Fundamentals of Mechanics of Materials

In this lecture the fundamental definitions and concepts used in the Mechanics of Materials are presented.

In this textbook only the time independent loads, the static loads, are considered to act on the three-dimensional solid. Consequently, the inertial forces are zero and the necessary conditions for equilibrium are expressed as:

(1)

(2)

where the notation stands for:

- is the number of concentrated forces;

- is the force resultant vector and its components;

are the magnitude of the components ofalong the coordinate axes;

- is a particular force;

- are the magnitude of the components ofalong the coordinate axes;

- are the Cartesian unit vectors of the system;

- is the moment resultant vector and its components calculated at the arbitrary point, the origin of the coordinate system;

- are the magnitude of the components of along the coordinate axes;

- is the position vector extending from the coordinate origin to the point where the concentrated force is applied.

The vectorial equations (1) and (2) can be re-written in a scalar form of six algebraic equations:

(3)

(4)

(5)

(6)

(7)

(8)

The equations of equilibrium (3) through (8) were previously developed and extensively used in the Classical Mechanics course, where the three-dimensional rigid solid was studied.

The mathematical statements, (1) and (2), imply that the three-dimensional solid body is free of constraints, a situation contradictory to the norm where the constraints are present. It should be noted that the equilibrium equations do not refer in any way to the nature of the material present in the solid or to any type of deformation. Consequently, those equations also apply fully in the general case of a deformable three-dimensional solid.

To utilize the equations (1) and (2) under the usual conditions, the three-dimensional deformable solid is released from its constraints, and these are replaced with the corresponding reaction forces. As described in section 1.3, the drawing that illustrates the geometry of the three-dimensional solid, the exterior actions, and the unknown reaction forces is called a free-body diagram. A generic example of a free-body diagram is shown in Figure 1.

Figure 1 Generic Free-Body Diagrams

(a)    Constrained Body, (b) Free Body

The attempt to write the equations of equilibrium, (3) through (8), using the active forces and the reaction forces, results in a system of six algebraic equations with constant coefficients, containing the reaction forces as unknown quantities. This system of equations may be solved using any linear algebra method of solution for algebraic equation systems with constant coefficients. If the equations contain only six unknown reactions it is called a statically determinate system and, consequently, can be solved. When the number of unknown reactions is higher than six, the equation system is called a statically indeterminate system and additional equations are required for the reactions to be calculated. If a number of supports can be removed without destroying the static equilibrium of the body, those supports may be identified as redundant supports and the corresponding reaction forces are called redundants.

In the particular case of the plane linear beam when the exterior acting loads and the reaction forces are contained in the vertical plane only three equations of equilibrium from the initial six are required for solution as follows:

(9)

(10)

(11)

Note: It can be shown that only the number of equilibrium equation is fixed at six, but the option of writing these equations vary. For the plane case the equations (9) through (11) may be replaced, for example, by one force equation and two moment equations.

The external loads applied on the three-dimensional deformable body are carried to the supports, which are the points where the interaction with the environment (other bodies) is considered to take place. At the supporting points the displacements, represented by a change in the position of the point, are known quantities with prescribed value. In general, the supported point is constrained against movement in some direction and thus, the displacement in that direction is zero. There are also cases where a displacement of known magnitude and/or direction is imposed a particular support point. Due to the Newtons first law, reaction forces are developed at the supporting points. These reaction forces are vector quantities with known direction, but unknown intensity.

If the studied system is composed of more than one single body, the common points between each adjoining body are called connections. To write the equilibrium equations for a multi- body system the connections should also be replaced by reaction forces. A free-body diagram and a set of equilibrium equations may then be constructed for each body allowing for solution of all external support reaction as well as for all internal connection reactions.

Tables 1 through 3 contain the most common idealized types of supports and their corresponding reaction forces. The two-dimensional cases are illustrated in Tables 1 and 2 which are directly related with the behavior of the linear plane beam, the case of interest for this course. The supports and reactions pertinent to the three-dimensional case, only occasionally considered in this course, are depicted in Table 3 for completeness.

