Effect of dentin tubules on the mechanical properties of dentin. Part I: Stress-strain relations and strength criterion

As known, there is a large number of dentin tubules in dentin. These tubules have varying radii and are shaped into radially parallel pattern. The anisotropy of microstructure of dentin shows that dentin should be treated as a material of varying transverse isotropy. In this Part, the elastic stress-strain relations and the quadratic strength criterion are established in the form of having varying transverse isotropy, in the framework of micromechanics to take into account of the effect of the microstructures-dentin tubules. Simplified forms for isotropic and homogeneous cases, as well as the corresponding plane stress form of the stress-strain relations are also given. These theoretical models are very well supported by the experiments shown later in the continued paper (Part II). 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)     

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.     

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.     

Theoretical statistical solution and numerical simulation of heterogeneous brittle materials

The analytical stress-strain relation with heterogeneous parameters is derived for theheterogeneous brittle materials under a uniaxial extensional load, in which the distributions of theelastic modulus and the failure strength are assumed to be statistically independent. This theoreticalsolution gives an approximate estimate of the equivalent stress-strain relations for 3-D heterogeneousmaterials. In one-dimensional cases it may provide comparatively accurate results. The theoreticalsolution can help us to explain how the heterogeneity influences the mechanical behaviors, Further, anumerical approach is developed to model the non-linear behavior of three-dimensional heterogeneousbrittle materials. The lattice approach and statistical techniques are applied to simulate the initialheterogeneity of heterogeneous materials. The load increment in each loading stage is adaptivelydetermined so that the better approximation of the failure process can be realized. When the maximumtensile principal strain exceeds the failure strain, the elements are considered to be broken, which canbe carried out by replacing its Young‘s modulus with a very small value. A 3-D heterogeneous brittlematerial specimen is simulated during a full failure process. The numerical results are in good agreementwith the analytical solutions and experimental data.     

基于BP神经网络与小冲杆试验确定在役管道钢弹塑性性能方法研究

为了能够在不停输油气工况下获得在役管道材料的弹塑性力学性能, 提出了一种人工智能BP (back-propagation)神经网络、小冲杆试验与有限元模拟相结合,通过确定材料真应力-应变曲线从而获得材料弹塑性力学性能的方法. 首先,通过系统改变Hollomon公式中的参数$K$, $n$值,获得457组具有不同弹塑性力学性能的假想材料本构关系, 其次,将得到的本构关系代入经试验验证的含有Gurson-Tvergaard-Needleman(GTN)损伤参数的小冲杆试验二维轴对称有限元模型,通过有限元计算得到了与真应力-应变曲线一一对应的457条不同假想材料的载荷-位移曲线,最终将两组数据作为数据库输入BP神经网络进行训练,建立了同种材料小冲杆试验载荷-位移曲线与真应力-应变曲线之间的关联关系.通过此关联关系,可利用试验得到的小冲杆载荷-位移曲线获取在役管道钢的真应力-应变曲线,从而确定其弹塑性力学性能.通过对比BP神经网络得到的X80管道钢真应力-应变曲线与单轴拉伸试验的结果以及引用现有文献中不同材料的试验数据对此关系进行验证,证明了该方法的准确性与广泛适用性.      

压杆/试样表面接触变形对SHPB实验应变测量的影响

采用SHPB（split Hopkinson pressure bar）实验技术测量了3种不同尺寸纯铁试样的动态压缩应力应变关系,根据实验结果提出一个经验模型,定量分析了SHPB实验中压杆/试样表面接触变形对应变测量的影响。分析表明,在轴向应力平衡条件下,表面的接触变形对弹性段的应变测量影响显著;对塑性段应变测量的影响与试样的强化模量和长度相关,当试样强化模量较大而长度较小时,这种影响将不可忽略,可根据影响量的经验分析模型对应变进行修正。     

A variational definition of the strain energy for an elastic string

Using the variational point of view, the constitutive equations of an elastic one-dimensional string are deduced from the stress-strain relations of nonlinear three-dimensional elasticity, by passing to the limit when the other dimensions go to zero. The assumptions made on the three-dimensional model are not very restrictive.     

准静态压缩应力-应变曲线测量方法的探索

本文介绍了用应变片直接测量材料的准静态应力-应变曲线的试验研究。在MTS810材料试验机上分别对93W、G50、砂浆等几种材料进行了准静态压缩试验。由于仪器的系统误差不能由MTSCod规准确的得到材料的真实的弹性变形。为此在试件的中部贴应变片得到材料的弹性变形,塑性变形仍旧由MTSCod规记录,从而得到试样的真实的应变,准确获得准静态压缩应力-应变曲线。试验结果表明:在试件中部贴应变片的方法能够准确得到该材料的杨氏模量;在试件两端垫块的刀口上安置Cod规可得到材料的应力应变曲线。两者的合理组合即可得到准确而完整的准静态压缩应力-应变曲线。试验中还发现准静态实验中试件的断面加工不平,偏心压缩等都会影响E的准确测量。     

Non-linear stress-strain equations for incompressible transversely isotropic elastic materials

Non-linear stress-strain equations for incompressible, transversely isotropic elastic materials are developed. In order to obtain these equations, the expressions for a strain energy function is found. The derivation of the strain energy function follows a geometrical approach and a method suggested by Mooney. These stress-strain relations are expressed in terms of three principal stretches to the sixth order.     

