Chapter 1
1.1 Introduction
The soft fingertips which deform during contact increase the ability to create a stable grasp. This is because when soft fingertips deform during contact they provide a large space for frictional forces and moments than their rigid counterparts. This is true not only for human grasping, but also for robotic hands using fingertips made of soft materials [26].
The superiority of human fingertips as compared to hard robot gripper fingers for grasping and manipulation has led to a number of investigations with robot hands employing soft materials such as fluids or powders beneath a membrane at the fingertips.
When the fingers are soft, during holding and for manipulation of the object, their property of softness maintains the area contact between, the fingertips and the object, which restraints the object and provides stability. In human finger there is a natural softness which is a combination of elasticity and damping. This combination of elasticity and damping is produced by nature due to flesh and blood beneath the skin. This keeps the contact firm and helps in holding the object firmly and stably.
A human finger with its components is shown in the Figure 1.1. A robotic finger can be constructed in a similar manner as shown in Figure 1.1. The robotic fingertips can be constructed with hard materials with soft coverings. A variety of materials can be used for soft covering e.g. Plastic, Rubber, and Sponge, Electro-rheological fluids such as Fine Powder, Paste and Gel etc.
1.2 Types of contact
There are three types of contact between robotic finger and grasped object
Point contact with friction (Hard finger contact): in this type of contact the end effectors can exert a force directed inside the friction cone at contact (the force includes both normal and tangential components).
Point contact without friction: in this type of contact the end effectors can only exert force at the point of contact.
Surface contact (soft finger contact): in this type of contact the robot can exert force and torque along the area of contact (a pure torsional movement about the common normal at the point contact is allowed).
1.3 Main problems occur during contact
Three potential problems
Impact force during contact
Conformability
Dissipation of strain energy
occur in multifinger hand grasping. If they are not dissipated, the manipulation become jerky, the functioning of fingertip sensor gets affected, and the life of finger’s skeletal structure may even become shorter.
Impact force: Impact force is a force that delivers a shock or high impact in a relatively short period of time. It occurs when two entities collide. Impact force results during each instant of grasping a rigid object can affect the functioning of the fingertip sensor.
Conformability: A hand with hard fingers cannot securely grasp objects that have uneven surfaces due to the poor conformability of the fingers.
Strain energy: The strains are induced into the fingers all throughout a manipulation task
1.4 Soft Materials for Robotic Fingers
Human fingertips are fleshy, soft, and deformable. They locally mould to the shape of a touched or grasped object, and for these reasons, are capable of extremely dextrous manipulation tasks. Until recently, the most robot fingers have been crude and pincer-like, and therefore rather limited in capability. This realisation is attracting the attention of researchers to investigate robotic manipulation with soft human-like fingers.
What is the best “skin” material for the contact areas of a robotic hand? The answer depends on many factors, including the tasks the hand must perform and environmental factors such as temperature, dirt and moisture. From a design standpoint, most of these factors can be reduced to requirements in
friction and adhesion under the range of expected gripping and handling conditions
mechanical properties such as resilience and elasticity
Durability, resistance to abrasion and chemical attack suitability for tactile sensing and compatibility with various tactile sensors.
1.5 Advantages of using soft material at finger tip
A robotic fingertip with soft elastic cover provides following advantages over conventional hard fingertips:
Manipulating a fragile object by defusing exerted force at the contact.
Dynamically adjusting friction according to a task context.
Achieving different types of contact according to the shape of an object surface: planer surface and edge.
1.6 Contact Mechanics
When two bodies at least one with curved surface are in contact under a force, the point or line contact between them changes to area contact, and contact stresses are developed. In 1882, Hertz derived and experimentally validate a relationship (based on linear elastic contact mechanics model) between growths of contact area as a function of normal force N. That is,
Where
a – radius of contact
N – Normal force
c – Constant of proportionality (depends on curvature and material)
To include nonlinear materials Nicholas Xydas and Imin Kao in 1998 [11] extend the linear elastic model which represent more realistic soft finger. They give a relation between radius of contact and normal force. That is,
Where
The experimental setup used for the investigation is shown in Figure 1.5. The finger was mounted on a vertical pole and an electronic scale was used to display the normal force which was placed under the fingertip. When fingertip comes in contact with paper surface it leaves an imprint on it because paper was coated with fine tonner powder. Various experiments were conducted with different hemispherical artificial fingertip materials at the range of normal force from 0 to 90N.
