Abstract

A novel application of the laser ultrasonic technique for the detection of drilling-induced delamination in composite components of aircrafts is proposed. Numerous key components of aircrafts are made of composite materials, and drilling is often a final operation during assembly. Drilling-induced delamination significantly reduces the structural reliability, and it is rather difficult to be detected effectively and automatically. The laser ultrasonic technique is a promising method to solve the problem. This paper investigates the characterization of drilling-induced delamination in composites by a noncontact laser ultrasonic method. A carbon fiber reinforced plastic laminate with drilling holes is prepared as a specimen. The characterization of drilling-induced delamination with laser-generated ultrasonic waves is investigated theoretically and experimentally, and the morphology features of the delamination are obtained by laser ultrasonic C-scan testing. The results prove that the laser ultrasonic technique is effective for the detection of drilling-induced delamination in composite components, and it is a feasible solution for evaluating the drilling quality during assembly.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
    [CrossRef]
  2. C. C. Tsao, H. Hocheng, and Y. C. Chen, “Delamination reduction in drilling composite materials by active backup force,” CIRP Ann. 61, 91–94 (2012).
    [CrossRef]
  3. T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
    [CrossRef]
  4. H. Hocheng and C. C. Tsao, “Effects of special drill bits on drilling-induced delamination of composite materials,” Int. J. Mach. Tools Manuf. 46, 1403–1416 (2006).
    [CrossRef]
  5. C. C. Tsao and H. Hocheng, “Computerized tomography and C-Scan for measuring delamination in the drilling of composite materials using various drills,” Int. J. Mach. Tools Manuf. 45, 1282–1287 (2005).
    [CrossRef]
  6. U. A. Khashaba, “Delamination in drilling GFR-thermoset composites,” Comp. Struc. 63, 313–327 (2004).
    [CrossRef]
  7. J. P. Davim, J. C. Rubio, and A. M. Abrao, “A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates,” Compos. Sci. Technol. 67, 1939–1945 (2007).
    [CrossRef]
  8. M. Dubois and T. E. Drake, “Evolution of industrial laser-ultrasonic systems for the inspection of composites,” Nondestructr. Test. Eval. 26, 213–228 (2011).
    [CrossRef]
  9. J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
    [CrossRef]
  10. W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
    [CrossRef]
  11. R. Chona, C. S. Suh, and G. A. Rabroker, “Characterizing defects in multi-layer materials using guided ultrasonic waves,” Opt. Laser Eng. 40, 371–378 (2003).
    [CrossRef]
  12. C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
    [CrossRef]
  13. J. H. Lee and C. P. Burger, “Finite element modeling of laser-generated lamb waves,” Comput. Struct. 54, 499–514 (1995).
    [CrossRef]
  14. J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
    [CrossRef]

2013 (1)

T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
[CrossRef]

2012 (2)

C. C. Tsao, H. Hocheng, and Y. C. Chen, “Delamination reduction in drilling composite materials by active backup force,” CIRP Ann. 61, 91–94 (2012).
[CrossRef]

C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
[CrossRef]

2011 (2)

M. Dubois and T. E. Drake, “Evolution of industrial laser-ultrasonic systems for the inspection of composites,” Nondestructr. Test. Eval. 26, 213–228 (2011).
[CrossRef]

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

2009 (1)

N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
[CrossRef]

2007 (2)

J. P. Davim, J. C. Rubio, and A. M. Abrao, “A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates,” Compos. Sci. Technol. 67, 1939–1945 (2007).
[CrossRef]

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

2006 (1)

H. Hocheng and C. C. Tsao, “Effects of special drill bits on drilling-induced delamination of composite materials,” Int. J. Mach. Tools Manuf. 46, 1403–1416 (2006).
[CrossRef]

2005 (1)

C. C. Tsao and H. Hocheng, “Computerized tomography and C-Scan for measuring delamination in the drilling of composite materials using various drills,” Int. J. Mach. Tools Manuf. 45, 1282–1287 (2005).
[CrossRef]