Table 1 Plane Supports and Reactions

Table 2 Plane Connections and Reactions

Table 3 Three-Dimensional Supports and Reactions

When subjected to external actions, the three-dimensional solid deforms and the internal original equilibrium is disturbed. To determine the internal forces located in the volume of the deformed body the method of sections is employed. The application of the methodology is depicted in Figures 2 and 3.

Figure 2 Three-Dimensional Free-Body Sectioned by a Cutting Plane

Practically, the original body is divided into two parts, each one carrying its portion of exterior and reaction forces, by an arbitrarily orientated plane. In a manner of speaking, the cutting plane plays, the role of the member connection. As in the connection case, the broken continuity of the body is replaced by internal forces acting at every point of the common surface.

Figure 3 Lower Half of the Sectioned Original Body

The internal forceis called the stress vector or traction and acts on the elementary surfacelocated in the vicinity of point (see Figure 3). Using a local coordinate system attached to the current point, where represents the unit vector normal to the cutting plane, the stress vector is decomposed into three components,, and . The decomposition process is illustrated in Figure 4. The first component of the stress vector,, is orientated parallel to the surface outside normal unit vector, while the other two components, and, are located in the cutting plane.

Note: The normal is univocally defined by the choice of cutting plane, while the other two directions, and are arbitrarily chosen under the condition that the three are mutually perpendicular.

Figure 4 Stress Vector and Components

The components of the stress tensor located in a three-dimensional solid body are defined through a limiting process.

The position of the point is uniquely defined by its position vector. The stress tensor is called normal stress and acts normal to the cutting plane. The in-plane stress tensors and are called shear stresses.

Note: The normal and shear stresses belong to a special mathematical category of algebra named tensors. It is beyond the scope of this course to elaborate on the mathematical properties of tensors, but it must be emphasized that they may not be manipulated as vectors. The stress tensor is transformed into a corresponding stress vector through multiplication with an area and, only then, manipulated through the familiar vectorial algebra.

Some clarifications regarding the sign convention of the stress tensors are necessary. The subscript indicates that the stresses are pertinent to the surface which has the outward normal vector . It is understood that the normal stress is positive when parallel to , while the in-plane shear stresses and are positive when they are parallel to the and axes, respectively.

If a new stress vector pertinent to the surface characterized by the outward normal is defined the following vectorial equation can be written:

(15)

After the algebraic simplification the fundamental relation between the surface stress tensor and the normal and shear tensors can be expressed as:

(16)

If a set of three mutually orthogonal planes, defined by a Cartesian general system and passing through the arbitrarily chosen point are used, as shown in Figure 5, then three corresponding sets of stress tensors are obtained by successively considering the unit vectors, and as the normal vector of the cutting plane.

Figure 5 Set of Three Mutually Orthogonal Planes

Consequently, the equation (16) is written for each one of the three orthogonal planes as follows:

(17)

(18)

(19)

The resulting stress tensors are depicted in Figure 6.

The following close examination and discussion of the orientation of the three sets of tensors shown in Figure 6 defines the sign convention used.

Figure 6 Three-Dimensional Stress Tensors (Cartesian Axes)

The normal stresses are positive when orientated parallel to the outward normal of the surface. They tend to pull the plane and consequently are said to be positive in tension. The shear stress tensor is considered to be positive when orientated parallel to and in the positive direction of the corresponding in-plane axis. These shear stress tensors are identified by two subscripts: the first subscript indicates the plane on which the shear stress acts by the axis defining its normal; the second subscript indicates the shear stress direction by identifying the coordinate system axis to which the shear tensor is aligned.

In Figure 6, for clarity reasons, only the stress tensors pertinent to the visible faces are shown. For the parallel planes having negative axes as normal the stresses have opposite signs. Why? The infinitesimal volume has to be in equilibrium.

Figure 7 Three-Dimensional Free-Body Diagram of an Infinitesimal Volume

Employing the notation convention described above, it is observed that the shear stress tensors converging towards any particular edge have the subscript order reversed. Consider the situation depicted in Figure 7, where for clarity only the stress tensors which can be used in writing a moment equation around the oz axis are retained.

