Stresses occur in any material that is subject to a load or any applied force. There are many types of stresses, but they can all be generally classified in one of six categories: residual stresses, structural stresses, pressure stresses, flow stresses, thermal stresses, and fatigue stresses.Residual StressesResidual stresses are due to the manufacturing processes that leave stresses in a material. Welding leaves residual stresses in the metals welded. Stresses associated with welding are further discussed later in this module.Structural StressesStructural stresses are stresses produced in structural members because of the weights they support. The weights provide the loadings. These stresses are found in building foundations and frameworks, as well as in machinery parts.Pressure StressesPressure stresses are stresses induced in vessels containing pressurized materials. The loading is provided by the same force producing the pressure. In a reactor facility, the reactor vessel is a prime example of a pressure vessel.Flow StressesFlow stresses occur when a mass of flowing fluid induces a dynamic pressure on a conduit wall. The force of the fluid striking the wall acts as the load. This type of stress may be applied in an unsteady fashion when flow rates fluctuate. Water hammer is an example of a transient flow stress.Thermal StressesThermal stresses exist whenever temperature gradients are present in a material. Different temperatures produce different expansions and subject materials to internal stress. This type of stress is particularly noticeable in mechanisms operating at high temperatures that are cooled by a cold fluid. Thermal stress is further discussed in Module 3.Fatigue StressesFatigue stresses are due to cyclic application of a stress. The stresses could be due to vibration or thermal cycling. Fatigue stresses are further discussed in Module 4.The importance of all stresses is increased when the materials supporting them are flawed. Flaws tend to add additional stress to a material. Also, when loadings are cyclic or unsteady, stresses can effect a material more severely. The additional stresses associated with flaws and cyclic loading may exceed the stress necessary for a material to fail.Types of Applied StressStress intensity within the body of a component is expressed as one of three basic types of internal load. They are known as tensile, compressive, and shear. Figure 1 illustrates the different types of stress. Mathematically, there are only two types of internal load because tensile and compressive stress may be regarded as the positive and negative versions of the same type of normal loading.However, in mechanical design, the response of components to the two conditions can be so different that it is better, and safer, to regard them as separate types.As illustrated in Figure 1, the plane of a tensile or compressive stress lies perpendicular to the axis of operation of the force from which it originates. The plane of a shear stress lies in the plane of the force system from which it originates. It is essential to keep these differences quite clear both in mind and mode of expression.FORCE FORCE

Figure 1 Types of Applied StressTensile StressTensile stress is that type of stress in which the two sections of material on either side of a stress plane tend to pull apart or elongate as illustrated in Figure 1(a).

Theories of FailureFailure is generally perceived to be fracture or complete separation of a member. However, failure may also occur due to excessive deformation (elastic or inelastic) or a variety of other reasons.

During the latter part of the 19th century and continuing up to the present, a number of basic failure theories were proposed and tested on a few materials.1 Most of the theories were based on the assumption that failure occurs when some physical variable such as stress, strain, or energy reaches a limiting value.Deformation: Elastic deformation is temporary (reversible) and involves bond stretching. Plastic deformation is permanent (irreversible), and involves bond breaking. Fracture is catastrophic.Maximum Principal Stress Theory (Rankine, Lam) Applied satisfactorily to many brittle materials, the theory is based on a limiting normal stress. Failure occurs when the normal stress reaches a specified upper limit.1Failure is predicted when either of the principal stresses, 1 or 2, equals or exceeds the yield strength, yp, of the material.3

1 < yp2 < yp

ExamplesClick on image for full size.

Maximum Shear Stress Theory (Tresca, Guest, Coulomb) Applied satisfactorily to ductile materials, the theory is based on the concept of limiting shearing stress at which failure occurs.1 Failure by yielding in a more complicated loading situation is assumed to occur when the maximum shearing stress in the material reaches a value equal to the maximum shearing stress in a tension test at yield. This yield criterion gives good agreement with experimental results for ductile materials; because of its simplicity, it is the most often used yield theory.2 The main objection to this theory is that it ignores the possible effect of the intermediate principal stress, 2. However, only one other theory, the maximum distortional strain energy theory, predicts yielding better than does the Tresca theory, and the differences between the two theories are rarely more than 15%.Failure is predicted when any of the three shear stresses corresponding to the principal stresses, 1,2, equals or exceeds the shear stress corresponding to the yield strength, yp, of the material in uniaxial tension or compression.3

Maximum Octahedral Shearing Stress Theory Failure by yielding in a more complicated loading situation is assumed to occur when the octahedral shearing stress in the material reaches a value equal to the maximum octahedral shearing stress in a tension test at yield.Plane stress caseUniaxial stress case

Note: This theory gives the same results as the maximum distortion energy theory.Maximum Principal Strain Theory (Saint-Venant) The theory is based on the assumption that inelastic behavior or failure is governed by a specified maximum normal strain.1 Failure will occur at a particular part in a body subjected to an arbitrary state of strain when the normal strain reaches a limiting level.Failure is predicted when either of the principal strains, resulting from the principal stresses, 1,2, equals or exceeds the maximum strain corresponding to the yield strength, yp, of the material in uniaxial tension or compression.3

