Geometrijsko in materialno nelinearna dinamika nosilcev z lokalizacijo deformacij : doktorska disertacija

Kusuma Chandrashekhara, Sudhanva (author), Zupan, Dejan (mentor), Bokan-Bosiljkov, Violeta (member of the commission for defense), Trtnik, Gregor (member of the commission for defense), Brojan, Miha (member of the commission for defense), Jelenić, Gordan (member of the commission for defense)

Modelling instabilities in structures undergoing complex deformation is usually reflected in sensitive behaviour of numerical solution methods and thus represents a significant challenge in the numerical modeling. In this dissertation, we present a numerical framework for the analysis post-critical structural response of geometrically and materially nonlinear spatial frame-like structures. Special emphasis is laid on the analysis of the phenomenon of material softening which often occurs in brittle heterogeneous materials. The onset of stress dependent critical condition at a material point of a solid body results in non-uniqueness of strain measures when evaluated from the known stresses. The onset of the critical condition results in localisation of strains and accelerated local damaging of the material. The post-peak softening response results in decrease in load bearing capacity with increasing deformation. These phenomena are addressed within the velocity-based framework for the geometrically exact spatial beams where the tangent space of the nonlinear configuration space is spanned using only additive quantities which are velocities in fixed basis and angular velocities in local basis. The proposed framework builds upon the original velocity-based formulation upgraded with the consideration of material nonlinearities, addressing critical points and describing the phenomenon of strain softening and subsequent localisation of strains. First, we introduce the novel velocity-based path-following technique to address the singularity of the global system at the onset of critical conditions in quasi-static problems. The proposed velocity-based constraint aligns with the differential form of arc-length constraint with time serving as the arc-length parameter. We will additionally show that the rotational degrees of freedom can also be members of the control parameters within the constraint and with very little modification, the same constraint can be used for the analysis of bifurcation points. The problems associated with buckling and post-buckling behaviour is substantially reduced due to the constraint that naturally fits with the velocity-based formulation. Several benchmarks within the elastic cases will be demonstrated to show the robustness and efficiency of the proposed continuation technique. The formulation is further updated for the consideration of material nonlinearities by considering the nonlinear fiber-based constitutive relationship at the cross-sectional level that takes into account also plasticity and softening. The stress-resultant quantities and the corresponding constitutive tangent matrix are obtained using a two-dimensional Gaussian quadrature rule. In the analysis of material softening, the formulation is additionally updated with the detection of critical load and critical cross-section. This detection is achieved through the monitoring the determinant of the cross-sectional constitutive tangent matrix, with the singularity of the tangent matrix indicating the onset of material softening. The localisation of strains at the onset of softening is addressed using two approaches: (i) non-local approach, (ii) method of embedded discontinuities. The non-local approach here is based on the assumption that the strains are localised within a finite but short length rather than at a point. The strain localisation at the critical cross-sections is resolved using short, lower-order elements. The proposed technique shows an exceptional performance even in the neighbourhood of critical points and serves as a comparison for the latter approach. The method of embedded discontinuities embraces the philosophy of representing the localisation of strains with the enhancement of primary interpolated variables. Thus the interpolated velocities and angular velocities are introduced with additional jump-like variables at the same level and enable the description of point-wise discontinuities. The additional jump-like variables are supplemented by a modified consistency condition derived using the method of weighted residuals which is in complete accordance with the theoretical concept of strong discontinuity. The computational advantages and the efficiency of both approaches in handling post-critical responses is demonstrated through demanding set of numerical examples.

• Language: English
• Keywords: Built Environment, civil engineering, doctoral thesis, geometrically exact beam, three-dimensional rotations, material softening, strain localization, arc-length method, embedded discontinuity, critical points
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FGG - Faculty of Civil and Geodetic Engineering
• Place of publishing: Ljubljana
• Publisher: [S. Kusuma Chandrashekhara]
• Year: 2024
• Number of pages: VI, 117 str., [18] str. pril.
• PID: 20.500.12556/RUL-159721-8cedf84a-de85-c413-04ca-73b2807d3700
• ISSN: 202448387
• UDC: 539.3:624.046(043)
• COBISS.SI-ID: 202448387
• Publication date in RUL: 19.07.2024

Secondary language

• Language: Slovenian
• Title: Geometrically and materially nonlinear beam dynamics with strain localization : doctoral dissertation

