Journal of the Mechanics and Physics of Solids

Published by Elsevier
Online ISSN: 0022-5096
Publications
Surface plot of the free energy rates associated with tumor growth. All units are W m À 3 . (a) The rate of change of chemical free energy density stored in the cells. (b) The rate of change of free energy density stored in newly formed cells. (c) The rate of change of free energy density stored in newly produced ECM. (d) The rate at which free energy density is dissipated into work done as the tumor spheroid grows against stress. (e) The rate at which free energy density is dissipated due to cell motion. (f) The rate of change of free energy density due to glucose consumption. See Narayanan et al. (2010).
Growth stresses in the young sunflower head. (a) The head was cut along two orthogonal lines. The cuts gape widely in the central region while they remain closely appressed in the peripheral region where new organs are being formed. This gaping pattern demonstrates the presence of radial tension and circumferential compression in the head. (b) Typical arcuate crack created when the surface of the head is put in tension. The crack propagates predominantly in the circumferential direction thus releasing the tensile stresses in the radial direction. (c) and (d) The same head before (c) and after (d) reducing the turgor pressure of the cells. The characteristic gaping of the cut has been lost after decreasing the pressure. (In all images, the head is about 5 mm in diameter.)
Isotropic growth based on phenomenological growth laws (Allen et al., 2001). Patient-specific virtual stent implantation. CT of human aorta, Bezier spline interpolation, solid aorta model, finite element discretization, and simulation of wall growth and in-stent restenosis (from left to right).
Anisotropic growth based on microscopically-motivated growth laws. Ventricular growth and remodeling. Pressure overload-induced hypertrophy in response to aortic stenosis, normal heart, and volume overload-induced dilation in response to myocardial infarction. Sections from Hunter et al. (1999) (left) and finite element simulation (right).
Article
The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.
 
Article
When a tensile strain is applied to a film supported on a compliant substrate, a pattern of parallel cracks can channel through both the film and substrate. A linear-elastic fracture-mechanics model for the phenomenon is presented to extend earlier analyses in which cracking was limited to the film. It is shown how failure of the substrate reduces the critical strain required to initiate fracture of the film. This effect is more pronounced for relatively tough films. However, there is a critical ratio of the film to substrate toughness above which stable cracks do not form in response to an applied load. Instead, catastrophic failure of the substrate occurs simultaneously with the propagation of a single channel crack. This critical toughness ratio increases with the modulus mismatch between the film and substrate, so that periodic crack patterns are more likely to be observed with relatively stiff films. With relatively low values of modulus mismatch, even a film that is more brittle than the substrate can cause catastrophic failure of the substrate. Below the critical toughness ratio, there is a regime in which stable crack arrays can be formed in the film and substrate. The depth of these arrays increases, while the spacing decreases, as the strain is increased. Eventually, the crack array can become deep enough to cause substrate failure.
 
Article
Electrical stimulation is currently the gold standard treatment for heart rhythm disorders. However, electrical pacing is associated with technical limitations and unavoidable potential complications. Recent developments now enable the stimulation of mammalian cells with light using a novel technology known as optogenetics. The optical stimulation of genetically engineered cells has significantly changed our understanding of electrically excitable tissues, paving the way towards controlling heart rhythm disorders by means of photostimulation. Controlling these disorders, in turn, restores coordinated force generation to avoid sudden cardiac death. Here, we report a novel continuum framework for the photoelectrochemistry of living systems that allows us to decipher the mechanisms by which this technology regulates the electrical and mechanical function of the heart. Using a modular multiscale approach, we introduce a non-selective cation channel, channelrhodopsin-2, into a conventional cardiac muscle cell model via an additional photocurrent governed by a light-sensitive gating variable. Upon optical stimulation, this channel opens and allows sodium ions to enter the cell, inducing electrical activation. In side-by-side comparisons with conventional heart muscle cells, we show that photostimulation directly increases the sodium concentration, which indirectly decreases the potassium concentration in the cell, while all other characteristics of the cell remain virtually unchanged. We integrate our model cells into a continuum model for excitable tissue using a nonlinear parabolic second order partial differential equation, which we discretize in time using finite differences and in space using finite elements. To illustrate the potential of this computational model, we virtually inject our photosensitive cells into different locations of a human heart, and explore its activation sequences upon photostimulation. Our computational optogenetics tool box allows us to virtually probe landscapes of process parameters, and to identify optimal photostimulation sequences with the goal to pace human hearts with light and, ultimately, to restore mechanical function.
 
Article
The goal of this manuscript is to establish a novel computational model for stretch-induced skin growth during tissue expansion. Tissue expansion is a common surgical procedure to grow extra skin for reconstructing birth defects, burn injuries, or cancerous breasts. To model skin growth within the framework of nonlinear continuum mechanics, we adopt the multiplicative decomposition of the deformation gradient into an elastic and a growth part. Within this concept, we characterize growth as an irreversible, stretch-driven, transversely isotropic process parameterized in terms of a single scalar-valued growth multiplier, the in-plane area growth. To discretize its evolution in time, we apply an unconditionally stable, implicit Euler backward scheme. To discretize it in space, we utilize the finite element method. For maximum algorithmic efficiency and optimal convergence, we suggest an inner Newton iteration to locally update the growth multiplier at each integration point. This iteration is embedded within an outer Newton iteration to globally update the deformation at each finite element node. To demonstrate the characteristic features of skin growth, we simulate the process of gradual tissue expander inflation. To visualize growth-induced residual stresses, we simulate a subsequent tissue expander deflation. In particular, we compare the spatio-temporal evolution of area growth, elastic strains, and residual stresses for four commonly available tissue expander geometries. We believe that predictive computational modeling can open new avenues in reconstructive surgery to rationalize and standardize clinical process parameters such as expander geometry, expander size, expander placement, and inflation timing.
 