1 - (2 + 3) < yp

Total Strain Energy Theory (Beltrami-Haigh) Applicable to many types of materials, the theory predicts failure or inelastic action at a point when the strain energy per unit volume exceeds a specified limit.1Failure is predicted when the total strain energy associated with the principal stresses, 1,2, equals or exceeds the total strain energy corresponding to that for the yield strength, yp, of the material in uniaxial tension or compression.3

12 + 22 + 32 -2(12 + 23 + 13) < yp2

Maximum Distortion Energy Theory (Huber-Henky-von Mises) The theory is based on a limiting energy of distortion, i.e. energy associated with shear strains.1 Strain energy can be separated into energy associated with volume change and energy associated with distortion of the body. The maximum distortion energy failure theory assumes failure by yielding in a more complicated loading situation to occur when the distortion energy in the material reaches the same value as in a tension test at yield. This theory provides the best agreement between experiment and theory and, along the Tresca theory, is very widely used today.2 Note: This theory gives the same results as the octahedral shear stress theory.Failure is predicted when the distortional energy associated with the principal stresses, 1,2, equals or exceeds the distortional energy corresponding to that for the yield strength, yp, of the material in uniaxial tension or compression.3

0.5 ( (1 - 2)2 + (2 - 3)2 + (3 - 1)2) < yp2

Below is a summary of two of most popular theories of failure applied to a simple uniaxial stress state and to a pure shear stress state.

Failure Criteria

TheoryLoadingRelationship

UniaxialPure Shear

Maximum principal stressmax = YPmax = YPYP = YP

Maximum principal strainmax = YP / Emax = 5YP / 4EYP = 0.8 YP

Maximum octahedral shear stressYP = 0.577 YP

Maximum distortional energy densityYP = 0.577 YP

Maximum shear stressmax = YP / 2max = YPYP = 0.5 YP

Types of Failure

There are two basic types of material failure: Fracture - or brittle failure - occurs when a material breaks in two after only a small amount, if any, plastic deformation. Ceramics (e.g., silicon carbide), concrete and glass are examples of materials that exhibit brittle failure. Yielding - or ductile failure - occurs when a material undergoes permanent (plastic) deformation. Metals such as aluminum, steel and copper are examples of materials that exhibit such failure.A state of plane stress has three non-zero stress values: x, y, and xy. While we can assess when a material fails when subjected to a uniaxial stress (x-only), or for shear-stress-only (xy) - see Chapt.3- we do not yet have a good model to determine when a material point fails under a general state of stress (x, y, and xy). Three failure theories are now introduced: Maximum Normal Stress Failure Theory (brittle materials) Maximum Shear Stress Failure Theory (ductile materials) von Mises Failure Theory (ductile materials)

Maximum Normal Stress

The Maximum Normal Stress criterion states that when the maximum normal stress in any direction of a brittle material reaches the strength of the material - the material fails. Thus, finding the Principal Stresses at critical locations is important. Failure occurs when: I > Su or II > Su (tension)

|I| > |Sc| or |II| > |Sc| (compression)

Su is the ultimate strength in tension Sc is the ultimate strength in compression Sc > Su for brittle materials; a typical ratio is Sc ~ 10SuThe above plot is a Failure Map. If the Principal Stresses fall outside of the shaded area, failure occurs.

Tresca - Maximum Shear Stress - for Plane Stress only

The Tresca Yield states that a ductile materials yields (fails) when the Maximum Shear Stress exceeds the shear strength y the material yields. The Maximum In-Plane Shear Stress is the average of the in-plane Principal Stresses.

The TWO Maximum Out-of-Plane Shear Stresses are:Note that the Out-of-Plane Principal Stress (III) for the plane stress condition is zero.Failure occurs when the maximum of the three Maximum Shear Stresses

reaches the shear yield stress, y.The above plot is a Failure Map. If the in-plane Principal Stresses lie outside of the shaded zone, failure occurs. Under a uniaxial load, II = III = 0. Thus, the axial yield stress isI = SY = 2y. The Maximum Shear Stress Theory predicts that the Shear Yield Stress is half the Axial Yield Stress. When the In-Plane Principal Stresses are the same sign (1st and 3rd quadrant), the Maximum Shear Stress in the system is Out-of-Plane.When the In-Plane Principal Stresses are opposite sign (2nd and 4th quadrant), the Maximum Shear Stress in the system is In-Plane.

von Mises - Maximum Distortion Energy

The von Mises Yield Criterion states that a ductile material fails (yields) when the von Mises Stress o exceeds the yield strength Sy. The von Mises Stress (or Equivalent Stress) is defined by:

When o = Sy the material is deemed to have yielded.For plane stress (z = xz = yz = 0) the von Mises Failure Criterion reduces to:

Using the general relationship with x = y = 0, the von Mises criterion predicts that ratio of the axial yield strength to the shear yield strength is: Sy = 1.732y.From the Tresca condition the predicted relationship between the yield strength and shear yield strength is: Sy = 2y.In general, metals tend to follow the axial yield strength-shear yield strength relationship of von Mises, making von Mises more accurate. However, von Mises is slightly harder to use, and any system that falls within the Tresca boundary, also falls within the von Mises boundary.

von Mises Failure Surface The above plot is a Failure Map. If the In-plane Principal Stresses lie outside the shaded zone, failure occurs. The dashed lines indicate the Tresca failure surface.

Mohrs CircleStep by step is available in http://web.mst.edu/~medialab/fipse/preview/Philpot/mohr_stress/a_draw.htm