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Cracks usually propagate dynamically that makes them so dangerous. However, most crack simulations are based on quasi-static analyses because they are simpler than the dynamic ones. Is it correct to use quasi-static analyses instead of the dynamic ones? Will the quasi-static and dynamic simulations provide similar results? We try to answer these questions in the present work. We compare results of quasi-static and dynamic simulations of crack propagation in aneurysm material. We use the material-sink (MS) approach, which is based on the notion of the diffused bond breakage. The latter feature implies a local loss of material and, consequently, decrease of mass density, which, in its turn, means that both stiffness and inertia go down in the damaged zone. The cancellation of inertia is an important feature of the MS approach in contrast to more formal regularization theories as phase field, gradient damage, and other nonlocal formulations. The MS approach is implemented within commercial finite-element software abaqus. A reduced mixed finite-element formulation is adopted to circumvent the volumetric locking and an implicit staggered solution algorithm is developed via the user-defined element subroutine UEL. Considered examples show that the onset of crack instability under static loads is followed by the dynamic rather than quasi-static crack propagation. Moreover, dynamic and quasi-static simulations, generally, provide different results.

This note discusses the Timoshenko–Ehrenfest beam theory and the Griffith fracture theory. Both were annunciated in the West in 1921, exactly a century ago. Much progress has been made in these fields. Discussing the deficiencies of the theories might pave way ahead.

Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangian strain tensor referred to the initial configuration and Lagrangian structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. In the present note, we show that the described two-step procedure is not necessary, and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is applied to the particular case of the transverse isotropy for the sake of illustration.

A micromechanical analysis is proposed for the establishment of the macroscopic constitutive relations for viscoelastic composite materials undergoing large deformations. The composites are assumed to possess a triply periodic microstructure and their viscoelastic constituents are modeled by the incorporation of the viscoelastic effects with an arbitrary chosen hyperelastic strain energy function. Furthermore, an energy limiter is introduced which enforces the saturation of the viscoelastic strain energy function. The value of the strain energy at the saturation corresponds to the failure energy of the viscoelastic constituent. In conjunction with the derived micromechanical analysis, the occurrence of the energy saturation of the viscoelastic constituent predicts the composite failure. Applications are given for the determination of the macroscopic (overall) response and creep of a viscoelastic unidirectional composite, and the behavior of viscoelastic porous materials. In all cases, failure occurrences of the unidirectional composite and porous materials are predicted.

In this note, we develop simple analytical formulas to estimate the effect of residual stresses on the pulse wave velocity in blood vessels. We combine these formulas with three constitutive models of the arterial wall. Particularly, we consider the Fung model and two models accounting for the dispersion of collagen fibers via 8 and 16 structure tensors accordingly. The residual stresses come into play with a description of the initial kinematics—the opening angle. Our numerical examples reveal that residual stresses reduce the pulse wave velocity. The latter effect becomes especially pronounced at high values of blood pressure.

In many practical applications rubber components are exposed to relatively high operating temperatures yet very few studies can be found in the literature that deal with the temperature influence on deformation and, especially, failure of rubberlike materials. In the present work, we partly fill this gap providing the experimental results on uniaxial tension and bulge tests for nitrile butadiene rubber, neoprene, and silicone for various temperatures in the range from 20°C to 90°C. Based on the results of the tests we develop novel thermoelastic constitutive models, which also incorporate a failure description via the method of energy limiters. Using the developed models, we study the cavitation problem under the elevated temperatures. We find that, generally, the heating might lower material strength while the material stiffness is not necessarily sensitive to the temperature alterations.

The remarkable phenomenon of the drag reduction via addition of small amounts of polymer molecules to a Newtonian solvent was observed experimentally long ago. However, the theoretical explanations of this observation are not overwhelming yet. In this note, we present a possible theoretical account of the phenomenon. It is based on the use of the Navier–Stokes model with viscous strength for the solvent and the upper-convected Maxwell model for the polymer solute. Simple analytical calculation shows that the laminar flow of the solvent is stabilized by an addition of the polymer solute and, thus, the transition to the chaotic and slower on average turbulent motion is suppressed.