Article
Biological tissues have unique mechanical properties due to the wavy fibrous collagen and elastin microstructure. In inflation, a vessel easily distends under low pressure but becomes stiffer when the fibers are straightened to take up the load. The current microstructural models of blood vessels assume affine deformation; i.e., the deformation of each fiber is assumed to be identical to the macroscopic deformation of the tissue. This uniform-field (UF) assumption leads to the macroscopic (or effective) strain energy of the tissue that is the volumetric sum of the contributions of the tissue components. Here, a micromechanics-based constitutive model of fibrous tissue is developed to remove the affine assumption and to take into consideration the heterogeneous interactions between the fibers and the ground substance. The development is based on the framework of a recently developed second-order homogenization theory, and takes into account the waviness, orientations, and spatial distribution of the fibers, as well as the material nonlinearity at finite-strain deformation. In an illustrative simulation, the predictions of the macroscopic stress-strain relation, and the statistical deformation of the fibers are compared to the UF model, as well as finite-element (FE) simulation. Our predictions agree well with the FE results, while the UF predictions significantly overestimate. The effects of fiber distribution and waviness on the macroscopic stress-strain relation are also investigated. The present mathematical model may serves as a foundation for native as well as for engineered tissues and biomaterials.
 
Article
A magnetic resonance measurement technique was developed to characterize the transient mechanical response of a gel cylinder subjected to angular acceleration. The technique employs tagged magnetic resonance imaging (MRI) synchronized to periodic impact excitation of a bulk specimen. The tagged MRI sequence provides, non-invasively, an array of distributed displacement and strain measurements with high spatial (here, 5 mm) and temporal (6 ms) resolution. The technique was validated on a cylindrical gelatin sample. Measured dynamic strain fields were compared to strain fields predicted using (1) a closed-form solution and (2) finite element simulation of shear waves in a three-parameter "standard" linear viscoelastic cylinder subjected to similar initial and boundary conditions. Material parameters used in the analyses were estimated from measurements made on the gelatin in a standard rheometer. The experimental results support the utility of tagged MRI for dynamic, non-invasive assays such as measurement of shear waves in brain tissue during angular acceleration of the skull. When applied in the inverse sense, the technique has potential for characterization of the mechanical behavior of gel biomaterials.
 
Article
If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically-induced compressive stresses.
 
Conference Paper
The existence of one-component surface waves in materials with a plane of symmetry normal to the direction of propagation is discussed, with particular attention given to transversely isotropic materials in this class. A unique feature of these waves is that the traction vector which vanishes at the free surface is zero at all depths, implying that one-component surface waves are also pure modes of slabs. All of the `theoretical materials' which support the wave have the unusual property that at least one Poisson's ratio must be negative
 
Article
A two-dimensional computer code, using a multi-material Eulerian finite element formulation, was used to investigate the dynamic micromechanical behavior of granular material. The main results are:The strain-rate insensitive material model provides the correct increase in the dissipated energy with an increase of shock pressure;An increase in the initial porosity or the pressure results in the transition from the quasistatic to the dynamic regime of particle deformation, which can be characterized by the intensive localized plastic flow on the particles' interfaces. A new space scale is introduced into the system—the width of localized plastic flow;The macro and micro-scale responses of the granular material do not depend on the particle size for a rate independent material model;The energy of the shock wave compression at high pressures cannot be completely dissipated during the pore collapse at the shock front;The transition pressure from the quasistatic to the dynamic deformation regime does not depend on the density of the solid material for a given porosity with the other material properties fixed;A well developed dynamic regime correlates with a critical value of the microkinetic energy, which is comparable to the geometrically necessary energy for complete pore collapse.The results of computer calculations are in qualitative agreement with the experiments.
 
Article
Multidirectional compression testing of 1100 aluminum cubes at room temperature in three orthogonal directions developed saturation flow stresses at very large accumulated strains. The saturation stress was found to be a function of the strain increment used, and followed a power-law relationship. The results correlated well with fatigue tests of aluminum and alpha iron. Copper data showed a similar but more pronounced behavior. The presence of dislocation cells, subgrains and dislocation tangles dominated the microstructure as observed by transmission electron microscopy. The microstructure changed in a systematic manner with accumulated straining. Significant differences in sizes and concentrations of cells and subgrains were found for unidirectional compared with multidirectional straining. These features were correlated using generally accepted relationships between individual substructural configurations and flow stress.
 