Abstract

Modeliranje nestabilnosti v konstrukcijah, ki so podvržene zahtevnejšim deformacijam, se običajno odraža v občutljivem obnašanju numeričnih metod reševanja in tako predstavlja pomemben izziv v numeričnem modeliranju. V disertaciji predstavljamo numerično formulacijo za analizo postkritičnega obnašanja geometrijsko in materialno nelinearnih prostorskih okvirnih konstrukcij. Poseben poudarek posvečamo analizi mehčanja materiala, ki je pogosto prisotno v krhkih heterogenih materialih. Ob nastopu določenih kritičnih pogojev v materialni točki trdnega telesa določitev deformacij iz znanih napetosti ni več enolična. Pojav takšnega kritičnega stanja vodi v lokalizacijo deformacij in pospešeno lokalno poškodbo materiala. Za postkritični odziv je značilno zmanjšanje nosilnosti konstrukcije z izrazitimi lokalnimi deformacijami. Ti pojavi so modelirani z uporabo na hitrostih osnovane formulacije geometrijsko točnih prostorskih nosilcev. S tem je tangentni prostor nelinearnega konfiguracijskega prostora napet zgolj na aditivnih količinah, to so hitrosti v fiksni bazi in kotne hitrosti v lokalni bazi. Predlagani pristop razširi originalno formulacijo, ki temelji na hitrostih, z upoštevanjem materialnih nelinearnosti, obravnavo kritičnih točk in natančnim opisom pojava mehčanja deformacij ter posledične lokalizacije deformacij. Najprej predstavimo nov postopek sledenja obtežno-deformacijski poti za obravnavo singularnosti globalnega sistema ob pojavu kritičnih pogojev v kvazistatičnih problemih, ki je usklajen s formulacijo, osnovano na hitrostih. Predlagana vezna enačba je neposredno izražena s hitrostmi in se tako ujema s kriteriji ločne dolžine v diferencialni obliki, čas pa služi kot parameter dolžine obtežno-deformacijske poti. Dodatno pokažemo, da so z našim pristopom lahko rotacijske prostostne stopnje povsem enakovreden del vezne enačbe, z majhno modifikacijo pa lahko vez uporabimo tudi za analizo bifurkacijskih točk. Pogoste numerične težave ob prisotnosti nestabilnosti in v postkritičnem obnašanju so pomembno zmanjšane zaradi izbranega pristopa in vezne enačbe, ki je povsem usklajena z izbrano formulacijo. Številni numerični primeri, kjer je upoštevan elastičen materialni model, prikazujejo robustnost in učinkovitost predlagane metode.Formulacija omogoča upoštevanje materialnih nelinearnosti na nivoju konstitucijskih lastnosti vlaken prečnega prereza, pri čemer zajamemo tudi plastičnost in mehčanje. Rezultantne napetostni prečnega prereza določimo z uporabo dvodimenzionalne Gaussove integracije. Za analizo mehčanja materiala formulacijo dodatno opremimo z zaznavanjem kritične obtežbe in kritičnega prereza. To dosežemo s spremljanjem determinante konstitutivne tangentne matrike prečnega prereza, pri čemer singularnost te matrike kaže na začetek mehčanja materiala. Lokalizacija deformacij ob začetku mehčanja je obravnavana z dvema pristopoma: (i) nelokalnim pristopom in (ii) metodo vgrajenih nezveznosti. Nelokalni pristop temelji na predpostavki, da so deformacije namesto v točki lokalizirane znotraj končne, vendar kratke dolžine. Lokalizacijo deformacij tako modeliramo z uporabo kratkih elementov nižjih redov. Predlagana tehnika kaže izjemne računske lastnosti tudi v bližini singularnosti, uporabimo pa jo tudi tudi kot za primerjavo z metodo vgrajenih nezveznosti. Metoda metoda vgrajenih nezveznosti opiše lokalizacije deformacij z izboljšanimi interpolacijskimi nastavki. Tako vpeljemo dodatne diskretne vrednosti hitrosti in kotnih hitrosti, ki omogočajo opis točkovnih nezveznosti. Dodatne skakalne spremenljivke določimo z modificiranim konsistenčnim pogojem, izpeljanim z metodo uteženih ostankov v skladu s teoretičnim konceptom krepke nezveznosti. Računalniške prednosti in učinkovitost obeh pristopov pri obravnavanju postkritičnih odzivov so prikazane skozi številne numerične primere.

• Keywords: doktorske disertacije, gradbeništvo, grajeno okolje, geometrijsko točen nosilec, tridimenzionalne rotacije, mehčanje materiala, lokalizacija deformacij, metoda ločne dolžine, vgrajena nezveznost, kritične točke