Rubber-like materials are widely used in several industrial applications. In these applications, rubber components are largely subjected to biaxial loading at a range of temperatures. In this work, we study the effect of short-term temperature on the ultimate properties of rubber materials, particularly, their strength. Such studies are lacking in the literature. For this purpose, we consider three different rubber-like materials; Nitrile Butadiene Rubber (NBR), Neoprene and Silicone. These rubber materials are tested under equi-biaxial tension using the bulge test. Tests are conducted till failure under a constant temperature. Four different temperatures are considered; 25 ∘C, 50 ∘C, 70 ∘C and 90 ∘C. Experiments are modeled using a finite element method. A constitutive model which includes the description of failure through energy limiters is calibrated against the bulge experiments. It is found that while the material stiffness is not significantly affected by temperature the ultimate stress and stretch, as well as the energy limiter for NBR and Neoprene greatly depend upon temperature. Stress carrying capacity for NBR and Neoprene decreases drastically at the highest temperature considered as compared to their values at room temperature (25 ∘C). Properties of Silicone are not affected significantly because of its temperature resistance. A new constitutive function is developed for the energy limiter, which allows unifying the description of different materials.

Remarkable mechanical properties of biocomposites (bone, teeth, shell, antler etc.) are usually attributed to their special design where staggered mineral platelets are embedded in a protein matrix. Because of the high aspect ratio of the platelet the soft protein deforms in the shear mode predominantly providing the linkage for the hard inclusions. Mimicking Nature one might design materials with a similar architecture. By employing a micromechanical analysis, we study in the present work the strength of a bio-inspired composite in which hard platelets are embedded in a soft matrix made of the vulcanized natural rubber. We perform simulations of uniaxial tension of the composite material based on a continuum mechanics formulation and the high-fidelity generalized method of cells. The use of the energy limiters in the constitutive model for rubber at finite strains allows us to model failure and arrive at the overall strength of the composite. We find that the overall strength of the composite depends on the deformation and failure of soft matrix in tension and shear. Moreover, we find that the strength of the composite cannot exceed the strength of the matrix. The latter observation is noteworthy because it is qualitatively different from the previous experimental results with biocomposites which show a dramatic (ten times) increase of the strength of the material as compared to the strengths of its constituents. We illustrate these analytical and numerical findings by our experiments on 3D printed composite materials.

There are two polar contemporary approaches to the constitutive modeling of arterial wall with anisotropy induced by collagen fibers. The first one is based on the angular integration (AI) of the strain energy on a unit sphere for the analytically defined fiber dispersion. The second one is based on the introduction of the generalized structure tensors (GST). AI approach is very involved computationally while GST approach requires somewhat complicated procedure for the exclusion of compressed fibers. We present some middle ground models, which are based on the use of 16 and 8 structure tensors. These models are moderately involved computationally and they allow excluding compressed fibers easily. We use the proposed models to study the role of the fiber dispersion in the constitutive modeling of the arterial wall. Particularly, we study the auxetic effect which can appear in anisotropic materials. The effect means thickening of the tissue in the direction perpendicular to its stretching. Such an effect was not observed in experiments while some simple anisotropic models do predict it. We show that more accurate account of the fiber dispersion suppresses the auxetic effect in a qualitative agreement with experimental observations.

Cracks are created by massive breakage of molecular or atomic bonds. The latter, in its turn, leads to the highly localized loss of material, which is the reason why even closed cracks are visible by a naked eye. Thus, fracture can be interpreted as the local material sink. Mass conservation is violated locally in the area of material failure. We consider a theoretical formulation of the coupled mass and momenta balance equations for a description of fracture. Our focus is on brittle fracture and we propose a finite strain hyperelastic thermodynamic framework for the coupled mass-flow-elastic boundary value problem. The attractiveness of the proposed framework as compared to the traditional continuum damage theories is that no internal parameters (like damage variables, phase fields, etc.) are used while the regularization of the failure localization is provided by the physically sound law of mass balance.

Traditional hyperelastic models usually obey requirement of material stability in various forms: Baker-Ericksen inequalities, strong ellipticity, polyconvexity etc. It is reasonable, of course, to require stability for the intact behavior of materials. However, all materials fail and a description of failure should be incorporated in the constitutive law. A simple version of hyperelasticity with failure can be formulated based on the introduction of the limiter in the strain energy density. The limited strain energy bounds maximum achievable stress automatically. Evidently, the elasticity with the energy limiter should exhibit material instability. This work addresses two practically interesting calculations concerning the onset of material instability via the loss of ellipticity. First, we consider simple shear of natural rubber. We find the direction of failure localization, which is in perfect qualitative correspondence with fracture observations in rubber bearings after earthquakes. Interestingly, the direction of failure localization is different from the one predicted by the criterion of maximum tension stress or stretch. Second, we consider equibiaxial tension of a sheet of aneurysm material. We find that the isotropic aneurysm material exhibits infinitely many possible directions of failure localization in equibiaxial tension. The latter means that the random direction of cracks in equibiaxial tension experiment, e.g. membrane inflation, can be an indicator of material isotropy. Accordingly, a preferable direction of the crack alignment can be interpreted as an indicator of the aneurysm anisotropy.