不同应变率下聚乙烯材料的压缩力学性能

为了研究聚乙烯材料在不同应变率下的压缩力学性能，通过准静态实验和动态实验获得聚乙烯材料不同应变率下的应力应变曲线，分析发现:聚乙烯的弹性模量和屈服强度随应变率增大而增大，具有明显的黏弹塑性；聚乙烯材料进入塑性阶段，其应力应变曲线在不同应变率下具有相近的变化趋势，即塑性切向模量近似相同。根据聚乙烯材料的压缩力学性能，建立了弹性区、屈服点和塑性区的分段本构模型。该模型的屈服点和塑性段与实验结果吻合较好，由于弹性段采用线弹性模型，与实验结果存在一定偏差，可近似描述材料的弹性行为。     

Photoelastic investigation of large deflections by low-modulus photoelastic materials

Information is given to enable the experimental analysis of large deformations by application of photoelastic techniques. Promising photoelastic materials having low elastic modulus and high optical sensitivity were obtained for practical use at room temperature. The time and temperature dependence of the stress-fringe order and stress-strain relations of one of these viscoelastic materials is discussed. Photoelastic investigations of large deflected straight and circular beams are reported. Photoelastic technique seems to be a useful method for analysis of large deformations of models with complicated shapes.     

Three-dimensional Maxwell stress-strain relations in viscoelasticity

Conclusion In this paper three-dimensional Maxwell stress-strain relations were deduced phenomenologically.In the first place we applied the Hamilton's principle to the viscoelastic deformation, and obtained the variational equation with respect to the elastic potential and the dissipation function.Then we assumed that the elastic potential is a function only of the stress, and the dissipation function is a function of stress and rate of stress. By the above variational equation of the virtual stress satisfying the equilibrium equation and the boundary conditions, we obtained the relations to be satisfied by the elastic potential and the dissipation function, and the conditions to be satisfied by the dissipation function.From these relations we obtained the required three-dimensional Maxwell stress-strain relations in viscoelasticity. These relations indicate that the strain is the sum of the internal elastic strain and the internal viscous strain.If a given substance is isotropic with respect to stress, the stress-strain relations are expressed by a linear Maxwell model consisting of Hookian spring in series with a Newtonian dashpot.It is the main result of this paper that the three-dimensional Maxwell stress-strain relations in viscoelasticity are deduced from physically appropriate assumptions.     

Effect of creep prestrain on subsequent plastic deformation

The effect of creep prestrain on subsequent plastic deformation is experimentally investigated. The experiments are performed by subjecting thin-walled tubular specimens of stainless steel SUS 304 after creep prestraining to combined axial load and torsion at room temperature to 600°C. The stress-strain relations subsequent to creep prestrain are determined under combined stress state with and without temperature changes in prestraining and subsequent plastic straining. On the experimental results, the plastic hardening effects by creep prestrain are discussed under various temperature conditions. The subsequent stress-strain relations are compared with the calculated results on the equi-plastic strain surfaces.     

Elasto-plastic constitutive relations of unidirectional fiber composites for cyclic loadings

The clasto-plastic constitutive behaviors of continuous fiber reinforced composites under cyclic loadings are studied by the micromechanics method in which the equal-strain model is used in the fiber direction, the equal-stress model in the other directions. It is supposed that fiber is linearly elastic and matrix is clastic-viscoplastic. The constitutive equations of the matrix are described by Bodner-Partom's unified constitutive theory. Boron/Aluminum composite, as an example, is investigated in detail for an understanding of the stress-strain relations and initial yield behaviors of metal matrix composites. Present results are compared with the experimental data.The project was supported by the Chinese National Natural Science Foundation.     

Plastic stress-strain history at notch roots in tensile strips under monotonic loading

Various authors have proposed analytic relations for predicting the plastic behavior at the root of a notch. Among these are relations by Neuber and by Hardrath and Ohman, the latter having generalized on earlier work by Stowell. These methods have had limited experimental confirmation and it was the objective of the investigation reported herein to assess the predictive value of each of the two theories by comparison with test data on externally notched tensile specimens, monotonically loaded to large strains. Since maximum notch-root strain and net-section stress are the only parameters which can be directly measured in a test, theoretical predictions of these same parameters were developed. This was done using a piece-wise analytical representation of the stress-strain curve. Computer programs were developed for analyzing the stress-strain data and for computing the theoretical results. A comparison of theory with tests on flat, notched specimens of AISI 4340 steel, heat treated, with initial elastic concentration factors of 1.5 to 2.0 showed a systematic discrepancy which is attributed in part to notch strengthening due to triaxial stress. The discrepancy is of the order of 5 percent for the Neuber theory and larger for the Hardrath-Ohman theory for notch strains less than of 0.015 in./in. and becomes progressively larger for both theories at notch strains in excess of 0.015 in./in. For the mild notches studied here, the Neuber theory has better predictive value.     

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.     

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.     

金属材料的强度与应力-应变关系的球压入测试方法

压入法获取材料单轴应力-应变关系和抗拉强度对服役结构完整性评价有重要的基础意义.假定材料均匀连续、各向同性、应力应变关系符合Hollomon律,基于能量等效假定,即代表性体积单元(representativevolume element, RVE)的vonMises等效和有效变形域内能量中值等效假定,本文提出了关联材料载荷、深度、球压头直径和Hollomon律的四参数半解析球压入(semi-analyticalspherical indentation,SSI)模型.通过球压入载荷-深度试验关系获得材料的应力-应变关系和抗拉强度.考虑压入过程中的损伤效应,针对金属材料提出了用于球压入测试的材料弹性模量修正模型.对11种延性金属材料完成了球压入试验,采用本文提出的球压入试验方法测到的弹性模量、应力-应变关系和抗拉强度与单轴拉伸试验结果吻合良好.