Article
Composites whose response can be described in terms of a convex potential function are discussed. Bounds are constructed for the overall, or effective, potential of the composite, given the individual potentials of its constituents. Steady-state creep is considered explicitly but the results apply equally well to physically nonlinear elasticity, or deformation-theory plasticity, if strain-rate is reinterpreted as infinitesimal strain. Earlier work employed a linear “comparison” medium. This permitted the construction of only one bound—either an upper bound or a lower bound—or even in some cases no bound at all. Use of a nonlinear comparison medium removes this restriction but at the expense of requiring detailed exploration of the properties of the trial fields that are employed. The fields used here—and previously—have the property of “bounded mean oscillation”; the use of a theorem that applies to such fields permits the construction of the bounds that were previously inaccessible. Illustrative results, which allow for three-point correlations, are presented for an isotropic two-phase composite, each component of which is isotropic, incompressible and conforms to a power-law relation between equivalent stress and equivalent strain-rate. Generalized Hashin-Shtrikman-type bounds follow by allowing the parameter corresponding to the three-point correlations to take its extreme values.
 
Article
The phenomena occurring during rapid crack propagation in brittle single crystals was studied by cleaving strip-like silicon specimens along the {1 1 1} low-energy cleavage plane under bending. The experiments reveal phenomena associated with rapid crack propagation in brittle single crystals not previously reported, and new crack path instabilities in particular. In contrast to amorphous materials, the observed instabilities are generated at relatively low velocity, while at high velocity the crack path remains stable. The experiments demonstrate that crack velocity in single crystals can attain the theoretical limit. No evidence for mirror, mist, and hackle instabilities, typical in amorphous materials, was found. The important role played by the atomistic symmetry of the crystals on controlling and generating the surface instabilities is explained; the importance of the velocity and orientation-dependent cleavage energy is discussed. The surface instabilities are generated to satisfy minimum energy dissipation considerations. These findings necessitate a new approach to the fundamentals of dynamic crack propagation in brittle single crystals.
 
Article
In this paper, the crack-tip fields in a general nonhomogeneous material are summarized. The fracture toughness and R-curve of functionally graded materials (FGMs) are studied based on the crack-bridging concept and a rule of mixtures. It is shown that the fracture toughness is significantly increased when a crack grows from the ceramic-rich region into the metal-rich region in an alumina-nickel FGM. By applying the concept of the toughening mechanism to the study of the strength behavior of FGMs, it is found that the residual strength of the alumina-nickel FGM with an edge crack on the ceramic side is quite notch insensitive.
 
Article
Rubbery polymers are subjected to severe environmental conditions under service. As a consequence of various ageing mechanisms, the outer surface of rubber components hardens in time and cracking occurs as a result of combined mechanical and chemical processes. Conventional phenomenological hyperelastic constitutive models do not account for material softening. Consequently, the stored energy and stresses tend to infinity as stretch increases. In this contribution, a network alteration for the ageing mechanism of rubber-like materials is introduced along with a micromolecular description of material failure. The proposed micro-continuum material model is based on a serial construction of a Langevin-type spring representing the energy storage owing to conformational changes induced by deformation, to a bond potential representing the energy stored in the polymer chain due to the interatomic displacement. For the representation of the micro–macro transition, the non-affine kinematics of the micro-sphere model is used. The Morse potential is utilized for the interatomic bond, which describes the energetic contribution to rubber-like materials and governs the failure of the polymer chain in terms of bond rupture. A novel numerical scheme for the FE implementation of the proposed model is demonstrated. The hardening phenomenon as a result of diffusion limited oxidation of rubber is explained by the principle of mass conservation which dictates simultaneous modulus hardening along with decrease in ultimate stretch observed in aged rubbery polymers.
 
Article
This paper focuses on the unification of two frequently used and apparently different strain gradient crystal plasticity frameworks: (i) the physically motivated strain gradient crystal plasticity models proposed by Evers et al. [2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids 52, 2379–2401; 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures 41, 5209–5230] and Bayley et al. [2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structure 43, 7268–7286; 2007. A three dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine 87, 1361–1378] (here referred to as Evers–Bayley type models), where a physical back stress plays the most important role and which are further extended here to deal with truly large deformations, and (ii) the thermodynamically consistent strain gradient crystal plasticity model of Gurtin (2002–2008) (here referred to as the Gurtin type model), where the energetic part of a higher order micro-stress is derived from a non-standard free energy function. The energetic micro-stress vectors for the Gurtin type models are extracted from the definition of the back stresses of the improved Evers–Bayley type models. The possible defect energy forms that yield the derived physically based micro-stresses are discussed. The duality of both type of formulations is shown further by a comparison of the micro-boundary conditions. As a result, this paper provides a direct physical interpretation of the different terms present in Gurtin's model.
 
Article
A general methodology to develop hyper-elastic membrane models applicable to crystalline films one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The effectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed configurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by finite elements.
 
Article
Solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain, are frequently modeled by gradient type non-local constitutive laws, i.e. continuum theories that include higher order deformation gradients. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent investigation of their localization and stability behavior under finite strains.In the interest of simplicity, the microscopic model is a discrete, periodic, non-linear elastic lattice structure in two or three dimensions. The corresponding macroscopic model is a continuum constitutive law involving displacement gradients of all orders. Attention is focused on the simplest such model, namely the one whose energy density includes gradients of the displacements only up to the second order. The relation between the ellipticity of the resulting first (local) and second (non-local) order gradient models at finite strains, the stability of uniform strain solutions and the possibility of localized deformation zones is discussed. The investigations of the resulting continuum are done for two different microstructures, the second one of which approximates the behavior of perfect monatomic crystals in plane strain. Localized strain solutions based on the continuum approximation are possible with the first microstructurc but not with the second. Implications for the stability of three-dimensional crystals using realistic interaction potentials are also discussed.
 