In this chapter we review some recent approaches to modeling failure and fracture of soft materials. By failure we mean the onset of damage via material instability. By fracture we mean further localization of damage into cracks with their subsequent propagation.

We propose a theoretical approach to predict the onset of cracks in arterial wall. The arterial wall is a soft composite made up of hydrated ground matrix of proteoglycans reinforced by elastin and collagen fibers spatially dispersed in the matrix. Like any other material, the arterial tissue cannot store and dissipate strain energy above a certain threshold. This threshold value is introduced in the constitutive theory via energy limiters. The limiters naturally constrain reachable stresses and enable analysis of mathematical condition of strong ellipticity. Loss of the strong ellipticity corresponds to the juncture when superimposed waves cease to propagate due to localization of material failure into cracks perpendicular to a possible wave direction. Thus, the direction in which crack starts to appear can be analyzed in addition to its inception. We enrich the recently developed constitutive theories that account for fiber dispersion of the arterial wall by including 8 and 16 structure tensors with energy limiters. We analyze the loss of strong ellipticity in uniaxial tension in circumferential and axial directions of the arterial wall. We find that cracks appear in the direction perpendicular to tension, when the speed of the superimposed longitudinal wave vanishes. We also find that the appearance of cracks is predicted in the direction inclined (non-perpendicular) to tension, when the speed of the superimposed transverse wave vanishes.

Most soft materials resist volumetric changes much more than shape distortions. This experimental observation led to the introduction of the incompressibility constraint in the constitutive description of soft materials. The incompressibility constraint provides analytical solutions for problems which, otherwise, could be solved numerically only. However, in the present work, we show that the enforcement of the incompressibility constraint in the analysis of the failure of soft materials can lead to somewhat nonphysical results. We use hyperelasticity with energy limiters to describe the material failure, which starts via the violation of the condition of strong ellipticity. This mathematical condition physically means inability of the material to propagate superimposed waves because cracks nucleate perpendicular to the direction of a possible wave propagation. By enforcing the incompressibility constraint, we sort out longitudinal waves, and consequently, we can miss cracks perpendicular to longitudinal waves. In the present work, we show that such scenario, indeed, occurs in the problems of uniaxial tension and pure shear of natural rubber. We also find that the suppression of longitudinal waves via the incompressibility constraint does not affect the consideration of the material failure in equibiaxial tension and the practically relevant problem of the failure of rubber bearings under combined shear and compression.

A constitutive model of a human artery adventitia is presented and calibrated in uniaxial tension tests in longitudinal and circumferential directions. Both the experiment and the model describe the descending branch of the stress-stretch curve and, thus, failure becomes a part of the constitutive law. The theoretical failure description is provided by the introduction of energy limiters and it does not use internal damage variables. The latter feature allows for an easy analysis of material instability and the onset of failure. Particularly, the failure envelope is created in biaxial tension under various stretch ratios. In addition, the loss of ellipticity is analyzed in equibiaxial stretch and pure shear and it is found that the localized failure—crack—is approximately aligned with the directions of collagen fibers in accordance with experimental observations.

Aneurysms are abnormal dilatations of vessels in the vascular system that are prone to rupture. Prediction of the aneurysm rupture is a challenging and unsolved problem. Various factors can lead to the aneurysm rupture and, in the present study, we examine the effect of calcification on the aneurysm strength by using micromechanical modeling. The calcified tissue is considered as a composite material in which hard calcium particles are embedded in a hyperelastic soft matrix. Three experimentally calibrated constitutive models incorporating a failure description are used for the matrix representation. Two constitutive models describe the aneurysmal arterial wall and the third one - the intraluminal thrombus. The stiffness and strength of the calcified tissue are simulated in uniaxial tension under the varying amount of calcification, i.e. the relative volume of the hard inclusion within the periodic unit cell. In addition, the triaxiality of the stress state, which can be a trigger for the cavitation instability, is tracked. Results of the micromechanical simulation show an increase of the stiffness and a possible decrease of the strength of the calcified tissue as compared to the non-calcified one. The obtained results suggest that calcification (i.e. the presence of hard particles) can significantly affect the stiffness and strength of soft tissue. The development of refined experimental techniques that will allow for the accurate quantitative assessment of calcification is desirable.