Note: The Figure 7 is simply a free-body diagram for an elementary volume located around a point P.

The sign convention previously explained is used. The usage of the superscript or (-) indicates the variation of the same stress tensor in-between two parallel faces. The following equations relating identical stress tensors located on parallel faces, where the position vector has been drop in the notation, are written as:

(20)

(21)

(22)

(23)

(24)

(25)

Note: The variation of the stresses between the parallel faces is due to the existence of the exterior actions.

For the infinitesimal volume to be in equilibrium the total moment induced by the stress vectors must be zero.

(26)

(27)

(28)

The moment component written about the axis considering the right-hand rule is expressed as:

(29)

Substituting the relations (20) through (25) and (29) into the equilibrium equation (28), equation (30) is obtained:

(30)

Note: The final expression of the equation (30) is obtained by neglecting the product of the change in stress ( or) and the geometrical elementary quantities (, and) as being very small quantity as, and tends to zero.

The equation (30) indicates that for the equilibrium to be enforced, the edge converging shear stress tensors are equal. The equality of the edge converging shear tensors is called the duality principle of the shear stress tensors and is expressed as:

(31)

It can be concluded that even in the presence of the normal stresses the shear stresses must satisfy the equation (31)

In a similarly manner, the other two equilibrium equations (26) and (27) involving moments around and axes are written and the duality of the other shear stresses is obtained:

(32)

(33)

The stress tensors can be collected in a tabular form called the generalized stress tensor:

(34)

The generalized stress tensor, which plays an important role in the deformable solid theory, has nine stress tensors components, but due to the duality principle only six are independent.

Note: It has to be emphasized that the general stress tensor is point dependent.

The Saint Venants Principle is not a rigorous law of Mechanics of Materials, but has a great practical significance in the analysis of beams and other deformable bodies.

The theoretical explanation of the Saint Venants Principle is depicted in the Figure 8.

Figure 8 Saint Venants Principle

The original free-body diagram of the three-dimensional body in equilibrium is shown in Figure 8.a. The three-dimensional solid is loaded with a forcedistributed over a small surface of area. Suppose that the generalized stress tensor is known for any current point of the solid body, identified by its position vector. Two concentrated loads of magnitude opposing each other and acting collinearly with the resultant of the distributed force are superimposed on the three-dimensional body as shown in Figure 8.b. Their magnitude is obtained by integrating the variation of the distributed force over area:

(35)

Proceeding in this manner, the equilibrium of the three-dimensional solid body remained unchanged and consequently, the generalized stress tensorremains unaffected. The new loading situation shown in Figure 28.b may then be divided into two load cases as indicated in Figures 8.c and 8.d. The original generalized tensor is written as the sum of two new generalized stress tensors and, each one corresponding to the loading situation illustrated in Figure 8.c and 8.d, respectively.

(36)

Note: This relation is valid only under the restrictions of the Principle of Linear Superposition discussed in section 8.

Because the loads shown in Figure 8.c are self-equilibrated (force and moment resultants are zero) the generalized stress tensor calculated in points away from the loading is expected to tend toward a zero value.

(37)

Consequently, the original generalized tensor is approximated by the generalized stress tensor pertinent to the loading situation contained in Figure 28.d:

(38)

Equation (34) provides validation for Saint Venants Principle.

The geometrical description of the linear member or beam was given in Section 1.3. Here, it is re-emphasized that the length of the member is much larger than the other two dimensions. Of principal importance in characterizing the behavior of the beam is a geometrical description of its cross-section. The geometrical properties of the cross-section and its theoretical role in the development of the beam theory will be elaborated on during the entire length of these lectures.

Consider the case of a beam with a general, undefined, geometrical cross-section as shown in Figure 9. The volume of the beam was sectioned by a normal cutting plane with its outward normal parallel to the longitudinal local axis . The general coordinate system, previously used, is identical now with the local coordinate system attached to the beam.

