Investigation of shear damage considering the evolution of anisotropy

The damage that occurs in shear deformations in view of anisotropy evolution is investigated. It is widely believed in the mechanics research community that damage (or porosity) does not evolve (increase) in shear deformations since the hydrostatic stress in shear is zero. This paper proves that the above statement can be false in large deformations of simple shear. The simulation using the proposed anisotropic ductile fracture model (macro-scale) in this study indicates that hydrostatic stress becomes nonzero and (thus) porosity evolves (increases or decreases) in the simple shear deformation of anisotropic (orthotropic) materials. The simple shear simulation using a crystal plasticity based damage model (meso-scale) shows the same physics as manifested in the above macro-scale model that porosity evolves due to the grain-to-grain interaction, i.e., due to the evolution of anisotropy. Through a series of simple shear simulations, this study investigates the effect of the evolution of anisotropy, i.e., the rotation of the orthotropic axes onto the damage (porosity) evolution. The effect of the evolutions of void orientation and void shape onto the damage (porosity) evolution is investigated as well. It is found out that the interaction among porosity, the matrix anisotropy and void orientation/shape plays a crucial role in the ductile damage of porous materials.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

The dynamic response of an edge crack in a functionally graded orthotropic strip

The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs).     

Thermodynamically based fictitious crack/interface model for general normal and shear loading

For small deformations a crack/interface model that considers general 3D normal and shear loading is proposed. It involves elasticity, plasticity and damage and it is thermodynamically based. An essential feature of the model is its consistency with the concepts behind the fictitious crack model. In particular, no crack deformation occurs before the crack is initiated and when a crack has just been initiated the proposed model provides an unloading stiffness that is infinitely large. For the same set of parameters, it is demonstrated that the proposed model is able to provide predictions that are in close agreement with experimental data for concrete for a wide range of loading situations.     

A rheological equation for anisotropic–anelastic media and simulation of field seismograms

In many cases, geological formations are composed of layers of dissimilar properties whose thicknesses are small compared to the wavelength of the seismic signal, as for instance, a sandstone formation that has intra-reservoir thin mudstone layers. A proper model is represented by an anisotropic (transversely isotropic) and viscoelastic stress–strain relation. In this work, we consider a sandstone reservoir, such as the Utsira formation, saturated with CO₂ and use White’s mesoscopic model to describe the energy loss of the seismic waves. The mudstone layers are assumed to be isotropic, poroelastic and lossless. Then, Backus averaging provides the complex and frequency-dependent stiffnesses of the transversely isotropic (TI) long-wavelength equivalent medium. We obtain the associated wave velocities and quality factors as a function of frequency and propagation direction, while the synthetic seismograms are computed with a finite-element (FE) method in the space-frequency domain. In this way, the frequency-dependent properties of the medium are modeled exactly, without the need of approximations with viscoelastic mechanical models. Numerical simulations of synthetic seismograms show results in agreement with the predictions of the theories and significant differences due to attenuation and anisotropic effects compared to the ideal isotropic and lossless rheology.     

Incremental Magnetoelastic Deformations,with Application to Surface Instability

In this paper the equations governing the deformations of infinitesimal (incremental) disturbances superimposed on finite static deformation fields involving magnetic and elastic interactions are presented. The coupling between the equations of mechanical equilibrium and Maxwell’s equations complicates the incremental formulation and particular attention is therefore paid to the derivation of the incremental equations, of the tensors of magnetoelastic moduli and of the incremental boundary conditions at a magnetoelastic/vacuum interface. The problem of surface stability for a solid half-space under plane strain with a magnetic field normal to its surface is used to illustrate the general results. The analysis involved leads to the simultaneous resolution of a bicubic and vanishing of a 7×7 determinant. In order to provide specific demonstration of the effect of the magnetic field, the material model is specialized to that of a “magnetoelastic Mooney–Rivlin solid”. Depending on the magnitudes of the magnetic field and the magnetoelastic coupling parameters, this shows that the half-space may become either more stable or less stable than in the absence of a magnetic field.      