Phase-field or gradient-damage approaches offer elegant ways to model cracks. Material stiffness decreases in the cracked region with the evolution of the phase-field or damage variable. This variable and, consequently, the decreased stiffness are spatially diffused, which essentially means the loss of the internal links and the bearing capacity of the material in a finite region. Considering the loss of material stiffness without the loss of inertial mass seems to be an incomplete idea when dynamic fracture is considered. Loss of the inertial mass in the damaged material region may have significant effect on the dynamic failure processes. In the present work, dynamic fracture is analyzed using a theory, which takes into account the local loss of both material stiffness and inertia. Numerical formulation for brittle fracture at large deformations is based on the Cosserat point method, which allows suppressing the hourglass type deformation modes in simulations. Based on the developed algorithms, the effect of the material inertia around a crack tip is studied. Two different problems with single and multiple cracks are considered. Results suggest that in dynamic fracture the localized loss of mass plays an important role at the crack tip. It is found, particularly, that the loss of inertia leads to lower stresses at the crack tip and, because of that, to narrower cracks as compared to the case in which no inertia loss is considered. It is also found that the regularized problem formulation provides global convergence in energy under the mesh refinement. At the same time, the local crack pattern might still depend on the geometry of the unstructured mesh.

Rubber bearings, used for seismic isolation of structures, undergo large shear deformations during earthquakes as a result of the horizontal motion of the ground. However, the bearings are also compressed by the weight of the structure and possible traffic on it. Hence, failure analysis of rubber bearings should combine compression and shear. Such combination is considered in the present communication. In order to analyze failure, the strain energy density is enhanced with a limiter, which describes rubber damage. The inception of material instability and the onset of damage are marked by the violation of the condition of strong ellipticity, which is studied in the present work. Results of the studies suggest that horizontal cracks should appear because of the dominant shear deformation in accordance with the experimental observations. It is remarkable that compression delays failure in terms of the critical stretches.

In this article we couple the previously developed theory of elasticity with energy limiters for modeling material failure with a description of thermal processes. We present a generalized formulation of the thermoelasticity with energy limiters. We illustrate the new theory via the analytically tractable example of uniaxial tension. The theory predicts the existence of the thermoelastic inversion point observed in experiments. Besides, it predicts the experimentally observed stiffening of rubberlike materials under heating. We find above all that the tensile strength increases as a result of the stiffening while the critical rupture stretch is almost unaffected by heating. It is hoped that the developed coupled theory can be useful for modeling failure in rubberlike solids.

Aneurysm formation and growth is accompanied by microstructural alterations in the arterial wall. Particularly, the loss of elastin may lead to tissue disintegration and appearance of voids or cavities at the micron scale. Unstable growth and coalescence of voids may be a predecessor and trigger for the onset of macroscopic cracks. In the present work, we analyze the instability of membrane (2D) and bulk (3D) voids under hydrostatic tension by using two experimentally calibrated constitutive models of abdominal aortic aneurysm enhanced with energy limiters. The limiters provide the saturation value for the strain energy, which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. We find that the unstable growth of voids can start when the critical stress is considerably less than the aneurysm strength. Moreover, this critical stress may even approach the arterial wall stress in the physiological range. This finding suggests that cavitation instability can be a rational indicator of the aneurysm rupture.

Aneurysms are abnormal dilatations of vessels in the vascular system. They exist in two major forms: fusiform and saccular. Fusiform aneurysms are found in the human abdominal aorta while saccular aneurysms are found in cerebral blood vessels. The growth and rupture of aneurysms is driven by micro-structural alterations of the vessel, yet precise mechanisms underlying the process remain to be uncovered. Medical treatments of aneurysms are both expensive and dangerous thus a biomechanical approach can be valuable in assisting medical doctors in the process of making decisions. Biomechanical theories ofaneurysm development should include a description of growth and failure. We argue that the growth description can be done within the framework of continuum mechanics based on the one-to-one mapping of material configurations during the tissue evolution. The latter evolution is accompanied by the alteration of mass density which, in turn, triggers deformations and changes of tissue shape. We also consider a constitutive description of failure based on the concept of energy limiters. The latter means that the strain energy density must be bounded by definition and, thus, the amount of energy that can be stored and dissipated by an infinitesimal material volume must be finite. We discuss some results of the aneurysm simulation based on the described approach.

Dynamic cavitation is known to be a typical failure mechanism in rubber-like solids. While the mechanical behaviour of these materials is generally rate-dependent, the number of theoretical and numerical works addressing the problem of cavitation using nonlinear viscoelastic constitutive models is scarce. It has been only in recent years when some authors have suggested that cavitation in rubber-like materials is a dynamic fracture process strongly affected by the rate-dependent behaviour of the material because of the large strains and strain rates that develop near the cavity. In the present work we further investigate previous idea and perform finite element simulations to model the dynamic expansion of a spherical cavity embedded into a rubber-like ball and subjected to internal pressure. To describe the mechanical behaviour of the rubber-like material we have used an experimentally calibrated constitutive model which includes rate-dependent effects and material failure. The numerical results demonstrate that inertia and viscosity play a fundamental role in the cavitation process since they stabilize the material behaviour and thus delay failure.