Figure 9 Cross-Section Beam Stresses

In this section the general formulas, expressed in equations (12) through (14), will be applied to the case in point of this textbook, the linear beam.

The normal stress is renamed as , while the shear stresses and became and. The local position vector is written as:

(39)

where , andare the unit vectors of the local Cartesian coordinate system.

The stress tensor components , and can be recomposed in the corresponding stress vectors,, and , respectively, by multiplying them with the elementary surface . By integrating over the entire surface of the cross-section the components of the resultant force and moment particular to the cross-section are obtained. They are named cross-sectional resultants and are expressed as:

(40)

(41)

(42)

(43)

(44)

(45)

The cross-section resultants are depicted in Figure 10.

Fig 10 Cross-Sectional Resultants

The cross-section forces expressed in the equations (40) through (42) are called force resultants. Force is named cross-section normal force or cross-section axial force, while forces and are called cross-section shear forces. The cross-section normal force is positive if orientated parallel to the outward normal of the cross-section, while the cross-section shear forces are positive when orientated parallel to the and positive axes. The normal force is called the tension or compression force if the action is in the positive or negative direction of the cross-section normal, respectively.

The equations (43) through (45) are written considering that the local coordinate system follows a right-hand rule and they are called the moment resultants. The moment vector parallel to axis is named cross-sectional torsion moment or torque. It has a twisting action on the cross-section. The other two cross-sectional moments, and , acting parallel with and positive axes are named cross-sectional bending moments.

When subjected to exterior actions the deformable solid body exhibits volume and shape changes. Geometrically speaking, this means that the geometry of the body before the action starts differs from the final geometry. During the deformation process all of the geometrical characteristics (line segments, angles, surface and volume) of the solid body are altered. A particular pointmoves at the end of the deformation process to a new position. Similarly, the point, located in the vicinity of point, is relocated in the new position. The initial undeformed and final deformed conditions of the three-dimensional solid body are depicted in Figures 11.a and 11.b, respectively.

Figure 11 Deformation of the Infinitesimal Line Segment

(a)    Undeformed Body and (b) Deformed Body

The infinitesimal arc, which has arclength , is deformed into a new infinitesimal arc with arclength . The definition of the extensional strain is given below.

The modification of the original right angle defined by the intersection of two line segments as a result of the deformation process is called shear strain and is schematically shown in Figure 1

Figure 12 Modification of the Right Angle

(a)    Undeformed Body and (b) Deformed Body

The original right angle changes after the deformation into a new angle.

Definition 2.6 The shear strain at point, identified by its position vector , represents the change of the right angle between two line segments attached to the point and orientated in directions and, is defined as: (2.47)

The extensional and shear strain definitions are generalized for three mutually orthogonal planes in a similar manner to the development given for the generalized stress tensor in Section 3. By choosing the Cartesian unit vectors successively as normal to the cutting planes, the generalized strain tensor is obtained:

(48)

where:

), and are the elongation strain measured in the , and , respectively;

, and - are the shear strains.

Situations involving specific the use of specific components of the generalized strain tensor are commonly encountered and thus, the subject will be revisited in the following lectures.

The rational used in the development of the Saint Venants principal considered only the generalized stress tensor, but a similar argument can be made for the case of the generalized strain tensor.

The functional expressions relating the generalized stress tensor components to the generalized strain tensor components are called constitutive equations. They reflect the nature of the material contained in the deformable body. Mathematically, a general expression such as equation (45) may be written to define the aforementioned functional relationship:

(49)

where - is the component of the generalized stress tensor;

- is the component of the generalized strain tensor;

- is a given function;

- is the time derivative

The functions are chosen such that they satisfy the Laws of Thermodynamics and with parameters established by laboratory testing to give reliable results in physical usage. The change in the subscripts, from and to and , indicates that in general a stress tensor component may be a function of all the strain tensor components. Only in a few simple cases does a one-to-one relation exist.

7.1 Tension Tests

In a laboratory control environment, using a hydraulically actuated testing machine, a material specimen is subjected to a tension test. The schematics of the theoretical tension test are shown in Figure 13.

