基于概率配点法的岩土材料参数随机场及其响应分析

概率配点法是进行不确定性问题分析的一种有效方法。通过对输入参数场进行Karhunen-Loeve展开,将其表示为一系列独立随机变量在不同权重下的线性组合,再以与之相同的随机变量组合形成混沌多项式展开对输出随机场进行分解,通过某种算法选取随机变量的值,将其作为插值点的组合(配点),在这些配点上,概率方程演化为一个确定性问题方程。由此,可以直接利用现有软件或者确定性问题计算程序进行求解,生成混沌多项式的系数矩阵后,即可得到该随机问题的各阶统计矩,从而实现参数随机场的不确定性分析。本文将该方法引进岩土工程材料参数随机场,将体积模量视为输入随机场,位移视为输出场,分别进行了弹性及塑性变形计算。结果表明该方法极大地降低了随机问题的求解难度,与MC法(Mento Carlo)相比,减少了运算消耗,提高了计算效率,具有明显的优越性。     

A simple framework for full-network hyperelasticity and anisotropic damage

A formulation of a constitutive behaviour law is proposed for hyperelastic materials, such that damage induced anisotropy can be accounted for continuously. The full-network approach with directional damage is adopted as a starting point. The full-network law with elementary strain energy density based on the inverse Langevin is chosen as a reference law which is cast into the proposed framework. This continuum formalism is then rewritten using spherical harmonics to capture damage directionality. The proposed formalism allows for an efficient (and systematic) expansion of complex non-linear anisotropic constitutive laws. A low order truncated expression of the resulting behaviour is shown to reproduce accurately the stress-strain curves of the exact behaviour laws.     

Fracture toughness from the standpoint of softening hyperelasticity

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack/notch is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture where the crack sharpness is of minor importance.¹To explain the experimental observations qualitatively we simulate tension of a thin plate with a small crack of a finite and varying sharpness. In simulations, we introduce the average bond energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find, in agreement with experiments, that the magnitude of the fracture toughness cannot be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack, the lower is the toughness.Based on the obtained results, we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance, etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of continuum fails.     

各向同性超弹性本构模型数值计算及验证

现有多种形式的橡胶本构模型试图预测橡胶力学性质,其中部分模型已写入有限元软件中用于仿真计算,还存在较多拟合性较好的模型无法在有限元材料库中直接获得。本文详述了由不变量和主伸长率描写的各向同性超弹性本构模型的数值实现方法,并结合最新的本构模型开发了UHYPER和UMAT子程序。将UHYPER用于有限元实现对多孔橡胶板的拉伸仿真,对比仿真和试验结果,验证子程序的正确性以及评估本构模型预测复杂应变场的准确性;将UMAT用于单轴、等双轴和剪切拉伸的有限元仿真,对比仿真和本构模型理论结果,验证子程序的可靠性。结果表明,有限元仿真结果与理论结果拟合较好,子程序能够契合本构模型的力学描述,所述方法可以用于超弹性材料的数值计算。     

Antiplane elastic wave cloaking using metamaterials,homogenization and hyperelasticity

We consider the problem of how to cloak objects from antiplane elastic waves using two alternative techniques. The first is the use of a layered metamaterial in the spirit of the work of Torrent and Sanchez-Dehesa (2008) who considered acoustic cloaks, motivated by homogenization theories, whilst the second is the use of a hyperelastic cloak in the spirit of the work of Parnell et al. (2012). We extend the hyperelastic cloaking theory to the case of a Mooney–Rivlin material since this is often considered to be a more realistic constitutive model of rubber-like media than the neo-Hookean case studied by Parnell et al. (2012), certainly at the deformations required to produce a significant cloaking effect. Although not perfect, the Mooney–Rivlin material appears to be a reasonable hyperelastic cloak. This is clearly encouraging for applications. We quantify the effectiveness of the various cloaks considered by plotting the scattering cross section as a function of frequency, noting that this would be zero for a perfect cloak.     