After the experiments graph (Figure 1.6) was plotted by taking contact radius as ordinate and and normal force as abscissa. It is observed from the curves that the radius of contact is exponent function of normal force N. Silicon and rubber were used for fingertips having same diameter and identical shape.
The results for silicon and rubber are:
a=2.10N^0.25 For silicon
a=1.85N^0.26 For rubber
The linear elastic model given by Hertz is shown as upper limit (γ =1/3) in the graph. The ideal soft finger has exponent of γ =0.
Chapter 2
Literature review
2.1 Introduction
In this chapter, a literature reviews reported keeping in view the scope and objectives of the study. Many researchers and academicians of international repute have probed into the related work, whose name and work abstract has been discussed in the coming section.
2.2 Related Work
Mark R. Cutkosky, John M. Jourdain and Paul K. Wright (1987) [1] tested compliant materials both under clean, dry conditions and environmental contamination to find most suitable material for the contact area of robotic hand and to establish models of the materials friction behaviors. Skin materials were mounted on the tip of a liver and pressed against glass, metal and plastic surfaces. The hard materials generally obeyed the coulomb’s law and soft materials showed substantial adhesion. The various materials they have tested were Poron, PDS polyurethane, Sensoflex, Tacky foam rubber.
Tsuneo Yoshikawa and Kiyoshi Nagai (1987) [2] discussed the manipulating force and grasping force for dexterous manipulation of objects by three fingered robot hand and gave a new representation of the internal force among the finger which is used to define the grasp force as an internal force which satisfies the static friction constraint. And they also introduced a mode of grasping. Following conditions were satisfied by manipulating force which is defined as fingertip force: i) a specified resultant force is produced by it, ii) fingertip force is not in reverse direction of grasping force, iii) It does not contain any grasping force component. They also gave an algorithm for decomposing a given fingertip force into manipulating and grasping forces.
Prasad Akella and Mark Cutkosky (1989) [3] studied fingertips, assert on energy loss mechanisms and their relation to the dynamics of manipulation. They made a soft fingertip filled with Electro-rheological fluid. They propose rolling models for fingertip adapted from hot metal rolling and extrusion. To help in choice of an appropriate model they presented sensitivity study by comparing point-contact, elastic rolling and fluid filled fingertips in terms of manipulation dynamics.
Gary L. Kenaley, Mark R. Cutkosky (1989) [4] fabricated and tested the prototype fingertips using electro rheological fluids with elementary tactile sensing. The fingertips consist of a layer of electro rheological fluid which is sandwiched between a grounded elsatomer (skin) and positively charged electrode. The arrangement forms a capacitor whose value increases as elastomer is deflected towards the positive electrode. The electro rheological fingertip can be put on a robotic gripper, or the finger of a dextrous robot hand. They examine the improved lifting forces available with electro rheological fingertips over passive designs and discussed the compatibility between skin materials and electro rheological fluids and the cause and prevention of fluid stratification.
Peamath Raj Sinha, Jacob M. Abel (1992) [5] Present an analytical investigation between two or more fingers in contact with an arbitrary rough object, with intent to focus on the transfer of object from one manipulator to another. For this they examine the finger/object interaction as a contact problem under distributed normal force. They formulate a relationship between normal and tangential forces.
K.B.Shimoga, A.A.Goldenberg (1992) [6] experimentally compare the ability of six different materials (plastic, rubber, sponge, a fine powder, a paste, a gel) to overcome the three potential problems (a. Impact force attenuation, b. Conformability c. Respective strain dissipation) exists in multifinger hand grasping. Three experiments are done on each material these are the impact experiment, the conformability measurements and cyclic load experiments.
Imin Kao (1994) [7] Investigate the problem of calibrating the stiffness of human and robotic hands using least-squares approach. Force and displacement data for human grasps are used to find the stiffness matrix of grasp by least least-square fit method. The stiffness matrix consisting of both active and passive components, correlate external disturbance forces with small displacements. As a result, they relate the force/moment on each fingertip in terms of the change in position and orientation of fingertip with respect to specific grasp configuration and orientation. The characteristics and stability of grasp is determined by using eigen values of stiffness matrix. Three different configurations of the constraints are studies (Handle support, Vacuum Cast, Free motion), which show that kinematics and close-loop chain of grasps have significant effects on the stiffness of a grasp.
