On controllable states of stress for every compressible, isotropic, elastic material

Summary.  A much shorter proof is given here of the result that the controllable states of stress for every compressible, isotropic, elastic material are only the homogeneous states; and a duality between controllable stress states and controllable deformation states is established. Received October 24, 2002 Published online: March 20, 2003     

Sensitivity analysis and shape optimization in nonlinear solid mechanics

The objective of this paper is twofold. First, it presents a boundary element formulation for sensitivity analysis for solid mechanics problems involving both material and geometric nonlinearities. The second focus is on the use of such sensitivities to obtain optimal design for problems of this class. Numerical examples include sensitivity analysis for small (material nonlinearities only) and large deformation problems. These numerical results are in good agreement with direct integration results. Further, by using these sensitivities, a shape optimization problem has been solved for a plate with a cutout involving only material nonlinearities. The difference between the optimal shapes of solids, undergoing purely elastic or elasto-viscoplastic deformation is shown clearly in this example.     

The mechanical response of glutaraldehyde-fixed bovine pericardium to uniaxial load

Bovine pericardium, treated with glutaraldehyde, is used in the construction of heart valve substitutes. This study examines the mechanical properties of this tissue by using a continuum physics approximation of the material. A consideration of the relative magnitudes of the characteristic deformation time of a heart valve leaflet and the measured relaxation time of the tissue suggests that it can be effectively represented by a non-linear elastic solid. A compressible isotropic strain energy function is used to characterize the homogeneous deformation of the tissue when it is subjected to uniaxial load. The initial elastic material which is characterized by only two elastic constants, undergoes a transition to a second elastic material which is governed by a strain energy function of different magnitude by the same functional form as that associated with the initial elastic solid. This model is used to investigate the pericardial sac-to-sac and within-sac directional variation of the response to load in the unstrained state. Analysis of variance shows that glutaraldehyde treated pericardium possesses no preferred directional strength properties in the unstrained state. Any observed differences in the mechanical properties of different test specimens can be attributed to random biological variation alone.     

Analysis of intrinsic stability criteria for isotropic third-order Green elastic and compressible neo-Hookean solids

Internal stability of isotropic nonlinear elastic materials under homogeneous deformation is studied. Results provide new insight into various intrinsic stability measures, first proposed elsewhere, for generic nonlinear elastic solids. Three intrinsic stability criteria involving three different tangent elastic stiffness matrices are considered, corresponding to respective increments in strain measures conjugate to thermodynamic tension, first Piola–Kirchhoff stress, and Cauchy stress. Primary deformation paths of interest include spherical (i.e., isotropic) deformation, uniaxial strain, and simple shear; unstable modes are not constrained to remain along primary deformation paths. Effects of choices of second- and third-order elastic constants on intrinsic stability are systematically studied for physically realistic ranges of constants. For most cases investigated here, internal stability according to strain increments conjugate to Cauchy stress is found to be the most stringent criterion. When third-order constants vanish, internal stability under large compression tends to decrease as Poisson’s ratio increases. When third-order constants are nonzero, a negative (positive) pressure derivative of the shear modulus often promotes unstable modes in compression (tension). For large shear deformation, larger magnitudes of third-order constants tend to result in more unstable behavior, regardless of the sign of the pressure derivative of the shear modulus. A compressible neo-Hookean model is generally much more intrinsically stable than second- and third-order elastic models when Poisson’s ratio is non-negative.     

Elastic/plastic indentation hardness and indentation fracture toughness: The inclusion core model

A new model for determining elastic/plastic indentation is presented. This model generalizes Johnson's incompressible core model to a compressible material and allows the indentation pressure to be transmitted via a misfitted inclusion core beneath the indenter which is surrounded by a hemispherical plastic zone. The internal stress field inside the core is obtained by applying Eshelby's spherical inclusion problem together with Hill's spherical-cavity expansion analysis. The plastic deformation considered here exactly ensures compatibility between the volume of a material displaced by the indenter and that accommodated by expansion. The analysis explains the essential relationships between the dimensions of the indentation and plastic zone and the dominant material properties; yield stress, hardness and elastic modulus. The solution is extended to evaluate the indentation fracture toughness by taking into account the reduced half-space constraint by the image force.     