Handedness-dependent hyperelasticity of biological soft fibers with multilayered helical structures

Collagen fibrils with multilayered helical structures widely exist in biological soft tissues, e.g., blood vessels, tendons, and ligaments. Understanding the mechanical properties of this kind of chiral materials is not only essential for evaluating the mechanical behaviors of the host tissues but also of significance for medical engineering, clinical diagnosis, and surgical operation. In this paper, a theoretical model is presented to investigate the hyperelasticity of biological soft fibers with multilayered helical structures. The effects of the initial helical angle, number and handedness of the fibers in each ply on the mechanical response of the material are examined. Our analysis reveals a switch of contact modes between two neighboring layers, which may greatly alter the overall non-linear response of the material. The Poisson׳s ratio of such a multilayered string can be greater than 0.5. The obtained results agree with relevant experiments of soft tissues. This work sheds light on the non-linear mechanics of chiral materials and may also guide the design of biomimetic materials.     

A stochastic model for the temporal aspects of flow intermittency in micropillar compression

Systematic experimental investigations have demonstrated that the plastic deformation of micropillar proceeds through a sequence of intermittent bursts, the sizes of which follow power-law statistics. In this study, a stochastic model based on the power-law distribution of burst size is formulated in the framework of crystal plasticity in order to investigate the temporal aspects of flow intermittency in micropillar compression. A Monte Carlo simulation scheme is developed to determine the burst size when a burst activity is captured. This burst size is considered as the displacement boundary condition of burst deformation. Three-dimensional finite element analysis of the model is performed and its predictions are validated by comparison with results from both micro-compression experiments and simulation tests of bulk crystals using the classic crystal plasticity finite element method (CPFEM). The model provides a reasonable prediction of stress–strain responses both at the macroscopic and microscopic scales. Finally, the capability of this model is shown with applications to the intermittent plastic deformation in micropillar compressions, in particular for their burst time durations and burst velocities. The results from such stochastic finite element analysis are shown to be consistent with earlier experimental findings and results of mean-field theory.     

On the structure of the group g1 appearing in hyperelasticity

In the representation theorem for the stored energy function of a hyperelastic material, it is necessary to determine a subgroup g^*₁ of the symmetry group g of the stress tensor. It is the purpose of this note to study the construction of g^*₁ and its relation to the other subgroups ot g, viz. the commutator subgroup g_c and the orthogonal subgroup g₀.     

A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.     

An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.     

Probabilistic norm of operators and resonance theorems

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Rupture of rubber

Summary The theory of the tearing of rubber has been further confirmed by experiments on three test pieces of widely different shapes, for each of which the tearing energy can be calculated. This rupture concept has been used to interpret measurements of out growth under conditions of both static and dynamic loading.

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Zusammenfassung Die Theorie des Reißens von Gummi wird durch Versuche an drei Proben von wesentlich verschiedener Gestalt, für die jeweils eine Reiß-Energie berechnet werden kann, weiter erhärtet. Diese Vorstellung über den Bruchvorgang wird dazu verwendet, Meßergebnisse bezüglich des Einschnittwachstums (Rißvergrößerung) sowohl bei statischer als auch bei dynamischer Versuchsführung zu interpretieren.

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Presented at a meeting on Flow, Fatigue, and Failure held in Leeds on January 8th and 9th, 1959, by the British Society of Rheology.     

Stability of stochastic Leipholz column with stochastic loading

Summary The Leipholz column which is having the Young modulus and mass per unit length as stochastic processes and also the distributed tangential follower load behaving stochastically is considered. The non self-adjoint differential equation and boundary conditions are considered to have random field coefficients. The standard perturbation method is employed. The non self-adjoint operators are used within the regularity domain. Full covariance structure of the free vibration eigenvalues and critical loads is derived in terms of second order properties of input random fields characterizing the system parameter fluctuations. The mean value of critical load is calculated using the averaged problem and the corresponding eigenvalue statistics are sought. Through the frequency equation a transformation is done to yield load parameter statistics. A numerical study incorporating commonly observed correlation models is reported which illustrates the full potentials of the derived expressions.

