Search

BackgroundWith limited agricultural land and increasing human population, it is essential to enhance overall photosynthesis and thus productivity. Oxygenic photosynthesis begins with light absorption, followed by excitation energy transfer to the reaction centres, primary photochemistry, electron and proton transport, NADPH and ATP synthesis, and then CO2 fixation (Calvin–Benson cycle, as well as Hatch–Slack cycle). Here we cover some of the discoveries related to this process, such as the existence of two light reactions and two photosystems connected by an electron transport ‘chain’ (the Z-scheme), chemiosmotic hypothesis for ATP synthesis, water oxidation clock for oxygen evolution, steps for carbon fixation, and finally the diverse mechanisms of regulatory processes, such as ‘state transitions’ and ‘non-photochemical quenching’ of the excited state of chlorophyll a.ScopeIn this review, we emphasize that mathematical modelling is a highly valuable tool in understanding and making predictions regarding photosynthesis. Different mathematical models have been used to examine current theories on diverse photosynthetic processes; these have been validated through simulation(s) of available experimental data, such as chlorophyll a fluorescence induction, measured with fluorometers using continuous (or modulated) exciting light, and absorbance changes at 820 nm (ΔA820) related to redox changes in P700, the reaction centre of photosystem I.ConclusionsWe highlight here the important role of modelling in deciphering and untangling complex photosynthesis processes taking place simultaneously, as well as in predicting possible ways to obtain higher biomass and productivity in plants, algae and cyanobacteria.

In this chapter the constitutive equations for linear elasticity in which the free energy is a quadratic function, and the stress a linear function, of the small strain tensor are introduced. In general, the fourth-order linear elasticity tensor which relates the stress to the strain has twenty-one independent elastic constants. Most solids, however, exhibit some symmetry, the effect of which is to reduce the number of elastic constants. Forms of the linear elasticity tensor for some anisotropic materials, as well as the widely-used forms for an isotropic material, which has only two independent elastic constants, are discussed. The engineering elastic moduli known as the shear modulus and the bulk modulus, as well as the Young’s modulus and the Poisson’s ratio for isotropic linear elastic materials are introduced. Considerations of temperature changes necessitate the introduction of another tensorial material property called the thermal expansion tensor, which for an isotropic material reduces to a scalar coefficient of thermal expansion.

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

This chapter presents the specialization of chemoelasticity to energy storage materials (batteries). The theory presented is quasi-linear in that the constitutive relations are modelled as kinematically linear but nonlinearly dependent on the concentration of the chemical species. This permits the proper modeling of large changes in the concentration of the chemical species, but restricted to the small deformation regime. The model is also specialized to the case of isotropic response.

This chapter discusses the differentiation of scalar, vector, and tensor fields with respect to position in order to calculate the gradients of such fields. The important differential operators: divergence, curl, and Laplacian are defined. The differentiation of scalar, vector, or tensor-valued functions whose arguments are scalars, vectors, and tensors are also discussed. In addition to these differential operations, integral operations are discussed, and two important integral theorems, namely the divergence theorem, and Stokes' theorem --- which are central to the formulation of the basic laws of continuum mechanics are stated.

This chapter presents and discusses the solution of several classical problems in linear elastostatics, including thick-walled spheres and cylinders under external and internal pressure; bending and torsion of prismatic bars of arbitrary cross section; and the use of Airy’s stress function method to solve several two-dimensional plane strain and plane stress traction boundary value problems, including a demonstration of the extent of the Saint-Venant effect. The discussion also includes an analysis of the asymptotic stress and deformation fields near the tips of sharp cracks, and a discussion of stress intensity factors which are of importance in linear elastic fracture mechanics.

For solids it is often a good approximation to assume that the displacement and displacement gradient are small everywhere in the body. In this setting the basic laws for balance of mass, the balance laws for forces and moments, and the two laws of thermodynamics --- the balance of energy and the imbalance of entropy --- take on simpler forms which are derived and discussed in this chapter.

This chapter presents several technologically important constitutive relations for elastomeric materials. In particular, the Neo-Hookean, Mooney-Rivlin, Ogden, Arruda-Boyce, and Gent free energy functions are discussed in the context of incompressible response. Extensions to the slightly compressible case are also detailed, this includes a presentation of a number of possible volumetric response relations and their properties.

This chapter introduces the notion of rigid-perfect plasticity, and provides proofs of the extremum principles of applied loads and velocities. The variational principles are applied to generate the upper- and lower-bound theorems for collapse loads in rigid-perfect plasticity models. The method of sliding rigid blocks is presented for the solution of practical problems in forming operations and structural collapse. To facilitate such modelling the hodograph graphical device is also discussed.

Continuum mechanics of Solids presents a unified treatment of the major concepts in Solid Mechanics for beginning graduate students in the many branches of engineering. The fundamental topics of kinematics in finite and infinitesimal deformation, mechanical and thermodynamic balances plus entropy imbalance in the small strain setting are covered as they apply to all solids. The major material models of Elasticity, Viscoelasticity, and Plasticity are detailed and models for Fracture and Fatigue are discussed. In addition to these topics in Solid Mechanics, because of the growing need for engineering students to have a knowledge of the coupled multi-physics response of materials in modern technologies related to the environment and energy, the book also includes chapters on Thermoelasticity, Chemoelasticity, Poroelasticity, and Piezoelectricity. A preview to the theory of finite elasticity and elastomeric materials is also given. Throughout, example computations are presented to highlight how the developed theories may be applied.

