact are discussed. Small-mass impacts on composite laminates are shown to be more critical than large-mass impacts of the same energy. Use of mass criteria for selecting analytical response models and considerations for sandwich plates are also discussed.

Mass criterion for wave controlled impact response of composite plates

Axial crush of metallic honeycombs

Force reconstruction: analysis and regularization of a deconvolution problem

A method to identify both the impact location and the transverse impact force history from the strain responses at certain points on a rectangular plate is presented. The governing equations of the plate were obtained by applying the Reissner-Mindlin plate theory and the Rayleigh-Ritz method. The strain response was related to the impact force by solving the above equations using the eigenmode expansion method. A mutuality relationship among any pairs of strain responses was used to find the impact location without knowing in advance the impact force history. The force history was subsequently determined after the impact location was identified. The conjugate gradient method was adopted to search for the optimal impact location as well as the force history. Numerical verification was performed using randomly generated impact locations and force histories to simulate impact events. The excellent agreement showed the effectiveness and the validity of the proposed method.

On the Inverse Problem of Rectangular Plates Subjected to Elastic Impact, Part I: Method Development and Numerical Verification

This paper presents an analytical method for the identification of force histories on a rectangular laminated plate when it is struck by foreign objects at multiple locations. The governing equations of motion of the plate are constructed by using the classical lamination theory and the Rayleigh-Ritz method. The modal superposition method is then employed to obtain Green's function in a transformed time domain. To obtain the impact force histories exerted at multiple locations on the laminate, a constrained optimization method has been employed. The method has been verified using both randomly generated numerical signals with a signal-to-noise ratio of 1.0 and experimental data. The agreement is satisfactory in all of the comparisons. The method is then applied to further identify the actual impact points among several locations on a laminate that may be hit. With minor modifications, this method is also applied to reconstruct the strain histories, a portion of which are accidently lost in an impact event. ECAUSE of the tendency for invisible damage to occur inside structures made of advanced composite materials, study of the low-velocity-impact response of composite laminates has attracted the attention of many researchers during recent decades. Most of the work has involved analyzing the responses of a laminated structure when the initial configurations of the structure and the impactor are known.1 To obtain the relationship between the impactor movement and the plate motion at the impact location, the force-indentation relationship has been frequently applied. This relationship may be obtained by experiment,1'2 by Hertz contact law,3 or by more rigor- ously theoretical derivations.4'5 In addition, the laminates analyzed may be subjected to in-plane prestressed loading,6 large deflection,7 or various types of damage.8 On the other hand, experimental results have been reported for the investigation of the responses of lami- nated structures subjected to single1'2'8 or repeated2'9 impact load- ing at low striking velocities. The relationship between a laminated plate subjected to low-velocity impact and to quasistatic contact was also reported.2'10 When the initial striking velocity of the projectile was higher, the focus was on investigation of the ballistic resistance capability of the composite materials.11'12 An alternate approach to understanding the impact behavior of a composite structure is to use the inverse method. When the impact duration is short such that wave motion dominates the impact re- sponse, the generalized ray theory is frequently employed.13'14 As the impact duration becomes longer, such as in low-velocity impact by a foreign object, the generalized ray theory becomes not applica- ble because stress waves are reflected many times from the bound- aries of a structure. To overcome this problem, Doyle has proposed a frequency domain method using a complicated filtering procedure to suppress the effect of this wave reflection and has successfully identified the impact force of a duration longer than 1 ms.15'16 Another method for identifying the impact force history with a du- ration on the order of 1 ms is to use the modal superposition method. Hollandsworth and Busby17 have identified the impact force exerted on a cantilever beam using the response recorded by an accelero- meter as the input data. For a circular plate struck at its center, Wu et al.18 have constructed Green's functions from a series of Bessel functions. An optimization method was employed to search for the optimal impact force history. In the same paper, they also proposed an experimental method, by which the impact force history could be identified through a simple deconvolution process. This empirical method can thus be applied to structures with complicated material properties or boundary conditions. However, responses of a structure to impact of foreign objects acting simultaneously at multiple loca- tions have not been addressed by any of the aforementioned studies. In this paper, an analytical method has been developed to iden- tify the force histories of a composite laminate when it is struck by foreign objects at multiple locations. This situation may be encoun- tered when a flying vehicle is passing through a region with rain, dust, or hail. In other situations, multiple dents may be found on the surface of a laminated structure such that it is hard to determine whether single or multiple impact occurred. For all of these situa- tions, the method developed in this paper can be used to identify the impact force histories from the recorded strain histories. Further, this method can be extended to reconstruct the strain histories when a portion of the strains are not recorded due either to an improper trig- gering setting or to unexpectedly large signals occurring in an impact event. To develop this analytical method, a rectangular composite laminated plate will be modeled by using the classical lamination theory, and the Rayleigh-Ritz method will be employed to form the governing equations of motion of the plate. The modal superposition method will then be applied to construct Green's functions, which will be used to relate the forces exerted at multiple locations to the responses of the plate. Meanwhile, the gradient projection method will be applied to search for the optimal impact force histories. Both numerical data with a signal-to-noise ratio of 1.0 and experimental data will then be used to verify the proposed method.