A simple analytical study of a single-atom-thick sheet of graphene under biaxial tension is presented. It is based on the combination of the approaches of continuum and molecular mechanics. On the molecular level the Tersoff-Brenner potential with a modified cut-off function is used as an example. Transition to a continuum description is achieved by employing the Cauchy–Born rule. In this analysis the graphene sheet is considered as a crystal composed of two simple Bravais lattices and the mutual atomic relaxation between these lattices is taken into account. Following this approach a critical failure surface is produced for strains in biaxial tension. The adopted methodology is discussed in the context of the alternative approaches.

In this paper we develop a new constitutive model to describe the viscoelastic response of elastomers subjected to high strain rates. The key and original feature of the model is that it takes into account the failure of the material using an energy limiter. We calibrate the constitutive model for various strain rates using the experimental data reported by Hoo Fatt and Ouyang (2008) and show the capacity of the proposed formulation to describe the rate-dependent behavior of styrene butadiene rubber. In addition, we implement the model into ABAQUS/Explicit using a simple scheme for the temporal integration of the constitutive equations. Finally, we show sample numerical simulations to illustrate the joint performance of the constitutive model and the integration algorithm. DAI, GV and JARM are indebted to the Spanish Ministry of Economy and Competitiveness (Projects EUIN2015-62556 and DPI2014-57989-P) for the financial support received which allowed conducting part of this work. KYV acknowledges support from the Israel Science Foundation, grant No. ISF-198/15. Publicado

The arterial wall is a composite where the preferred orientation of collagen fibers induces anisotropy. Though the hyperelastic theories of fiber-reinforced composites reached a high level of sophistication and showed a reasonable correspondence with the available experimental data they are short of the failure description. Following the tradition of strength of materials the failure criteria are usually separated from stress analysis. In the present work we incorporate a failure description in the hyperelastic models of soft anisotropic materials by introducing energy limiters in the strain energy functions. The limiters provide the saturation value for the strain energy which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. By using some popular constitutive models enhanced with the energy limiters we analyze rupture of a sheet of arterial material under the plane stress state varying from the uniaxial to equal biaxial tension. We calculate the local failure criteria including the maximum principal stress, the maximum principal stretch, the von Mises stress, and the strain energy at the moment of the sheet rupture. We find that the local failure criterion in the form of the critical strain energy is the most robust among the considered ones. We also find that the tensile strength-the maximum principal stress-that is usually obtained in uniaxial tension tests might not be appropriate as a failure indicator in the cases of the developed biaxiality of the stress-strain state.

Rubber materials and structures can fracture because tensile deformation and growth of small pre-existing voids become unstable, leading to failure localization and crack propagation. Thus, it is important to predict the onset of static instability of the growing voids. We consider two typical cases of interest: the instability of 3D voids under the remote hydrostatic tension in the bulk and the instability of 2D voids under the remote biaxial tension in the membrane. For the purpose of analysis, we use constitutive models of natural and styrene-butadiene rubbers with the failure description enforced by energy limiters. The limiters provide the saturation value for the strain energy which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. We find that the unstable growth of a 3D bulk void can start when the remote hydrostatic tension reaches the value of ~2 ÷ 3 MPa and the unstable growth of a 2D membrane void can start when the remote biaxial tension reaches the value of ~50 ÷ 60 MPa.

Strain energy increases unlimitedly with the increase of deformation according to traditional hyperelastic models of materials. This ‘growth condition’ is evidently unphysical because no real material can sustain large enough deformations without failure. To introduce failure in hyperelasticity we propose to replace the strain energy of any intact material, W , with a modified expression: ψ = Φ { Γ (1/ m , 0) − Γ (1/ m , W m / Φ m )}/ m , where Φ is a material constant which sets a limit for the energy that can be accumulated during deformation, Γ is the upper incomplete gamma function, and m is a material constant controlling the sharpness of the transition to failure. The new formula for the strain energy is used to model failure of natural (NR) and styrene–butadiene (SBR) rubbers under plane stress conditions and the results are compared to the available experiments and other theories. The comparisons show that the proposed approach can be efficient for modeling failure in solids. The new theory allows reassessing the local criteria of material failure.

