Effect of dentin tubules to the mechanical properties of dentin. Part II: Experimental study

To verify the theoretical models of varying transversely isotropic stress-strain relations of dentin established in the preceding work (Part I), we perform a set of experiments. Because of the very fine tooth size, it usually seems to be difficult to directly measure the inhomogeneous and anisotropic parameters of dentin. In this paper, by the digital speckle correlation method, tensile experiments are made on the small dentin samples either parallel or perpendicular to the dentin tubules. With the theoretically predicted elastic stress-strain relations, an optimization method is proposed to fit the strain curve adapted to the experimental data. The results show that the theoretical elastic stress-strain relations coincides very well with the experimental observations. The determined Young's modulus and Poisson's ratio of dentin matrix are 29.5 GPa and 0.44, respectively, in the optimization sense. The project supported by the National Natural Science Foundation of China (19525207)     

Effect of dentin tubules on the mechanical properties of dentin Part III: Numerical analysis

Imitating a real tooth and the periodontal supporting tissues, we have established a 2D finite element model and carried out a numerical analysis based on the inhomogeneous and anisotropic (IA) stress-strain relation and strength model of dentin proposed in the preceding Parts I and II, and the conventional homogeneous and isotropic (HI) model, respectively. Quite a few cases of loadings for a non-defected and a defected tooth are considered. The numerical results show that the stress level predicted by the IA model is remarkably higher than that by the HI model, revealing that the effect of the dentin tubules should be taken into a serious consideration from the viewpoint of biomechanics. The project supported by the National Natural Science Foundation of China (19525207, 19891180) and the Tsinghua University Fundamental Research Foundation (Jc1999033)     

Importance of aging to dehydration shrinkage of human dentin

There is an increase in the mineral content of human dentin with aging.Due to the consequent changes in the mineral to the collagen ratio,this process may influence the degree of hydrogen bonding that occurs with the loss of water and the extent of shrinkage as a result of dehydration.Thus,the objective of this investigation is to quantify the differences in the dehydration shrinkage of human dentin with patient age.Specimens of coronal dentin are prepared from the molars of young(23 age 34) and old(52 age 62) patients,and then maintained in storage solutions of water or hanks balanced salt solutions(HBSS).Dimensional changes of the dentin specimens occurring over periods of free convection are evaluated by using the microscopic digital image correlation(DIC).The results distinguish that the shrinkage of the young dentin is significantly larger than that of the old dentin,regardless of the orientation and period of the storage.The strains parallel to the tubules increase with proximity to the dentin enamel junction(DEJ),whereas the shrinkage strains in the transverse direction are the largest in the deep dentin(i.e.,near the pulp).The degree of anisotropy in the shrinkage increases from the pulp to the DEJ,and is the largest in the young dentin.     

Region dependent fracture resistance behavior of human dentin based on numerical simulation简

Dentin has a hierarchical structure and is composed of numerous tubules whose diameters and densities vary with the distances to the dentin-enamel junction. The unique structure determines the mechanical performance of dentin. In this study, a multiscale model, which is based on the combination of the virtual multidimensional internal bond (VMIB) theory and the Monte Carlo method, is used to simulate the fracture behavior of human dentin. Numerical simulations reveal that human dentin exhibits a graded resistance curve (R-curve). Among the three regions of dentin, superficial dentin shows the strongest resistance to crack propagation, and deep dentin has the weakest resistance. In addition, the predictions of fracture toughness of middle dentin agree well with the experimentally reported values, suggesting that the proposed model can be used to characterize the fracture behavior of human dentin comprehensively and properly.     

