Non-elliptic deformation field near the tip of a mixed-mode crack in a compressible hyperelastic material

This paper gives an asymptotic analysis of the deformation field near the tip of an arbitrary mixed-mode crack in a compressible hyperelastic harmonic material which loses ellipticity at sufficiently large deformations. It is found that the near-tip deformation field is characterized by a localized non-elliptic deformation band issuing from the crack-tip and bounded by two curves of discontinuous deformation gradient. Explicit expression for the near-tip deformation field is obtained both inside and outside the localized deformation band. In particular, a simple relation is derived that determines the orientation of the deformation band in terms of two complex governing parameters of the near-tip fields inside and outside the deformation band, respectively.     

Phase field modeling of fracture in anisotropic brittle solids

A phase field model of fracture that accounts for anisotropic material behavior at small and large deformations is outlined within this work. Most existing fracture phase field models assume crack evolution within isotropic solids, which is not a meaningful assumption for many natural as well as engineered materials that exhibit orientation-dependent behavior. The incorporation of anisotropy into fracture phase field models is for example necessary to properly describe the typical sawtooth crack patterns in strongly anisotropic materials. In the present contribution, anisotropy is incorporated in fracture phase field models in several ways: (i) Within a pure geometrical approach, the crack surface density function is adopted by a rigorous application of the theory of tensor invariants leading to the definition of structural tensors of second and fourth order. In this work we employ structural tensors to describe transverse isotropy, orthotropy and cubic anisotropy. Latter makes the incorporation of second gradients of the crack phase field necessary, which is treated within the finite element context by a nonconforming Morley triangle. Practically, such a geometric approach manifests itself in the definition of anisotropic effective fracture length scales. (ii) By use of structural tensors, energetic and stress-like failure criteria are modified to account for inherent anisotropies. These failure criteria influence the crack driving force, which enters the crack phase field evolution equation and allows to set up a modular structure. We demonstrate the performance of the proposed anisotropic fracture phase field model by means of representative numerical examples at small and large deformations.     

Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress

A new method is developed to determine the dominant asymptotic stress and deformation fields near the tip of a Mode-I traction free plane stress crack. The analysis is based on the fully nonlinear equilibrium theory of incompressible hyperelastic solids. We show that the dominant singularity of the near tip stress field is governed by the asymptotic solution of a linear second order ordinary differential equation. Our method is applicable to any hyperelastic material with a smooth work function that depends only on the trace of the Cauchy-Green tensor and is particularly useful for materials that exhibit severe strain hardening. We apply this method to study two types of soft materials: generalized neo-Hookean solids and a solid that hardens exponentially. For the generalized neo-Hookean solids, our method is able to resolve a difficulty in the previous work by Geubelle and Knauss (1994a). Our theoretical results are compared with finite element simulations.     

A transient solution method for the finite element incompressible Navier-Stokes equations

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a direct time integration method which has unique features in its formulation as well as in its evaluation of the contribution of external functions. Particular processes regarding the continuity conditions and the boundary conditions lead to a set of non-linear recurrence equations which represent evolution of the velocities and the pressures under the incompressibility constraint. An iteration process as to the non-linear convective terms is performed until the convergence is achieved in every integration step. Excessively artificial techniques are not introduced into the present solution procedure. Numerical examples with vortex shedding behind a rectangular cylinder are presented to illustrate the features of the proposed method. The calculated results are compared with experimental data and visualized flow fields in literature.     

Segregated finite element algorithms for the numerical solution of large-scale incompressible flow problems

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.     

Saint-Venant decay rates for a non-homogeneous isotropic mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity on the decay of Saint-Venant end effects in anti-plane shear deformations of linear isotropic mixtures of elastic solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear isotropic mixtures of elastic solids.     

Finite deformations of a hyperelastic, compressible and fibre reinforced tube

Finite torsion and axial stretch of a long, hyperelastic, compressible and circular tube is studied for the design of a prototype of small diameter vascular prosthesis. The analysis is carried out in the context of the finite elasticity theory by using a class of Ogden strain energy function augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. The highly non-linear differential equations with variable coefficients governing the problem are solved numerically using a Runge–Kutta method. For different prestresses supported by the tube, the effects of the combined deformation on the stress distributions are presented.     

Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations

In this paper, a segregated finite element scheme for the solution of the incompressible Navier-Stokes equations is proposed which is simpler in form than previously reported formulations. A pressure correction equation is derived from the momentum and continuity equations, and equal-order interpolation is used for both the velocity components and pressure. Algorithms such as this have been known to lead to checkerboard pressure oscillations; however, the pressure correction equation of this scheme should not produce these oscillations. The method is applied to several laminar flow situations, and details of the methods used to achieve converged solutions are given.     

ELECTROELASTIC FIELD FOR AN IMPERMEABLE ANTI-PLANE SHEAR CRACK IN A PIEZOELECTRIC CERAMICS PLATE

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Electroelastic field for an impermeable anti-plane shear crack in a piezoelectric ceramics plate

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate are obtained. The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h → ∞. Biography: LI Xian-fang (1964-)     

A general solution method for an anti-plane shear crack dynamically accelerating along a bimaterial interface

We develop a general solution method for a dynamically accelerating crack under anti-plane shear conditions along the interface between two different homogeneous isotropic elastic materials. The crack is initially at rest, and after loading is applied the crack-tip speed which may accelerate up to the shear wave speed of the more compliant material. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time-dependent crack-face traction, as well as expressions for computing the crack-face displacements and the stress in front of the crack. We also present some numerical examples for fixed loads and for loads moving with the crack tip, using a stress intensity factor fracture criterion, in order to examine the predicted effect of material mismatch on interfacial fracture.     

