Stability of active muscle tissue

In the sintering of advanced ceramics, a digital twin consisted of a finite element model can predict the final shape and microstructure of a sintered ceramic. However, to minimise uncertainties, it is essential to continually update the mechanical properties in the digital twin using data collected from the manufacturing process. One promising approach for achieving this is through artificial neural networks (ANN). This study introduces a machine learning strategy to update the constitutive behaviour of advanced ceramics in finite element analysis of sintering deformation. A major challenge in implementing machine learning in material processing is the huge amount of data required by the training and validation of an ANN, which are often unavailable or incomplete in a real manufacturing process. This study demonstrates that the data requirement can be reduced by employing a two-step training technique. Firstly, the ANN is trained using a nonlinear constitutive law, which describes a general relationship between the strain rates and stresses. Subsequently, the weights and biases of the ANN are transferred for the retraining using limited experimental data for an actual ceramic. It is shown that such approach can successfully capture the nonlinear constitutive behaviour of fine-grained alumina without demanding for large amount of experimental data. A case study is provided, highlighting the feasibility of implementing the ANN in a commercial finite element package, to replace the constitutive law and predict the shrinkage and distortion of a sintering part. In particular, the sintered dumb-bell shape part, simulated using the retrained ANN, showed grain size and relative density that markedly different from those using the nonlinear constitutive law. It is important to note that the proposed methodology is generic and can be used to create ANNs to replace constitutive laws in finite element analysis in digital twins for a wide range of other engineering processes.

On finite strain micromorphic elastoplasticity

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process.The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form.In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation.As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor.The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration.Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways.An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), "not closed" stress trajectories and nonzero stress work for a closed deformation trajectory.In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor.When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations.The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case -elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures.Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various

The famous bifurcation analysis performed by Flügge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Flügge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Flügge's formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Flügge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.

Constitutive Model for the Subsequent Time-Dependent Deformations of Type 304 Stainless Steel at Room Temperature

Variability response functions for beams with nonlinear constitutive laws

Introduction of (1) material frame indifference, (2) indifference with respect to rigid-body rotations of the reference configuration, and (3) material symmetry classification guides the development of non-linear (both Cauchy and hyper-) elastic constitutive laws for isotropic and anisotropic behaviour (for materials with a micro-structure). Incremental elasticity is introduced with reference to the relative Lagrangean description, and the Biot framework for incompressible elasticity is derived. Hypo-elasticity is also briefly included. Broadly speaking, constitutive laws set a bridge between strain and stress, so they have to keep into account the behaviour of the specific material under consideration. This behaviour can be fully reversible and described by elasticity (e.g., in the case of rubber) or can be partially irreversible and described by elastoplasticity (e.g., in the case of mild steel), or it may be time-dependent and described by viscous laws (e.g., in the case of a Newtonian fluid). We exclude for the moment the elastoplastic behaviour, and we present constitutive laws in this chapter for anisotropic elastic materials and for incrementally deformed solids (elastoplasticity is deferred to Chapter 8). The introduction of constitutive laws for materials subject to large strains requires preliminary statements of general principles, such as the so-called material frame indifference and indifference with respect to rigid-body rotations of the reference configuration. When these concepts are specified, the development of constitutive equations to model different materials is greatly simplified. After introduction of these general principles, we conclude the present chapter with the formulation of elastic constitutive laws, generalising concepts proposed in Chapter 4, now to includemicro-structural anisotropy.

