As scientists and engineers, we make sense of the world around us through observation and experimentation. Using mathematics, we attempt to describe our observations andmake useful predictions based on these observations. For example, a simple experimental observation that the distance traversed by an object traveling at a constant velocity is linearly related to both the velocity and the time can be formalized using the relation, d = vt, where d is the distance vector, v is the velocity vector, and t is the time. The distance, velocity, and time are physical quantities that can be measured or controlled. Physical quantities such as distance, velocity, and time are represented mathematically as tensors. A scalar, for example, is a zeroth-order tensor. Only amagnitude is required to specify the value of a zeroth-order tensor. In our previous example, time is such a quantity. If you are told that the duration of an event was 3 seconds, you need no other information to fully characterize this physical quantity. Velocity, on the other hand, requires both a magnitude and a direction to specify its meaning. The velocity would be represented using a first-order tensor, also known as a vector. The internal stress in a material is a second-order tensor, which requires a magnitude and two directions to specify its value. You may recognize that the two required directions are the normal of the surface on which the stress acts and the direction of the traction vector on this surface.

In this chapter, we introduce the use of internal variables. Specifically, we will consider a thermo-mechanical solid, which grows in response to the stress state within the material. This could be used as a simple model for growing or remodeling biological tissue. Internal variables can be used to capture a wide range of phenomena such as material damage, plasticity, and crystallinity. This chapter is meant to illustrate the use of an internal variable and the numerical methods used to implement such a model. The internal variable in the model outlined in this chapter is used to account for the amount of growth within the material. A simple evolution equation that couples the internal variable representing growth to the stress state of the material is written. The formulation of this model is significantly different from the previous models in that mass may be introduced or removed from the system. As mass is removed at any given point within the system, the momentum and energy associated with that mass is also removed. In effect, the mass instantaneously disappears. When mass is added at a given point, it is introduced with the same velocity, temperature, and energy of the mass that currently occupies the point.

Forces and Fields

In this chapter, we consider a thermo-mechanical solid model, which includes an internal variable accounting for the growth of the solid. Growth is captured in this model by introducing the rate of mass change per unit volume, ɸg (x, t).

Kinematics is the study of motion without regards to the forces responsible for that motion. Intuitively, we know that the application of a force can lead to the movement of an object. The equations characterizing this movement are called the equations of motion. Perhaps, we might compute the displacement of an object by measuring how far it has moved from its initial location. In this chapter, we will build on these intuitive concepts to explore the kinematics of deformable continua. We will show how simple geometric relations allow us to compute the deformation and strain from the equations of motion. Similarly, the velocity and acceleration fields may be determined by differentiation the equations of motion.

Configurations

We know that matter consists of atoms, which consist of protons, electrons, and neutrons, all of which consist of quarks. However, this level of detail can often be ignored when mathematically modeling a macroscopic object's response to external fields. The true discreet nature of material can be modeled as a continuous distribution of mass and the atomic or subatomic structure can be ignored. Within this representation, an object is no longer made up of a finite set of atoms each with its own mass or charge but instead consists of an infinite number of material points or particles. Instead of defining atomic mass or charge, we define a density and a charge density field.

When discussing the development of constitutive models for the ideal gas, fluids, and elastic solids, we had restricted the discussion to materials that consist of a single phase. However, there are many applications where a single material model may be used to represent the behavior of multiple interacting phases. For example, consider biological tissue, which we may think of as being made up of a solid component that consists of the extracellular matrix, a fluid component that contains significant quantities of water and smaller amounts of charged particles, and living cells that secrete substances and react to chemical and mechanical stimuli. In addition, cells undergo growth or death and cause remodeling of the extracellular matrix in response to external stimuli. It is the interaction among all of these phenomena over time that gives biological materials such a diverse and interesting response.

Using continuum mixture theory, we can model the extracellular matrix and the permeating fluid with a single model. This model consists of two continuously distributed phases that interact with one another through the transfer of momentum and energy. In reality, the solid phase and the fluid phase cannot occupy the same region of space. The true microstructure of the material may consist of an extracellular network with channels through which fluid is transported, but the model treats both phases as coexisting at the same spatial points (Figure 8.1). The continuum multiphase model performs well at capturing the aggregate behavior for many materials when the alternative, which is to model the detailed microscopic structure and phase interactions directly, is computationally intractable.