A finite element solution of the time-dependent incompressible Navier–Stokes equations using a modified velocity correction method

A finite element solution of the two-dimensional incompressible Navier–Stokes equations has been developed. The present method is a modified velocity correction approach. First an intermediate velocity is calculated, and then this is corrected by the pressure gradient which is the solution of a Poisson equation derived from the continuity equation. The novelty, in this paper, is that a second-order Runge–Kutta method for time integration has been used. Discretization in space is carried out by the Galerkin weighted residual method. The solution is in terms of primitive variables, which are approximated by polynomial basis functions defined on three-noded, isoparametric triangular elements. To demonstrate the present method, two examples are provided. Results from the first example, the driven cavity flow problem, are compared with previous works. Results from the second example, uniform flow past a cylinder, are compared with experimental data.     

Variable penalty method for finite element analysis of incompressible flow

A new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ?.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word-length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.     

Finite element solution of the streamfunction-vorticity equations for incompressible two-dimensional flows

The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework of the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution of two well-known benchmark problems: flow in a lid-driven cavity at Re ? 10,000 and flow over a backward-facing step at Re = 800.     

Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example

Summary This investigation aims at the elastostatic field near the edges (tips) of a plane crack of finite width in an all-round infinite body, which — at infinity — is subjected to a state of simple shear parallel to the crack edges. The analysis is carried out within the fully nonlinear equilibrium theory of homogeneous and isotropic, incompressible elastic solids. Further, the particular constitutive law employed here gives rise to a loss of ellipticity of the governing displacement equation of equilibrium in the presence of sufficiently severe anti-plane shear deformations.The study reported in this paper is asymptotic in the sense that the actual crack is replaced by a semi-infinite one, while the far field is required to match the elastostatic field predicted near the crack tips by the linearized theory for a crack of finite width. The ensuing global boundary-value problem thus characterizes the local state of affairs in the vicinity of a crack-tip, provided the amount of shear applied at infinity is suitably small.An explicit exact solution to this problem, which is deduced with the aid of the hodograph method, exhibits finite shear stresses at the tips of the crack, but involves two symmetrically located lines of displacement-gradient and stress discontinuity issuing from each crack-tip.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.     

Towards the solution of finite plane-strain problems for compressible elastic solids

A new approach to the solution of finite plane-strain problems for compressible Isotropie elastic solids is considered. The general problem is formulated in terms of a pair of deformation invariants different from those normally used, enabling the components of (nominal) stress to be expressed in terms of four functions, two of which are rotations associated with the deformation. Moreover, the inverse constitutive law can be written in a simple form involving the same two rotations, and this allows the problem to be formulated in a dual fashion.For particular choices of strain-energy function of the elastic material solutions are found in which the governing differential equations partially decouple, and the theory is then illustrated by simple examples. It is also shown how this part of the analysis is related to the work of F. John on harmonic materials.Detailed consideration is given to the problem of a circular cylindrical annulus whose inner surface is fixed and whose outer surface is subjected to a circular shear stress. We note, in particular, that material circles concentric with the annulus and near its surface decrease in radius whatever the form of constitutive law within the given class. Whether the volume of the material constituting the annulus increases or decreases depends on the form of law and the magnitude of the applied shear stress.     

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.     

A finite element for incompressible plane flows of fluids with memory

Flows of fluids with single-integral memory functionals are considered. Evaluation of the stress at a material point involves the deformation history of that point, and a dominant computational cost in finite element approximation is the construction of streamlines. It is shown that the simple crossed-triangle macro-element is in many ways an ideal finite element for the difficult non-linear, non-self-adjoint problem. The question as to whether this element produces convergent velocity and pressure solutions is addressed in the light of its failure to satisfy the discrete LBB condition. The effect of the element's ill-disposed (‘spurious’) pressure modes is discussed, and a pressure smoothing scheme is given which gives good results in Newtonian and non-Newtonian flows at various Reynolds and Deborah numbers. As an example of the element's success in modelling such flows, the problem of pressure differences in flows over transverse slots is studied numerically. The results are compared with experimental observations of such flows. The effect of fluid memory on the relation between first normal-stress differences and pressure differences is investigated.     

A nonlinear finite element model for the stress analysis of soft solids with a growing mass

The paper presents a new finite element (FE) model for the stress analysis of soft solids with a growing mass based on the work of Lubarda and Hoger (2002). Contrary to the traditional numerical methods emphasizing on the influence of growth on constitutive equations, an equivalent body force is firstly detected, which is resulted from the linearization of the nonlinear equation and acts as the driver for material growth in the numerical aspect. In the algorithm, only minor correction on the traditional tangent modulus is needed to take the growth effects into consideration and its objectivity could be guaranteed comparing with the traditional method. To solve the resulted equation in time domain, both explicit and implicit integration algorithms are developed, where the growth tensor is updated as an internal variable of Gauss point. The explicit updating scheme shows higher efficiency, while the implicit one seems to be more robust and accurate. The algorithm validation and its good performance are demonstrated by several two-dimensional examples, including free growth, constrained growth and stress dependent growth.