Separation of two particles is characterized by a magnitude of the bond energy, which limits the accumulated energy of the particle interaction. In the case of a solid composed of many particles a magnitude of the average bond energy – the failure energy – exists, which limits the energy that can be accumulated in an infinitesimal material volume under strain. The energy limiter controls material softening, which indicates failure. Thus, by limiting the stored energy density it is possible to include a description of material failure in the constitutive model. When the failure energy, i.e. the energy limiter, is introduced in the constitutive model it can be calibrated in macroscopic experiments. Traditional elasticity models do not have energy limiters and they allow for the unlimited energy accumulation under the strain increase, which is physically meaningless because no material can sustain large enough strains without failure. We use elasticity with energy limiters for modeling dynamic failure propagation in brittle solids. Two models of isotropic Hookean solids with energy limiters are introduced and examined in simulations of the penetration of a projectile into a brittle plate in the present work. The first model uses the energy limiter with the overall energy term while the second model has separate energy limiters for the volumetric and deviatoric components. The results of the penetration simulation obtained by using both models are similar qualitatively. It is remarkable that the penetration depth is mesh-independent for fine meshes even without the special regularization procedures. This is the first work where the methods of elasticity with energy limiters are used in dynamic analysis of brittle failure.

In this letter we calculate the characteristic length ~0.2 mm of natural rubber, which presents the width of a narrow zone where damage localizes initiating a crack. It is remarkable that our direct calculation is based on the results of the macroscopic experiments only.

Hyperelasticity with energy limiters is used to model high-velocity penetration in a sheet of natural rubber. Particularly, the Yeoh strain energy function is replaced with a modified expression, which sets a limit for the energy that can be accumulated during deformation. The material constants controlling failure are fitted to the tension tests. The modified strain energy function is plugged in ABAQUS for the three-dimensional explicit analysis of the edge penetration of a stiff elastic projectile into a thin sheet of the natural rubber and the time history of the failure process is tracked.

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture and Linear Elastic Fracture Mechanics which both ignore the crack sharpness. To explain the experimental observations qualitatively we earlier considered Mode I cracks [Volokh KY, Trapper P. Fracture toughness from the standpoint of softening hyperelasticity. J Mech Phys Solids 2008;56:2459–72.] and in the present work we extend our considerations to Mode II cracks. We simulate pure shear of a thin plate with a small crack of a finite and varying sharpness. In simulations we introduce the failure energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find that the magnitude of the fracture toughness can not be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack the lower is the toughness. Based on the obtained results we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of length-independent continuum fails.

Corrosion-induced cracking of the concrete cover poses a serious problem for serviceability of reinforced concrete structures. In the paper a new analytical model for predicting cover cracking due to corrosion of reinforcing steel is presented. In the model, concrete around a corroding reinforcing bar is considered as a thick-walled cylinder subjected to uniform pressure at its inner surface, which represents expansion caused by corrosion products. The pressure leads to formation of radial cracks near the inner surface of the cylinder. In order to account for it, the cylinder is divided into two parts—a partially cracked inner cylinder and an uncracked outer one. Cracks in the inner cylinder are taken into account by gradually reducing its tangential stiffness along the radial direction. The model ensures a consistent stress-strain description within both inner and outer cylinders and complete continuity of stresses and strains on the boundary between the cylinders that distinguish it from previously published analytical models. The model is calibrated using available experimental data and then employed to estimate the amount of corrosion products penetrated into concrete pores before full cracking of the concrete cover. Estimates obtained in the paper show that this amount may be higher than has been assumed previously.

Experiments [T. Mullin, R.R. Kerswell (Eds.), Laminar Turbulent Transition and Finite Amplitude Solutions, Proceedings of the UITAM Symposium, Springer, Bristol, UK, Dodrecht, 2005.] show that transition to turbulence can start at Reynolds numbers lower than it is predicted by the linear stability analysis – the subcritical transition to turbulence. To give a possible qualitative explanation of these observations we suggest that the onset of subcritical instability can be related to failure of the fluid viscosity: friction between fluid layers drops with the increase of the velocity gradient. To describe the drop of friction theoretically we relax the assumption of the stability of the fluid material and introduce a constant of the fluid strength . Particularly, we enhance the Navier–Stokes model with a failure description by introducing the fluid strength in the constitutive equation for the viscous stress. The classical model is obtained from the enhanced one when the strength goes to infinity. We use the modified Navier–Stokes model to analyze the Couette flow between two parallel plates and find that the lateral perturbations can destabilize the flow and the critical Reynolds number is proportional to the fluid strength. The latter means that the classical Navier–Stokes model of a stable material with the infinite strength does not capture the subcritical transition to turbulence while the modified model does.

