A new formulation of geometrically exact planar Euler-Bernoulli beam in multi-body dynamics is proposed. For many applications, the use of the Euler-Bernoulli model is sufficient and has the ...advantage of being a nodal displacement-only formulation avoiding the integration of rotational degrees of freedom. In this paper, an energy momentum method is proposed for the nonlinear in-plane dynamics of flexible multi-body systems, including the effects of revolute joints with or without torsional springs. Large rotational angles of the joints are accurately calculated. Several numerical examples demonstrate the accuracy and the capabilities of the new formulation.
This paper discusses the multiplicative decomposition of the deformation gradient into its volumetric and isochoric parts and its implications in the case of anisotropy. An analysis is carried out ...showing that the volumetric-isochoric split of the stored energy function can be justified and systematically derived on the basis of the physical assumption that the spherical part of the stress depends on the determinant of the deformation gradient without ad hoc introduction of the multiplicative split. The analysis shows that care must be exercised in the case of anisotropic material description in order not to violate certain physical requirements. Additive splits of the energy can be justified on the basis of certain physical observations and independent of the multiplicative decomposition of the deformation gradient. Specifically, it is shown that a spherical state of stress will cause even in the incompressible case, a change of shape. In fibre reinforced materials, the split of the stored energy function into a part related to the matrix and a part related to the fibre is considered, showing that the volumetric-isochoric split should be applied to the matrix part only.
There has been increasing experimental evidence of non-affine elastic deformation mechanisms in biological soft tissues. These observations call for novel constitutive models which are able to ...describe the dominant underlying micro-structural kinematic aspects, in particular relative motion characteristics of different phases. This paper proposes a flexible and modular framework based on a micromorphic continuum encompassing matrix and fibre phases. In addition to the displacement field, it features so-called director fields which can independently deform and intrinsically carry orientational information. Accordingly, the fibrous constituents can be naturally associated with the micromorphic directors and their non-affine motion within the bulk material can be efficiently captured. Furthermore, constitutive relations can be formulated based on kinematic quantities specifically linked to the material response of the matrix, the fibres and their mutual interactions. Associated stress quantities are naturally derived from a micromorphic variational principle featuring dedicated governing equations for displacement and director fields. This aspect of the framework is crucial for the truly non-affine elastic deformation description. In contrast to conventional micromorphic approaches, any non-local higher-order material behaviour is excluded, thus significantly reducing the number of material parameters to a range typically found in related classical approaches. In the context of biological soft tissue modelling, the potential and applicability of the formulation is studied for a number of academic examples featuring anisotropic fibre-reinforced composite material composition to elucidate the micromorphic material response as compared with the one obtained using a classical continuum mechanics approach.
•Finite strain micromorphic anisotropic approach separately considering matrix and fibre continua.•Preferred material directions linked to non-affine deforming micromorphic directors.•Elastic fibre–matrix bond with independent non-affine fibre and matrix motions.•Local micromorphic approach with a limited number of physically motivated parameters.•Relative non-affine matrix–fibre motion linked to dedicated governing equations.
In this research, a reduced order method (ROM) called the Proper Orthogonal Decomposition with Interpolation (PODI) is used to drastically reduce computation time of highly complex and non-linear ...problems as encountered in simulating the heart. The idea behind this method is to first construct a database of pre-computed full-scale solutions using the Element Free Galerkin method (EFG) and then project a selected subset of these solutions to a low dimensional space. Using the Moving Least Square method an interpolation is carried out for the problem at hand, before the resulting coefficients are projected back to the original high dimensional solution space. Computations are carried out on a bi-ventricle model to investigate the performance and accuracy varying the material parameters and to determine the sensitivity with respect to the parametric values. The PODI calculations are completed within 1.25 s on a normal desktop machine with the relative ℓ2 error norm not exceeding 2.5×10−3. Hence, it is demonstrated that real-time modelling of the heart can be successfully carried out at acceptable error levels.
This article describes a novel equilibrium‐based geometrically exact beam finite element formulation. First, the spatial position and rotation fields are interpolated by nonlinear ...configuration‐dependent functions that enforce constant strains along the element axis, completely eliminating locking phenomena. Then, the resulting kinematic fields are used to interpolate the spatial sections force and moment fields in order to fulfill equilibrium exactly in the deformed configuration. The internal variables are explicitly solved at the element level and closed‐form expressions for the internal force vector and tangent stiffness matrix are obtained, allowing for explicit computation, without numerical integration. The objectivity and absence of locking are verified and some important numerical and theoretical aspects leading to a computationally efficient strategy are highlighted and discussed. The proposed formulation is successfully tested in several numerical application examples.
