Topic
Tangent stiffness matrix
About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.
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TL;DR: In this article, a mixed formulation is adopted for the derivation of the local element tangent stiffness matrix and nodal forces based on a two-field Hellinger-Reissner variational principle.
Abstract: This paper presents a corotational formulation of a three-dimensional elasto-plastic mixed beam element that can undergo large displacements and rotations. The corotational approach applies to a two-noded element a coordinate system which continuously translates and rotates with the element. In this way, the rigid body motion is separated out from the deformational motion. In this paper, a mixed formulation is adopted for the derivation of the local element tangent stiffness matrix and nodal forces based on a two-field Hellinger-Reissner variational principle. The local beam kinematics is based on a low-order nonlinear strain expression using Timoshenko assumption. The warping effects are characterized by adopting Benscoter theory that describes the warping degree of freedom by an independent function. Shape functions that satisfy the nonlinear local equilibrium equations are selected for the interpolation of the stress resultants. This local element, together with the corotational framework, can be used to analyze the nonlinear buckling and postbuckling of thin-walled beams with generic cross-section. The mixed formulation solution is compared against the results obtained from a corotational displacement-based formulation having the same beam kinematics. The superiority of the mixed formulation is clearly demonstrated.
30 citations
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TL;DR: In this paper, the use of a penalty method to enforce the constraint of incompressibility in nonlinear elasticity is described and a theoretical analysis of the associated mixed method and a new equivalence theorem are seen to lead to a way to retain positive definiteness.
Abstract: This paper describes the use of a penalty method to enforce the constraint of incompressibility in nonlinear elasticity. As an example, a problem involving the use of the Newton–Raphson method in conjunction with incremental loading and a successive mesh refinement scheme is presented. It is shown that during the incremental loading phase and the Newton–Raphson refinement on a fixed mesh, all tangent stiffness matrices are positive definite for the chosen energy density and load increment. But when the mesh is refined and the solution is interpolated as a starting value on the new mesh, the tangent stiffness matrix is indefinite. A theoretical analysis of the associated mixed method and a new equivalence theorem are seen to lead to a way to retain positive definiteness. The key is the use of an equivalent tangent stiffness matrix which is the reduced Hessian matrix. The numerical example shows that both positive definiteness and the quadratic convergence rate of the Newton–Raphson method are obtained.
30 citations
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TL;DR: A mixed finite element for coupled thermo-hydro-mechanical (THM) analysis in unsaturated porous media is proposed in this article, where the weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu-Washizu three-filed variational principle.
Abstract: SUMMARY A mixed finite element for coupled thermo-hydro-mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three-filed variational principle. The proposed mixed finite element formulation is derived. The non-linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto-plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non-linearity, the co-rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright 2005 John Wiley & Sons, Ltd.
29 citations
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TL;DR: In this paper, the authors apply a thermo-mechanics based granular micromechanics constitutive relationship of cementitious materials to predict such failure phenomena and investigate the macro- and the micro-scale mechanisms that govern the predicted behavior.
29 citations
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29 citations