The onset of crack propagation in rubber is studied computationally by using the softening hyperelasticity approach. The basic idea underlying the approach is to limit the capability of a material model to accumulate energy without failure. The latter is done by introducing a limiter for the strain energy density, which results from atomic/molecular considerations and can be interpreted as the average bond energy or the failure energy. Including the energy limiter in a constitutive description of material it is possible to enforce softening and, consequently, allow tracking the onset of structural instability corresponding to the onset of material failure. Specifically, initiation of crack propagation is studied in the case of a thin sheet of a rubber-like solid under the hydrostatic tension. The large deformation neo-Hookean material model enhanced with the energy limiter is used for finding the critical tension corresponding to the onset of static instability of the sheet, i.e. the onset of fracture propagation. The influence of the crack sharpness and length on the critical load is analyzed. It is found that material is sensitive to the crack sharpness when the shear modulus is significantly greater than the average bond energy. The sensitivity declines when the value of the shear modulus approaches the value of the failure energy. Roughly speaking, softer materials are less sensitive to cracks than more brittle materials where the brittleness is defined as a ratio of the shear modulus to the failure energy. It is also found that the critical tension is proportional to the inverse square root of the crack length for more brittle materials. The latter means that the Griffith theory based on the linearized elasticity is also applicable to softer materials undergoing large deformations. Unfortunately, the applicability of the Griffith theory is restricted to cracks with equivalent sharpness only.

New models of viscoelastic solids at small and finite deformations are proposed that describe material failure by enforcing the energy limiter — the average bond energy. Basically, the bond energy defines the energy that is necessary to separate two attracting particles. In the case of a solid composed of many particles there exists a magnitude of the average bond energy that is necessary to separate particles in a small material volume. The average bond energy can be calculated if a statistical distribution of the bond density is known for a particular material. Alternatively, the average bond energy can be determined in macroscopic experiments if the energy limiter is introduced in a material constitutive model. Traditional viscoelastic models of materials do not have energy limiters and, consequently, they allow for unlimited energy accumulation under the strain increase. The latter is unphysical, of course, because no material can sustain large enough deformations without failure. The average bond energy is the energy limiter that controls material softening, which indicates failure. Thus, by limiting the stored energy we include a description of material failure in the constitutive model. Viscoelasticity including energy limiters can be called softening hyperviscoelasticity. We present two softening hyperviscoelasticity models for small and finite deformations. In all cases the elastic and viscoelastic responses are described by potentials with limiters, which control material softening. The models are studied in the case of simple shear and uniaxial tension. The results of the calculations show that softening hyperviscoelasticity can be used for analysis of rate-dependent failure of materials.

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack/notch is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture where the crack sharpness is of minor importance. 1 To explain the experimental observations qualitatively we simulate tension of a thin plate with a small crack of a finite and varying sharpness. In simulations, we introduce the average bond energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find, in agreement with experiments, that the magnitude of the fracture toughness cannot be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack, the lower is the toughness. Based on the obtained results, we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance, etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of continuum fails.

We present here a coupled mathematical model of growth and failure of the abdominal aortic aneurysm (AAA). The failure portion of the model is based on the constitutive theory of softening hyperelasticity where the classical hyperelastic law is enhanced with a new constant indicating the maximum energy that an infinitesimal material volume can accumulate without failure. The new constant controls material failure and it can be interpreted as the average energy of molecular bonds from the microstructural standpoint. The constitutive model is compared to the data from uniaxial tension tests providing an excellent fit to the experiment. The AAA failure model is coupled with a phenomenological theory of soft tissue growth. The unified theory includes both momentum and mass balance laws coupled with the help of the constitutive equations. The microstructural alterations in the production of elastin and remodeling of collagen are reflected in the changing macroscopic parameters characterizing tissue stiffness, strength and density. The coupled theory is used to simulate growth and rupture of an idealized spherical AAA. The results of the simulation showing possible AAA ruptures in growth are reasonable qualitatively while the quantitative calibration of the model will require further clinical observations and in vitro tests. The presented model is the first where growth and rupture are coupled.

Microscopic voids can irreversibly grow into the macroscopic ones under hydrostatic tension. To explain this phenomenon, it was suggested in the literature to use the asymptotic value of the hydrostatic tension in the plateau yieldlike region on the stress–stretch curve obtained for the neo-Hookean model. Such an explanation has two limitations: (a) it relies on analysis of only one material model and (b) the hyperelasticity theory is used for the explanation of the failure phenomenon. In view of the mentioned limitations, the objective of the present note is twofold. First, we simulate the cavity expansion in rubber by using various experimentally calibrated hyperelastic models in order to check whether the stress–stretch curves have the plateau yieldlike regions independently of the constitutive law. Second, we repeat simulations via these same models enhanced with a failure description. We find (and that was not reported in the literature) that the process of cavity expansion simulated via hyperelastic constitutive models exhibiting stiffening, due to unfolding of long molecules, is completely stable and there are no plateau yieldlike regions on the stress–stretch curves associated with cavitation. In addition, we find that the instability in the form of yielding observed in experiments does appear in all simulations when the constitutive laws incorporate failure description with energy limiters.

