Energy Methods Energy Methods Presented by Calvin M. Stewart, PhD MECH 5390-6390 Fall 2020 Outline
• Motivation • Energy Balance • Energy Release Rate, G • Fixed Grip and Load Conditions
• Relationship between G and KI Motivation Motivation
• In this previous lectures, we focused on the stress intensity factor, K approach where the factor is used to regularize the singular stress field. • Challenges • Must Derive the Stress Field Equations specific to the geometry and load conditions. • Mixed Loading (Mode I, II, and III) are difficult to describe analytically • Limited Plasticity Motivation
• Stress Intensity Factor (SIF) is a fracture mechanics parameter used to characterize the strength of the singularity at the crack tip, denoted by K. • The Critical Fracture Toughness is a cracked material’s ability to resist fracture, denoted by Kc. It is considered a material property and commonly used in design.
Stress Intensity Factor (SIF) Equation Critical Fracture Toughness Property KKIc
Equation. Function of Material Property Function of • Loading Conditions Environment • Body Geometry Surface Roughness • Crack Geometry Microstructural Mechanisms Motivation
According to the First Law of Thermodynamics, when a system goes from a nonequilibrium state to equilibrium, there will be a net decrease in energy. • AA. Griffith (1920), The Phenomena of rupture and flow in solids, Phys Transactions Vol 221. • It may be supposed for the present purpose, that the crack is formed by a sudden annihilation of the tractions acting on its surface. At the instant following this operation, the strain, and therefore the potential energy under consideration, have their original values; but in general, the new state is not one of equilibrium. If it is not a state of equilibrium, then, by the theorem of minimum potential energy, the potential energy is reduced by the attainment of equilibrium; if it is a state of equilibrium the energy does not change. Motivation
• A crack can form (or an existing crack can grow) only if such a process causes the total energy to decrease or to remain constant. Thus the critical conditions for fracture can be defined as the point where crack growth occurs under equilibrium conditions, with no net change in total energy. • The Energy Concept 1. Energy is needed to grow a crack 2. Energy is released by relaxing the body If 2>1 the crack will grow
Energy is a scalar and often easier to deal with then the stress fields Energy Balance Energy Balance
• In this section, the energy balance approach will be considered for an arbitrary loading condition.
• In the energy balance approach a cracked elastic plate and its loading system are considered.
• The combination of plate and loading system is assumed to be isolated from its surroundings, i.e. exchange of work can only take place between the two. Energy Balance
• The energy content of the plate plus the loading system, denoted as the total energy U, is written as Energy Balance
• In order to understand why the work F must be subtracted in equation (4.1), consider a plate placed in series with a spring between fixed grips. • The spring loads the plate in tension. This represents an arbitrary loading condition, since both load and displacement of the plate will change during the introduction of a crack. • Furthermore, it is clear that no work can be performed from outside the combination of spring and plate, since no external displacements are allowed. Energy Balance
• When a crack is introduced, the stiffness of the plate is reduced and the plate becomes somewhat longer. • Consequently the spring becomes shorter by the same amount. During this process, the spring performs an amount of work F on the plate. • This happens at the expense of the elastic energy content of the spring, which thus decreases. • Since the elastic energy content of the spring is part of the total energy U, it follows that F must be subtracted in equation (4.1) Energy Balance
• At this point we define the potential energy of an object. • Potential energy is either related to the position of an object in a conservative power field or due to its state. • An evident example of the first is the potential energy of a mass which is determined by its position (height) in a gravitational field. • Elastic strain energy is an example of potential energy which is due to the state of an object. • In all cases potential energy is characterized by its ability (potential) to perform work. Energy Balance
• Only part of the total energy U given by equation 4.1 performs work.
• This part will be designated as the potential energy, Up, of the elastic plate and its loading system and is equal to
• Note: the change in surface energy does not appear. Energy Balance
• In general, for a centre cracked plate under arbitrary loading conditions U will vary as a function of half crack length, a, according to the schematic plot. • Crack growth instability will occur as soon as U decreases with the crack length, i.e. after U has reached a maximum value. This condition is given by dU.
The variation of the total energy of a centre cracked plate, U, as a function of half crack length a. Energy Balance
• This crack instability condition is given by
• where since U0 is constants. Energy Balance
• Finally we arrive at the following inequality,
• The left-hand side is the decrease in potential energy if the crack were to extend by da. • During this same crack extension the surface energy would increase by an amount given by the right-hand side. • In other words, equation (4.5) states that crack growth will occur when the energy available for crack extension is larger than the energy required. Energy Release Rate, G Energy Release Rate, G
• Irwin defined the energy available per increment of crack extension and per unit thickness as the energy release rate, G. • When considering a central crack with length 2a, an increment of crack extension is d(2a) and therefore:
Note that for a central crack, with length 2a, G is found by differentiating Up to d(2a), while for an edge crack, with length a, differentiating to da would be sufficient. In both cases G is found as the negative derivative of the potential energy with respect to the newly formed crack area dA, where this area is defined as the projection normal to the crack plane of the newly formed surfaces. Energy Release Rate, G
• The energy required per increment of crack extension is defined as the crack resistance, R:
• Thus our inequality can be rewritten as Energy Release Rate, G
• Now let’s consider the elastic energy change in a remotely loaded centre crack plated. • In the LEFM II lectures, we derived an expression for the plane stress crack flank displacement of a central crack with length 2a in an infinite plate remotely loaded by a tensile stress.