Determination of elastic and plastic mechanical properties of dentin based on experimental and numerical studies

The aim of this study is to investigate the change of mechanical properties of human dentin due to aging and spatial variation. Sections of coronal dentin are made from human molars in three groups: young, mid-aged, and old patients. A nanoindentation test is conducted from regions near the pulp to the dentin-enamel junction (DEJ) to evaluate the load-depth indentation response and determine Young's modulus and hardness. Based on the loading and unloading load-displacement curves in nanoindentation, a numerical model of plastic damage is used to study the plastic and the damage behaviors and the contribution to the degradation in the unloading stiffness. The experimental results show that Young's modulus of the inner dentin is significantly lower than that of outer dentin in each age group. Compared with the young dentin, the old dentin has greater hardness and Young's modulus with similar spatial variations. The magnitudes of the yield strength and the damage variable are also affected by aging and vary with spatial locations. In the same age group, the yield strength in inner dentin is lower than those in middle and outer dentin, more damage occurs with similar spatial variations, and the yield strength of young dentin is generally lower and causes more damage compared with those in both the mid-aged and old groups.     

Fatigue and fracture of bovine dentin

In this paper, the fatigue and fracture properties of bovine dentin are evaluated usingin vitro experimental analyses. Double cantilever beam (DCB) specimens were prepared from bovine maxillary molars and subjected to zeroto-tension cyclic loads. The fatigue crack growth rate was evaluated as a function of the dentin tubule orientation using the Paris law. Wedge-loaded DCB specimens were also prepared and subjected to monotonic opening loads. Moiré interferometry was used to acquire the in-plane displacement field during stable crack growth, and the instantaneous wedge load and crack length were acquired to evaluate the crack growth resistance and crack tip opening displacement (CTOD) with crack extension. The rate of fatigue crack growth was generally larger for crack propagation occurring perpendicular to the dentin tubules. The Moiré fringe fields documented during monotonic crack growth exhibited non-linear deformation occurring within a confined region adjacent to the crack tip. Both the wedge load and CTOD response provided evidence that a fracture process zone contributes to energy dissipation during crack extension and that dentin exhibits a risingR-curve behavior. Results from this preliminary investigation are being used as a guide for an evaluation of the fatigue and fracture properties of human dentin.     

EXACT SOLUTIONS FOR FREE VIBRATIONS OF TRANSVERSELY ISOTROPIC CIRCULAR PLATES

By virtue of the general solution of dynamic elasticity equations for transverse isotropy as well as the variable separation method, three-dimensional exact solutions of circular plates are obtained under two types of boundary conditions. The solutions can consider both axisymmetric and non-axisymmetric cases. Solutions based on the classical plate theory and Mindlin plate theory are also presented under the corresponding boundary conditions. Numerical results are finally presented and comparisons between the three theories are made. The project is supported by the National Natural Science Foundation of China (No. 19872060).     

On the Bending, Vibration and Stability of Laminated Rectangular Plates with Transversely Isotropic Layers

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The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part II

Summary From the continuum mechanics points of view, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity in a continued work (Part III, forthcoming).The supports from the Alexander von Humboldt Foundation, Germany and the Research Foundation of Tsinghua University, P. R. China are acknowledged by the first author.     

Localized acoustic waves at twist boundaries in transversely isotropic media

Existence criteria and basic characteristics are analytically found for elastic waves localized at a twist boundary in transverse isotropic media. The boundary is formed by two identical semi-infinite bodies with non-collinear principal axes parallel to the interface. The analysis is based on the Stroh formalism specified to the case of transverse isotropy. The dispersion equation is presented in a general form and explicitly solved for small misorientations. The waves in the sector situated between the directions of transverse isotropy in the sub-media of the bicrystal are explicitly described.     

Exact relations for the anti-plane effective magneto-electro-elastic coefficients of two-phase fibrous composites

Two-phase fiber-reinforced magneto-electro-elastic composites are considered. The constituents exhibit transverse isotropy and the composite is assumed to have global monoclinic symmetry. The Milgrom–Shtrikman compatibility conditions are applied to obtain explicitly exact relations for the eighteen anti-plane effective coefficients. Such relations are written in terms of nine equalities of fourth-order determinants. These fourth-order determinants exhibit the regularity of a third-order minor formed by the response matrix of the matrix material and are completed by a row and/or column of the response matrices of the fibers material and the composite, respectively. Other two less explored alternative theories, namely, a second type of the Milgrom–Shtrikman conditions, which involve only effective coefficients, and Milgrom's version of the original Milgrom–Shtrikman conditions, are followed in order to derive twenty and forty exact relations, respectively. Particular and limit cases are recovered from the obtained relations.     