A new strain gradient plasticity theory is formulated to accommodate more than one material length parameter. The theory is an extension of the classical J2 flow theory of metal plasticity to the micron scale. Distinctive features of the proposed theory as compared to other existing theories are the simplicities of mathematical formulation, numerical implementation and physical interpretation. The proposed theory is based on the physical idea that the effective plastic strain is a measure of the defect 1 density. The latter allows interpreting the yield/hardening condition as an equation describing (quasi-static) evolution of defects. At the macroscopic scale, the defect diffusion is negligible and the yield condition is a purely algebraic equation. At the scale of microns, however, the defect diffusion is sound and it can be included in the yield condition through the divergence of the vector of the defect flux. We propose that the defect flux is linearly proportional to the gradient of the defect density, i.e. the accumulated plastic strain. This proportionality is provided by a second-order tensor of the defect conductivity. Such tensor is the main source of the material characteristic lengths. If, for example, this tensor is equivalent to the length-scaled identity tensor then we arrive at the earlier Aifantis theory of strain gradient plasticity where the defect diffusion is isotropic. It is reasonable, however, to assume that the defect diffusion is not purely isotropic and stresses induce anisotropy. The latter means that the defect conductivity tensor should depend on the scaled stress tensor. The corresponding formulation of strain gradient plasticity is developed in the present work. It appears that the proposed theory is, probably, the simplest one among the theories, which include more than one characteristic length. It is also important that in contrast to the majority of strain gradient theories no direct use of higher-order stresses, which can hardly be interpreted in simple physical terms, has been made in the present work.

Traditional hyperelastic models of materials allow for the unlimited increase of the strain energy under the strain increase. It is clear, however, that no real material can accumulate the energy unlimitedly sustaining large enough strains. In the present work, we introduce a limiter for the strain energy—the critical failure energy, which can be interpreted as a failure constant characterizing the material ‘toughness’. We show that the critical failure energy controls materials softening. The softening can enrich any existing model of the intact material with a failure description. We demonstrate the efficiency of the softening hyperelasticity approach on a variety of analytically tractable boundary value problems with a variety of material models. The proposed softening hyperelasticity approach is a possible alternative to the simplistic pointwise failure criteria of strength of materials on the one hand and the sophisticated approach of damage mechanics involving internal variables on the other hand.

Material failure analysis based on the constitutive model of isotropic softening hyperelasticity is presented. In addition to the bulk and shear moduli the model includes only one material constant of volumetric failure work. The latter is in contrast to the traditional damage theories, which include internal variables that are difficult to calibrate experimentally. The softening hyperelasticity model is used to analyze the critical hydrostatic tension corresponding to the onset of instability of spherical and cylindrical voids. It is shown that the critical tension predicted by the softening hyperelasticity model does not depend on the void size in agreement with the linear elasticity solution showing that the stress/strain state at the edge of the void does not depend on its size. This prediction stays in contrast to the prediction based on the Griffith energy method where the critical tension depends on the size of the void and tends to infinity when the void radius approaches zero. It is argued that the controversial results of the Griffith method are a consequence of a separation of stress analysis and criticality conditions. It is concluded, based on the considered examples, that a description of material failure should be an inseparable part of constitutive models of materials.

In previous publications, strain-energy functions with limiters have been introduced for the prediction of onset of failure in monolithic isotropic hyperelastic materials. In the present investigation, such enhanced strain-energy functions whose ability to accumulate energy is limited have been incorporated with a finite strain micromechanical analysis. As a result, macroscopic constitutive equations have been established which are capable to predict the onset of loss of static stability in a hyperelastic phase of composite materials undergoing large deformations. The details of the micromechanical analysis, based on a tangential formulation, for composites with periodic microstructure are presented. The derived micromechanical analysis includes the capability to model a possible imperfect bonding between the composite’s constituents and to provide the field distribution in the composite. The micromechanical method is verified by comparison with analytical and finite difference solutions for porous hyperelastic materials that are valid in some special cases. Results are given for a rubberlike matrix characterized by softening hyperelasticity, reinforced by unidirectional nylon fibers. The response of the composite to various types of loadings is presented up to the onset of loss of static stability at a location within the hyperelastic rubber constituent, and initial failure envelopes are shown.