Identification of impact forces at multiple locations on laminated plates

The contact behavior between a composite laminate and a rigid sphere is discussed in this paper using a method derived from the three-dimensional anisotropic elasticity theory. Response of the plate and the force-indentation ( F − α ) relationship were obtained by constructing an exact Green’s function of the laminate. Agreement between the prediction and the experimental data reported by others is excellent before laminates are damaged. The F - α relationship is approximately proportional to the out-of-plane Young’s modulus and is influenced by the plate thickness. The effects of changing the in-place Young’s moduli, all the shear moduli, stacking sequences and span of the laminates are insignificant unless identation is large. Further, the modified Hertz law underestimates the contact force for the same indentation.

The Contact Behavior Between Laminated Composite Plates and Rigid Spheres

Experimental verification of the method developed in Part 1 to identify both the impact location and the force history from strain responses on a rectangular plate was performed. Results showed the validity of the method in a real impact event. Also, a method was developed to further identify the initial velocity and the mass of an impactor by which a transverse impact was induced. This was accomplished by solving algebraic equations obtained from the assumption that the lateral displacements of both the impactor and the plate at the impact point were coincident during the contact period. Moreover, the inverse problem using incomplete response signals as the given data was investigated. A procedure to temporarily reconstruct the lost portions of the recorded signals was first presented, and the identification problem could then be solved by similar methods as that used for the complete response signals. Experimental verification was also performed. The agreement between the measured and the identified results was very satisfactory.

On the Inverse Problem of Rectangular Plates Subjected to Elastic Impact, Part II: Experimental Verification and Further Applications

Two methods are presented to detect the impact-force history on elastic plates. The first method involves construction of a Green’s function to relate the strain responses to an impact force acting axisymmetrically on a circular plate. The classical plate theory and a series of Bessel functions were used to obtain the Green’s functions in the time domain. The gradient projection method was employed to search for the optimal force history. Examples using circular plates with free and fixed boundary conditions are demonstrated. On the other hand, the second method is purely experiment-based and involves no Green’s function. Thus, this method can be applied to structures of various geometries, boundary conditions and material properties. The very satisfactory agreement between the measured and the detected force histories using both methods shows that they are reliable for impact-force determination purposes.

Ching-Shih Yen

Papers

On the Inverse Problem of Rectangular Plates Subjected to Elastic Impact, Part I: Method Development and Numerical Verification

Identification of impact forces at multiple locations on laminated plates

The Contact Behavior Between Laminated Composite Plates and Rigid Spheres

On the Inverse Problem of Rectangular Plates Subjected to Elastic Impact, Part II: Experimental Verification and Further Applications

Two methods for determining impact-force history on elastic plates