Green's function for incremental nonlinear elasticity: shear bands and boundary integral formulation

An elastic, incompressible, infinite body is considered subject to plane and homogeneous deformation. At a certain value of the loading, when the material is still in the elliptic range, an incremental concentrated line load is considered acting at an arbitrary location in the body and extending orthogonally to the plane of deformation. This plane strain problem is solved, so that a Green's function for incremental, nonlinear elastic deformation is obtained. This is used in two different ways: to quantify the decay rate of self-equilibrated loads in a homogeneously stretched elastic solid; and to give a boundary element formulation for incremental deformations superimposed upon a given homogeneous strain. The former result provides a perturbative approach to shear bands, which are shown to develop in the elliptic range, induced by self-equilibrated perturbations. The latter result lays the foundations for a rigorous approach to boundary element techniques in finite strain elasticity.     

Seepage consolidation of elastic half-space under an axisymmetric load

The process of seepage consolidation of elastic saturated half-space under the action of a normal load on its surface is investigated assuming both incompressibility of fluid and skeleton grains and independence of the total skeleton stresses on time. Analytic representations for the fluid pressure and the half-space surface settlement are found when the half-space is loaded by a concentrated force. The maximum settlement is also found for a uniform loading of the surface over the circle area.     

Modeling multi-axial deformation of planar anisotropic elasto-plastic materials,part I: Theory

In sheet metal forming processes local material points can experience multi-axial and multi-path loadings. Under such loading conditions, conventional phenomenological material formulations are not capable to predict the deformation behavior within satisfying accuracy. While micro-mechanical models have significantly improved the understanding of the deformation processes under such conditions, these models require large sets of material data to describe the micromechanical evolution and quite enormous computation expenses for industrial applications. To reduce the drawbacks of phenomenological material models under the multi-path loadings a new anisotropic elasto-plastic material formulation is suggested. The model enables the anisotropic yield surface to grow (isotropic hardening), translate (kinematic hardening) and rotate (rotation of the anisotropy axes) with respect to the deformation, while the shape of the yield surface remains essentially unchanged.Essentially, the model is formulated on the basis of an Armstrong–Frederick type kinematic hardening, the plastic spin theory for the reorientation of the symmetry axes of the anisotropic yield function, and additional terms coupling these expressions. The capability of the model is illustrated with multi-path loading simulations in ‘tension-shear’ and ‘reverse-shear’ to assess its performance with ‘cross’ hardening and ‘Bauschinger’ effects.     

Shear waves in an elastic-plastic material due to a supersonic surface step load

A step shear load moves steadily on the surface of an elastic-plastic half space at a speed exceeding the elastic shear wave speed of the material. The orientation of the shear traction is such that the deformation is two-dimensional antiplane strain. Two different representations of the rate independent elastic-plastic material response are considered. The first material model is based on the associated flow rule and the Mises yield condition with isotropic hardening, whereas the second model is based on a particular flow theory of plasticity which represents incremental behavior at a corner of the instantaneous yield surface. Both models predict the same response under the same proportional loading. The stress history experienced by a typical material particle during passage of the load step is determined, and the variation of final strain with the magnitude of the load step is calculated. One conclusion resulting from comparison of results for the two material models for this problem is that the influence of yield surface vertex formation is not significant.     

Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law

Starting from Eshelby’s solution of the equivalent inclusion problem, an approximate solution is proposed in order to model interface debonding of a spherical inhomogeneity isolated in a uniform matrix. Both phases are linear elastic but the interface traction-separation law is non-linear. A semi-analytical incremental model is developed which is suitable for any type of loading. For computational efficiency, the model relies on two simplifying assumptions: (i) the eigenstrain is uniform inside the inhomogeneity and (ii) the interface compliance is averaged over inhomogeneity’s surface when computing the average strain within the inhomogeneity. An extensive parametric study is conducted for three loading modes and 144 combinations of non-dimensional parameters. The predictions are assessed against full-field finite element solutions based on two error measures of the mean stress field inside the inhomogeneity. The results show that the mean error value is acceptable in all cases and indicate the parameter ranges for which the model is most accurate.     