The tension test is conducted in the following steps: (a) the original (undeformed) gage length and cross-sectional area of the specimen are measured before the test; (b) the ends of the specimen shown in Figure 13.a are gripped in the testing machine; (c) force is slowly applied to the ends of the specimen until the rupture of the material is obtained; (d) during the load application measurements of the gage length , cross-sectional area and the value of the applied forced are tabulated. The steps (c) and (d) are repeated at regular time intervals and the loading of the specimen is conducting slowly in order to avoid the rise in temperature and dynamic effects. When the rupture of the specimen occurs the area is measured once again. This kind of laboratory experiments will be carried out in the Department Laboratory during the course.

Figure 13 Tension Test of Structural Steel Specimen

(a)    Undeformed Specimen, (b) Deformed Specimen

Figure 14 shows the test results obtained from the testing of a structural steel specimen, characterized by low carbon content. To emphasize the test results obtained at lower strain values the beginning end of the test was plotted at a magnified scale. This type of steel is commonly used in structural applications (buildings, bridges, etc.).

Figure 14 Stress-Strain Diagram

The values of the normal stress and elongation strain can be calculated for each loading step as follows:

(50)

(51)

where (*) indicates a particular measurement.

The normal stress defined by the equation (50) is called engineering stress, while the extension strain expressed by equation (51) is known as the engineering strain. If the normal stress is calculated using the value of the step measured area , the value obtained is called the true stress and the corresponding strain is called the true strain. They are calculated as follows:

(52)

(53)

Note: The definition (53) is based on the fact that the volume remained unchanged.

The engineering strain and the true strain can are related as expressed by equation (54).

(54)

Note: Analyzing the expressions of the engineering and true stress and strain it has to be noted that the true values are larger than the engineering values.

7.2 Mechanical Properties of Materials

Several important material properties are obtained from the analysis of the stress-strain diagram. The theoretical behavior of a typical structural steel is illustrated by the stress-strain diagram shown in Figure 15.

The loading and deformation process begins at point A and continues until point B is reached. This is the elastic region and is characterized by a proportionality of the ratio between stress and strain. The point B is named proportional limit occurring when the stress reaches . The ratio of stress to strain in the linear range (elastic region shown) is called the modulus of elasticity or Youngs modulus and is designated by the letter.

when (55)

The units for the modulus of elasticity are stress units (F/L²). Typical units are ksi or GPa.

Figure 15 Theoretical Description of the Stress-Strain Diagram

Continuing beyond the stress and strain characterizing the proportional limit at point B, the specimen begins to yield. Smaller incremental loading steps are necessary to induce additional elongation. Two new points, C and D, named the upper yielding point,, and the lower yielding point, , are obtained. For practical purposes only the lower yielding point D is retained and will be called the yielding point. From point D to point E increased elongation of the specimen occurs with no increase in stress and consequently, the stress at point E has an identical value with the stress at point D. As shown in Figure 15, the diagram is horizontal and the region is called the perfectly plastic region.

The stress begins again to increase with increased in elongation after passing point E. The increase continues until point F, where the ultimate stress or the ultimate strength, is reached. The phenomenon and the zone are called strain hardening and strain hardening region, respectively.

If the elongation of the specimen continues a decrease in the stress is recorded and the so-called necking appears. The necking represents the visual reduction of the cross-section and continues until the fracture of the specimen is attended at point G. The stress calculated at point F, is the maximum stress characterizing the entire stress-strain diagram. This value is called the ultimate stress or ultimate strength and is designated by .

If the material stress and strain remain in the elastic region where the stress is proportional the strain, the material exhibits elastic behavior. If the material is loaded over the proportionality limit the material exhibits plastic behavior.

The material behavior described above is based on the engineering characterization of the stress and strain. For comparison the true stress and strain behavior is also plotted in Figure 15. The true stress-strain diagram is plotted using a dashed line, while the engineering stress-strain diagram is plotted with a heavy continuous line.