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

两种正交异性材料界面上裂纹冲击后的动态响应

本文研究了两种不同正交异性材料界面半无限长裂纹,在冲击荷载下的动态弹塑性响应。通过积分变换,Ｗｉｅｎｅｒ－Ｈｏｐｆ方法和Ｃａｇｎｉａｒｄ－ｄｅＨｏｏｐ反演围通技术,求得一般解析解,获得了该裂纹的动应力强度因子;通过采用Ｄｕｇｄａｌｅ模型,建立了裂纹尖端塑性区延伸速度与裂纹扩展速度的关系,以及动态ＣＯＤ与裂纹扩展速度的关系。     

On the Active Response of Soft Living Tissues

Soft tissues exhibit a nonlinear, essentially incompressible (visco-) elastic response; a key issue is the active nature of muscle fibres, in other words their ability to contract and relax in response to biochemical signals. Here we present a continuum model able to describe an active elastic medium.      

A comparative study of kinematic hardening rules at finite deformations

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong-Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong-Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).     

Splitting a Viscoelastic Orthotropic Body with a Rigid Wedge of Thickness Increasing with Time

A crack model with a constant-length prefracture zone and a critical opening criterion are used to solve the wedging problem for a linear elastic body split by a rigid wedge with thickness increasing with time. Governing equations describing the behavior of the crack during the incubation and transient stages are presented. An algorithm is proposed to solve numerically the governing equation of transient crack growth     

Finite viscoelasticity and viscoplasticity of semicrystalline polymers

Observations are reported on low-density polyethylene in uniaxial tensile and compressive tests with various strain rates and in tensile and compressive relaxation tests with various strains. A constitutive model is developed for the time-dependent response of a semicrystalline polymer at arbitrary three-dimensional deformations with finite strains. A polymer is treated as an equivalent network of chains bridged by junctions (entanglements between chains in the amorphous phase and physical cross-links at the lamellar surfaces). Its viscoelastic behavior is associated with separation of active strands from temporary junctions and merging of dangling strands with the inhomogeneous network. The viscoplastic response is attributed to sliding of junctions between chains with respect to their reference positions. Constitutive equations are derived by using the laws of thermodynamics. The stress–strain relations involve 6 material constants that are found by matching the observations.      

Macroscopic response of particle-reinforced elastomers subjected to prescribed torques or rotations on the particles

Particle-reinforced rubbers are composite materials consisting of randomly distributed, stiff fibers/particles in a soft elastomeric material. Since the particles are stiff compared to the embedding rubber, their deformation can be ignored for all practical purposes. However, due to the softness of the rubber, they can undergo rigid body translations and rotations. Constitutive models accounting for the effect of such particle motions on the macroscopic response under prescribed deformations on the boundary have been developed recently. But, in some applications (e.g., magneto-active elastomers), the particles may experience additional torques as a consequence of an externally applied (magnetic) field, which, in turn, can affect the overall rotation of the particles in the rubber, and therefore also the macroscopic response of the composite. This paper is concerned with the development of constitutive models for particle-reinforced elastomers, which are designed to account for externally applied torques on the internally distributed particles, in addition to the externally applied deformation on the boundary of the composite. For this purpose, we propose a new variational framework involving suitably prescribed eigenstresses on the particles. For simplicity, the framework is applied to an elastomer reinforced by aligned, rigid, cylindrical fibers of elliptical cross section, which can undergo finite rotations in the context of a finite-deformation, plane strain problem for the composite. In particular, expressions are derived for the average in-plane rotation of the fibers as a function of the torques that are applied on them, both under vanishing and prescribed strain on the boundary. The results of this work will make possible the development of improved constitutive models for magneto-active elastomers, and other types of smart composite materials that are susceptible to externally applied torques.