• Here we denoted the remote load as σ∞ giving the crack flank displacement, V as
• where by changing x we can find it anywhere along the crack.
Energy Release Rate, G
• Using this, we can find the change in elastic energy of the plate, Ua, by considering an open crack with stress-free crack flanks, and calculating how much work is involved in closing the crack.
• The situation of a closed crack resembles that without a crack, since in both cases a uniform stress field σ∞ is present.
• Furthermore, closing the crack increases the elastic energy by an amount equal to the work involved. Energy Release Rate, G
• For a flank length dx, the work involved in closing the crack is:
• In this calculation linear elastic material behaviour is assumed, i.e. when the stress σ increases from 0 to σ∞, the displacement ξ increases linearly from 0 to V. The work involved in closing the whole crack is found by integrating along the crack from -a to +a, which is the same as integrating twice from 0 to +a. Energy Release Rate, G
• The change in elastic energy, Ua, involved in creating a central crack with length 2a is the negative value of this work and is given by:
• For plane strain replace E with E/(1-v^2) Fixed Grip and Constant Load Conditions Fixed Grip and Constant Load Conditions
• In a centre cracked plate we have crack extension when, according to equations (4.6) -(4.8),
• A finite plate under fixed grip conditions resembles an infinite plate because no work is performed by external forces, i.e. F = constant during crack growth. • A difference is, however, that crack extension reduces the plate stiffness and so causes the load to drop. Fixed Grip Condition
Under fixed grip conditions with a displacement V, the load on the plate will drop from P to P + ΔP (i.e. ΔP < 0) when the crack extends by Δa at both tips. Fixed Grip and Constant Load Conditions
• Knowing this, it is to be expected that for a finite plate under fixed grip conditions dUa/d(2a) is not the same as for an infinite plate. • It can be argued that in a finite plate loaded under fixed grip conditions the change in Ua owing to crack extension approaches that of an infinite plate if the crack size 2a is small compared to the plate’s dimensions. • For such a plate we can write Fixed Grip and Constant Load Conditions
• The surface energy, Uγ, is equal to the product of the surface tension of the material, γe, and the surface area of the crack (two surfaces with length 2a):
• Therefore, the crack resistance reduces to • The Crack extension inequality becomes Fixed Grip and Constant Load Conditions
• In the case of constant load, then crack extension results in increased displacement owing to decreased stiffness of the plate.
• The situation is thus more complicated than that for the fixed grip condition.
• For a constant load condition with a load P, the same crack extension results in a displacement from V to V + ΔV (i.e. ΔV > 0). Constant Load Condition
For a constant load condition with a load P, the same crack extension results in a displacement from V to V + ΔV (i.e. ΔV > 0). Fixed Grip and Constant Load Conditions
• We will now consider the change in the work performed by external forces, ΔF, and the change in elastic energy, Δ Ua, involved in both fixed grip and constants load cases of crack extension.
• These changes are expressed in terms of areas in the load- displacement diagram Fixed Grip and Constant Load Conditions
• Realistically any amount of crack extension Δa indicates fracture. • If the crack extension Δa → 0, the area IV will become negligibly small compared to area I and Δ(F - Ua) will become equal for the two loading conditions.
• Thus for both fixed grip and constant load conditions the criterion for crack extension in a remotely loaded large plate with a small crack is Energy Release Rate, Criterion G >R
• For large plates with small central cracks G may be obtained as Relationship between G and KI Relationship between G and KI
• A relation of prime importance is obtained by substituting KI, an expression which also is valid only for a large plate with a small central crack, in the G equations:
• This direct relation between G and KI means that under LEFM conditions the prediction of crack growth and fracture is the same for both the energy balance and elastic stress field approaches. Irwin already demonstrated this equivalence Summary
➢ The Stress Intensity Factor approach can be difficult to derive for a specific geometry and loading condition. ➢ The Fracture Energy Approach provides a scalar quantity obtained from the Laws of Thermodynamics ➢ When explore fixed grip versus constant load conditions, energy release rate – crack resistance inequality remains the same. ➢ The energy release rate, G has a direct relationship to the Stress Intensity Factor, KI Homework 7
1. The total energy U of a loaded elastic body containing a crack is given by: U = Uo + Ua + Uγ – F a) What do the terms mean? b) What is the energy available for crack growth? c) What is the energy required for crack growth? Homework 7
2. Homework 7
3. References
• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd Edition, Spon Press • Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press. • Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall • Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture Mechanics of Engineering Materials, 5th Edition, Wiley. • https://www.fracturemechanics.org/ Calvin M. Stewart Associate Professor Department of Mechanical Engineering The University of Texas at El Paso CONTACT 500 W. University Ave, Suite A126, El INFORMATION Paso, TX 79968-0521 Ph: 915-747-6179 [email protected] me.utep.edu/cmstewart/