ANISOTROPIC STRENGTH CRITERION BASED ON T CRITERION AND THE TRANSFORMATION STRESS METHOD 1)

岩土材料通常呈现出成层水平分布特点, 即可将其视为横观各向同性材料, 横观各向同性对于岩土材料的变形以及强度值都会产生显著的影响. 基于已提出的t强度准则, t强度准则是基于各向同性单元体中存在有效滑移面来构建的, 并根据该空间有效滑移面上主剪应力与主法向应力的比值达到一定阈值为破坏条件. 在空间中存在有效滑移面与物理沉积面, 基于上述两个面在空间的位置关系, 用两面夹角作为表征横观各向同性对剪切强度影响程度的参量, 并假定当该夹角值越大, 则各向异性对强度贡献程度越大, 对应更大的应力比强度值, 反之, 则对应更小的应力比强度值. 基于上述思路并类比将其推广为正交三维各向异性准则, 基于三维各向异性材料的三维沉积面, 提出了三维特征沉积面的概念, 并基于空间滑移面与三维特征沉积面之间的夹角作为度量各向异性程度的变量, 提出了基于两面角作为参量考虑原生各向异性的应力比强度公式, 并利用该应力比强度公式来修正已提出的t强度准则, 最终建立了考虑各向异性影响的t准则公式. 在上述准则基础上, 考虑将各向异性应力空间转换为各向同性应力空间的思路, 在各向异性t准则基础上, 推导得到了基于各向异性强度t准则的变换应力公式, 利用变换应力公式可以将传统的以p, q为变量的各向同性本构模型转变为可考虑各向异性的三维本构模型. 通过对岩土材料的强度以及真三轴条件下的应力应变关系试验数据预测, 验证了所提的各向异性t准则及其变换应力公式的有效性及适用性.     

基于t准则的各向异性强度准则及变换应力法

岩土材料通常呈现出成层水平分布特点, 即可将其视为横观各向同性材料, 横观各向同性对于岩土材料的变形以及强度值都会产生显著的影响. 基于已提出的t强度准则, t强度准则是基于各向同性单元体中存在有效滑移面来构建的, 并根据该空间有效滑移面上主剪应力与主法向应力的比值达到一定阈值为破坏条件. 在空间中存在有效滑移面与物理沉积面, 基于上述两个面在空间的位置关系, 用两面夹角作为表征横观各向同性对剪切强度影响程度的参量, 并假定当该夹角值越大, 则各向异性对强度贡献程度越大, 对应更大的应力比强度值, 反之, 则对应更小的应力比强度值. 基于上述思路并类比将其推广为正交三维各向异性准则, 基于三维各向异性材料的三维沉积面, 提出了三维特征沉积面的概念, 并基于空间滑移面与三维特征沉积面之间的夹角作为度量各向异性程度的变量, 提出了基于两面角作为参量考虑原生各向异性的应力比强度公式, 并利用该应力比强度公式来修正已提出的t强度准则, 最终建立了考虑各向异性影响的t准则公式. 在上述准则基础上, 考虑将各向异性应力空间转换为各向同性应力空间的思路, 在各向异性t准则基础上, 推导得到了基于各向异性强度t准则的变换应力公式, 利用变换应力公式可以将传统的以p, q为变量的各向同性本构模型转变为可考虑各向异性的三维本构模型. 通过对岩土材料的强度以及真三轴条件下的应力应变关系试验数据预测, 验证了所提的各向异性t准则及其变换应力公式的有效性及适用性.      

