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.

In this chapter, we discuss constitutive model development for elastic materials. A material is considered purely elastic if it lacks the means to internally dissipate energy during deformation. Thus, a purely elastic material will not exhibit hysteresis or rate dependence. In addition, the deformation of an elastic material is reversible. Removal of load will allow the material to regain its original shape. A thermoelastic material model was used to introduce the concepts of constitutive modeling in Chapter 4. While many of the results from the analysis in Chapter 4 will be restated here, the reader may refer back to Chapter 4 for the detailed derivations.

While first developing the general framework for finite thermoelasticity, we will also discuss specialization of the model for isothermal finite elasticity, which assumes constant uniform temperature fields, hyperelastic materials for which there exists a strain energy function, and linear thermoelastic materials for which stress and strain are proportional. While the linear dependence between stress and strain works well for some materials, many engineering materials such as natural and synthetic polymers exhibit strong nonlinearity of the stress-strain curve even before yielding. This leads to a material nonlinearity in the constitutive response functions. In addition, subjecting any material model to large strains leads to a geometric nonlinearity. Large strain deformation is referred to as finite deformation. Interestingly, some of these polymers may be deformed to very large strains and recover their original shape when released.

Introduction

As explained in the previous chapter, damage affects the overall stress–strain response of the solid continuum body. Damage mechanics pertains to the study of this effect. Two widely different subfields have emerged over the years in this field. One concerns study of damage directly at the scale of formation of cracks, i.e., the microstructural scale, and hence can be called “micro-damage mechanics” (MIDM). The other approach, on the contrary, looks at the overall response at the macro or structural scale by using some internal variables to characterize damage, and thus can be termed as “macro-damage mechanics” (MADM). These terms were originally coined by Hashin [1]. MADM is the same as “continuum damage mechanics” (CDM), which is still the commonly used terminology.

MIDM for composite materials is derived from an older and more mature field called micromechanics that deals with overall properties of heterogeneous materials (see, e.g., [2]). In micromechanics one views heterogeneities such as inclusions and voids as “microstructure” and estimates overall properties by various methods, e.g., averaging schemes such as self-consistent and differential schemes, or variational methods to obtain bounds to average properties. Microcracks are treated as limiting geometry of microvoids, such as ellipsoidal voids with one dimension much smaller than the other two. As illustrated in the previous chapter, “damage” in composite materials has significant complexities concerning the geometry as well as evolution characteristics such as multiplication of cracks within a fixed volume. For these reasons a simple extension of micromechanics to damage in composites is generally not possible. A separate field identified as MIDM has therefore emerged. This chapter will treat the features of MIDM that have been developed to specifically treat certain cases of damage in composite materials. Since determining local (micro-level) stress or displacement fields is a necessary feature of micromechanics, it is expected that not all cases within the wide range of damage in composites can be handled by MIDM. However, this limitation can be alleviated by incorporating computational solutions of the local stress or displacement fields, thereby broadening classical micromechanics to include so-called computational micromechanics. In the most recent versions of MIDM this strategy has been used. More on this will be discussed toward the end of this chapter.

Preface

The field of composite materials has advanced steadily from the early developments during the 1970s when laminate plate theory and anisotropic failure criteria were in focus to today's diversification of composite materials to multifunctional and nanostructured composite morphologies. Throughout the 1970s and 1980s several books appeared along with courses that were developed and taught at advanced levels dealing with mechanics of composite materials and structures. The failure analysis was mostly limited to descriptions of strength that extended previous continuum descriptions of metal yielding and failure. Beginning around the mid-1980s, micromechanics and continuum damage mechanics were applied to multiple cracking observed in composite materials. Under the overall description of “damage mechanics” a flurry of activities took place as evidenced by conferences and symposia. Other than several conference proceedings that recorded such activities, a collection of seminal contributions to the field appeared in a volume (Damage Mechanics of Composite Materials, R. Talreja, ed., Composite Materials Series, R.B. Pipes, series ed., Vol. 9, Amsterdam: Elsevier Science Publishers, 1994). The two main avenues of approach to damage in composite materials and its effect on materials response, now referred to as micro-damage mechanics (MIDM) and macro-damage mechanics (MADM), were presented in a balanced form in that volume. In the years since then, many developments have taken place that have brought this field to such level of maturity that a book coherently presenting the material was felt to be timely. It is hoped that this book will help provide impetus for teaching advanced courses in composite damage at universities as well as support short courses for professional development of engineers in industry. The wealth of material covered can also help new researchers in advancing the field further. To this end, the last chapter provides some guidance in identifying gaps and needs for further work.