On Deformations with the Same Principal Stretches

In the context of the finite strain theory, plane isochoric homogeneous deformations are considered. Inspired by two examples (plane elliptical and plane hyperbolic deformations), it is seen that for any such isochoric deformation the corresponding principal stretches are equal to those of simple shear provided there is a certain relation between the amount of shear of the simple shear and the parameters of the general plane deformation. Then, the link is established between any two homogeneous deformations which have identical principal stretches. It involves two rotations, one in the undeformed state and the other in the deformed state. These rotations are determined explicitly for an arbitrary isochoric homogeneous deformation and the simple shear with the same principal stretches.     

A failure of a class of K-BKZ equations based on principal stretches

The invariants in the K-BKZ constitutive equation for an incompressible viscoelastic fluid are usually taken to be the trace of the Finger strain tensor and its inverse. The basis for this choice of invariants is not derived from the K-BKZ theory, but rather is due to the perception that this is the most natural choice. Research into using other sets of invariants in the K-BKZ equation, such as the principal stretches or the eigenvalues of the Finger strain tensor (i.e., the squares of the principal stretches) is relatively new. We attempt here to derive a K-BKZ equation based on the squares of the principal stretches that models the behavior of a low-density polyethylene melt in simple shear and uniaxial elongational deformation. In doing so, two assumptions are made as to the form of the strain-dependent energy function: first, that there is a function f(q) such that the energy function can be written as the sum of f(q _i ),i = 1, 2, 3, where the q _i 'sare the squares of the principal stretches, and second that f is a power law. We find that the K-BKZ equation resulting from these two assumptions is inadequate to describe both the shear and elongational behavior of our material and we conclude that the second of the above assumptions is not valid. Further investigation, including predictions of the second normal stress difference and some finite element calculations reveals that the first assumption is also invalid for our material.     

Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.     

On constitutive modelling of porous neo-Hookean composites

In this paper a hyperelastic constitutive model is developed for neo-Hookean composites with aligned continuous cylindrical pores in the finite elasticity regime. Although the matrix is incompressible, the composite itself is compressible because of the existence of voids. For this compressible transversely isotropic material, the deformation gradient can be decomposed multiplicatively into three parts: an isochoric uniaxial deformation along the preferred direction of the material (which is identical to the direction of the cylindrical pores here); an equi-biaxial deformation on the transverse plane (the plane perpendicular to the preferred direction); and subsequent shear deformation (which includes “along-fibre” shear and transverse shear). Compared to the multiplicative decomposition used in our previous model for incompressible fibre reinforced composites [Guo, Z., Peng, X.Q., Moran, B., 2006, A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. J. Mech. Phys. Solids 54(9), 1952–1971], the equi-biaxial deformation is introduced to achieve the desired volume change. To estimate the strain energy function for this composite, a cylindrical composite element model is developed. Analytically exact strain distributions in the composite element model are derived for the isochoric uniaxial deformation along the preferred direction, the equi-biaxial deformation on the transverse plane, as well as the “along-fibre” shear deformation. The effective shear modulus from conventional composites theory based on the infinitesimal strain linear elasticity is extended to the present finite deformation regime to estimate the strain energy related to the transverse shear deformation, which leads to an explicit formula for the strain energy function of the composite under a general finite deformation state.     

Strain Energy Functions for a Poisson Power Law Function in Simple Tension of Compressible Hyperelastic Materials

The most general strain energy function that yields a power law relationship between the principal stretches in the simple tension of nonlinear, elastic, homogeneous, compressible, isotropic materials is obtained. The approach taken generalises that used by Blatz and Ko. The strain energy function obtained depends on the choice of two stretch invariants. The forms of the strain energy function for a number of such choices are obtained. Finally, some consequences of the choice of strain energy function on the stress–strain relationship for uniaxial tension are investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Large-deformation properties of wheat dough in uni- and biaxial extension. Part I. Flour dough

Rheological and fracture properties of optimally mixed flour doughs from three wheat cultivars which perform differently in cereal products were studied in uniaxial and biaxial extension. Doughs were also tested in small angle sinusoidal oscillation. In accordance with previously published results the linear region was found to be very small. The rheological properties at small deformations hardly depended on the cultivar. A higher water content of the dough resulted in a lower value for the storage modulus and a slightly higher value for tan . For both uniaxial and biaxial extension a more than proportional increase in stress was found with increasing strain, a phenomenon called strain hardening. In uniaxial extension (i) stresses at a certain strain were higher and (ii) the stress was less dependent on the strain rate than in biaxial extension. This indicates that in elongational flow orientational effects are of large importance for the mechanical properties of flour dough. This conclusion is consistent with published data on birefringence of stretched gluten. Fracture stress and strain increased with increasing deformation rate. The observed time-dependency of fracture properties can best be explained by inefficient transport of energy to the crack tip. Presumably, this is caused by energy dissipation due to inhomogeneous deformation because of friction between structural elements, e.g. between dispersed particles and the network. Differences in the rheological properties at large deformations between the cultivars were observed with respect to (i) stress, (ii) strain hardening, (iii) strain rate dependency of the stress, (iv) fracture properties and (v) the stress difference between uniaxial and biaxial extension.     