Coefficients for analytic constitutive and equation of state models (EOS), which are used by many hydro codes at LLNL, are currently stored in a legacy material database (Steinberg, UCRL-MA-106349). Parameters for numerous materials are available through this database, and include Steinberg-Guinan and Steinberg-Lund constitutive models for metals, JWL equations of state for high explosives, and Mie-Gruniesen equations of state for metals. These constitutive models are used in most of the simulations done by ASC codes today at Livermore. Analytic EOSs are also still used, but have been superseded in many cases by tabular representations in LEOS (http://leos.llnl.gov). Numerous advanced constitutive models have been developed and implemented into ASC codes over the past 20 years. These newer models have more physics and better representations of material strength properties than their predecessors, and therefore more model coefficients. However, a material database of these coefficients is not readily available. Therefore incorporating these coefficients with those of the legacy models into a portable database that could be shared amongst codes would be most welcome. The goal of this paper is to describe the MatProp effort at LLNL to create such a database and associated access library that could be used by codes throughout the DOE complex and beyond. We have written an initial version of the MatProp database and access library and our DOE/ASC code ALE3D (Nichols et. al., UCRL-MA-152204) is able to import information from the database. The database, a link to which exists on the Sourceforge server at LLNL, contains coefficients for many materials and models (see Appendix), and includes material parameters in the following categories--flow stress, shear modulus, strength, damage, and equation of state. Future versions of the Matprop database and access library will include the ability to read and write material descriptions that can be exchanged between codes. It will also include an ability to do unit changes, i.e. have the library return parameters in user-specified unit systems. In addition to these, additional material categories can be added (e.g., phase change kinetics, etc.). The Matprop database and access library is part of a larger set of tools used at LLNL for assessing material model behavior. One of these is MSlib, a shared constitutive material model library. Another is the Material Strength Database (MSD), which allows users to compare parameter fits for specific constitutive models to available experimental data. Together with Matprop, these tools create a suite of capabilities that provide state-of-the-art models and parameters for those models to integrated simulation codes. This document is broken into several appendices. Appendix A contains a code example to retrieve several material coefficients. Appendix B contains the API for the Matprop data access library. Appendix C contains a list of the material names and model types currently available in the Matprop database. Appendix D contains a list of the parameter names for the currently recognized model types. Appendix E contains a full xml description of the material Tantalum.

Data-driven nonparametric identification of material behavior based on physics-informed neural network with full-field data

Propagation and radiation of electromagnetic waves in a lossless, reciprocal, chiral medium is studied in this thesis. Such a medium is described electromagnetically by the constitutive relations D = eE + iγB and H = iγE + (1/µ)B. The constants e, µ, γ are real and have values that are fixed by the size, shape, and the spatial distribution of the elements that collectively compose the medium. The plane wave propagation in an unbounded chiral medium is considered. The propagation constants are obtained and the polarization properties of electromagnetic waves in such a medium are discussed in detail. The problem of reflection from, and transmission through a semi-infinite chiral medium is solved by obtaining the Fresnel equations. The conditions for the total internal reflection of the incident wave from the interface, and the existance of the Brewster angle are obtained. The effects of the chirality on the polarization and intensity of the reflected wave from the chiral half-space are discussed and illustrated by employing the Stokes parameters. The propagation of electromagnetic waves through an infinite slab of chiral medium is formulated for oblique incidence and solved analytically for the case of normal incidence. The radiation emitted by an oscillating dipole in an unbounded, lossless, chiral medium is calculated. From the constitutive relations and from the time-harmonic Maxwell equations ∇ x E = iωB and ∇ x H = J - iωD, it is seen that the wave equation for such a medium is given by ∇ x ∇ x E - ω2µeE - 2ωµγ∇ x E = iωµJ where the source term J is the current density of the oscillating dipole and where E is the electric vector of the radiated field. The desired solution of this wave equation is found by the dyadic Green's function method, that is, by first constructing the dyadic Green's function Γ and then evaluating the expression E = iωµ∫Γ(r,r')•J(r')dV'. The dyadic Green's function Γ and the components of the radiated electric field E are obtained in closed form. The components of the radiated B, D, and H fields can be derived from knowledge of E by using the Maxwell equation B = (1/iω)∇ x E and the constitutive relations. The wave impedance of the medium and the radiation resistance of the dipole are also obtained. The effects of the chiral medium on the polarization and intensity of the dipole radiation are discussed.