CFD-DEM analysis of internal packing structure and pressure characteristics in compressible filter cakes using a novel elastic–plastic contact model

Cake-forming filtration is a proven method for separating particles from suspensions. Most filtration models are based on the simplification of incompressible and homogeneous cake structures. However, most filter cakes are in fact unevenly compressed by e.g. the high transmembrane pressures, leading to dense structures with high flow resistance at the filtration membrane. Experimental investigations of these inhomogeneous cakes are challenging due to mostly invasive procedures after filtration has already taken place. In contrast, numerical methods can provide extensive information about fluid flow, particle separation and cake formation during filtration. However, this requires that both elastic and plastic particle deformation and forces are modeled correctly. To achieve this, the present study implemented a novel elastic–plastic DEM model that only requires measurable material parameters and therefore does not need any fitting. Subsequently, previously measured material parameters for elastic–plastic cellulose-lactose pellets (MCC) were used to investigate the packing density, fluid pressure levels and contact forces inside compressible filter cakes using CFD-DEM coupling. A comparison with incompressible and elastically compressible filter cakes showed a significant difference in the filtration behavior. Due to plastic deformation, a strong increase of the packing density when nearing the filtration membrane was found, leading to higher flow resistance for the filtration process. For cyclic filtration events, only the plastically deformed cake showed reduced height recovery in a relaxed state.     

Enlargement of a circular hole in a disc of plastically compressible material

A semi-analytic solution for the elastic/plastic distribution stress and strain in a thin annular disc subject to pressure over its inner radius is presented. It is assumed that a pressure-dependent yield criterion and its associated flow rule are valid in the plastic zone. Thus, the material is plastically compressible, which is a distinguished feature of the solution. Also, in contrast to most studies on elastic/plastic deformation of thin plates and discs under plane stress conditions, the flow theory of plasticity is adopted in conjunction with a smooth yield surface. Numerical methods are only necessary to evaluate ordinary integrals and to solve simple transcendental equations. It is shown that the stress path is not proportional and, therefore, the application of deformation theories of plasticity widely used to calculate the distribution of stresses and strains in thin plates and discs is not justified.     

Self-activated fragmentation

We explore the consequences of a point of view stating that fracture in strained brittle solids results from the action of mechanically activated random fluctuations. As opposed to thermally activated fracture, the energy reservoir feeding the crack progression is given by the mean elastic energy stored in the cracked region, itself a sub-sample of the strained material. The crack and the material bulk exchange self-consistently energy and momentum via random elastic waves, which are themselves produced by the fracture progression. The statistics of the fluctuating energy, and of the random force directing the fracture are derived, as well as the corresponding Langevin dynamics of the crack position. The role of pre-existing defects in the material is shown to be secondary. We examine in particular the case of a bent beam, for which we determine the distributions of breakup times, fragments number and lengths as a function of its initial deformation and of the material elastic and cohesion properties. The corrections due to plasticity, the effect of a transiently applied load, and the special case of liquids are discussed as well.     

Comparative study of granular solid-structure interaction in a uni-axial compression system

Interaction between granular solids and confining structures is an elementary problem encountered in subsurface structural design and bulk solids storing and handling. A classic scenario is uni-axial compression of granular solids in a deformable cylindrical container. Despite being apparently simple in loading condition, the understanding of this scenario remains limited, mainly due to complex interactive deformation between the two components via frictional interfaces. This paper comparatively examines such a uni-axial compression particulate system by a laboratory experiment and two different numerical approaches, namely, continuum finite element method (FEM) and linked discrete-finite element method (linked DEM-FEM). In the continuum FEM approach, two intendedly chosen simple material models, linear elastic and porous elastic models, are attempted. The comparative study reveals that the majority of resultant characteristics show satisfactory agreement amongst the numerical predictions and the experimental measurements. The simple elastic continuum FEM models can hence be a useful alternative in modelling such problems with mild structural flexibility under a monotonic loading scenario. However, precise prediction of some characteristics, such as lateral pressure ratio, may demand more elaborated material model or parameter selection. The enhancements needed for each numerical approach in order to achieve an improved result are further discussed.     