This chapter discusses the first and second laws of thermodynamics. The first law represents a balance between the rate of change of the internal energy plus the rate of change of kinetic energy of a part of the body, and the rate at which energy in the form of heat is transferred to the part plus the mechanical power expended upon it. A part also possesses entropy, and the second law is the statement that the rate at which the net entropy of a part changes is greater than or at a minimum equal to the entropy flow into the part, resulting in a free energy imbalance known as the Clausius-Duhem inequality.

This chapter presents the theory of linear poroelasticity. The formulation is modeled after the theory of Biot and has applications in geomechanics, petroleum engineering, and hydrogeology. Relations for force and fluid balance are presented along, with constitutive relations coupling mechanical skeletal motion to fluid pore pressure and vice versa. Microstructural considerations are discussed, and the theory is demonstrated via application to geomechanical consolidation and to incompressible gels.

In this chapter the basic field equations in terms of displacement, strain, and stress, and typical boundary conditions which are necessary to formulate a complete three-dimensional boundary-value-problem for linear elasticity in the static case (i.e., neglect of inertial terms) are stated. The uniqueness of a solution to such a boundary value problem is discussed. There are a number of alternate ways that one can approach the statement of an elastostatic boundary-value-problem. The first major approach is obtained by reducing the set of field equations by expressing them solely in terms of the displacement field --- the Navier equations --- while in the second major approach the general system of equations may be reformulated by eliminating the displacement and strain fields and casting the system solely in terms of the stress field --- the Beltrami-Michell equations. The special formulation of idealized two-dimensional boundary value problems is presented the two basic theories of plane strain and plane stress for isotropic materials are discussed.

This chapter briefly considers linear elasticity under circumstances in which inertial effects are accounted for, and states the initial-boundary-value-problem of linear elastodynamics. Sinusoidal progressive waves form an important class of solutions to the equations of linear elastodynamics. Such waves for isotropic media in the absence of a conventional body force are considered and it is shown that for an isotropic medium only two types of sinusoidal progressive waves are possible: longitudinal and transverse.

This chapter presents a theory for the coupled thermal and mechanical response of solids under circumstances in which the deformations are small and elastic, and the temperature changes from a reference temperature are small --- a framework known as the theory of linear thermoelasticity. The basic equations of the fully-coupled linear theory of anisotropic thermoelasticity are derived. These equations are then specialized for the case of isotropic materials. Finally, as a further specialization a weakly-coupled theory in which the temperature affects the mechanical response, but the deformation does not affect the thermal response, are discussed; this is a specialization which is of importance for many engineering applications, a few of which are illustrated in the examples.

This chapter discusses the law of balance of mass, as well as the laws of balance of forces and moments. The important related concept of stress is this then presented as formalized by Cauchy in terms of a central theorem of Continuum Mechanics, which asserts that satisfaction of global balance of forces and moments is equivalent to the existence of a symmetric tensor field in the deformed body called the Cauchy stress, such that the traction vector acting across each oriented surface element at a point in the body is given by the Cauchy stress tensor operating linearly on the outward unit normal to the surface at that point. In addition, the stress tensor must satisfy a partial differential equation, known as the equation of motion, which asserts that the divergence of the stress tensor plus a body force per unit volume, is equal to the mass density times the acceleration.

This chapter presents a coupled theory for transport of a single atomic (or molecular) chemical species through a solid that deforms elastically. Consideration is limited to isothermal conditions and circumstances in which the deformations are small and elastic, and the changes in species concentration from a reference concentration are small --- a framework known as the theory of linear chemoelasticity. Underlying the presented approach is the notion that the solid can deform elastically but it retains its connectivity and does not itself diffuse. To account for the energy flow due to species transport, the notion of chemical potential of the species is introduced. First the basic equations of the fully-coupled linear theory of anisotropic linear chemoelasticity are derived, and then these equations are specialized for the case of isotropic materials.

This chapter provides an introduction to isotropic hardening plasticity models in the three-dimensional small strain setting. The additive decomposition of the strain is introduced along with the concepts of plastic strain and equivalent tensile plastic strain for three-dimensional problems. Plastic flow is discussed and defined, and a complete model of plasticity is formulated with Kuhn-Tucker loading/unloading conditions. Both rate independent and rate dependent (viscoplastic) models are discussed with an emphasis on Mises-Hill type theories which utilizes a Prandtl-Reuss flow rule. A variety of important flow models are presented.

This chapter provides an introduction to combined isotropic-kinematic hardening plasticity models in the three-dimensional small strain setting. The additive decomposition of the strain is introduced along with the concepts of plastic strain, equivalent tensile plastic strain, and back stress for three-dimensional problems. Plastic flow is discussed and defined, and a complete model of plasticity is formulated with Kuhn-Tucker loading/unloading conditions. The kinematic hardening model is based upon the Armstrong-Fredrick evolution law. Both rate-independent and rate-dependent (viscoplastic) models are discussed.

This chapter presents conditions for determining the limits of elastic behaviour for isotropic materials. The stress invariants of equivalent pressure, equivalent shear stress, and equivalent tensile stress are defined. These are then used to define common yield conditions, viz. the pressure-independent Mises and Tresca yield conditions, as well as the pressure-dependent Coulomb-Mohr and the Drucker-Prager yield conditions. Rankine’s failure criterion for brittle materials in tension, that is failure in a brittle material will initiate when the maximum principal stress at a point in the body reaches a critical value, is also discussed.