Ying Xue and Imin Kao (1994) [8] showed that the model of sliding analysis with minimized non rigid body norm can be extended to the motion planning over finite range of motions and trajectory planning for soft robotic fingertips. The result shows that: 1) the relative magnitudes of rigid body and non-rigid body motions in manipulation tasks can be used as an index to measure the overall motion of the grasped object.2) One can plan the more effective rigid body motions of the grasped object by careful planning of advantageous orientations of external forces/ motions.
Dean C. Chang and Mark R. Cutkosky (1995) [9] determine the kinematic effect of soft fingertips during manipulation with pure rolling without sliding. They have conducted experiments with a variety of soft material including rubber, foams and membrane fingertips, and found that in most cases, for modest deformations on the order of 10% of the under formed fingertip radius, the change is rolling distance is small but noticeable and can be either positive or negative depending on fingertip material. The principal cause for the difference in expected and actual trajectories is circumferential strain. Adding a thin, inelastic band around the fingers prevents circumferential strain and result in object rolling motions almost exactly identical to those predicted with rigid body rolling kinematics.
Tsuneo Yoshikawa, Yong Yu, Masashi Koike (1996) [10] Describe a technique that can estimate the contact position and geometric information between the object and environment from a number of positions and orientations of moving object, which are measured while object is slid and rotated on the environment around the contact position. The method proposed on the basis as follows: The object and environment are polyhydra, the position and orientation of grasped object with respect of hand frame are already known, the position and orientation of the robot end-effecter can be obtained from the robot internal sensor. They validate their proposed method by experiments using robotic arm.
Nicholas Xydas and Imim Kao (1998) [11] conduct experiments to validate the theory of relationship between normal force and area of contact for soft finger by considering the soft finger materials as nonlinearly elastic according to which the radius of contact is proportional to the normal force raised to the power γ which ranges from 0 to 1/3 for anthropomorphic soft fingers. That is,
For the experiment the artificial finger is mounted on a linear stage through a vertical pole. An electronic scale is placed under the fingertip with tray vertical to the lateral axis of the finger. When the fingertip comes in contact with surface of tray, the normal force is developed between tray and finger which is displayed on electronic scale. The area of contact measured directly from the finger imprint.
A metallic surface and a plexiglass were tested in this experiment. The fingertip used for the experiment was hemispherical therefore the shape of contact area was circular. A least square curve fitting algorithm is used to fit the experimental data to provide an empirical relationship between the normal forces and area of contact. The value of γ for various fingertip materials is calculated are as follows: Silicon = 0.25, Rubber = 0.26.
Because rubber is the harder material than silicon so it is concluded that harder materials tends to have higher value of exponent γ. Experimental data for materials fall within 0 ≤ γ ≤ 1/3. The lower bound γ = 0 corresponds to the ideal soft finger, while the upper bound γ = 1/3 corresponds to the linear elastic contact model derived by Hertz in 1982.
Nicholas Xydas, MilindBhagavat, and Imin Kao (2000) [12] study the soft-finger contact mechanics by employing the nonlinear finite element analysis. Two fingertips of the same material but with different sizes are analyzed. The results are compared with experiments to support the power-law theory ( ). Based on the power-law theory, the force-radius relationship simulated by the FEM analysis matches with the experimental results conducted. In all cases, the exponent of the power law ( ), for soft fingers is confirmed to be within the range of . The profiles of pressure distribution for the fingertips are obtained by the FEM analysis. The FEM model shows that the influence of friction for small deformations ( ) is negligible, but it becomes gradually significant for ( ).
Where
a – Radius of contact
Ro – Fingertip Radius
c – Constant depending upon material and geometry of fingertip
N – Normal force
γ – Constant ( )
Yanmei Li and Imin Kao (2001) [13] presented a progress in modelling of dexterous manipulation utilizing soft contacts and stiffness control. The result extends the hertzian contact model from linear elastic contacts to soft contacts. They also presented the CCT (conservative congruence transformation) for stiffness control in robotics, which are often used in dexterous grasping and manipulation.