Volume changes associated with the deformation of rubber-like solids

Volumechanges accompanying the deformation of rubber-like solids are analysed on the basis of isotropic elasticity theory. In particular, a simple, but general, result relating the volume and the stretch in simple tension is obtained. This is achieved by the introduction of certain modified principal stretches which allow the dependence of the strain energy on the isochoric and the dilatational parts of the deformation to be considered separately. The fact that volume changes in rubber-like solids are typically of order 0.01 % is used to linearize the stress-deformation relations in the dilatation. This enables the dilatation to be given explicitly as a function of the stretch in simple tension. Specific results are obtained for certain classes of constitutive law and good agreement with the experimental data for simple tension is demonstrated.Results for equi-biaxial tension and pure shear are also given in anticipation of further experimental data becoming available. The need for volume-change data for a wide variety of types of strains is emphasized.     

Constitutive branching analysis of cylindrical bodies under in-plane equibiaxial dead-load tractions

Finite homogeneous deformations of hyperelastic cylindrical bodies subjected to in-plane equibiaxial dead-load tractions are analyzed. Four basic equilibrium problems are formulated considering incompressible and compressible isotropic bodies under plane stress and plane deformation condition. Depending on the form of the stored energy function, these plane problems, in addition to the obvious symmetric solutions, may admit asymmetric solutions. In other words, the body may assume an equilibrium configuration characterized by two unequal in-plane principal stretches corresponding to equal external forces. In this paper, a mathematical condition, in terms of the principal invariants, governing the global development of the asymmetric deformation branches is obtained and examined in detail with regard to different choices of the stored energy function. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. With reference to neo-Hookean, Mooney-Rivlin and Ogden-Ball materials, a broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. Finally, using the energy criterion, a number of considerations are put forward about the stability of the computed solutions.     

Comparison of the elongational behavior of various polyolefins in uniaxial and equibiaxial flows

For processing operations with a pronounced elongational component, it was found that the uniformity of extruded items is improved by the presence of strain hardening usually measured in uniaxial elongation. Many processing operations such as foaming, film blowing, and blow molding are dominated by biaxial deformations, however, and therefore, the question arises how strain hardening in uniaxial and biaxial deformation compares. Besides a linear and long-chain branched PP, one classical LDPE, an HDPE pipe extrusion grade with a bimodal MMD, and a LCB-mPE were also characterized. For the measurements in uniaxial elongation the Münstedt tensile rheometer (MTR) and the ARES-EVF were used, while the lubricated flow method was applied for equibiaxial deformation. It was found that the strain hardening in uniaxial elongation is more pronounced. The dependence of strain hardening on strain rate is qualitatively the same in both modes. In the range of strain rates, the chosen long-chain branched LDPE and PP exhibit a strain hardening, which is approximately independent of the elongational rates applied, whereas for the HDPE it becomes smaller with increasing rate.

Constitutive structure in coupled non-linear electro-elasticity: Invariant descriptions and constitutive restrictions

A constitutive framework for electro-sensitive materials in the context of non-linear elasticity is analyzed. Constitutive equations are given in terms of energy functions that depend on several invariants. The study includes both the analysis of the invariants, which are present in the energy functions, and the analysis of constitutive restrictions that have to be obeyed by the constitutive functions. Isotropic as well as non-isotropic electro-sensitive elastomers are studied. The set of invariants that describe each material model is analyzed under two homogeneous deformations: (i) an uniaxial elongation and (ii) a simple shear deformation. These deformations are chosen since they are relevant to specific experiments, from which one may try to fit constitutive equations. The constitutive restrictions developed are based on classical ones used for isotropic non-linear elastic materials, in particular, are based on the Baker–Ericksen inequality and the ellipticity condition.     

Stability,bifurcation and post-critical behavior of a homogeneously deformed incompressible isotropic elastic parallelepiped subject to dead-load surface tractions

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions.     