Three-dimensional modeling of elastic guided waves excited by arbitrary sources in viscoelastic multilayered plates

This paper provides a modal solution for the three-dimensional modeling of Lamb and SH waves excited by sources of arbitrary shape. This solution is applicable to elastic and viscoelastic plates, in the far-field as well as in the near-field regions, under the assumption of transverse isotropy about the thickness direction. The theoretical developments are conducted based on a semi-analytical finite element formulation. This formulation yields a one-dimensional modal problem, fast from a computational point of view, and allows to readily handle heterogeneous materials having depth-varying properties (multilayered, piecewise or continuously varying, functionally graded). The modal solution is shown to be expressed in terms of Hankel functions of multiple order thanks to a proper application of inverse transforms and Cauchy residue calculus. The link between the proposed formulation and a fully analytical approach is discussed. The solution of this paper is then successfully compared to literature results and degenerates to the point source case. Formulas are presented to calculate point source excitabilities from lines sources. These formulas remain valid for non-propagating modes, viscoelastic materials and account for the near-field contribution. Finally, the example of a viscoelastic bilayer waveguide excited by a rectangular source is considered in order to check the theoretical results.     

Application of transversely isotropic materials to multi-layer shell elements undergoing finite rotations and large strains

The contribution deals with an extension of a classical Neo–Hookean model for compressible isotropic materials to transverse isotropy. With this enhancement for one preferred material direction there is a possibility to simulate large strains in volume changes of the isotropic basic continuum and supplementary in fiber direction. The integrity basis of polynomial invariants in case of transversely isotropic hyperelasticity consists of three principal invariants of the isotropic basic continuum and additionally of two principal invariants for the preferred material direction. The proposed stored energy function for transverse isotropy contains the classical theory near to the natural state and fulfills the restriction on polyconvexity and coerciveness.By numerical enforcement of the material model into shell kinematics without rotational variables a four-node isoparametric finite element is developed using special concepts to avoid locking. The capability of the algorithms proposed is demonstrated by a numerical example involving large strains as well as finite rotations.     

General stress-strain relations in non-linear theory of viscoelasticity for large deformations

Conclusion General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented.In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations.On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form.If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot.Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations.The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.Work supported in part by National Science Foundation Grant INT-9106519.     

Bending of functionally graded piezoelectric rectangular plates

By introducing two displacement functions as well as two stress functions, two independent state equations with variable coefficients are derived from the three-dimensional theory equations of piezoelasticity for transverse isotropy. A laminated approximation is used to transform the state equations to those with constant coefficients in each sub-layer. The bending problem of a functionally graded rectangular plate is then analyzed based on the state equations. Numerical results are presented and the effect of material gradient index is discussed. Supported by the National Natural Sciences Foundation of China (No. 10002016).     

Stress State of a Transversely Isotropic Ferromagnetic with an Elliptic Crack in a Homogeneous Magnetic Field

The magnetoelastic stress-strain problem for a transversely isotropic ferromagnetic body with an elliptical crack in the isotropy plane is solved explicitly. The body is in an external magnetic field perpendicular to the isotropy plane. The magnetic field induces elastic strains and an internal magnetic field in the body. The main characteristics of stress-strain state and induced magnetic field are determined and their features in the neighborhood of the crack are analyzed. Formulas for the stress intensity factors of the mechanical and magnetic fields near the crack tip are presented__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 48–59, January 2005.     

THE SHEAR LAG EFFECT OF BOX BEAMS WITH VARYING LATERAL LOADING LOCATIONS 1)

对箱梁各翼板(顶板、悬臂板、底板)分设不同剪力滞广义纵向位移,其横向分布均取二次抛物线形式,并引入载荷横向位置参数η,以分析载荷横向变位对剪力滞效应的影响.运用能量变分原理,建立剪力滞控制微分方程,求解了简支梁和悬臂梁在均布载荷作用下的控制微分方程的解.算例分析表明:载荷横向变位改变直接承受载荷的翼板的正负剪力滞特性,对非直接承载翼板只改变其应力幅度;箱梁横向框架效应对直接承载翼板纵向应力的贡献远远大于剪切变形.与块体有限元分析结果较吻合,表明该算法能较准确分析载荷横向变位作用下箱梁剪力滞的变化规律.