{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,25]],"date-time":"2026-10-25T13:41:36Z","timestamp":1792935696114,"version":"4.5.0"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100012165","name":"Key Technologies Research and Development Program","doi-asserted-by":"publisher","award":["2024GX-YBXM-178"],"award-info":[{"award-number":["2024GX-YBXM-178"]}],"id":[{"id":"10.13039\/501100012165","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["52475123"],"award-info":[{"award-number":["52475123"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100007128","name":"The Natural Science Foundation of Shaanxi Province","doi-asserted-by":"crossref","award":["2024JC-YBMS-370"],"award-info":[{"award-number":["2024JC-YBMS-370"]}],"id":[{"id":"10.13039\/501100007128","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-024-76751-w","type":"journal-article","created":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T08:29:40Z","timestamp":1730190580000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A constitutive model based on energy density for power-law variable thickness plates and its application"],"prefix":"10.1038","volume":"14","author":[{"given":"Miao-xia","family":"Xie","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qian-lang","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peng","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin-tao","family":"Ren","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feng-wei","family":"Gao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ling","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yue-ming","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,29]]},"reference":[{"key":"76751_CR1","unstructured":"Lu, X., Guo, H. & Du, X. Simulation study of wedge plate straightening process based on dynamic press down. Heavy (2022)."},{"issue":"5","key":"76751_CR2","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1016\/0045-7949(88)90008-9","volume":"28","author":"M Ohga","year":"1988","unstructured":"Ohga, M. & Shigematsu, T. Bending analysis of plates with variable thickness by boundary element-transfer matrix method. Comput. Struct. 28 (5), 635\u2013640 (1988).","journal-title":"Comput. Struct."},{"key":"76751_CR3","unstructured":"Fertis, D. G. Dynamics and Vibration of Structures (Robert E. Krieger Publishing Co, 1984)."},{"issue":"10","key":"76751_CR4","doi-asserted-by":"publisher","first-page":"2287","DOI":"10.1061\/(ASCE)0733-9399(1989)115:10(2287)","volume":"115","author":"DG Fertis","year":"1989","unstructured":"Fertis, D. G. & Mijatov, M. M. Equivalent systems for variable thickness plates. J. Eng. Mech. 115 (10), 2287\u20132300 (1989).","journal-title":"J. Eng. Mech."},{"key":"76751_CR5","doi-asserted-by":"crossref","unstructured":"Fertis, D. G. & Lee, C. Elastic and inelastic analysis of variable thickness plates, using equivalent systems*. Mech. Struct. Machines (1993).","DOI":"10.1080\/08905459308905187"},{"issue":"3","key":"76751_CR6","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/S0307-904X(02)00123-3","volume":"27","author":"MNM Allam","year":"2003","unstructured":"Allam, M. N. M. & Zenkour, A. M. Bending response of a fiber-reinforced viscoelastic arched bridge model. Appl. Math. Model. 27 (3), 233\u2013248 (2003).","journal-title":"Appl. Math. Model."},{"issue":"10","key":"76751_CR7","doi-asserted-by":"publisher","first-page":"1045389X1875818","DOI":"10.1177\/1045389X18758184","volume":"29","author":"X Li","year":"2018","unstructured":"Li, X. & Ding, Q. Analysis on vibration energy concentration of the one-dimensional wedge-shaped acoustic black hole structure. J. Intell. Mater. Syst. Struct. 29 (10), 1045389X1875818  (2018).","journal-title":"J. Intell. Mater. Syst. Struct."},{"issue":"5","key":"76751_CR8","doi-asserted-by":"publisher","first-page":"055001","DOI":"10.1088\/1361-665X\/abed32","volume":"30","author":"Q Fu","year":"2021","unstructured":"Fu, Q., Du, X., Wu, J. & Zhang, J. Dynamic property investigation of segmented acoustic black hole beam with different power-law thicknesses. Smart Mater. Struct. 30 (5), 055001 (2021).","journal-title":"Smart Mater. Struct."