Constitutive equations and anisotropic elasticity

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

Large strain deformations of elastic shells constitutive modelling and finite element analysis

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.

There are many fibrous materials, both natural and man made, ranging from hard material such as wood and carbon fibre composites to relatively soft materials such as rubber and softer materials such as biological tissues. With particular reference to fibre-reinforced soft materials, this chapter provides the theoretical background for analyzing the elastic properties of such materials based on nonlinear continuum mechanics through so-called structure tensors from which are constructed the deformation invariants that form the independent variables in the elastic constitutive laws. The theory is applied to some simple deformations by way of illustration of the material response. The notion of structure tensor is then used to incorporate residual stresses into the constitutive law and the important effects of residual stresses are illustrated in a prototype boundary-value problem related to the elastic behaviour of arteries.

Isothermal and adiabatic shear moduli have been measured countless times. Adiabatic shear moduli which are generally considered to be the same as isothermal shear moduli are shown here to be unstable, so they cannot exist. A constitutive law that avoids adiabatic moduli is found to apply to polycrystalline isotropic solids. The equation is a power law in the volume per unit mass time the shear modulus. The volume is in turn a function of temperature T and pressure p in the system. The adiabatic moduli are established in shear stress versus strain space using a construction from isothermal elastic shear moduli. The derivation of this law replaces adiabatic with isochoric conditions in linear elastic, sheared solids. It is widely reported that the shear thermal expansion coefficient is zero since isotropic solids do not change the shape when heated. All solids with applied shear stresses and temperature dependent shear moduli change the shape and have shear thermal expansion coefficients. The thermal expansion coefficient is coupled into the shear component of the entropy and the T dependence of the isothermal shear modulus. In a sheared solid, only the single point τ = 0, γ = 0 has no shear entropy contributions and no thermal expansion and is mechanically unstable in adiabatic constructions. The exact form of the proposed constitutive law is based on solutions from a second order differential equation based on shear contributions from the entropy.

The stability condition of cantilevered masonry piers subjected to their own weight and to a concentrated compressive top load is investigated, considering no-tension material with nonlinear stress-strain law in compression, of an experimental nature, and including softening behavior. The analysis is carried out by improving a numerical approach adopted in previous works where stability problems were solved assuming an infinitely elastic linear constitutive law in compression. Before the geometrical nonlinearity is considered, the limit equilibrium of a typical pier rectangular cross section is detected assuming unlimited available compressive strain. This preliminary analysis allows one to determine analytically the limit value that has to be imposed on the eccentricity of the resultant compressive force and to derive the moment-curvature relationships on which the second-order effects depend. Then the stability domains are derived in dimensionless form and their boundaries are modeled by analytical approximate expressions of practical use. Some numerical examples show that, depending on the average normal stress level, the assumption of an approximate linear constitutive law in compression, affected by the same elasticity modulus as that at the origin of the actual stress-strain law, can provide an unacceptable overestimate of the critical load.

In this contribution we present an extension of the multiscale Capriccio method towards inelasticity. This enables coupled simulations of a particle domain embedded into a continuum with particular focus on polymer systems. Starting from the method's initial implementation of pure elasticity, we substitute the nonlinear elastic continuum constitutive law by a recently developed viscoelastic‐viscoplastic one which is able to capture the mechanical behaviour of the particle system in a much larger strain range. Furthermore, we discuss numerical aspects like the choice of time steps and iteration numbers.

The appropriateness of several elastic constitutive laws for apple and potato cell walls is tested using uniform cell inflation data. Whole-tissue stress-strain behavior under uniaxial loading is predicted from an analysis of the compression of a conglomerate of cells in a simple arrangement.

Motivated by applications in orthopaedic and maxillo-facial surgery, the mechanical behaviour of cortical bone in cyclic overloads at physiological strain rates is investigated. To this end, a new one-dimensional rate-independent constitutive model for compact bone is proposed to simulate the damage accumulation occurring during tensile or compressive overloading. We adopted a macroscopic and phenomenological description of the mechanics of cortical bone. The mathematical formulation of the model is established within the framework of generalized standard materials and is based on the definition of three internal state variables: a tensile and a compressive damage variable that represent the microcrack density and a residual strain variable that represents the permanent strain associated with the sliding behaviour of these microcracks. Distinct damage threshold stresses are used in tension and compression. As the macroscopic mechanical behaviour of bovine cortical bone is very similar to that of human cortical bone and of much easier access, we first achieved the validation and identification of the material constants of the constitutive laws in tension using new uniaxial experimental results of bovine compact bone of our own. With adequate original hardening rules, the constitutive model is able to reproduce the main features of cortical bone behaviour under arbitrary cyclic tensile and compressive loadings. The proposed algorithm applies for the first time three distinct projections based on the relationship between the three internal variables and criteria. The predicted stress-strain curves exhibit a damaged reloading which is collinear with the origin as many cyclic overloading experiments on cortical bone have shown. Note that as our model was identified for physiological strain rates, it hardly can be applied in high strain rate situations like the ones involved in impact studies.

An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model

Helical buckling and postbuckling of pre-stressed cylindrical tubes under finite torsion

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.

Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory

<div class="section abstract"><div class="htmlview paragraph">Metal foil is a widely used material in the automobile industry, which not only is the honeycomb barrier material but is also used as current collectors in Li-ion batteries. Plenty of studies proved that the mechanical property of the metal foil is quite different from that of the metal sheet because of the size effect on microscopic scale, as the metal foil shows a larger fracture stress and a lower ductility than the metal sheet. Meanwhile, the fracture behavior and accurate constitutive model of the metal foil with the consideration of the strain rate effect are widely concerned in further studies of battery safety and the honeycomb. This article conducted experiments on 8011H18 aluminum foil, aiming to explore the quasi-static and dynamic tension testing method and the anisotropic mechanical behavior of the very thin foil. Two metal foil dog-bone specimens and three types of notched specimens were tested with a strain rate ranging from 2 × 10<sup>−4</sup>/<i>s</i> to 40/<i>s</i> and various stress states. It was shown that the metal foil has an obvious anisotropic behavior and the stress level increases with the strain rate. An anisotropic material model, which combined the Yld2000-2d plasticity model and the rate-dependent plasticity framework, was used to describe the mechanical behavior of the metal foil. Meanwhile, the fracture behavior of the metal foil was calibrated with an anisotropic fracture initiation model. The results of the constitutive model reproduced the strain rate effect and the anisotropic behavior of the metal foil, which have a good agreement with the experimental data.</div></div>

Geometric Continuum Mechanics ( GCM) is a new formulation of Continuum Mechanics ( CM) based on the requirement of Geometric Naturality ( GN). According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory. The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm ( GP) stating that push-pull transformations are the natural comparison tools for material fields. This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport. The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues. As a consequence, the notion of Material Frame Indifference ( MFI) is corrected to the new Constitutive Frame Invariance ( CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains. Geometrically consistent notions of Rate Elasticity ( RE) and Rate Elasto-Visco-Plasticity ( REVP) are formulated and consistent relevant computational methods are designed.