ES240 Solid Mechanics Fall 2011
7. Finite Deformation
References
G.A. Holzapfel, Nonlinear Solid Mechanics, Wiley, 2000.
T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley, 2000.
L.E. Malvern, Introduction to the Mechanics of Continuous Medium, Prentice-Hall, 1969.
C. Trusdell and W. Noll, The Non-linear Field Theories of Mechanics, 3rd edition, Springer, 2004.
C. Trusdell and R.A. Toupin, The Classical Field Theories, in Encyclopedia of Physics, Volume III/1, Pringer-Verlag, Berlin, 226-793 (1960).
L.R.G. Treloar, The Physics of Rubber Elasticity, 3rd ed. 1975. Reissued in 2005.
Be wise, linearize. Following the advice of George Carrier, we have been mostly looking at Hookean materials and infinitesimal deformation. We have mixed the 3 ingredients in solid mechanics (deformation geometry, momentum balance, material law) without fussing over subtleties. The results are fascinating and useful. Now we wish to go nonlinear, hopefully also with wisdom. We will refine the ingredients by considering non-Hookean materials and finite deformation. The two refinements need not be mixed. For example, a viscoelastic material is non-Hookean, but deformation of such a material can be infinitesimal. In this brief introduction to finite deformation, we will outline some of the fundamental considerations, and describe a few illustrative phenomena.
Finite deformation. When a structure deforms, Newton’s law holds true in every deformed state. We have often violated this law. For example, in analyzing a truss, we have balanced forces as if the truss did not deform.
You might think that a structure suffering a small strain, say less than 1%, entitles you to neglect the change in geometry when you balance forces. A counter example is familiar to you. Upon buckling, the strains in a column are indeed small, but you must enforce equilibrium in the deflected state of the column. Mechanics of deformation is a tricky business. We proceed with caution. The essential point is this: we must enforce Newton’s law in every deformed state, and justify any simplification on this basis.
Non-Hookean materials. Moving nonlinear and inelastic will take us in many directions. For example, when deformation is large, the force may vary nonlinearly with the elongation. As another example, we’ve already looked at time-dependent behavior of materials, such as viscoelasticity. we also have daily experience of metals. After elastic deformation, upon unloading, a metal recovers its shape. After plastic deformation, upon unloading, the metal does not fully recover its shape. If we apply an axial force to a metal bar, and measure its length, the force-length relation is linear for elastic deformation, and is nonlinear for plastic deformation. During unloading, the metal bar deforms elastically. After plastic loading and elastic unloading, the force-length relation is not a one-to-one relation, but is history-dependent. Of course, a viscoelastic material is also history-dependent.
To analyze finite, history-dependent deformation of a structure, a general approach is to evolve the state incrementally, and enforce Newton’s law in every state.
A rod under axial load. We also proceed with our subject incrementally, beginning with a rod in incremental states of uniaxial stress. Initially, the rod is unstressed, and has cross-sectional area and length . The rod is then subject to an axial force P, and deforms to cross-sectional area a and length l. We next examine the 3 ingredients in solid mechanics.
Strain measures. Any state can be used as a reference state. For example, we can take the initial, unstressed state as the reference state. Define the engineering strain by the elongation divided by the reference length:
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Another strain measure is defined as follows. Deform the material from a current length l by a small amount to . Define the increment in the strain, , as the increment in the length of the rod divided by the current length of the rod, namely,
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This equation defines the increment of natural strain. Integrating from to l, we obtain that
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Yet one more strain measure, the Lagrange strain, is defined as
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This definition is hard to motivate in 1D. But if you take the view that any increasing function of is a suitable measure of strain, then no motivation is really needed.
Indeed, even the ratio itself has a name: the stretch is defined as the length of the rod in the current state divided by the length of the rod in the reference state:
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There seems to be no lack of human ingenuity to form a dimensionless quantity out of two lengths L and l. Needless to say, all these strain measures contain the same information. For example, every one of the measures defined above is an increasing function of stretch:
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Because they are all one-to-one functions, any one measure can be taken to be “basic” and then used to express all the rest. For example, we can express all measures in terms of the engineering strain:
Thus, when we call e the engineering strain, we do not mean that e is unscientific or crude or unnatural strain. We just need a name. When the strain is small, namely, , the three measures are approximately equal, .
Later on, we will provide motivations for some of these definitions, but these motivations are probably elaborate ways to express preferences of individual people.
Stress measures. Work done by a force. When dealing with finite deformation, we must be specific about the area used in defining the stress. Define the nominal stress, s, as the force in the current state divided by the area in the reference state:
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When the rod elongates from length l to length , the force P does work . Recall that and , so that the work done by the force is
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Since AL is the volume of the rod in the reference state, we note that
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We say that the nominal stress and the engineering strain are work-conjugate. Also note that . Consequently, the stretch is also work-conjugate to the nominal stress.
Define the true stress, , as the force in the current state divided by the area in the current state, namely,
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Recall that . The work done by the force is
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Since is the current volume of the bar, we note that
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That is, the true stress is work-conjugate to the natural strain.
Given a measure of strain, we can define its work-conjugate stress. For example, consider the Lagrange strain, . Subject to an increment in the strain, , the force acting on the element does the work. Denote
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This expression defines a new measure of stress, S.