2004 (1)

U. A. Khashaba, “Delamination in drilling GFR-thermoset composites,” Comp. Struc. 63, 313–327 (2004).
[CrossRef]

2003 (1)

R. Chona, C. S. Suh, and G. A. Rabroker, “Characterizing defects in multi-layer materials using guided ultrasonic waves,” Opt. Laser Eng. 40, 371–378 (2003).
[CrossRef]

1996 (1)

W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
[CrossRef]

1995 (1)

J. H. Lee and C. P. Burger, “Finite element modeling of laser-generated lamb waves,” Comput. Struct. 54, 499–514 (1995).
[CrossRef]

Abrao, A. M.

J. P. Davim, J. C. Rubio, and A. M. Abrao, “A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates,” Compos. Sci. Technol. 67, 1939–1945 (2007).
[CrossRef]

Akhter, N.

N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
[CrossRef]

Burger, C. P.

J. H. Lee and C. P. Burger, “Finite element modeling of laser-generated lamb waves,” Comput. Struct. 54, 499–514 (1995).
[CrossRef]

Chang, H. S.

N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
[CrossRef]

Chen, Y. C.

C. C. Tsao, H. Hocheng, and Y. C. Chen, “Delamination reduction in drilling composite materials by active backup force,” CIRP Ann. 61, 91–94 (2012).
[CrossRef]

Chia, C. C.

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

Chona, R.

R. Chona, C. S. Suh, and G. A. Rabroker, “Characterizing defects in multi-layer materials using guided ultrasonic waves,” Opt. Laser Eng. 40, 371–378 (2003).
[CrossRef]

Davim, J. P.

T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
[CrossRef]

J. P. Davim, J. C. Rubio, and A. M. Abrao, “A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates,” Compos. Sci. Technol. 67, 1939–1945 (2007).
[CrossRef]

Dhital, D.

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

Drake, T. E.

M. Dubois and T. E. Drake, “Evolution of industrial laser-ultrasonic systems for the inspection of composites,” Nondestructr. Test. Eval. 26, 213–228 (2011).
[CrossRef]

Dubois, M.

M. Dubois and T. E. Drake, “Evolution of industrial laser-ultrasonic systems for the inspection of composites,” Nondestructr. Test. Eval. 26, 213–228 (2011).
[CrossRef]

Gachagan, A.

W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
[CrossRef]

Grilo, T. J.

T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
[CrossRef]

Guan, J. F.

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Hayward, G.

W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
[CrossRef]

Hocheng, H.

C. C. Tsao, H. Hocheng, and Y. C. Chen, “Delamination reduction in drilling composite materials by active backup force,” CIRP Ann. 61, 91–94 (2012).
[CrossRef]

H. Hocheng and C. C. Tsao, “Effects of special drill bits on drilling-induced delamination of composite materials,” Int. J. Mach. Tools Manuf. 46, 1403–1416 (2006).
[CrossRef]

C. C. Tsao and H. Hocheng, “Computerized tomography and C-Scan for measuring delamination in the drilling of composite materials using various drills,” Int. J. Mach. Tools Manuf. 45, 1282–1287 (2005).
[CrossRef]

Huh, Y. H.

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

Hutchins, D. A.

W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
[CrossRef]

Jung, H. C.

N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
[CrossRef]

Khashaba, U. A.

U. A. Khashaba, “Delamination in drilling GFR-thermoset composites,” Comp. Struc. 63, 313–327 (2004).
[CrossRef]

Kim, K. S.

N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
[CrossRef]

Lee, J. H.

J. H. Lee and C. P. Burger, “Finite element modeling of laser-generated lamb waves,” Comput. Struct. 54, 499–514 (1995).
[CrossRef]

Lee, J. R.

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

Lu, J.

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Ng, C. T.

C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
[CrossRef]

Ni, X. W.

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Paulo, R. M. F.