In this chapter we illustrate the use of the abstract results obtained in Chap. 4, in the study of three representative static frictional contact problems for deformable bodies. In the first two problems we model the material’s behavior with a nonlinear elastic constitutive law and with a viscoelastic constitutive law with long memory, respectively, and we describe the frictional contact with subdifferential boundary conditions. In the third problem the deformable body is assumed to be piezoelectric and, therefore, we model its behavior with an electro-elastic constitutive law. And, again, the contact conditions, including the electrical conditions on the contact surface, are of subdifferential type. For each problem we provide a variational formulation. For the first two problems it is in a form of a hemivariational inequality for the displacement field and, for the third one, it is in a form of a system of hemivariational inequalities in which the unknowns are the displacement and electric potential fields. Then, we use the abstract existence and uniqueness results presented in Chap. 4 to prove the weak solvability of the corresponding contact problems and, under additional assumptions, their unique weak solvability. Finally, we present concrete examples of constitutive laws and frictional contact conditions for which our results work and provide the related mechanical interpretation. Everywhere in this chapter we use the notation introduced in Chap. 6.

Application of the kinetic theory of gases to granular flows has greatly increased our understanding of ‘rapid’ granular flows. One of the underlying assumptions is that particles interact only through binary collisions. For a given set of material and flow parameters, as the concentration increases, the transition from a binary collision mode to other modes of interaction occurs. Kinetic theory can no longer be applied. A numerical model is utilized to simulate the mechanical behaviour of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow. There are two objectives: to obtain the ‘empirical’ constitutive law and to gain insight into the mechanisms that operate in the transitional and quasi-static regimes. In a simple shear flow, spatially and temporally averaged dimensionless stresses $\tau^*_{ij} = \tau_{ij}/(\rho_{\rm s}D^2\dot{\gamma}^2)$ are functions of the concentration C, the dimensionless shear rate $B =\dot{\gamma}/(K_n/m)^{\frac{1}{2}}$, and material parameters ζ n , K s /K n and μ. Here $\dot{\gamma}$ is the shear rate, K n is the normal stiffness of an assumed viscoelastic contact force model, K s /K n is the ratio of tangential to normal stiffness, ζ n is the normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass, respectively. The range of B from 0.001 to 0.0707 was investigated for C ranging from 0.5 to 0.9, with material constants fixed as ζ n = 0.0709 (corresponding to the restitution coefficient e = 0.8 in binary impacts), K s /K n = 0.8 and μ = 0.5. It is found that for lower concentrations (C < 0.75) dimensionless stresses τ* ij are nearly independent of B, while for higher concentrations (C > 0.75) τ* ij monotonically decreases as B increases. Moreover, their relationship in this regime is well approximated by power law: τ* ij ∝ B−n(C). The powers n ij range from nearly zero for C = 0.775 (corresponding to the familiar square power dependency of dimensional stresses on the shear rate in the rapid flow regime), to nearly two for C = 0.9 (corresponding to shear-rate independence in quasi-static regime). The intermediate concentration range corresponds to transition. Distinct mechanisms that govern transitional and quasi-static regimes are observed and discussed.

In this paper by the use of “convex analysis” the general nonlinear unilateral static problem of structural mechanics is formulated. By means of the “subdifferentiability” concept the unilateral nonlinear boundary conditions and the unilateral nonlinear constitutive laws are formulated as “subdifferential relations”, permitting the derivation of the variational inequalities of virtual and complementary virtual work. Thus the minimum potential and complementary energy theorems are proved. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with nonlinear unilateral constitutive laws and nonlinear unilateral boundary conditions illustrate the theory.

Artificial neural network (ANN) models are used to learn the nonlinear constitutive laws based on indirectly measurable data. The real input and output of the ANN model are derived from indirect data using a mechanical system, which is composed of several subsystems including the ANN model. As the ANN model is coupled with other subsystems, the input of the ANN model needs to be determined during the training. This approach integrates measurable data, mechanics, and ANN models so that the ANN models can be trained without direct data which is usually not available from experiments. Two examples are provided as an illustration of the proposed approach. The first example uses two-dimensional (2D) finite element (FE) analysis to train an ANN model to learn the nonlinear in-plane shear constitutive law. The second example applies a continuum damage model to train an ANN model to learn the damage accumulation law. The results show that the trained ANN models achieve great accuracy based on the proposed approach.

An asymptotic solution of an initial value problem for a non-linear viscoelastic rod