Article
The plastic behavior of micro- and nano-scale crystalline pillars is investigated under nominally uniform compression. The transition from forest hardening to exhaustion hardening dominated behavior is shown to emerge from discrete dislocation dynamics simulations upon reduction in the initial source density. The analyses provide new insight into the scaling of flow stress with specimen size and also highlight the connection between individual dislocation mechanisms, collective phenomena and overall behavior.
 
Article
Many metals which fail by a void growth mechanism display a macroscopically planar fracture process zone of one or two void spacings in thickness characterized by intense plastic flow in the ligaments between the voids; outside this region, the voids exhibit little or no growth. To model this process a material layer containing a pre-existing population of similar sized voids is assumed. The thickness of the layer, D, can be identified with the mean spacing between the voids. This layer is represented by an aggregate of computational cells of linear dimension D. Each cell contains a single void of some initial volume. The Gurson constitutive relation for dilatant plasticity describes the hole growth in a cell resulting in material softening and, ultimately, loss of stress carrying capacity. The collection of cells softened by hole growth constitutes the fracture process zone of length l1. Two fracture mechanism regimes can be identified corresponding to l1 ≈ D and l1 ⪢ D. The connection between these mechanisms and fracture resistance is discussed.
 
Article
Stress discontinuities in composite media can be treated systematically by splitting second-rank tensors at an interface into exterior and interior parts. With anisotropic response the decomposition needs support by an appropriate algebra of fourth-rank interfacial operators. This topic is reviewed here and some fresh standpoints are proposed. These lead to a reorganization of the algebra and its further development. The main new application is to elastoplastic composites whose response is subject to the normality flow-rule. Contact is also made with existing theories of incipient discontinuities in hyperbolic states of plastic deformation.
 
Article
The mechanical properties of an open-framework structure constructed of joints and beam members are strongly influenced by both its geometrical configuration and joint flexibility. This paper clarifies the relationship between joint flexibility and Poisson's ratio, which is a mechanical criterion for solid deformation, and discusses two types of in-plane anisotropic structures made up of four-coordinate flexible joints and elbowed beam members. Uniaxial tensile analyses estimate the linear and nonlinear elastic properties of these frameworks by applying straightforward joint modeling with multi-rotational degrees of freedom. The numerical results show that these proposed frameworks produce a variety of deformability dependent on the joint flexibility in auxetic deformation with a negative Poisson's ratio, the folding mechanics under kinematic indeterminacy, and the transition of Poisson's ratio between the positive and negative values. The geometrical and topological aspects of the obtained mechanical behaviors are discussed.
 
The lattice with a semi-infinite crack. The beams of bending stiffness EI are rigidly connected at the nodes.
Positive directions of the forces, moments, rotations and displacements.
The bending stiffness of the square lattice related anisotropic plate as a function of the polar angle.  
The bending moments in the beams ahead of the crack, mZ 0, and the rotation angles of the broken beams at y ¼ a=2 þ 0,mo 0 (x= am). Solid lines correspond to the long wave asymptotes for the crack line (49).  
The lattice under symmetric bending: w(x,y)/(2A) (56).  
Article
We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass–spring lattices. Considering the static problems we use the discrete Fourier transform and the Wiener–Hopf technique. Monotonically distributed bending moments ahead of the crack are determined for the symmetric mode, and a self-equilibrated transverse force distribution is found for the antisymmetric mode. It is shown that in the latter case only the crack-front beam resists to the fracture development, whereas the forces in the other beams facilitate the fracture. In this way, the macrolevel fracture energy is determined in terms of the material strength. The macrolevel energy release is found to be much greater than the critical strain energy of the beam, especially in the hyper-singular mode. In both problems, it is found that among the beams surrounding the crack the crack-front beam is maximally stressed, and hence its strength defines the strength of the structure.
 
Article
The viscoplasticity theory based on overstress (VBO) is used to predict ratchetting tests reported by Ruggles and Krempl in Part I (J. Mech. Phys. Solids38, 575, 1990). The VBO predicts the influence of ratchetting rate correctly. An increase in ratchet stress rate reduces the ratchet strain. Quantitative predictions are reasonable for tests with no prior cyclic hardening. For tests with prior hardening the theory predicts too large a ratchet strain. It is shown that this can be corrected by making the shape function dependent on history. Theory and experiment do not exhibit a stress rate history effect.
 