Shih-Feng Chen, Yanmei Li and Imin Kao (2001) [14] presented a theory of stiffness control for modelling dexterous manipulation and its applications in grasping. The fingers of a dexterous hand manipulate an object by exerting both forces and moments at the contact. The stiffness control indicates how the fingertip forces and moments change when the fingertips are moved by small amounts. In this literature, the geometrical method provides a systematic way of constructing 6 x 6 Cartesian stiffness matrices in robotic stiffness control and manipulation/grasping.
Tsutomu Hasegawa and Kyuhei Honda (2001) [15] proposed a method of detection and measurement of fingertip slip on the surface of manipulated object in a multi-fingered precision manipulation with rolling contact. Tangential slip displacement of the fingertip contact is reliably detected and measured from noisy real time data by multi-sensor fusion of the stereo vision, joint-encoders, and the fingertip force/torque sensors.
Kwi-Ho Park, Byoung-Ho Kim, and Shinichi Hirai (2003) [16] analyze the geometrical relation of the soft fingertip when it is deformed and investigate the force distribution of the soft fingertip by using a compressional strain mechanism. And also, propose a nonlinear model of the soft fingertip with the help of which they obtain the total contact force at the contact surface of each finger in manipulating tasks. A hemisphere-shaped soft fingertip for soft fingers was developed and a nonlinear force function of a soft fingertip according to the deformation was modelled by considering the force distribution in the contact surface. Model was considered in the one-dimensional contact of a finger. Through experimental evaluations, the proposed force function was verified, where a tactile sensor and a tactile sensor signal processing system were used to measure the contact force distribution in the contact surface and its total force.
Takahiro Inoue and Shinichi Hirai (2003) [17] propose a simple contact model of a soft fingertip based on geometrical analysis and statics of the fingertip. They describe a soft fingertip by a collection of elastic cylindrical components. Material used to make soft hemispheric fingertip is polyurethane plastic material. The following assumptions were made during modelling: i) the object is rigid body and the contact plane between a finger and the object is planar. ii) External force exerts on the object along the normal vector of the sensing plane. iii) Young’s modulus of soft fingertip is constant. iv) Elastic force generated by a component acts along the central axis of the component. By Collecting the deformation of individual components, describe the deformation of a soft fingertip during the manipulation process of an object. Then formulate pressure distribution along a bottom plane of a soft fingertip. Based on the formulated distribution, compute the total force exerted on a fingertip and the pressure center of the distribution. After that they verify the proposed model experimentally by comparing the simulated pressure distribution and the measured distribution.
LiorKogut and IzhakEtsion (2003) [18] presented an improved elastic-plastic model for the contact of rough surfaces that is based on an accurate FEA solution of a single asperity contact. It predicts the contact parameters, such as separation, real area of contact and real contact pressure as function of plasticity index and contact load. This model is based on constitutive laws appropriate to any regime of deformation, be it elastic or plastic.
Imin Kao and Fuqian Yang (2004) [19] Studies the nonlinear stiffness of contact for soft fingers, commonly used in robotic grasping and manipulation, under a normal load and Experimental results are used to validate the theoretical analysis.
Qiao Lin, Joel W. Burdick, and Elon Rimon (2004) [20] computes and analyzes the natural compliance of fixturing and grasping arrangements. They derived a closed-form formula for the stiffness matrix of multiple contact arrangements that admits a variety of nonlinear contact models, including Hertz model.
Luigi Biagiotti, Claudio Melchiorri, Paolo Tiezzi and Gabriele Vassura (2005) [21] Performed the static and dynamic characterization of viscoelastic pads for robotic hands. Develop a dynamic model for viscoelastic pads. Two different materials (a polyurethane gel and a silicon rubber) were tested to evaluate the capability of the model to reproduce the soft fingertip behaviour, which show behaviour similar to that of human pads and seem very suitable for robotic applications. Then they extend the model with the use of digital filters to make clear-cut identification process.
Takahiro Inoue and Shinichi Hirai (2005) [22] propose a static elastic model of a hemispherical soft fingertip in a physically reasonable form suitable for theoretical analysis of robotic handling motions and also validate the static elastic model by conducting a compression test of the hemispherical soft fingertip and comparing the results.