This new stress measure does not have a “simpler” interpretation than its status as the work conjugate to the Lagrange strain. Indeed, if we are liberal about the definition of strain measures, without being obsessive about “motivating” each measure, we may as well take a liberal view to call the work conjugate of each strain a stress measure, and name the stress measure after a mechanician who can no longer protest. You can easily invent and name other stress measures, but the above stress measures have already got names:
· : true stress or the Cauchy stress.
· : nominal stress or the first Piola-Kirchhoff stress.
· : the second Piola-Kirchhoff stress.
Recall the relations among the measures of strain:
We obtain the relations among their increments:
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Consequently, the three measures of stress are related as
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All stress measures are linear in force applied to the cross-sectional area, but differ by geometric factors.
Material laws. For a metal undergoing large, plastic deformation, the stress-strain curve (without unloading) is often fit to a power law in terms of the true stress and the natural strain:
where K and N are parameters to fit experimental data. Some representative values: N = 0.15-0.25 for aluminum, N = 0.3-0.35 for copper, N = 0.45-0.55 for stainless steel. K has the dimension of stress; it represents the true stress at strain . Representative values for K are 100 MPa – 1GPa. At large deformation, volumetric strain is negligible compared to tensile strain. Consequently, the material is often taken to be incompressible.
Rubbers are often assumed to obey the neo-Hookean law (more details later). For a rubber rod in uniaxial states of stress, the stress-strain data are fit to
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Recall that . For small strains, , the above reduces to
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Thus, we interpret as Young’s modulus and the shear modulus. Rubbers are nearly incompressible, so that Poisson’s ratio is taken to be ½. Representative values for are 1 MPa – 100 MPa.
Are these alternatives necessary? Now we have described the 3 ingredients for a rod under uniaxial tension. Even in this simplest setting, for each ingredient we have given several alternative descriptions. Some alternatives are necessary; for example, metals and rubbers behave differently. But the difference in their force-displacement relations does not justify us to use different stress and strain measures to describe different materials. In fact, to see the difference in material behavior, we would like to use the same stress and strain measures for both materials. For example, we can use the natural strain to describe the stress-strain relation for rubbers:
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This change of variable immediately brings out a key insight: in tension, the stress in rubbers rises more steeply than in metals. We will return to this insight shortly.
Are these alternative stress and strain measures necessary? I have my own thoughts, but you should form your own opinions. The question perhaps boil down to something no more profound than asking, “Is it necessary to know many alternative roads to Boston Common?” Whatever your opinions are, however, it may alleviate some of your pains in studying the subject by knowing that textbooks of nonlinear continuum mechanics are full of equivalent alternatives at every turn. These alternatives often hide behind forests of notation and verbiage, and may offer some tantalizing sights. You will just have to look beyond them for matters of consequence.
Exercise. Use the 3 ingredients outlined about to obtain the force-deflection relation for the truss sketched in the beginning of the notes. Assuming all three members of the truss are made of rubber bands, and that deformation is large.
Necking in a bar. Considère condition. Let us try to apply the newly refined 3 ingredients to a specific phenomenon: necking. Subject to a tensile force, a metal bar first elongates uniformly and then, at some strain, a small part of the rod starts to thin down preferentially, forming a neck. By contrast, a rubber band under tension usually does not form a neck. We would like to interpret these observations. To do so we must be explicitly specify the measures of stress and strain.
Here is a summary of the 3 ingredients, using a specific set of alternatives:
· Force balance:
· Material law: For a metal bar under uniaxial tension, the true stress relates to the natural strain as .
· Deformation geometry: . We will assume that the volume of the rod is constant during deformation, , or .
Put the three ingredients together, and we obtain the force as a function of strain:
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Plot P as a function of e. In plotting the figure, I’ve set , with N = 0.5. Observe the two competing factors: material hardening and geometric softening. As the bar elongates, the material hardens, as reflected by the hardening exponent in the stress-strain relation . At the same time, the elongation reduces the cross-sectional area, an effect known as geometric softening. For small deformation, as ; material hardening prevails, and the force increases as the bar elongates. For large deformation, so long as the stress-strain relation increases slower than , as , geometric softening prevails, and the force drops as the bar elongates.
To determine the peak force, note that
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Consequently, the force P peaks when the true stress equals the tangent modulus:
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This equation, known as the Considère condition, determines the strain at which the force peaks. For the power-law material, the force peaks at the critical strain
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When the metal bar is loaded beyond this critical strain, deformation becomes nonuniform, with a segment of the bar elongates at a higher strain than the rest of the bar. That is, a neck forms in the bar. For an analysis of the nonuniform deformation, see Needleman (1972, A numerical study of necking in circular cylindrical bar, Journal of the Mechanics and Physics of Solids, 20, 111). ABAQUS can be used to study the necking process.
For a rubber band, assume the material is Neo-Hookean:
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Thus, at a large tensile strain, the true stress increases exponentially with the natural strain, so that the force always increases with the strain. The rubber band will not form a neck under uniaxial tension.
Hyperelastic materials. We next explore nonlinear stress-strain relations under multiaxial states of stress. Of course, the only way to really know such relations is to run tests for a given history of state of stress, but this would be too time-consuming and quickly become impractical. We’ll have to reduce the number of tests by some approximations. The art of making such compromise between accuracy and labor is known as formulating constitutive laws. As an example, here we attempt to describe this art for rubbers.
A rubber rod, length L and cross-sectional area A in the unstressed state, is stretched by force P to length and cross-sectional area a. When the rod extends from length l to length , the force does work . Recall that and , so that the work is