Modeling elasto-plastic indentation on layered materials using the equivalent inclusion method

This paper develops a fast semi-analytical model for solving the three-dimensional elasto-plastic contact problems involving layered materials using the Equivalent Inclusion Method (EIM). The analytical elastic solutions of a half-space subjected to a unit surface pressure and a unit subsurface eigenstrain are employed in this model; the topmost layer is simulated by an equivalent inclusion with fictitious eigenstrain. Accumulative plastic deformation is determined by a procedure involving an iterative plasticity loop and an incremental loading process. Algorithms of the fast Fourier transform (FFT) and the Conjugate Gradient Method (CGM) are utilized to improve the computation efficiency. An analytical elastic solution of layered body contact (O’Sullivan and King, 1988) and an indentation experiment result involving a layered substrate (Michler et al., 1999) are used to examine the accuracy of this model. Comparisons between numerical results from the present model and a commercial FEM software (Abaqus) are also presented. Case studies of a rigid ball loaded against a layered elasto-plastic half-space are conducted to explore the effects of the modulus, yield strength, and thickness of the coating on the hardness, stiffness, and plastic deformation of the composite body.     

A plastic flow rule at a yield vertex

A yield vertex having three distinct facets at the uniaxial stress point is considered. An associated plastic flow rule is constructed using a new 3 × 3 coupled hardening matrix. This has the property that the incipient shear modulus is less than the elastic value. The approach is novel and distinct from previous work having that conclusion. In particular, all the relevant incipient moduli governing fully active in-plane loading can be fitted, if desired, to those values which J₂-deformation theory would require. To that extent the proposed incremental theory therefore legitimizes the use of the latter moduli, for example in certain ‘paradoxical’ buckling problems. When the moduli of the incremental theory are so chosen, the domain of stress-rate vectors enforcing loading is a calculable pyramid which contracts from the Mises half-space as the stress increases beyond yield. This domain becomes wider, at a given stress, as the initial smooth curvature of the stress-strain curve is imagined to become sharper.     

Deformation of porous Cosserat elastic bars

This paper is concerned with the linear theory of porous Cosserat elastic solids. We study the equilibrium of a cylindrical bar which is subjected to resultant forces and resultant moments on the ends, to body loads and to surface tractions on the lateral surface. The Almansi problem, where the body loads and the surface loading on the lateral surface are polynomials in the axial coordinate, is considered. The bar is made of an inhomogeneous and isotropic material whose constitutive coefficients are independent of the axial coordinate. The problem is reduced to the study of two-dimensional problems. The results are used to study two practical applications concerning the deformation of a circular rod. It is shown that a uniform pressure on the lateral surface produces an extension, a uniform change of the porosity, and a plane deformation. The bending by terminal couples produces a non-uniform variation of the porosity and a microrotation of the material particles.     

Irreversible deformation with subsequent unloading of a spherical viscoelastoplastic layer

Analytic solutions are obtained for a sequence of one-dimensional quasistatic problems describing viscoelastic deformation processes in the material of a hollow ball and the plastic flow nucleation and evolution processes occurring in the ball as the pressure on the outer boundary increases. The unloading process under slow removal of the loading pressure is considered as well. The stress fields and the elastic and plastic strain fields in the spherical layer material, the law of motion of the elastoplastic boundary, and the residual stress level and distribution are computed. It is assumed that at the stage preceding the plastic flow the material obeys the viscoelastic Voigt model and the loading surface is determined by the von Mises plastic flow condition.     