Article
The spuniaxial viscoplastic behavior of AISI Type 304 stainless steel was investigated by tensile tests at various strain-rates (10−8−10−2s−1), and by short-term creep and relaxation tests up to 5 h. Instantaneous large changes in strain-rate were also performed during monotoniC and cyclic loading. A servocontrolled testing machine and displacement measurement on the specimen gage length were used for all tests.The results show significant rate-sensitivity, creep and relaxation. Test histories involving loading and unloading with positive loads up to 15% strain show that the relaxation behavior in the plastic range depends only on the strain-rate preceding the relaxation test and is independent of the strain magnitude. Also, the relaxation behavior is uniquely related to the stress changes corresponding to instantaneous large changes in the strain-rate during tensile tests. Completely reversed strain-controlled loading gradually changes the stress change/strain-rate change behavior. Annealed specimens and specimens loaded to a cyclic steady-state differ not only in their work-hardening characteristics but also in their rate-dependent behavior. In the cyclic steady-state, different hysteresis loops are traced for different strain-rates with fully reversible transitions from one hysteresis loop to another under strain-rate changes. These results support the notion that the viscoplastic behavior can be represented by piecewise nonlinear viscoelasticity.
 
Article
Following previous work (Krempl, 1979), a servocontrolled testing machine and strain measurement at the gage length were used to study the uniaxial rate(time)-dependent behavior of AISI Type 304 stainless steel at room temperature. The test results show that the creep strain accumulated in a given period of time depends strongly on the stress-rate preceding the creep test. In constant stress-rate zero-to-tension loading the creep strain accumulated in a fixed time-period at a given stress level is always higher during loading than during unloading. Continued cycling causes an exhaustion of creep ratchetting which depends on the stress-rate. Periods of creep and relaxation introduced during completely reversed plastic cycling show that the curved portions of the hysteresis loop exhibit most of the inelasticity. In the straight portions, creep and relaxation are small and there exists a region commencing after unloading where the behavior is similar to that at the origin for virgin materials. This region does not extend to zero stress.The results are at variance with creep theory and with viscoplasticity theories which assume that the yield surface expands with the stress. They support the theory of viscoplasticity based on total strain and overstress.
 
Article
In many materials, including tin–lead eutectic solder, a substantial portion of their fatigue life is spent in accumulation of randomly distributed microcracks. An assembly of discrete interconnected elements (e.g. grains or phases) approximates the solid in this paper. Percolation theory is employed to derive critical microcrack density and to predict size effect. Self-similarity in microcrack nucleation leads to a simple power law for lifetime prediction. Its application to the fatigue of tin–lead eutectic solder shows a good agreement with experimental data.
 
Article
Mode I crack initiation and growth under plane strain conditions in tough metals is computed using an elastic-plastic continuum model which accounts for void growth and coalescence ahead of the crack tip. The material parameters are the Young's modulus, yield stress and strain hardening exponent of the metal, along with the parameters characterizing the spacing and volume fraction of voids in material elements lying in the plane of the crack. For a given set of these parameters and a specific specimen, or component, subject to a specific loading, relationships among load, load-line displacement and crack advance can be computed with no restrictions on the extent of plastic deformation. Similarly, there is no limit on crack advance, except that it must take place on the symmetry plane ahead of the initial crack. Suitably defined measures of crack tip loading intensity, such as those based on the J-integral, can also be computed, thereby directly generating crack growth resistance curves. In this paper, the model is applied to five specimen geometries which are known to give rise to significantly different crack tip constraints and crack growth resistance behaviors. Computed results are compared with sets of experimental data for two tough steels for four of the specimen types. Details of the load, displacement and crack growth histories are accurately reproduced, even when extensive crack growth takes place under conditions of fully plastic yielding.
 
Article
We present an asymptotic algorithm for analysis of a singularly perturbed problem in a domain containing an interfacial crack. The crack is assumed to be flat and its front, initially straight, is perturbed in the plane containing the crack. The aim of the work is to determine the asymptotic representation of the stress-intensity factors near the edge of the crack. Mathematically, the limit problem is reduced to the analysis of a matrix, 3×3, Wiener-Hopf problem, and its solution generates the “weight matrix-function” characterised by a special singular solution near the crack edge. The two-term asymptotic representation for the weight function components is required by the asymptotic algorithm, together with two-term asymptotics for stress components associated with the physical fields near the edge of the crack. The particular feature of the solution is the coupling between the normal opening mode (Mode-I), and the shear modes (Mode-II and Mode-III), and the oscillatory behaviour of certain stress components near the crack edge. Explicit asymptotic formulae for the stress-intensity factors are obtained at the edge of a “wavy crack” at an interface.
 
Article
Our understanding of the elasticity and rheology of disordered materials, such as granular piles, foams, emulsions or dense suspensions relies on improving experimental tools to characterize their behaviour at the particle scale. While 2D observations are now routinely carried out in laboratories, 3D measurements remain a challenge. In this paper, we use a simple model system, a packing of soft elastic spheres, to illustrate the capability of X-ray microtomography to characterise the internal structure and local behaviour of granular systems. Image analysis techniques can resolve grain positions, shapes and contact areas; this is used to investigate the material's microstructure and its evolution upon strain. In addition to morphological measurements, we develop a technique to quantify contact forces and estimate the internal stress tensor. As will be illustrated in this paper, this opens the door to a broad array of static and dynamical measurements in 3D disordered systems Comment: 11 pages
 