The response of a fluid depends on the rate of deformation. In this chapter, we present the development of the constitutive law for a Newtonian fluid, the formulation of the field equations, and methods for determining the material parameters within the Newtonian fluid constitutive equations. The compressible and incompressible Navier-Stokes equation and Bernoulli's equation are derived from the constitutive equations and the balance laws for a Newtonian fluid. Finally, we include a brief discussion of non-Newtonian fluid models.

The balance between molecular interactions and thermal energy determine the state of matter. In a fluid, thermal energy is sufficient for atoms or molecules to slide relative to one another. Because of the low barrier to relative motion, the fluid cannot sustain shear stress in its equilibrium state. This leads to the familiar consequence that the fluid will flow to take the shape of the container it occupies. However, unlike a gas, the attractive interactions between atoms or molecules in a fluid are sufficient to maintain a constant density. In other words, the fluid will not expand to fill the volume of its container.

Although the fluid may not sustain an equilibrium shear stress, the molecules within a deforming fluid may interact with one another giving rise to internal friction. The viscosity of the fluid is a measure of the internal friction between molecules, which leads to transient shear stresses within the deforming fluid.

As engineers, we seek to develop mathematical models that allow us to predict a system's response to external stimuli. For example, one might want to predict the strain that results when an object is subjected to a set of prescribed forces. In this text, we will discuss the development of a set of equations that describe the relationship between applied forces, thermodynamic variables, and deformation. The procedure presented for building a practical model of a material system consists of four major steps. First, we must identify the forces, fields, and thermodynamic variables that we would like to model. For example, we might be interested in modeling the material's response to changes in temperature and electric field. In nature, there are many forces and fields which influence the behavior of materials. A model that captures the coupling between all of these fields would be exceedingly complex. Instead, we must select the forces and field which are of primary interest or restrict the applicability of the mathematical model to a narrow range of external forces to simplify the model. Second, the balance laws and constitutive model must be formulated given the relevant variables and material characteristics determined in step one. The result of this second step is a set of mathematical equations describing the connections between the selected forces and fields in the given material system. Third, a strategy for parameterizing the constitutive model must be developed.

As we have seen in this textbook, constitutive models have many material parameters that must be determined experimentally. These parameters are often found by fitting the model's predicted behavior to the experimentally observed behavior. It is critical that the uncertainty in these material parameters be reported along with the values of the parameters. In this chapter, we introduce the methodology for providing uncertainty estimates for experimental measurements and for parameters obtained from curve fitting.

Propagation of Error

Experimental measurements suffer from both systematic and random error. For example, measurements from a force transducer used to measure load have systematic and random error due to the physical sensor and the data acquisition system used to acquire data. The error in the force measurements due to the sensor is a combination of systematic uncertainty due to nonlinearity and hysteresis of the sensor and random uncertainties due to thermal-stability error and repeatability error. The error from the data acquisition system is a combination of systematic uncertainty due to nonlinearity and gain error and random uncertainty due to quantization and noise. Often these errors are well documented by the producers of the measurement equipment. However, one must often take experimentally determined values and combine or manipulate them to report calculated quantities. For example, one might report a stress that was computed using a force measurement and measurement of the cross-sectional area of a specimen. In order to compute the error for a computed quantity that is a function of the distance or force measurement, we will need to propagate these errors through the equations used to compute the desired quantities.

This textbook is designed to give students an understanding and appreciation of continuum-level material modeling. The mathematics and continuum framework are presented as a tool for characterizing and then predicting the response of materials. The textbook attempts to make the connection between experimental observation and model development in order to put continuum-level modeling into a practical context. This comprehensive treatment of continuum mechanics gives students an appreciation for the manner in which the continuum theory is applied in practice and for the limitations and nuances of constitutive modeling.

This book is intended as a text for both an introductory continuum mechanics course and a second course in constitutive modeling of materials. The objective of this text is to demonstrate the application of continuum mechanics to the modeling of material behavior. Specifically, the text focuses on developing, parameterizing, and numerically solving constitutive equations for various types of materials. The text is designed to aid students who lack exposure to tensor algebra, tensor calculus, and/or numerical methods. This text provides step-by-step derivations as well as solutions to example problems, allowing a student to follow the logic without being lost in the mathematics.

The first half of the textbook covers notation, mathematics, the general principles of continuum mechanics, and constitutive modeling. The second half applies these theoretical concepts to different material classes. Specifically, each application covers experimental characterization, constitutive model development, derivation of governing equations, and numerical solution of the governing equations.