Working Content> MacroModels > Solids  

Prerequisites

• Solids
• Young's modulus
• Shear modulus
• Bulk modulus -- solids

Although we often talk about normal matter coming in three forms, solids, liquids, and gases, a more careful discussion of matter shows a more nuanced story. Both solids and liquids deform and rearrange their atoms and molecules in response to forces. A liquid is a material that changes its shape easily (flows) in response to forces. A solid on the other hand is defined as a material that can respond to forces but not by much. While an external force on a solid will deform it, we call a solid an elastic solid if it returns to its original size and shape when the force is removed. This is what we typically observe for small deformations of most solid materials we encounter in everyday life. 

However, many materials in nature do not behave in such clearcut ways. Even objects that appear to be solid (such as rocks and mountains) may deform if considered under situations of unusually large forces or long time scales. Rheology is the measurement and study of the relationship between the deformation of a material (measured by its strain -- the fractional deformation) and its mechanical forces (measured by its stress -- the force per unit area).

Idealized solids are characterized by a number of parameters that specify how stress and strain are to be measured, including directions. The three primary parameters, called moduli are:

• Young's modulus (E) describes the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of normal stress to normal strain. It is often referred to simply as the elastic modulus.
• The shear modulus or modulus of rigidity (G) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces. It is defined as ratio of shear stress to shear strain.
• The bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform in all directions when squeezed uniformly such as by being placed deep underwater. The bulk modulus is essentially an extension of Young's modulus to three dimensions.

Idealized liquids are also called "Newtonian" liquids.  Their response to force is to flow.  The flow speed in response to a force is not the same for all liquids, but is characterized by its viscosity.  For Newtonian liquids this parameter depends on the material, and can also depend on temperature, but it does not depend on how rapidly we apply a force on the material.

Many materials we encounter in our daily life are more complex in their response to forces than ideal liquids or solids.  Often we encounter soft or squishy materials, which are now being called soft matter.   Typical soft matter encompasses materials that really cannot to classified as either solids or liquids. For example, glue, ketchup, pastes, Jell-O – are they solid or liquid? They  seem to have properties of both. Pierre-Gilles de Gennes, who won the Nobel Prize in physics in 1991 for his work on such materials defined them to be materials on which a mild external influence has a large effect. Typically these are materials that are held together by weak inter-molecular interactions, such as van der Waals forces. Gels such as Jell-O are a common type of soft matter. Gels have a solid phase such as a three-dimensional network of crosslinked (interconnected) filaments interspersed within a liquid. Many of the gels found in biological systems, such as actin networks in cells, are comprised of networks of biological polymers that have both viscous and elastic properties.

A network of filaments of the protein actin which is a major component of all cells. Actin networks are viscoelastic. (Howard VIndin, wikimedia commons)

From our analysis of density, Young's modulus, and other properties, we know that many properties of objects can be defined as shape-independent parameters (or moduli) that only depend on the material of which the object is made. To get the response property of the particular object, you have to multiply that modulus by geometric parameters (such as volume to get mass from density, or area over length to get a spring constant from a Young's modulus).

To compare how “soft” materials are different from “hard” materials, take the case of an iron bar compared to an ordinary cup of yogurt.  As we may expect, the soft matter requires much less force to deform. Yoghurt turns out to be about 10⁹ times softer than the iron bar. Why is that so? We can actually zoom in and pinpoint the culprit: there are stronger forces between atoms in the iron bar than the forces in yoghurt. Indeed, in an iron bar the dominant forces are due to positive charges and negative charges interacting. In soft materials on the other hand the dominant forces come from weaker effects in particular polarization in one molecule inducing polarization in another (the so called van der Waals forces).  Thus, for most soft materials, the interaction forces are relatively weak. 

This has two effects.  First, the materials are more easily deformable.  Second, the materials may already be deformed by the small random forces that naturally occur due to temperature (and e.g. lead to Brownian motion). As a result, these materials do not behave like crystalline solids or free-flowing liquids. A hallmark of soft materials is that the macroscopic material properties can be independent of their microscopic chemical identities, instead being set by the interaction strengths and typical length scales.

Many soft materials behave in a way that combines the viscous and elastic response depending on the rate of deformation or the timescales involved. This is known as viscoelasticity. A common example is Jello. When you set it down on a plate, it oscillates like a solid (the elastic response) but only for a few times before stopping. This is because the internal friction (viscosity) strongly damps the motion. Another example is silly putty. If one applies a stress on a slow timescale this material flows like a very viscous liquid, but if one rolls it up and drops it on to a hard surface, it bounces like an elastic solid. This type of behavior is known as viscoelasticity. A viscoelastic material responds to an applied stress in a time-dependent manner.

In general, if the deformation is rapid (e.g. a quick tap) viscoelastic materials behave like a solid but if the deformation is slow (slow shear), they behave like a liquid. For a constant applied stress, a viscoelastic material first responds in an elastic manner with a constant strain. After a certain time (τ) it begins to flow like a liquid, with the strain increasing linearly with time (see Figure 2). The time τ is the relaxation time, which separates the solid-like behavior from the liquid-like behavior. If a stress is applied on time scales shorter than the relaxation time, the material will behave like a solid, as the silly putty does when we bounce it on the floor. For stresses applied at longer times, the material flows. If we define an elastic modulus G0, which characterizes the elastic response of the material at times shorter than the relaxation time, and we characterize the viscous behavior at longer time by a viscosity η, we can write the following relation: η ~ G0 τ

Schematic strain response of a viscoelastic material to a shear stress applied at time t=0 and then held constant.

Arpita Upadhyaya 10/22/13

Wolfgang Losert 10/23/13