Article
The theory of first strain gradient elasticity (SGE) is widely used to model size and non-local effects observed in materials and structures. For a material whose microstructure is centrosymmetric, SGE is characterized by a sixth-order elastic tensor in addition to the classical fourth-order elastic tensor. Even though the matrix form of the sixth-order elastic tensor is well-known in the isotropic case, its complete matrix representations seem to remain unavailable in the anisotropic cases. In the present paper, the explicit matrix representations of the sixth-order elastic tensor are derived and given for all the 3D anisotropic cases in a compact and well-structured way. These matrix representations are necessary to the development and application of SGE for anisotropic materials
 
Article
A tomographic method for identification of stress fields based on 3D photoelasticity has been developed. A second order tensor field tomographic method based on the general inverse problem of 3D photoelasticity, previously developed by the authors, is found to be highly sensitive to errors in photoelastic observations. In this study a new tomographic method for stress field with fairly high robustness to errors in photoelastic observations has been developed by introducing both equilibrium condition and linear elasticity to the previously developed general tensor field tomographic method. This new stress field tomographic method expands unknown 3D stress distributions as a linear combination of independent set of basis functions and a new inverse problem is posed: identify the amplitudes of basis functions based on photoelastic observations. Just as the inverse problem of 3D photoelasticity, this newly posed inverse problem is also nonlinear and ill posed. Unlike conventional approaches to 3D photoelasticity, both these nonlinearity and ill-posedness are properly treated using a load incremental approach. Load incremental approach chops the nonlinear solution space into segments with unique solutions by conducting photoelastic observations at sufficiently small increments in external load. Validating both numerically and experimentally, it is shown that this new stress field tomographic method has sufficient robustness against errors in photoelastic observations and is applicable to experimental stress measurements.
 
Article
Prior experiments have revealed exceptionally high values of the work of fracture (0.4–) in carbon/epoxy 3D interlock woven composites. Detailed destructive examination of specimens suggested that much of the work of fracture arose when the specimens were strained well beyond the failure of individual tows yet still carried loads . A mechanism of lockup amongst broken tows sliding across the final tensile fracture surface was suggested as the means by which high loads could still be transferred after tow failure. In this paper, the roles of weave architecture and the distribution of flaws in the mechanics of tow lockup are investigated by Monte Carlo simulations using the so-called Binary Model. The Binary Model was introduced previously as a finite element formulation specialised to the problem of simulating relatively large, three-dimensional segments of textile composites, without any assumption of periodicity or other symmetry, while preserving the architecture and topology of the tow arrangement. The simulations succeed in reproducing all qualitative aspects of measured stress–strain curves. They reveal that lockup can indeed account for high loads being sustained beyond tow failure, provided flaws in tows have certain spatial distributions. The importance of the interlock architecture in enhancing friction by holding asperities on sliding fibre tows into firm contact is highlighted.
 
Article
Self-assembly of complex structures is common in nature. Self-assembly principles provide a promising way to fabricate three-dimensional, micro- or millimeter scale devices. In the present paper, we present a generalized analytical study of the self-folding of thin plates into deterministic 3D shapes through fluid–solid interactions. Based on the beam theory, a mechanics model is developed, incorporating the two competing components—a capillary force promoting folding and the bending rigidity of the foil that resists folding into a 3D structure. Through an equivalence argument of thin foils of different geometry, an effective folding parameter, which uniquely characterizes the driving force for folding, has been identified. A criterion for spontaneous folding of any shaped 2D patterned foil based on the effective folding parameter is thus established. The model predictions show excellent agreement with experimental measurements made on a variety of materials, indicating that the assumptions used in the analysis arevalid.
 
Article
Dynamic crack propagation experiments have been performed using wedge loaded double cantilever beam specimens of an austenitized, quenched and tempered 4340 steel. Measurements of the dynamic stress intensity factor have been made by means of the optical method of caustics. The interpretation of experimental data, obtained from the shadow spot patterns photographed with a Cranz-Schardin high speed camera, is based on an elastodynamic analysis. The instantaneous value of the dynamic stress intensity factor KdI is obtained as a function of crack tip velocity. Finally, the interaction of reflected shear and Rayleigh waves with the moving crack tip stress field is considered.
 
Article
Dynamic (γ≈103⧹sec) torsional experiments were performed to investigate the process of initiation and formation of adiabatic shear bands in Ti-6Al-4V alloy. In this study, thin-wall tubular specimens were deformed dynamically in a torsional Kolsky bar (torsional split Hopkinson bar) . Through high-speed photography of a grid pattern previously printed on the specimens outer surface, the local strain and the local strain rate were found to be in the range of 75%–350% and 8.0×104⧹sec, respectively. The width of the shear bands ranged from 12–55 μm. In addition, an array of infrared detectors was employed to measure the local temperature rise during the deformation process. A peak temperature of 440–550°C was found in the various tests. The fracture surface of the shear band material was characterized by (1) regions of elongated dimples within which no second phase particles were observed, and (2) regions with a relatively flat and smeared appearance. There was no clear evidence based either on the appearance of the shear band in SEM or the measured temperature rise to suggest that the material within the shear band had undergone a phase transformation.
 