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Stabilität eines Leipholz-Stabes mit stochastischen Eigenschaften und Belastungen

Übersicht Behandelt wird der Leipholz-Stab, dessen Elastizitätsmodul, Masseverteilung und tangential folgende Streckenlast stochastisch sind. Die nicht selbstadjungierte Differentialgleichung und die Randbedingungen werden als solche mit Zufallskoeffizienten betrachtet und die übliche Störungsmethode benutzt. Im Regularitätsbereich werden die nicht selbstadjungierten Operatoren benutzt. Hergeleitet wird die vollständige Kovarianz-Struktur der Eigenwerte der freien Schwingung und kritischen Lasten als Funktionen der Eigenschaften zweiter Ordnung der zufälligen Eingangsgrößen, die die Schwankungen der Systemparameter charakterisieren. Der Mittelwert der kritischen Last wird aus dem gemittelten Problem berechnet und die zugehörige Eigenwert-Statistik wird gesucht. Über die Frequenzgleichung wird eine Transformation vorgenommen, um die Statistik des Lastparameters zu erhalten. Eine numerische Studie illustriert die Leistungsfähigkeit der hergeleiteten Ausdrücke.

     

Stiffness and damping coefficients of rubber

Summary An analytical-experimental study has been carried out for the estimation of the stiffness and damping coefficients of rubber. A given specimen of a rubber pad (Neoprene GN) in the shape of a solid circular cylinder is subjected to a vibrating force created by an electrodynamic shaker together with a vibration exciter.In the analytical investigation, a mathematical model is developed from which the stiffness and damping coefficients for rubber as well as the damped energy can be evaluated. From the stimulated mathematical model together with the experimental data, the stiffness and damping coefficients could be estimated. Subsequently, the solution for the heat conduction equation describing the temperature field in the rubber speciemen can be formulated. The analytical behaviour of the temperature is found to agree fairly well with the measured temperatures at various locations inside the rubber specimen. This renders the simulating model to be significant with potential for further extension.

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Steiügkeits- und Dämpfungskoeffizienten von Gummi

Übersicht In einer analytisch-experimentellen Arbeit wird das Verhalten einer zylindrischen Probe aus Gummi unter pulsierender Belastung untersucht. Mit Hilfe eines mathematischen Modells und experimenteller Daten können die Steifigkeits- und Dämpfungsparameter von Gummi bestimmt werden. Anschließend wird die Lösung der Wärmeleitungsgleichung für das Temperaturfeld in der Probe angegeben. Die analytische Lösung stimmt mit den Meßwerten gut überein.

     

Capturing anisotropic constitutive models with WYPiWYG hyperelasticity; and on consistency with the infinitesimal theory at all deformation levels

The elastic nonlinear behavior of fiber-reinforced materials and soft biological tissues is analyzed using anisotropic hyperelastic models. Frequently, these models are not compatible with the corresponding infinitesimal theory, but some of them may be modified to accommodate that theory in the limit. WYPiWYG hyperelasticity is compatible with the infinitesimal theory at all deformation levels and capable of capturing exactly a complete set of experimental data, which reproduces all deformation modes at every strain level, under homogeneous deformations. In this work we study the relevance of recovering the infinitesimal theory at every deformed configuration and also the performance of the WYPiWYG method in predicting the behavior of anisotropic materials at large strains under nonhomogeneous deformations.     

应变疲劳强度概率分布

利用概率论方法建立了应变疲劳强度概率分布与应变疲劳寿命概率分布之间的数学关系式，并根据应变－寿命转换关系，由已知的给定应变水平下的疲劳寿命概率分布求出给定寿命下的应变疲劳强度分布函数，便于进行可靠性分析。