},{"key":"76751_CR9","doi-asserted-by":"crossref","unstructured":"Wu, L. et al. Experimental and finite element analysis on the sound absorption performance of wedge-like knitted composite. Thin-Walled Struct. (2023).","DOI":"10.1016\/j.tws.2022.110289"},{"key":"76751_CR10","doi-asserted-by":"crossref","unstructured":"Choi, K. K. et al. Parametric Design Sensitivity Analysis of high frequency structural-acoustic problems using energy finite element Method. Int. J. Numer. Methods Eng. 62.1 (2005).","DOI":"10.1002\/nme.1190"},{"issue":"1","key":"76751_CR11","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1115\/1.3269830","volume":"111","author":"DJ Nefske","year":"1989","unstructured":"Nefske, D. J. & Sung, S. H. Power flow finite element analysis of dynamic systems: basic theory and application to beams. J. Vib. Acoust. 111 (1), 94 (1989).","journal-title":"J. Vib. Acoust."},{"key":"76751_CR12","unstructured":"Lase, Y. & Jezequel, J. Analysis of a dynamic system based on a new energetic formulation. International Congress on Intensity Techniques, France:145\u2013150P (1990)."},{"issue":"3","key":"76751_CR13","doi-asserted-by":"publisher","first-page":"616","DOI":"10.2514\/3.10964","volume":"30","author":"OM Bouthier","year":"1992","unstructured":"Bouthier, O. M. & Bernhard, R. J. Models of space-averaged energetics of plates. AIAA J. 30 (3), 616\u2013623 (1992).","journal-title":"AIAA J."},{"key":"76751_CR14","unstructured":"Xie, M., Chen, H. & Wu, J. Energy finite element analysis to high-frequency bending vibration in cylindrical shell.Hsi-An Chiao Tung Ta Hsueh\/Journal of Xi\u2019an Jiaotong University (2008)."},{"key":"76751_CR15","doi-asserted-by":"publisher","unstructured":"Liu, Z. Improved model of structural vibration energy flow method and its application to functional gradient structures. Shandong University. https:\/\/doi.org\/10.27272\/d.cnki.gshdu.2021.005776 (2021).","DOI":"10.27272\/d.cnki.gshdu.2021.005776"},{"key":"76751_CR16","doi-asserted-by":"crossref","unstructured":"Xie, M. et al. Energy finite element model for predicting high frequency dynamic response of taper beams. Arch. Appl. Mech. 94(8), 2335\u20132353 (2024).","DOI":"10.1007\/s00419-024-02638-x"},{"key":"76751_CR17","doi-asserted-by":"crossref","unstructured":"Xie, M. et al. Energy Flow analysis of high-frequency Flexural Vibration of Wedge Beam structures. Shock Vib, 2022, 1\u201314 (2022).","DOI":"10.1155\/2022\/2935655"},{"key":"76751_CR18","doi-asserted-by":"publisher","DOI":"10.1115\/1.3423344","author":"JD Achenbach","year":"1974","unstructured":"Achenbach, J. D. & Author, S. Wave propagation in elastic solids. J. Appl. Mech,\u00a041(2), 544.\u00a0https:\/\/doi.org\/10.1115\/1.3423344 (1974).","journal-title":"J. Appl. Mech."},{"key":"76751_CR19","unstructured":"Yang, G. Elasticity (Higher Education Press, 1998)."},{"key":"76751_CR20","unstructured":"Zhu, Y. & Zhou, J. Nonlinear Vibration and Motion Stability (Xi\u2019an Jiaotong University, 1992)."},{"key":"76751_CR21","unstructured":"Liu, Y. & Chen, L. Nonlinear Vibration, 57109 (Higher Education Press, 2001)."},{"key":"76751_CR22","unstructured":"Zheng, G. A study of some problems of energy finite element methods. Shanghai Jiao Tong Univ. (2013)."},{"issue":"3","key":"76751_CR23","first-page":"119","volume":"37","author":"D Wang","year":"2018","unstructured":"Wang, D., Zhu, X., Li, T., Heng, X. & Gao, S. Vibration analysis of a fgm beam based on energy finite element method. Zhendong Yu Chongji\/Journal Vib. Shock. 37 (3), 119\u2013124 (2018).","journal-title":"Zhendong Yu Chongji\/Journal Vib. Shock"}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-024-76751-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-024-76751-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-024-76751-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T10:06:45Z","timestamp":1730455605000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-024-76751-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,29]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["76751"],"URL":"https:\/\/doi.org\/10.1038\/s41598-024-76751-w","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,29]]},"assertion":[{"value":"22 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 October 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"25908"}}