复杂加载下混凝土的弹塑性本构模型

混凝土材料在不同应力路径下或复杂加载条件下会表现出差异性显著的应力应变关系,在小幅循环加载条件下,其应力应变关系会表现出类似于弹性变形的滞回曲线.在不同应力水平下,混凝土的应力应变关系以及破坏特性都具有静水压力相关特点,即随着静水压力增大,各向异性强度特性弱化.此外,混凝土受压及受拉破坏机理不同,因而对应于混凝土硬化损伤亦有不同,即可分为受压硬化损伤,受拉硬化损伤及两者的混合硬化损伤类型.基于Hsieh模型,对该模型进行了三点改进.（1）针对小幅循环加载下混凝土无塑性变形的试验规律,而模型中在应力水平较低的循环加载条件下始终存在塑性变形的预测问题,采用在边界面模型框架下,设置了应力空间的弹性域,初始屈服面与后续临界状态屈服面几何相似的假定.（2）基于广义非线性强度准则将原模型采用变换应力方法将其推广为三维弹塑性本构模型,采用变换后模型可合理的考虑不同应力路径对于子午面以及偏平面上静水压力效应形成的影响,并避免了边界面应力点奇异问题.（3）分别对拉压两种加载损伤模式建议了相应的硬化参数表达式,可分别用于描述上述加载中产生的应变软化及强度退化行为.基于多种加载路径模拟表明:所建立的三维弹塑性本构模型可合理地用于描述混凝土的一般应力应变关系特性.      

分级加载条件下紫色泥岩三轴蠕变特性研究

流变性质是岩石材料的重要力学特性,以紫色泥岩为例,在分级加载条件下研究了其三轴蠕变特性。采用自行研制的重力杠杆式岩石蠕变实验机,并配备三轴压力室,对紫色泥岩实施分级加载三轴蠕变实验。实验结果表明,紫色泥岩在相同轴压条件下,围压越大,瞬时弹性变形和蠕变变形越小;在相同围压条件下,轴压越大,二者变形越大。紫色泥岩的蠕变速率具有相同的变化趋势,且在相同轴压条件下,围压越大,稳定蠕变阶段的蠕变速率越不明显。另外,应力与瞬时应变线性拟合结果表明,瞬时轴向应变与轴向荷载近似成比例增长。通过分析紫色泥岩蠕变曲线变化趋势,认为采用H/M模型进行模拟较为合理。并且采用MATLAB软件非线性回归分析进行模型参数拟合,最后得到实验曲线与理论曲线的最大误差在2e-5左右。证明采用的H/M模型能较好地描述紫色泥岩的蠕变特性。     

Finite deformation analysis of linearly viscoelastic simple structures2

In order to determine the effect of finite deformations on the stability and non-linear time-deflection behaviour of linearly viscoelastic uniaxially stressed structures, a series of simple rigid-bar-spring dashpot models were analysed ‘exactly’. The material representation was also kept as simple as possible using the standard three-element solid model.Results obtained indicate that the relaxation behaviour of such a structure depends only on its material properties. The creep response is influenced not only by the load level but most significantly by the instantaneous non-linear elastic characteristics of the structure. For structures exhibiting instantaneous elastic local instability a ‘critical time’ may be defined beyond which equilibrium is impossible. The definition for ‘safe-load-limit’ or viscoelastic critical force usually used in linear stability analyses of viscoelastic columns is generalized.     

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

It is proved that fractional exponential hereditary kernels of nonlinear viscoelasticity can be used to evaluate creep strains and stress relaxation. A nonlinear theory of viscoelasticity with time-independent nonlinearity described as a nonlinear curve of instantaneous elastoplastic deformation is used. The calculated results are validated against experimental data on the viscoelastic deformation of laminated and unidirectional fibrous composites and their components under the conditions of constant stresses, complete unloading, incremental loading, pure torsion, and constant strains     

The Thermoviscoelastoplastic Axisymmetric Stress–Strain State of Laminated Flexible Orthotropic Shells

A technique is proposed to determine the thermoviscoelastoplastic axisymmetric stress–strain state of laminated shells made of isotropic and orthotropic materials. The paper deals with processes of shell loading such that both instantaneous elastoplastic and creep strains occur in isotropic materials and elastic and creep strains in orthotropic materials. The technique is developed within the framework of the Kirchhoff–Love hypotheses for a stack of layers with the use of the equations of the geometrically nonlinear theory of shells in a quadratic approximation. The deformation of isotropic materials is described by the equations of the theory of deformation along slightly curved trajectories, while the deformation of orthotropic materials is described by Hooke's law with additional terms allowing for creep. A numerical example is given     

A nonlinear,transient finite element method for coupled solvent diffusion and large deformation of hydrogels