The area contained under the stress-strain diagram represents the energy necessary to deform the specimen. It can be noted, from Figures 14 and 15, that the energy characterizing the elastic deformation is considerable smaller than the energy characterizing the plastic deformation. This is an important characteristic typical of ductile materials.

The behavior of the structural steel, as it was described above, can be explained without going into the theoretical details by behavior of the metal micro-structure. Obviously, different metals exhibit different type of stress-strain behavior when subjected to a tension test. To emphasize this observation, the stress-strain diagrams for low carbon steel, three high-strength alloy steels and three aluminum alloys are presented in Figures 16.a and 16.b, respectively.

Figure 16 Examples of Stress-Strain Diagrams

(a)    High-Strength Alloy Steels, (b) Aluminum Alloys

Materials, such as structural steel, which undergo large strain before fracture are called ductile materials. The ductility factor is represented by the ratio of the strain measured at rupture (point G) and that corresponding to the point B in Figure 15. For metals used in structural engineering the ductility factor can reach values between 10 and 20.

The materials which fail at small strain value are called brittle materials. A qualitative difference between a ductile and brittle material are shown in Figure 17. Typical examples of brittle behavior are the following materials: ceramics, glass, cast iron. The welding material, when the weld is improperly made, also exhibits brittle behavior which in many cases compromises the quality of the entire structural ensemble. Drastic change in temperature, pressure and load application can very much influence the material behavior.

Figure 17 Ductile and Brittle Stress-Strain Diagrams

The temperature is an important factor in the material behavior. Same materials can change from ductile to brittle behavior as a function of the temperature.

Figure 18 Variation of the Stress-Strain Diagram with Temperature

(a)    Low Temperature, (b) High Temperature

In most common applications the materials used in structural engineering are not subjected to temperature extremes great enough to adversely affect the ductile behavior. For the special situations when the material is subjected to high or low temperature corresponding stress-strain diagrams must be constructed. Figures 18.a and 18.b illustrate the variation of the stress-strain diagram with temperature for a type of stainless steel. Note that the strain increases for a given stress with increase in temperature and vice versa.

The American Society of Mechanical Engineers (ASME) publishes a handbook containing a large range of stress-strain diagrams for different metal composition and temperatures.

7.3 Elasticity, Plasticity and Creep

The stress-strain diagram described above has been obtained experimentally by continuous loading of the specimen. If at some time during the loading the load applied to the specimen is reversed, theoretically following the same increments as were used during loading, the process is called un-loading.

If the un-loading stress-strain diagram follows the same path as the loading branch the material is said to be an elastic material. Obviously, the elasticity of the material is manifested only if the un-loading occurs before the stress and strain reach the elastic limit, represented by the stress-strain point C shown in Figure 19.

As described previously for a typical structural steel, the stress is proportional to strain (the modulus of elasticity has a constant value) and the behavior is called linear-elastic. There are materials characterized by a nonlinear-elastic behavior. An example of non-linear elastic material is shown in Figure 19.

Figure 19 Non-Linear Elastic Behavior

If the stress-strain values corresponding to point C are exceeded the material exhibits a plastic behavior and is said to undergo a plastic flow. A typical loading and un-loading cycle from a plastic range is schematically shown in Figure 20. Noted that during un-loading the material behave as a linear-elastic material.

Figure 20 Plastic Behavior

For theoretical applications, the stress-strain diagram shown in Figure 15 is replaced by an artificial but useful stress-strain diagram named Prandtls diagram. This type of diagram is shown in Figure 21 and represents a material with a linear-elastic and perfectly plastic behavior.

Figure 21 Prandtls Stress-Strain Diagram

The experimental loading and un-loading of the specimen being discussed here are conducted in a relatively short time interval (a few minutes). An interesting phenomenon occurs if the specimen is loaded and then left under a constant load for a long period of time. If the initial strain in the specimen is calculated and the calculation is repeated at intervals of a few days, an increase in the strain is observed. This phenomenon is called creep. The result of a creep experiment is shown in Figure 2

Figure 22 Creep

Similarly, if the specimen is kept under constant strain for a long period of time a reduction in the stress value can be observed. This phenomenon is known as relaxation. A relaxation diagram is depicted in Figure 23.