Three-dimensional Full-field Measurements of Large Deformations in Soft Materials Using Confocal Microscopy and Digital Volume Correlation

A three-dimensional (3-D) full-field measurement technique was developed for measuring large deformations in optically transparent soft materials. The technique utilizes a digital volume correlation (DVC) algorithm to track motions of subvolumes within 3-D images obtained using fluorescence confocal microscopy. In order to extend the strain measurement capability to the large deformation regime (>5%), a stretch-correlation algorithm was developed and implemented into the Fast Fourier Transform (FFT)-based DVC algorithm. The stretch-correlation algorithm uses a logarithmic coordinate transformation to convert the stretch-correlation problem into a translational correlation problem under the assumption of small rotation and shear. Estimates of the measurement precision are provided by stationary and translation tests. The proposed measurement technique was used to measure large deformations in a transparent agarose gel sample embedded with fluorescent particles under uniaxial compression. The technique was also employed to measure non-uniform deformation fields near a hard spherical inclusion under far-field uniaxial compression. Introduction of the stretch-correlation algorithm greatly improved the strain measurement accuracy by providing better precision especially under large deformation. Also, the deconvolution of confocal images improved the accuracy of the measurement in the direction of the optical axis. These results shows that the proposed technique is well-suited for investigating cell-matrix mechanical interactions as well as for obtaining local constitutive properties of soft biological materials including tissues in 3-D.     

Size dependent response of large shell elements under in-plane tensile loading

Large-scale thin-walled structures with a low weight-to-stiffness ratio provide the means for cost and energy efficiency in structural design. However, the design of such structures for crash and impact resistance requires reliable FE simulations. Large shell elements are used in those simulations. Simulations require the knowledge of the true stress–strain response of the material until fracture initiation. Because of the size effects, local material relation determined with experiments is not applicable to large shell elements. Therefore, a numerical method is outlined to determine the effect of element size on the macroscopic response of large structural shell elements until fracture initiation. Macroscopic response is determined by introducing averaging unit into the numerical model over which volume averaged equivalent stress and plastic strain are evaluated. Three different stress states are considered in this investigation: uniaxial, plane strain and equi-biaxial tension. The results demonstrate that fracture strain is highly sensitive to size effects in uniaxial tension whereas in plane strain or equi-biaxial tension size effects are much weaker. In uniaxial and plane strain tension the fracture strain for large shell elements approaches the Swift diffuse necking condition.     

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.     

A constitutive model in finite viscoelasticity

A new constitutive model is suggested for the viscoelastic behavior of rubber-like materials at finite strains. The model treats a viscoelastic medium as a system with a variable number of purely elastic links, which can arise and collapse due to micro-Brownian motion of molecules.Assuming that the processes of birth and death for elastic links are independent of stresses, we obtain operator linear constitutive equations in finite viscoelasticity. According to this model, elastic and viscous effects may be distinguished and described independently of each other by a relaxation measure and a strain energy density.The potential energy of deformations is assumed to depend on the principal invariants of the relative Finger tensor of strains. Unlike the standard approach, we do not suggest any expression for the strain energy densitya priori, but suppose that this function is presented as a sum of two functions of one variable which are found by fitting experimental data.The proposed approach allows results of several experiments (uniaxial tension, biaxial tension, and torsion) for styrene butadiene rubber and butyl rubber to be predicted correctly.     

Large deformation extensional rheology of bread dough

The stress–strain curves of bread dough were derived under uniaxial compression, uniaxial tension and equi-biaxial tension loading conditions. In uniaxial compression, a lubricant was used to eliminate frictional effects between the loading platens and the sample. In uniaxial tension, cylindrical samples with thin flat discs at both ends (‘I’ samples) were tested. The discs at both ends were allowed to air-dry and were subsequently glued onto the loading platens. In equi-biaxial tension, a thin disc of dough was inflated into a bubble using pressurised air. The thickness at the top of the bubble was measured by shining a light through the walls of the bubble and recording the change in light intensity as the wall becomes thinner. All methods ensured that uniform deformation was obtained. Stress and strain were accurately evaluated using image analysis techniques. The tests were performed at various strain rates and speeds that defined the time dependence of the material. A non-linear viscoelastic model based on the Prony series and Van der Waals hyperelasticity was used to predict all test data. The model had a total of five material parameters and two time constants, which were set to represent the actual time scales of the experiments. A reasonable agreement between the experimental data and the chosen material model was observed.     

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.     

On the failure of ellipticity of the equations for finite elastostatic plane strain

Summary In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.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.     

Constitutive equations for hot-working of metals

Elevated temperature deformation processing - “hot-working,” is an important step during the manufacturing of most metal products. Central to any successful analysis of a hot-working process is the use of appropriate rate and temperature-dependent constitutive equations for large, interrupted inelastic deformations, which can faithfully account for strain-hardening, the restoration processes of recovery and recrystallization and strain rate and temperature history effects. In this paper we develop a set of phenomenological, internal variable type constitutive equations describing the elevated temperature deformation of metals. We use a scalar and a symmetric, traceless, second-order tensor as internal variables which, in an average sense, represent an isotropic and an anisotropic resistance to plastic flow offered by the internal state of the material. In this theory, we consider small elastic stretches but large plastic deformations (within the limits of texturing) of isotropic materials. Special cases (within the constitutive framework developed here) which should be suitable for analyzing hot-working processes are indicated.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.