Biological materials: Structure and mechanical properties

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed It is based on the nonlocal effects of the strain field and first gradient strain field This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

A microstructure-dependent nonlinear Euler–Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Karman geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler–Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.

Microstructure-dependent couple stress theories of functionally graded beams

A nonlocal beam theory for bending, buckling, and vibration of nanobeams

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

The equations of motion of the Euler–Bernoulli and Timoshenko beam theories are reformulated using the nonlocal differential constitutive relations of Eringen [International Journal of Engineering Science 10, 1–16 (1972)]. The equations of motion are then used to evaluate the static bending, vibration, and buckling responses of beams with various boundary conditions. Numerical results are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies of carbon nanotubes.

Nonlocal continuum theories of beams for the analysis of carbon nanotubes

Because the uncertainty in current empirical safety factors for structural strength is far larger than the relative errors of structural analysis, improvements in statistics offer great promise. One improvement, proposed here, is that, for quasibrittle structures of positive geometry, the understrength factors for structural safety cannot be constant but must be increased with structures size. The statistics of safety factors has so far been generally regarded as independent of mechanics, but further progress requires the cumulative distribution function (cdf) to be derived from the mechanics and physics of failure. To predict failure loads of extremely low probability (such as 10 - 6 to 10 - 7 ) on which structural design must be based, the cdf of strength of quasibrittle structures of positive geometry is modelled as a chain (or series coupling) of representative volume elements (RVE), each of which is statistically represented by a hierarchical model consisting of bundles (or parallel couplings) of only two long sub-chains, each of them consisting of sub-bundles of two or three long sub-sub-chains of sub-sub-bundles, etc., until the nano-scale of atomic lattice is reached. Based on Maxwell–Boltzmann distribution of thermal energies of atoms, the cdf of strength of a nano-scale connection is deduced from the stress dependence of the interatomic activation energy barriers, and is expressed as a function of absolute temperature T and stress-duration τ (or loading rate 1 / τ ). A salient property of this cdf is a power-law tail of exponent 1. It is shown how the exponent and the length of the power-law tail of cdf of strength is changed by series couplings in chains and by parallel couplings in bundles consisting of elements with either elastic–brittle or elastic–plastic behaviors, bracketing the softening behavior which is more realistic, albeit more difficult to analyze. The power-law tail exponent, which is 1 on the atomistic scale, is raised by the hierarchical statistical model to an exponent of m = 10 to 50, representing the Weibull modulus on the structural scale. Its physical meaning is the minimum number of cuts needed to separate the hierarchical model into two separate parts, which should be equal to the number of dominant cracks needed to break the RVE. Thus, the model indicates the Weibull modulus to be governed by the packing of inhomogeneities within an RVE. On the RVE scale, the model yields a broad core of Gaussian cdf (i.e., error function), onto which a short power-law tail of exponent m is grafted at the failure probability of about 0.0001–0.01. The model predicts how the grafting point moves to higher failure probabilities as structure size increases, and also how the grafted cdf depends on T and τ . The model provides a physical proof that, on a large enough scale (equivalent to at least 500 RVEs), quasibrittle structures must follow Weibull distribution with a zero threshold. The experimental histograms with kinks, which have so far been believed to require the use of a finite threshold, are shown to be fitted much better by the present chain-of-RVEs model. For not too small structures, the model is shown to be essentially a discrete equivalent of the previously developed nonlocal Weibull theory, and to match the Type 1 size effect law previously obtained from this theory by asymptotic matching. The mean stochastic response must agree with the cohesive crack model, crack band model and nonlocal damage models. The chain-of-RVEs model can be verified and calibrated from the mean size effect curve, as well as from the kink locations on experimental strength histograms for sufficiently different specimen sizes.

http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/464.pdf

Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture

Enhancement of barrier properties of cement mortar with graphene nanoplatelet

Improvement in concrete resistance against water and chloride ingress by adding graphene nanoplatelet

This study investigates the use of Graphene Nanoplatelets (GNP) in cement composite to quantify the material damage extent. The damage sensing capability of this new cement composite is demonstrated by experimentally measuring the electric potential across prisms with a known notch depth and comparing it with the finite element simulations. Meanwhile, the damage extent is directly related to the change in electric potential based on a mathematical analogy between the electrostatic field and the elastostatic field under anti-plane shear loading. It is shown that the fractional change in electric potential arising from damage is equivalent to the fractional change in elastic compliance, which can be exploited for structural health evaluation.

Sze Dai Pang

Papers

Nonlocal continuum theories of beams for the analysis of carbon nanotubes

Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture

Enhancement of barrier properties of cement mortar with graphene nanoplatelet

Improvement in concrete resistance against water and chloride ingress by adding graphene nanoplatelet

Use of 2D Graphene Nanoplatelets (GNP) in cement composites for structural health evaluation