Figure 23 Relaxation

7.4 Linear Elasticity, Hooks Law and Poissons Ratio

The elastic behavior of materials is important for the structural engineering point of view. From the explanations regarding the stress-strain diagram it is clear that if the material is loaded beyond the elastic limit permanent deformations will be present in the solid body. In general, structural engineers design their structures to behave elastically.

Analyzing the stress-strain diagram interval between points A and B illustrated in Figure 15 the proportionality between the stress and strain is evident. Mathematically, the relationship is expressed as:

for (56)

Equation (56) is known as Hooks law. As noted earlier, the constant is named modulus of elasticity or Youngs modulus. Structural steels typically have modulus of elasticity varying around the value of 30000 ksi or 210 GPa.

During the elongation of the tensile specimen, a reduction of the dimensions in the other two directions normal to the deformation direction is observed. This transversal contraction, mathematically represented by equation (57), is illustrated in Figure 24 at an exaggerated scale:

(57)

The constant is called Poissons ratio. This constant is non-dimensional and varies for structural steel around the value of 1/3 value. Theoretically, the Poissons ratio is limited to 0.5. The limits of the Poissons ratio are treated in Section 5.4.3.

Figure 24 Elongation and Transversal Contraction

7.5 Generalized Hooks Law for Isotropic Materials

The relationship between stress and strain described for the simple case of uni-axial elongation will be extended to the general case involving generalized stress and strain generalized tensors. The material constants and , together with the thermal coefficient of expansion are the most relevant material characteristics.

Figure 25 depicts the general case of the three-axial elongation. Under the influence of applied load and temperature change, the total extensional deformation can be decomposed into four independent effect components: (a) the deformation resulting from the change in temperature, (b), (c) and (d) deformations due to individual application of the three-directional normal stresses. This approach is called the Principle of Linear Superposition.

Note: The Principle of Linear Superposition is discussed in Section 8. The viability of this principle is restricted to the case of small deformation and linear elastic materials.

Figure 25 Deformation Induced by a Change in Temperature and

Normal Stress Components

Mathematically, the phenomena illustrated in Figure 25 are described with the following equation:

(57)

(58)

(59)

where the notation stands for:

- is the total elongation strain in direction;

- is the elongation strain induced by the thermal expansion;

- are the elongation strains induced by the action of the normal strains , respectively, in the direction

The total strains and are similarly defined by substitution of the appropriate subscripts.

The existence of the normal stress acting along the ox direction (see Figure 25.c) introduces, as described in the previous section, the following elongation strains:

(60)

(61)

(62)

Analogously, the formulae (60) through (62) are written for the case when the normal stress is considered (see Figure 25.d) as:

(63)

(64)

(65)

Considering the application of the normal stress depicted in Figure 25.e the following equations are written:

(66)

(67)

(68)

The thermal strains, which are only elongations, are expressed as:

(69)

where:

- is the coefficient of thermal expansion;

- is the change in temperature.

Introducing the formulae (60) through (69) into (57) through (59) equations (70) through (72) are obtained as:

(70)

(71)

(72)

For the case of the isotropic linear material the shear strains are related to the shear stresses by the following relationships:

(73)

(74)

(75)

Figure 26 Deformation induced by the Shear Stress Components

Equations (73) through (75) are illustrated in Figure 26. The constant is the shear modulus and is obtained from and using the following relation:

(76)

Equations (70) through (75) are named the Generalized Hooks Law and are widely used in structural engineering applications. By algebraic manipulations equation (70) through (75) may be re-written with the stress tensor components expressed as functions of the strain tensor components. The resulting equations are:

(77)

(78)

(79)

(80)

(81)

(82)

Equations (77) through (82) represent the second form of Hooks law.

The methods employed in Mechanics of Materials are based on three basic hypotheses. They are as follows:

(H1) The solid body is continuous and remains continuous when subjected to exterior actions or changes in temperature;

(H2) Hooks Law;

(H3) There exists a unique unstressed state of body to which the body returns whenever all the exterior actions are removed.

