Laminated multilayered composite and sandwich plates and shells are widely used for military and civilian aircraft, aerospace vehicles, and naval and civil structures because of their high specific stiffness and strength and tailoring capabilities. Moreover, composite materials are increasingly adopted for primary load-bearing structures in the form of thick laminates with a large number of layers. Those structures are subjected to complicated load paths and may exhibit critical deformation effects due to transverse shearing and transverse-normal stretching, with the consequential susceptibility to environmental and inter-laminar damage effects. One of the key aspects of the analysis of composite structures is the accurate prediction of the stress field; it can be obtained employing several numerical methods that can provide very detailed results at an high computational cost. This approach is well suited for small parts but cannot be applied to complex structures or for damage-tolerance analysis. Thus, accurate and computationally efficient models are required for the design and analysis of multilayered composites. Moreover, multilayered composites are affected by intelaminar damage that is one of the major reasons of failure. Since the delamination's presence has to be taken into account, damage tolerance analysis have to be carried out in order to understand the structure's capabilities to continue bearing loads without incurring in critical failure. The crack-growth simulation is a challenging task for the finite element method because it involves domain's separation and stress singularities at the crack tip. It is therefore necessary to introduce methods that account for crack-tip singularities and that allow to model both the initiation and the crack-growth process. The present work is divided into two main parts: in the fist one, going from chapter 2 to chapter 5, Refined Zigzag Theory, RZT, is introduced and employed formulate and develop plate and shell finite elements, in the second one, from chapter 6 to 7, the Discontinuous Galerkin method applied to structural problems is presented, a beam and a shell DG elements are formulated and the DG method is used besides a cohesive zone model to the study of the delamination's propagation process in laminate composite materials. In the following chapter Refined Zigzag theory is introduced starting from the Mindlin displacement model and passing from the Di Sciuva displacement field; then, the piece-wise linear zigzag functions that characterize RZT are obtained according to appropriate kinematic assumptions. In chapter 3 the equilibrium equations and the virtual work statement are derived for flat and curved shell elements for linear static analysis. The RZT displacement model for the plate element is obtained from the Reissner-Mindlin displacement field, while the displacement model employed for the curved RZT shell is built up on the Naghdi's displacement field. The tangent matrix computation is then modified to take into account for geometric non-linearities and the Element Independent Corotational Framework, that is used for geometric non-linear analysis in presence of small strains, is presented. The virtual work statement is then used in chapter 4 where three finite elements are developed in the RZT framework. The first one is a triangular plate finite element, the second one is a three-node flat shell finite element with drilling degree of freedom and the last one is a bilinear four-node curved shallow shell element.The two three-node elements employ the so called anisoparametric or interdependent interpolation to remove the shear locking effect and particular attention has been payed to the interaction of this strategy with the RZT displacement field. Moreover, the three-node flat shell element formulation has been extended to the geometric non-linear analysis. The four-node shallow shell element is used instead to understand the issues arising from the extension of RZT to curved manifolds. The two shell elements are built up paying particular attention to the ABAQUS user element specifications. The RZT-based accuracy has been assessed by way of numerical experiments and the results are presented in chapter 5. When available, analytical solutions have been employed as reference results, for complex problems the reference solution has been, instead, obtained from ABAQUS shell or solid elements. In the following two chapters the Discontinuous Galerkin method applied to elliptic problems is introduced and is employed to formulate two finite elements. In chapter 6 the DG formulation of the variational principle is presented starting from an intuitive point of view and then the rigorous mathematical of the method is derived from the equilibrium equations. The DG method is then specialized for two types of numerical flux and its formulation is extended, using the Cohesive Zone Model(CZM), to non-linear problems that involve interface separation and crack propagation. In chapter 7 two DG finite elements, a Timoshenko beam and a six-node curved shell, are developed and their accuracy is assessed by means of numerical experiments. Moreover, the shell element is based on an hybrid DG-CZM formulation and its performances have been validated employing two single mode tests: the double cantilever beam test(DCB) and the end notched flexure test(ENF).

Refined theories and Discontinuous Galerkin methods for the analysis of multilayered composite structures / Versino, Daniele. - (2012).