T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
[CrossRef]

Rabroker, G. A.

R. Chona, C. S. Suh, and G. A. Rabroker, “Characterizing defects in multi-layer materials using guided ultrasonic waves,” Opt. Laser Eng. 40, 371–378 (2003).
[CrossRef]

Rose, L. R. F.

C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
[CrossRef]

Rubio, J. C.

J. P. Davim, J. C. Rubio, and A. M. Abrao, “A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates,” Compos. Sci. Technol. 67, 1939–1945 (2007).
[CrossRef]

Shen, Z. H.

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Shin, H. J.

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

Silva, C. R. M.

T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
[CrossRef]

Suh, C. S.

R. Chona, C. S. Suh, and G. A. Rabroker, “Characterizing defects in multi-layer materials using guided ultrasonic waves,” Opt. Laser Eng. 40, 371–378 (2003).
[CrossRef]

Tsao, C. C.

C. C. Tsao, H. Hocheng, and Y. C. Chen, “Delamination reduction in drilling composite materials by active backup force,” CIRP Ann. 61, 91–94 (2012).
[CrossRef]

H. Hocheng and C. C. Tsao, “Effects of special drill bits on drilling-induced delamination of composite materials,” Int. J. Mach. Tools Manuf. 46, 1403–1416 (2006).
[CrossRef]

C. C. Tsao and H. Hocheng, “Computerized tomography and C-Scan for measuring delamination in the drilling of composite materials using various drills,” Int. J. Mach. Tools Manuf. 45, 1282–1287 (2005).
[CrossRef]

Veidt, M.

C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
[CrossRef]

Wang, C. H.

C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
[CrossRef]

Wang, J. J.

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Wright, W. M. D.

W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
[CrossRef]

Xu, B. Q.

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Yoon, D. J.

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

CIRP Ann. (1)

C. C. Tsao, H. Hocheng, and Y. C. Chen, “Delamination reduction in drilling composite materials by active backup force,” CIRP Ann. 61, 91–94 (2012).
[CrossRef]

Comp. Struc. (1)

U. A. Khashaba, “Delamination in drilling GFR-thermoset composites,” Comp. Struc. 63, 313–327 (2004).
[CrossRef]

Compos. Sci. Technol. (1)

J. P. Davim, J. C. Rubio, and A. M. Abrao, “A novel approach based on digital image analysis to evaluate the delamination factor after drilling composite laminates,” Compos. Sci. Technol. 67, 1939–1945 (2007).
[CrossRef]

Composites, Part B (1)

T. J. Grilo, R. M. F. Paulo, C. R. M. Silva, and J. P. Davim, “Experimental delamination analyses of CFRPs using different drill geometries,” Composites, Part B 45, 1344–1350 (2013).
[CrossRef]

Comput. Struct. (1)

J. H. Lee and C. P. Burger, “Finite element modeling of laser-generated lamb waves,” Comput. Struct. 54, 499–514 (1995).
[CrossRef]

Int. J. Mach. Tools Manuf. (2)

H. Hocheng and C. C. Tsao, “Effects of special drill bits on drilling-induced delamination of composite materials,” Int. J. Mach. Tools Manuf. 46, 1403–1416 (2006).
[CrossRef]

C. C. Tsao and H. Hocheng, “Computerized tomography and C-Scan for measuring delamination in the drilling of composite materials using various drills,” Int. J. Mach. Tools Manuf. 45, 1282–1287 (2005).
[CrossRef]

J. Sound Vib. (1)

C. T. Ng, M. Veidt, L. R. F. Rose, and C. H. Wang, “Analytical and finite element prediction of Lamb wave scatting at delaminations in quasi-isotropic composite laminates,” J. Sound Vib. 331, 4870–4883 (2012).
[CrossRef]

Nondestructr. Test. Eval. (1)

M. Dubois and T. E. Drake, “Evolution of industrial laser-ultrasonic systems for the inspection of composites,” Nondestructr. Test. Eval. 26, 213–228 (2011).
[CrossRef]