Article
A new experimental strategy for measuring the tensile strength of ice coatings to structural surfaces is presented. In this experiment, a laser-induced compressive stress pulse travels through a 1 mm-thick substrate disc that has a layer of ice grown on its front surface. The compressive stress pulse reflects into a tensile wave from the free surface of the ice and pulls the ice\interface apart, given a sufficient amplitude. The interface strength was calculated by recording the free surface velocity of an Al substrate using a Doppler interferometer and calculating the stress at the interface using a finite-difference elastic wave mechanics simulation with the free surface velocity as an input. The test procedure was used to study ice adhesion on 6061 aluminum alloy sheets. It was found that the adhesion strength of ice to unpolished aluminum substrates was 274 MPa at -10°C. This value decreased with temperature, down to 179 MPa at -40°C. Interestingly, this decrement in the tensile strength could be directly related to the existence of a liquid-like layer that is known to exist on the surface of solid ice till -30°C. The interface strength was also shown to decrease by polishing the Al substrate surface or by adding thin polymer coatings on the unpolished Al substrate. The sensitivity of the technique to such microstructural changes in the interfacial region is indicative of the experiments ability to provide basic adhesion data, which in turn, can be used to solve the deicing problem from a fundamental standpoint. 1998 Elsevier Science Ltd. All rights reserved.
 
Article
Uniaxial compression tests were performed on cylindrical samples made from AL-6XN stainless steel using an Instron servohydraulic testing machine and the enhanced Hopkinson technique. Tests were also conducted involving temperature and low and high strain rates to understand the underlying deformation mechanism. The microstructure of the undeformed and deformed samples was observed by optical microscopy. A physically based model was developed for the deformation behavior of the sample, including the effect of viscous drag on the motion of dislocations. Good agreement between the model predictions and the results of the experiments was obtained.
 
Article
Periodic density functional theory (DFT) and DFT+U calculations are used to evaluate various mechanical properties associated with the fracture of chromia (Cr2O3) along the [0 0 0 1] and directions. The properties investigated include the tensile strength, elastic constants, and surface energies. The tensile strengths are evaluated using an ideal tensile test, which provides the theoretical tensile strength, and by fitting the calculated data to universal binding energy relationships (UBER), which permit the extrapolation of the calculated results to arbitrary length scales. The results demonstrate the ability of the UBER to yield a realistic estimate of the tensile strength of a 10-μm-thick sample of Cr2O3 using data obtained through calculations on nanoscopic systems. We predict that Cr2O3 will fracture most easily in the direction, with a best estimate for the tensile strength of 386 MPa for a 10 μm grain, consistent with flexural strength measurements for chromia. The grain becomes considerably stronger at the nanoscale, where we predict a tensile strength along the same direction of 32.1 GPa for 1.45 nm crystallite. The results also provide insight into the origin of the direction dependence of the mechanical properties of Cr2O3, with the differences in the behavior along different directions being related to the number of Cr–O bonds supporting the applied tensile load. Additionally, the results shed light on various practical aspects of modeling the mechanical properties of materials with DFT+U calculations and in using UBERs to estimate the mechanical properties of materials across disparate length scales.
 
Article
The absorption of vibrational energy in the form of elastic waves generated by a transient localized force acting normally to the free surface of a semi-infinite solid is calculated in terms of the Fourier components of the force. The result is applied to tile Hertzian collision of a small body with the plane surface of a massive specimen. It is concluded that for impact velocities small compared with the propagation velocity of elastic waves in the specimen, a negligible proportion of the original kinetic energy of the small body is transferred to the specimen by the collision. The result is of some interest in justifying tlie validity of the Hertz theory for collisions between a small and a massive body.While experimental evidence for the impact of steel ball bearings on massive glass specimens shows some measure of agreement with these theoretical conclusions, results obtained for steel specimens are somewhat at variance, the observed values of the coefficient of restitution (0.90–0.95) indicating absorption of 10 %–20 % of the initial kinetic energy of the ball. A possible explanation of this anomaly is the existence, at the high rates of strain involved in such collisions, of either anelastic or viseous (dissipative) forces in the steel.
 
Article
Conical frustra made from leaded gun-metal have been compressed axially. Collapse is either by a travelling plastic hinge or by tearing. An analytical model is developed for the travelling plastic hinge in a rigid, ideally plastic solid; its predictions are compared with the observed response, and with those of an axisymmetric finite element analysis. The travelling hinge mechanism is also observed in the compressive collapse of an egg-box material comprising a square array of conical frustra. Collapse mechanism maps are constructed for the egg-box material, and they show the regimes of dominance of elastic buckling, material tearing and the travelling plastic hinge. The maps are useful for selecting egg-box geometries that maximise the energy absorption per unit mass at any prescribed value of collapse stress. The optimisation indicates that the egg-box material has a similar energy absorption capacity to that of hexagonal honeycombs and is superior to that of metal foams.
 
Article
The aim of this paper is to present a streamlined and fully three-dimensional version of the quasicontinuum (QC) theory of Tadmor et al. (Philos. Mag. A 73 (1996) 1529; Langmuir 12 (1996) 4529) and to analyze its accuracy and convergence characteristics. Specifically, we assess the effect of the summation rules on accuracy; we determine the rate of convergence of the method in the presence of strong singularities, such as point loads; and we assess the effect of the refinement tolerance, which controls the rate at which new nodes are inserted in the model, on the development of dislocation microstructures.
 