Hydrogels are capable of coupled mass transport and large deformation in response to external stimuli. In this paper, a nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on a nonlinear continuum theory that is consistent with classical theory of linear poroelasticity. A mixed finite element method is implemented with implicit time integration. The incompressible or nearly incompressible behavior at the initial stage imposes a constraint to the finite element discretization in order to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition for stability of the mixed method, similar to linear poroelasticity as well as incompressible elasticity and Stokes flow; failure to choose an appropriate discretization would result in locking and numerical oscillations in transient analysis. To demonstrate the numerical method, two problems of practical interests are considered: constrained swelling and flat-punch indentation of hydrogel layers. Constrained swelling may lead to instantaneous surface instability for a soft hydrogel in a good solvent, which can be regulated by assuming a stiff surface layer. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, in comparison with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. It is concluded that the present finite element method is robust and can be extended to study other transient phenomena in hydrogels.     

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研究了埋置于弹性地基内充液压力管道中非线性波的传播. 假设管壁是线弹 性的,地基反力采用Winkler线性地基模型,管中流体为不可压缩理想流体. 假定系统初始 处于内压为$P_0$的静力平衡状态,动态的位移场及内压和流速的变化是叠加在静 力平衡状态上的扰动. 基于质量守恒和动量定理,建立了管壁和流体耦合作用的非 线性运动方程组; 进而用约化摄动法, 在长波近似情况下得到了KdV方程,表征 着系统有孤立波解.     

On the mechanics of solids with a growing mass

A general constitutive theory of the stress-modulated growth of biomaterials is presented with a particular accent given to pseudo-elastic soft living tissues. The governing equations of the mechanics of solids with a growing mass are revisited within the framework of finite deformation continuum thermodynamics. The multiplicative decomposition of the deformation gradient into its elastic and growth parts is employed to study the growth of isotropic, transversely isotropic, and orthotropic biomaterials. An explicit representation of the growth part of the deformation gradient is given in each case, which leads to an effective incremental formulation in the analysis of the stress-modulated growth process. The rectangular components of the instantaneous elastic moduli tensor are derived corresponding to selected forms of the elastic strain energy function. Physically appealing structures of the stress-dependent evolution equations for the growth induced stretch ratios are proposed.     

Micromechanical approach to the behavior of poroelastic materials

A micromechanical analysis of the behavior of saturated elastic porous media is presented. Pores are connected and saturated with a fluid at uniform pressure. The non-linear macroscopic response of the porous medium is assumed to be related to the existence of a network of microcracks in the solid phase. The progressive closure of cracks during loading is viewed to control the material non-linearity.Using averaging schemes based on both the classical and modified secant methods, the non-linearity of the overall response is shown to be sometimes controlled by an effective stress. The modified secant method accounts in a certain way for the non-uniform distribution of strains within the solid phase. Different definitions of the effective stress are found, each of them being a combination of the overall stress with the fluid pressure. The hypothesis of connexion between pores and microcracks is discussed in details. The consequences of this hypothesis on the non-linear overall response are illustrated and comparisons are made with experimental data.     

基于SPH方法的湿润性固壁边界条件模拟研究

固体壁面由于表面特殊结构和材料属性,时常表现出对交界面上水体的吸附作用,而这一特征对微小水体作用尤为明显。本文提出了一种湿润性固壁边界条件的计算方法,即假设壁面粒子的亲水性以及毛细吸附作用统一表现为对支持域内流体粒子的吸附力。基于光滑粒子流体动力学（SPH）方法,模拟了静态液滴在不同湿润性壁面上的变形至稳定过程。模拟了液滴撞击疏水壁面的过程,将液滴的运动过程分为碰撞、铺展、回缩和回弹四个阶段,分析各阶段壁面受力分布情况。研究表明：根据模拟液滴静态接触角的变化特点,本文湿润性固壁边界条件可以较好的反映出壁面湿润性;液滴撞击输水表面的模拟数据与试验结果趋势上吻合良好;壁面压力波伴随着液滴的铺展和回缩传播并衰减;只有在回弹后期液滴即将脱离壁面时壁面拉力起主导作用,其余各时刻壁面均以压力为主。