Note A note on finite amplitude transverse waves in a thermoelastic solid

Summary. If thermal and mechanical coupling is neglected, there is a class of isothermal strain energy functions for isotropic compressible hyperelastic solids, that admits the propagation of a finite amplitude transverse wave, without a coupled longitudinal wave, and a subclass that admits the simultaneous propagation of an uncoupled finite amplitude transverse wave and a longitudinal wave [1]. Several examples of this class of strain energy function are discussed. When thermal and mechanical coupling is considered the solid is described as thermoelastic. The purpose of this paper is to investigate the possibility of the propagation of a finite amplitude transverse wave without a coupled longitudinal wave, or the uncoupled simultaneous propagation of a finite amplitude transverse wave and a longitudinal wave, in an isotropic thermoelastic solid that has no underlying deformation. It is shown that an extensive class of isotropic thermoelastic solids does not admit the propagation of an uncoupled finite amplitude transverse wave, with or without an uncoupled longitudinal wave, even if the corresponding hyperelastic solid does.     

A VDQ-based multifield approach to the 2D compressible nonlinear elasticity

A numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu-Washizu principle, considering displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix-vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ-based approach. Some well-known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.     

Boundary element method for finite elasticity

This paper describes some integral formulations and implementations of a Boundary Element Method to solve two- and three-dimensional finite deformation problems of rubber-like materials. The integral equations are formulated in terms of unknown incremental displacement and total boundary traction fields, or alternatively in terms of the incremental displacement and incremental boundary traction fields. The elastic material is either compressible or incompressible with given constitutive equations. Both formulations are implemented and tested. The uniaxial elongation and simple shear deformations of a model material are successfully simulated by both formulations. Some non-trivial examples are performed using the first formulation.     

Second order effects in an elastic half-space acted upon by a non-uniform shear load

Summary A closed form solution to the second order elasticity problem, when an isotropic compressible elastic half-space undergoes a deformation owing to a non-uniformly distributed shear load, is presented. The method of integral transform is employed to determine the solutions. An example is discussed in detail to illustrate the second order effects. Numerical calculations for the second order elastic material for thez-direction displacement and the stresst _rz are carried out. It is found that the second order effect is to reduce thez-direction displacement and to dereaset _rz inside the circle but to increase its value outside the circle.     

Study on cavitated bifurcation problems for spheres composed of hyper-elastic materials

In this paper, spherical cavitated bifurcation problems are examined for incompressible hyper-elastic materials and compressible hyper-elastic materials, respectively. For incompressible hyper-elastic materials, a cavitated bifurcation equation that describes cavity formation and growth for a solid sphere, composed of a class of transversely isotropic incompressible hyper-elastic materials, is obtained. Some qualitative properties of the solutions of the cavitated bifurcation equation are discussed in the different regions of the plane partitioned by material parameters indicating the degree of radial anisotropy in detail. It is shown that the cavitated bifurcation equation is equivalent, by use of singularity theory, to a class of normal forms with single-sided constraint conditions at the critical point. Stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed by using the minimal potential-energy principle. For compressible hyper-elastic materials, a group of parameter-type solutions for the cavitated deformation for a solid sphere, composed of a class of isotropic compressible hyper-elastic materials, is obtained. Stability of the solutions is also discussed.     

Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids

In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible non-linearly elastic solids under uniaxial plane deformations. We consider first fiber reinforcement that endows the material with additional stiffness only in the fiber direction. It is shown, in particular, that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and may require concavity of the Cauchy stress–stretch curve. Secondly we examine fiber reinforcement that introduces additional stiffness under shear deformations. In this case we find that loss of ellipticity again occurs at a turning point of the nominal stress, in contrast to the situation for incompressible materials.     

Dynamic frictional slip along an interface between plastically compressible solids

Dynamic frictional slip along an interface between plastically compressible solids is analyzed. The plane strain, small deformation initial/boundary value problem formulation and the numerical method are identical to those in Shi et al. (Int J Fract 162:51, 2010) except that here the material constitutive relation allows for plastic compressibility. The interface is characterized by a rate and state dependent friction law. The specimens have an initial compressive stress and are subject to shear loading by edge impact near the interface. Two loading conditions are analyzed, one giving rise to a crack-like mode of slip propagation and the other to a pulse-like mode of slip propagation. In both cases, the initial compressive stress is taken to vary with plastic compressibility such that the associated initial effective stress is the same for all values of plastic compressibility. The volume change for the crack-like slip mode is mainly plastic while the elastic volume change plays a larger role for the pulse-like mode. For the crack-like slip mode, the proportion of plastic dissipation in the material increases with the increasing plastic compressibility, but the effect of plastic compressibility on the energy partitioning for the pulse-like slip mode is much smaller. The predicted propagation speeds approach a speed about the dilational wave speed for both the crack-like and pulse-like slip modes and this speed is not sensitive to the value of the plastic compressibility parameter. Plastic dissipation is found to be mainly associated with the deformation induced by the loading wave rather than with the deformation arising from slip propagation. The amplitude of the slip rate in the slip pulses is found to be largely governed by the value of the initial compressive stress regardless of the value of plastic compressibility.

     

Compressive failure of microcracked porous brittle solids

Constitutive equations for porous, brittle solids are developed based on the damage mechanics of elastic materials containing cavities and microcracks. For homogeneous deformation modes, microcrack growth from pores causes changes in the average elastic compliance of the material. Failure criteria in terms of bifurcations of the constitutive paths are established by examining the properties of the evolving tangent stiffness tensor. Limit points as well as localized shear band failure modes are addressed. The influence of moderate levels of lateral stresses is studied for biaxial stress states.     

Adiabatic shear failure in brittle solids

Recent studies have revealed microscopic amorphous lamella resulting from inelastic deformation in the ballistic impact of boron carbide ceramic. The possibility that these deformation features are a consequence of adiabatic shear deformation in the impact event is explored. An early theory of adiabatic shear that was limited to the response of rigid-plastic deformation is expanded to include elastic strain energy. The study reveals that elastic strain energy is commonly a small, but not negligible, contribution to impact-induced adiabatic shear in metals. Elastic strain energy is paramount in brittle solids. Relations are developed from the theory to predict the nominal width and spacing of adiabatic shear-bands in brittle solids. Comparisons of the theoretical predictions are consistent with observations of impact-induced deformation features in boron carbide.     

A class of universal relations for constrained,isotropic elastic materials

Summary A class of universal relations for all kinematically constrained, isotropic, elastic materials is described by the equationSB=BS relating the symmetric extra stress and the Cauchy-Green deformation tensors. This rule generates easily at most three universal relations for all kinematically admissible deformations of any constrained, isotropic body for which these tensors are nondiagonal. New universal formulae for homogeneous, compressible and incompressible materials reinforced by inextensible fibers in a variety of arrangements are presented for several kinds of homogeneous and nonhomogeneous, controllable universal deformations.     

An eigen expression for piezoelectrically stiffened elastic and dielectric constants based on the static theory

The piezoelectrically stiffened elastic and dielectric constants are studied here by the eigen mechanical and electric theory under physical presentation. The complete set of uncoupled expressions of the equivalent elastic and dielectric constants for various piezoelectric solids is obtained. In final part of this paper, the equivalent constants for piezoelectric material of class 6 mm are discussed.