Article
We develop a general solution method for a dynamically accelerating crack under anti-plane shear conditions along the interface between two different homogeneous isotropic elastic materials. The crack is initially at rest, and after loading is applied the crack-tip speed which may accelerate up to the shear wave speed of the more compliant material. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time-dependent crack-face traction, as well as expressions for computing the crack-face displacements and the stress in front of the crack. We also present some numerical examples for fixed loads and for loads moving with the crack tip, using a stress intensity factor fracture criterion, in order to examine the predicted effect of material mismatch on interfacial fracture.
 
Article
Sudden jumps in the crack tip velocity were revealed by numerical simulation (in both continuum/cohesive element and molecular dynamics approaches) and experiments for rapid shear cracking. The cracking velocity may accelerate from a sub-Rayleigh speed to the intersonic range, or from an intersonic speed to a higher one, when the reflected impact wave reloads the crack tip. On the other hand, the cracking velocity may decelerate from an intersonic speed to a lower one or recede to the sub-Rayleigh range when the fracture driving force declines. The velocity change encountered during intersonic cracking plays a different role from that in the acceleration or deceleration of a subsonic crack. A crack propagating at an intersonic speed would leave a shear wave trailing behind. When the crack decelerates or accelerates, the effect of the trailing wave will lead to a transition period from one steady-state solution of crack tip singularity to another. This investigation aims at quantifying these processes. The full field solution of an intersonic mode II crack whose speed changed suddenly from one velocity (intersonic or subsonic) to another (intersonic or subsonic) is given in closed form. The solution is facilitated via superposing a series of propagating crack problems that are loaded by dislocations to seal the unwanted crack-face sliding or by concentrated forces moving at various speeds to negate the crack-face traction. In contrast to the subsonic solution, the results in the intersonic case indicate that the elastic fields around the crack tip depend on the deceleration or acceleration history that is traced back over a long time. Singularity matching dictates the jump that may actually take place.
 
Article
By the term ‘acceleration wave’ is meant an isolated geometric surface that moves relative to the material and across which the acceleration (but not the velocity) is discontinuous. The class of materials considered is characterized by a homogeneous linear tensor relation between stress-rate and strain-rate, arbitrary except for a symmetry restriction on the coefficients.A general matrix equation is obtained for the possible wave speeds and polarizations (with modifications when the material is incompressible). Calculations are carried out in detail for a wide range of elastic/plastic solids.Other topics include stationary discontinuities; the relation with vibration analysis; a physical interpretation for the matrix equation; and connexions between the theories of waves, stability, and uniqueness.
 
Article
Transient hydrogen diffusion and elastically accommodated hydride formation coupled with material elastic deformation are studied in a hydride forming system. The constitutive behavior of the material is modeled as isotropically linear elastic and account is taken of the effect of the dilatational strain induced by the solute hydrogen and formed hydride. The concept of terminal solid solubility of hydrogen as affected by stress is described and the mode of hydrogen diffusion through the two-phase material (matrix + hydride) is discussed. Probabilistic precipitation of hydride is modeled in the neighborhood of a stationary sharp crack tip under mode I plane strain loading, fixed hydrogen concentration on the crack surfaces and the outer boundary, and a uniform initial hydrogen concentration below the stress-free terminal solid solubility. A full transient finite element analysis allows for numerical monitoring of the development and expansion of the hydride zone. Information about the shape, size and density of the hydride in the hydride zone is obtained. The mechanistic effects of the solute hydrogen and hydride formation on the stress intensity at the crack tip are analyzed and their consequence on the fracture toughness resistance of the material is discussed.
 
Article
Imagine a residual glide twin interface advancing in a grain under the action of a monotonic stress. Close to the grain boundary, the shape change caused by the twin is partly accommodated by kinks and partly by slip emissions in the parent; the process is known as accommodation effects. When reached by the twin interface, slip dislocations in the parent undergo twinning shear. The twinning shear extracts from the parent dislocation a twinning disconnection, and thereby releases a transmuted dislocation in the twin. Transmutation populates the twin with dislocations of diverse modes. If the twin deforms by double twinning, double-transmutation occurs even if the twin retwins by the same mode or detwins by a stress reversal. If the twin deforms only by slip, transmutation is single. Whether single or double, dislocation transmutation is irreversible. The multiplicity of dislocation modes increases upon strain, since the twin finds more dislocations to transmute upon further slip of the parent and further growth of the twin. Thus, the process induces an increasing latent hardening rate in the twin. Under profuse twinning conditions, typical of double-lattice structures, this rate-increasing latent hardening combined with crystal rotation to hard orientations by twinning is consistent with a regime of increasing hardening rate, known as Regime II or Regime B. In this paper, we formulate governing equation of the above transmutation and accommodation effects in a crystal plasticity framework. We use the dislocation density based model originally proposed by Beyerlein and Tomé (2008) to derive the effect of latent hardening in a transmuting twin. The theory is expected to contribute to surmounting the difficulty that current models have to simultaneously predict under profuse twinning, the stress-strain curves, intermediate deformation textures, and intermediate twin volume fractions.
 
Top-cited authors
James R. Rice
  • Harvard University
Huajian Gao
  • Brown University
Norman Fleck
  • University of Cambridge
W.D. Nix
  • Stanford University
Alan Needleman
  • Texas A&M University