Opt. Laser Eng. (3)

J. R. Lee, H. J. Shin, C. C. Chia, D. Dhital, D. J. Yoon, and Y. H. Huh, “Long distance laser ultrasonic propagation imaging system for damage visualization,” Opt. Laser Eng. 49, 1361–1371 (2011).
[CrossRef]

N. Akhter, H. C. Jung, H. S. Chang, and K. S. Kim, “Location of delamination in laminated composite plates by pulsed laser holography,” Opt. Laser Eng. 47, 584–588 (2009).
[CrossRef]

R. Chona, C. S. Suh, and G. A. Rabroker, “Characterizing defects in multi-layer materials using guided ultrasonic waves,” Opt. Laser Eng. 40, 371–378 (2003).
[CrossRef]

Opt. Laser Technol. (1)

J. J. Wang, Z. H. Shen, B. Q. Xu, X. W. Ni, J. F. Guan, and J. Lu, “Numerical simulation of laser-generated ultrasound in nonmetallic material by the finite element method,” Opt. Laser Technol. 39, 806–813 (2007).
[CrossRef]

Ultrasonics (1)

W. M. D. Wright, D. A. Hutchins, A. Gachagan, and G. Hayward, “Polymer composite material characterization using a laser/air-transducer system,” Ultrasonics 34, 825–833 (1996).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

Schematic diagram of the CFRP composite laminate with drilling holes and artificial defect.

Fig. 2.
Fig. 2.

Schematic diagram of the experimental setup used for detection of drilling-induced delamination.

Fig. 3.
Fig. 3.

Laser-generated broadband ultrasonic wave in composite laminate: (a) time domain signal and (b) its frequency spectrum.

Fig. 4.
Fig. 4.

FE results of time snapshots of laser ultrasonic wave propagation in (a) intact and (b) delaminated composite laminate with drilling hole: concerning the interaction of laser ultrasonic wave with drilling-induced delamination.

Fig. 5.
Fig. 5.

Experimental results of laser ultrasonic signals (3–5 MHz) in intact and delaminated area: (a) reflected wave signals monitored at the center of the source and (b) transmitted wave signals monitored at the bottom.

Fig. 6.
Fig. 6.

FE results of laser ultrasonic signals (3–5 MHz) in intact and delaminated area: (a) reflected wave signals monitored at the center of the source and (b) transmitted wave signals monitored at the bottom.

Fig. 7.
Fig. 7.

(a) Reflection and (b) attenuation coefficient spectra for drilling-induced delamination in composite laminate: open rectangle, experiment measurement and open triangle, FE prediction.

Fig. 8.
Fig. 8.

Results of scanning composite laminate with drilling-induced delamination based on (a),(b),(c) transmission and (d),(e),(f) pulse echo method at (a),(d) 2–4 MHz, (b),(e) 4–6 MHz, and (c),(f) 6–8 MHz. Each figure is normalized by its maximum amplitude.

Fig. 9.
Fig. 9.

Laser ultrasonic C-scan image of CFRP composite laminate specimen with drilling-induced delamination: (a) transmission method and (b) pulse echo method.

Fig. 10.
Fig. 10.

C-scan image of the analysis area (Fig. 2) with laser ultrasonic based on (a) transmission and (b) pulse echo method.

Tables (2)

Tables Icon

Table 1. Thermal Properties of the CFRP Prepreg Unidirectional Carbon/Epoxy Lamina Used in FE Calculation

Tables Icon

Table 2. Elastic Properties of the CFRP Prepreg Unidirectional Carbon/Epoxy Lamina Usedin FE Calculation

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

K2TρCT/t=q,
(λ+2μ)(·U)μ××Uρ2U/t2=α(3λ+2μ)T,
[K]{T}+[C]{T˙}={Rq}+{RQ},
[M]{U¨}+[K]{U}={Rext},
Ve[B]T[E]{ε0}dV,

Metrics