The body satisfying those thee hypotheses is defined as a linear elastic body. Under the hypotheses (H1) the real microscopic atomic structure of the solids is ignored and the solid body is idealized as a geometrical copy in the Euclidian space whose points are identical with the material points of the body. This way the continuity is simulated and no cracks or holes may open in the interior of the solid during the exterior actions. A material satisfying the hypotheses (H1) is called a continuum. The hypotheses (H2) represent the mathematical base of the definition of material elasticity. The direct implication of the Hooks Law is the Principle of Linear Superposition, a principle frequently used in the Mechanics of Materials.

The fundamental concepts presented in the preceding sections are grouped in three fundamental equations extensively used in Mechanics of Materials: (1) the equilibrium equations, (2) the geometry of deformation and (3) the material behavior.

9.1 Stress Distribution and Stress Resultants in a Rectangular Cross-Section

The cross-section characteristics and the stresses acting on it are depicted in Figure 27. The origin of the coordinate system passes though the center of the rectangular cross-section.

Figure 27 Example 9.1

The stresses existing in the cross-section are expressed by the following functions:

(83)

(84)

(85)

Substituting the relations (83) through (85) into equations (40) through (45) the stress resultant forces and moment are calculated as:

(86)

(87)

(88)

(89)

(90)

(91)

For numeric application the following cross-section dimensions and constant coefficients are assigned:

The area and the stress resultants obtained by replacing the above values into the equations (86) through (91) are:

9.2 Extensional and Shear Strain

The deformation of a thin square plate with an edge length is illustrated in Figure 28. The cornermoved horizontally by the length into a new position, while the other three corners kept their original location. It is supposed that the edge remains linear through the deformation.

The extensional strain in the direction, accordingly to the definition (46) is:

(92)

To determine the lengths and two points and as shown in Figure 28 are selected. After the deformation of the plate they are located in the new positions and.

The original area of the plate is theoretically divided in both directions in equally spaced divisions and the mesh shown in the Figure 28 is obtained. For clarity, in Figure 28 the number of division was limited only to five (). The point is located at node of the mesh, while the point is located at node. The original length is obtained as:

(93)

Figure 28 Elongation Strain in Square Plate

(a) Undeformed and (b) Deformed Shapes

After the deformation the segment is shorter and is calculated as:

(94)

The extensional strain is obtained as:

(95)

It can be concluded that the elongation strain is a function only of variable and varies between values of zero and .

The shear strain which reflects the right angle change is calculated following a similar rationale. The original right angle is defined by the segments and as shown in Figure 29.a. After the deformation, as illustrated in Figure 29.b, the angle changes to angle defined between the segments and .

Figure 29 Elongation Strain in Square Plate

(a)    Undeformed and (b) Deformed Shapes

From definition 6 the shear strain is given by:

(96)

The following geometric relations can be written:

(97)

(98)

(99)

(100)

(101)

The angle is calculated using the above formulae:

(102)

The shear strain is obtained as:

(103)

The shear strain is dependent only on variable and varies from to zero value.

9.3 Volumetric Strain

The rectangular parallelepiped of isotropic linear elastic material is subjected to three normal stresses, and as shown in Figure 30.

The absence of the shear stress indicates that only volumetric changes result without shape change. Consequently, the undeformed volume is modified and after the deformation the solid body has a new volume. The volumetric deformation is called dilatation. The volumetric strain is defined by the following ratio:

(104)

Using the notation shown in Figure 30 the following relations are written:

(105)

(106)

Figure 30 Volumetric Strain

The change in the dimensions is expressed as:

(107)

(108)

(109)

Substituting the above expressions into the expression of the volumetric strain the following equation is obtained:

(110)

If the strains are small the products can be neglected and the volumetric strain becomes:

(111)

Substituting the strain equations (70) through (72) into equation (111) and for the condition the volumetric strain is written as a function of stresses:

(112)

Equation (112) indicates that the volumetric strain is always a positive value in the presence of three dimensional tensions, which geometrically represents a volume increase.