### Refined theories and Discontinuous Galerkin methods for the analysis of multilayered composite structures

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*VERSINO, DANIELE*

##### 2012

#### Abstract

Laminated multilayered composite and sandwich plates and shells are widely used for military and civilian aircraft, aerospace vehicles, and naval and civil structures because of their high specific stiffness and strength and tailoring capabilities. Moreover, composite materials are increasingly adopted for primary load-bearing structures in the form of thick laminates with a large number of layers. Those structures are subjected to complicated load paths and may exhibit critical deformation effects due to transverse shearing and transverse-normal stretching, with the consequential susceptibility to environmental and inter-laminar damage effects. One of the key aspects of the analysis of composite structures is the accurate prediction of the stress field; it can be obtained employing several numerical methods that can provide very detailed results at an high computational cost. This approach is well suited for small parts but cannot be applied to complex structures or for damage-tolerance analysis. Thus, accurate and computationally efficient models are required for the design and analysis of multilayered composites. Moreover, multilayered composites are affected by intelaminar damage that is one of the major reasons of failure. Since the delamination's presence has to be taken into account, damage tolerance analysis have to be carried out in order to understand the structure's capabilities to continue bearing loads without incurring in critical failure. The crack-growth simulation is a challenging task for the finite element method because it involves domain's separation and stress singularities at the crack tip. It is therefore necessary to introduce methods that account for crack-tip singularities and that allow to model both the initiation and the crack-growth process. The present work is divided into two main parts: in the fist one, going from chapter 2 to chapter 5, Refined Zigzag Theory, RZT, is introduced and employed formulate and develop plate and shell finite elements, in the second one, from chapter 6 to 7, the Discontinuous Galerkin method applied to structural problems is presented, a beam and a shell DG elements are formulated and the DG method is used besides a cohesive zone model to the study of the delamination's propagation process in laminate composite materials. In the following chapter Refined Zigzag theory is introduced starting from the Mindlin displacement model and passing from the Di Sciuva displacement field; then, the piece-wise linear zigzag functions that characterize RZT are obtained according to appropriate kinematic assumptions. In chapter 3 the equilibrium equations and the virtual work statement are derived for flat and curved shell elements for linear static analysis. The RZT displacement model for the plate element is obtained from the Reissner-Mindlin displacement field, while the displacement model employed for the curved RZT shell is built up on the Naghdi's displacement field. The tangent matrix computation is then modified to take into account for geometric non-linearities and the Element Independent Corotational Framework, that is used for geometric non-linear analysis in presence of small strains, is presented. The virtual work statement is then used in chapter 4 where three finite elements are developed in the RZT framework. The first one is a triangular plate finite element, the second one is a three-node flat shell finite element with drilling degree of freedom and the last one is a bilinear four-node curved shallow shell element.The two three-node elements employ the so called anisoparametric or interdependent interpolation to remove the shear locking effect and particular attention has been payed to the interaction of this strategy with the RZT displacement field. Moreover, the three-node flat shell element formulation has been extended to the geometric non-linear analysis. The four-node shallow shell element is used instead to understand the issues arising from the extension of RZT to curved manifolds. The two shell elements are built up paying particular attention to the ABAQUS user element specifications. The RZT-based accuracy has been assessed by way of numerical experiments and the results are presented in chapter 5. When available, analytical solutions have been employed as reference results, for complex problems the reference solution has been, instead, obtained from ABAQUS shell or solid elements. In the following two chapters the Discontinuous Galerkin method applied to elliptic problems is introduced and is employed to formulate two finite elements. In chapter 6 the DG formulation of the variational principle is presented starting from an intuitive point of view and then the rigorous mathematical of the method is derived from the equilibrium equations. The DG method is then specialized for two types of numerical flux and its formulation is extended, using the Cohesive Zone Model(CZM), to non-linear problems that involve interface separation and crack propagation. In chapter 7 two DG finite elements, a Timoshenko beam and a six-node curved shell, are developed and their accuracy is assessed by means of numerical experiments. Moreover, the shell element is based on an hybrid DG-CZM formulation and its performances have been validated employing two single mode tests: the double cantilever beam test(DCB) and the end notched flexure test(ENF).##### Pubblicazioni consigliate

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`https://hdl.handle.net/11583/2497085`

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