The conventional representation of isotropic hyperelastic strain energy densities as functions of scalar invariants of finite deformation tensors does not naturally extend to the field of anisotropic ...mechanics. Formulating an invariant-free representation of the strain energy function, fourth-order Orthotropic Lamé tensors define the constitutive law whilst naturally collapsing to the transversely isotropic and fully isotropic case where necessary simply as a by-product of known material symmetries. In this study, a simple linear isoparametric hexahedral finite element capable of describing anisotropic invariant-free hyperelasticity is presented. Careful conversion of the fourth-order tensor operations present in the strain energy function to computational arrays then applying to the principle of virtual work generates a weak formulation for finite element analyses. The finite element is then applicable to materials of any degree of anisotropy or compressibility and is particularly useful for predicting highly nonlinear responses such as the stiffening of fibrous biological tissues. A discussion of simple shear experimentation and modelling follows, as well as remarks on modelling nearly-incompressible materials.
•A new time integration scheme that converse the mechanical energy has been proposed for co-rotating planar beams.•In absence of external loads, the linear and angular momenta remain constant with ...the proposed scheme.•Formal proofs of conservation properties are given.•Stability and accuracy are achieved in long term dynamics.•Four numerical examples highlight the merits of the new integration scheme.
This article presents an energy-momentum integration scheme for the nonlinear dynamic analysis of planar Euler-Bernoulli beams. The co-rotational approach is adopted to describe the kinematics of the beam and Hermitian functions are used to interpolate the local transverse displacements. In this paper, the same kinematic description is used to derive both the elastic and the inertia terms. The classical midpoint rule is used to integrate the dynamic equations. The central idea, to ensure energy and momenta conservation, is to apply the classical midpoint rule to both the kinematic and the strain quantities. This idea, developed by one of the authors in previous work, is applied here in the context of the co-rotational formulation to the first time. By doing so, we circumvent the nonlinear geometric equations relating the displacement to the strain which is the origin of many numerical difficulties. It is rigorously shown that the proposed method conserves the total energy of the system and, in absence of external loads, the linear and angular momenta remain constant. The accuracy and stability of the proposed algorithm, especially in long term dynamics with a very large number of time steps, is assessed through four numerical examples.
In this paper, an energy-momentum method for geometrically exact Timoshenko-type beam is proposed. The classical time integration schemes in dynamics are known to exhibit instability in the ...non-linear regime. The so-called Timoshenko-type beam with the use of rotational degree of freedom leads to simpler strain relations and simpler expressions of the inertial terms as compared to the well known Bernoulli-type model. The treatment of the Bernoulli-model has been recently addressed by the authors. In this present work, we extend our approach of using the strain rates to define the strain fields to in-plane geometrically exact Timoshenko-type beams. The large rotational degrees of freedom are exactly computed. The well-known enhanced strain method is used to avoid locking phenomena. Conservation of energy, momentum and angular momentum is proved formally and numerically. The excellent performance of the formulation will be demonstrated through a range of examples.
•Geometrically exact In-plane Timoshenko-like beam dynamics.•Large deformation and large rotation.•Energy-momentum method.•Stable time integration scheme conserving energy, linear and angular momentum.•Formal and numerical proof of conservation properties.
The
Cosserat continuum falls into the group of so-called generalized continua which have the capacity to consider internal lengths and so describe a certain type of size effects. The paper addresses ...some aspects of the non-linear formulation of the
Cosserat continuum and its meshfree approximation. The use of moving least square approximations P. Lancaster, K. Salkauskas, Mathematics of Computations 37(155) (1981) 141–158 in a non-linear
Cosserat continuum-based formulation gives rise to certain implications which are related to the essential boundary condition enforcement as well as to the updating of the rotation field. Here, the enforcement of the displacement boundary conditions is accomplished by modifying the initial variational principle or weak form of equations such that the essential boundary conditions appear as Euler–Lagrange equations. The updating of the rotational degrees of freedom in the meshfree code adopts a multiplicative scheme which is based on the spinor theory and has been already successfully applied to finite elements by the first author. The suitability of the proposed methodology is exemplified in modelling size-scale effects in elasticity. The impact of the
Cosserat continuum-based formulation on small-scale structures is demonstrated and the comparison with a classical
Green strain tensor-based approach reveals significant differences.
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