1235 多結晶性強誘電体の二次元非線形有限要素解析法(J10-4 マイクロ・ナノ材料システムの力学と強度・機能評価(4) 形状記憶・電磁力学特性評価,ジョイントセッション,21世紀地球環境革命の機械工学:人・マイクロナノ・エネルギー・環境)

In this paper by the use of “convex analysis” the general nonlinear unilateral static problem of structural mechanics is formulated. By means of the “subdifferentiability” concept the unilateral nonlinear boundary conditions and the unilateral nonlinear constitutive laws are formulated as “subdifferential relations”, permitting the derivation of the variational inequalities of virtual and complementary virtual work. Thus the minimum potential and complementary energy theorems are proved. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with nonlinear unilateral constitutive laws and nonlinear unilateral boundary conditions illustrate the theory.

In the sintering of advanced ceramics, a digital twin consisted of a finite element model can predict the final shape and microstructure of a sintered ceramic. However, to minimise uncertainties, it is essential to continually update the mechanical properties in the digital twin using data collected from the manufacturing process. One promising approach for achieving this is through artificial neural networks (ANN). This study introduces a machine learning strategy to update the constitutive behaviour of advanced ceramics in finite element analysis of sintering deformation. A major challenge in implementing machine learning in material processing is the huge amount of data required by the training and validation of an ANN, which are often unavailable or incomplete in a real manufacturing process. This study demonstrates that the data requirement can be reduced by employing a two-step training technique. Firstly, the ANN is trained using a nonlinear constitutive law, which describes a general relationship between the strain rates and stresses. Subsequently, the weights and biases of the ANN are transferred for the retraining using limited experimental data for an actual ceramic. It is shown that such approach can successfully capture the nonlinear constitutive behaviour of fine-grained alumina without demanding for large amount of experimental data. A case study is provided, highlighting the feasibility of implementing the ANN in a commercial finite element package, to replace the constitutive law and predict the shrinkage and distortion of a sintering part. In particular, the sintered dumb-bell shape part, simulated using the retrained ANN, showed grain size and relative density that markedly different from those using the nonlinear constitutive law. It is important to note that the proposed methodology is generic and can be used to create ANNs to replace constitutive laws in finite element analysis in digital twins for a wide range of other engineering processes.

Variability response functions for beams with nonlinear constitutive laws

Artificial neural network (ANN) models are used to learn the nonlinear constitutive laws based on indirectly measurable data. The real input and output of the ANN model are derived from indirect data using a mechanical system, which is composed of several subsystems including the ANN model. As the ANN model is coupled with other subsystems, the input of the ANN model needs to be determined during the training. This approach integrates measurable data, mechanics, and ANN models so that the ANN models can be trained without direct data which is usually not available from experiments. Two examples are provided as an illustration of the proposed approach. The first example uses two-dimensional (2D) finite element (FE) analysis to train an ANN model to learn the nonlinear in-plane shear constitutive law. The second example applies a continuum damage model to train an ANN model to learn the damage accumulation law. The results show that the trained ANN models achieve great accuracy based on the proposed approach.

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function , we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

Helical buckling and postbuckling of pre-stressed cylindrical tubes under finite torsion

Improved nonlinear rotational constitutive law for predicting critical loads in elastomeric isolation bearings

The famous bifurcation analysis performed by Flügge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Flügge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Flügge's formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Flügge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.

The notion of material stability is examined in the context of active muscle tissue modeling, where the nonlinear constitutive law is dependent both on the physiologically driven muscle contraction and finite mechanical deformation. First, the governing equations and constitutive laws for a general active-elastic material with a single preferred direction are linearized about a homogeneous underlying configuration with respect to both the active contraction and deformation gradient. In order to obtain mathematical restrictions analogous to those found in elastic materials, stability conditions are derived based on the propagation of homogeneous plane waves with real wave speeds, and the generalized acoustic tensor is obtained. Focusing on 2D motions, and considering a simplified, decoupled transversely isotropic energy function, the restriction on the active acoustic tensor is recast in terms of a generally applicable constitutive law, with specific attention paid to the fiber contribution. The implication of the material stability conditions on material parameters, active contraction, and elastic stretch is investigated for prototype material models of muscle tissue.

An asymptotic solution of an initial value problem for a non-linear viscoelastic rod

Accurate modeling of many engineering systems requires the integration of multibody system and large deformation finite element algorithms that are based on general constitutive models, account for the coupling between the large rotation and deformation, and allow capturing coupled deformation modes that cannot be captured using beam formulations implemented in existing computational algorithms and computer codes. In this investigation, new three-dimensional nonlinear dynamic rubber chains and belt drives models are developed using the finite element absolute nodal coordinate formulation (ANCF) that allows for a straight forward implementation of general linear and nonlinear material models for structural elements such as beams, plates, and shells. Furthermore, this formulation, which is based on a more general kinematic description, can be used to predict the cross section deformation and its coupling with the extension and bending of the belt drives and rubber chains. The ANCF cross section deformation results are validated by comparison with the results obtained using solid finite elements in the case of a simple tension test problem. The effect of the use of different linear and nonlinear constitutive laws in modeling belt drive mechanisms is also examined in this investigation. The finite element formulation presented in this paper is implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing detailed models of mechanical systems subject to general loading conditions, nonlinear algebraic constraint equations, and arbitrary large displacements that characterize belt drives and tracked vehicle dynamics. The successful integration of large deformation finite element and multibody system algorithms is shown to be necessary in order to be able to study the dynamics of complex tracked vehicles with rubber chains. A computer simulation of a three-dimensional multibody tracked vehicle model that consists of twenty rigid bodies and two flexible rubber chains is used in order to demonstrate the use of the formulations presented in this investigation.

A non-linear fracture mechanics approach was used to predict the failure response of complex cement-bone constructs. A series of eight mechanical tests with a combination of tensile and shear loading along the cement-bone interface was performed. Each experiment was modeled using the finite element method with non-linear constitutive models at the cement-bone interface. Interface constitutive parameters were assigned based on the quantity of bone interdigitated with the cement. There was a strong correlation (r2 = 0.80) between experimentally measured and finite element predicted ultimate loads. The average error in predicted ultimate load was 23.9 percent. In comparison to the ultimate load predictions, correlations and errors for total energy to failure (r2 = 0.24, avg. error = 38.2 percent) and displacement at 50 percent of the ultimate load (r2 = 0.27, avg. error = 52.2 percent) were poor The results indicate that the non-linear constitutive laws could be useful in predicting the initiation and progression of interface failure of cemented bone-implant systems. However improvements in the estimation of post-yield interface properties from the quantity of bone interdigitated with cement are needed to enhance predictions of the overall failure response.

Fully coupled thermo-viscoplastic analysis of composite structures by means of multi-scale three-dimensional finite element computations

VAM applied to dimensional reduction of non-linear hyperelastic plates

Effect of variable periodontal ligament thickness and its non-linear material properties on the location of a tooth’s centre of resistance