Giovanni Berselli and Gabriele Vassura (2009) [23] proposes differentiated layer design, that is the adoption of a single elastic material, dividing the overall thickness of the pad into layers with different structural design (e.g. a continuous skin layer coupled with an internal layer with voids). Four basic design patterns were developed and tested, (i) Pattern with equally spaced hemispherical protrusions. (ii) Pattern with equally spaced hemispherical voids. (iii) Pattern with circumferential ribs connecting the core to the external layer. (iv) Pattern with a series of inclined micro-beams, fundamentally subjected to bending. Two types of elastic materials were used: (i) Tango Gray (with tensile strength of 4.36 MPa) (ii) Tango plus (with tensile strength of 1.50 MPa). Their compressive behaviors are tested and comparatively evaluated.
AnandVaz and Aman Kumar Maini (2009) [24] integrate the Bond Graph technique with Finite element method and modeled a system dynamics during the soft contact interaction between a rigid body and a soft material.
Elango. N and Marappan. R (2010) [25] investigates the suitability of different finger configurations, diameters and soft materials which are applied to power grasping. They investigate three different finger configurations ((i) a semi cylindrical solid finger made of the soft material (ii) A cylindrical solid finger made of the soft material (iii). A cylindrical core member covered by a skin like soft material), three different diameters (17.8mm, 16.5mm, 16mm) and three different hyper elastic materials (Viton E-60C, Silicon R401/70 and Neoprene W). The experiments were carried out against Mild steel and Polycarbonate surfaces and deformation on fingers were measured.
Khurshid, A., Ghafoor, A. and Malik, M. A. (2011) [26] designed a robotic gripper with soft fingers using bond graph modeling technique to obtain mathematical model of two soft contact robotic fingers.
Sadeq H. Bakhy, Shaker S. Hassan, Somer M. Nacy, K. Dermitzakis and Alejandro Hernandez Arieta (2012) [27] studied the nonlinear contact mechanics of hemicylindrical soft fingers through theoretical modelling and also validate the result experimentally. They derive a relationship between the normal force and the half width contact area assuming that the materials of anthropomorphic hemi cylindrical soft finger with different silicone-based materials are nonlinear elastics. And experimentally determine the growth of the contact area with respect to the normal force for typical anthropomorphic hemi cylindrical soft fingers.
Anil Kumar Narwal, AnandVaz, K.D. Gupta (2014) [28] Modeled the dynamics of contact between a rigid body rolling on a soft material using multi bond graph approach integrated with Finite Element Method.
Anil Kumar Narwal, AnandVaz, K.D. Gupta (2014) [29] developed a bond graph model for soft contact interaction between soft material and a rigid body and simulate it for non-circular rigid body. This model is applicable to all rigid bodies of different geometries, in contact with soft material. The approach facilitates the determination of the contact area, and the distribution of forces at the contact interface, during dynamic contact interaction.
Amin Fakhari, Mehdi Keshmiri, Mohammad Keshmiri (2014) [30] Studied the dynamic modeling and slippage analysis of a three-link soft finger manipulating a rigid object on a horizontal surface.
Anil Kumar Narwal, AnandVaz, K.D. Gupta (2014) [31] Developed a bond graph model for the soft contact and simulated it for a non-circular rigid body. The model is applicable to all geometries of the rigid body.
Arvind Kumar Pathak, Neeraj Mishra, AnandVaz (2015) [32] presented a methodology for modelling and simulating the dynamics of a three-joint prosthetics finger actuated by remaining natural finger joint based on string-tube mechanism using multibond graph.
Amin Fakhari (2015) [33] proposed a more accurate model to describe the asymmetry of the pressure distribution in the contact interface of a hemispherical soft finger under both normal and tangential forces.
MohitSachdeva, Anil Kumar Narwal, AnandVaz (2015) [34] Modeled the impact rolling contacts between a rigid sphere and a soft material, using multibond graph.
Chapter 3
Finite element Method (Two dimensional triangular elements)
3.1 Introduction
FEM is a technique used to solve the complex physical problems. Complex problems have complex equations, FEA divide the complex physical problems into small parts. These finite parts are called finite elements. More number of elements will give more accurate result but calculation will become complex. Each element has nods which are connected to next element. Each node in an element is solving for its equation, which together forms a matrix known as stiffness matrix for element. Each element is solved for its own stiffness matrix, at the end all the stiffness matrixes are combined into a large matrix, which represent the stiffness of whole system. FEM can be applied on 1D, 2D and 3D.
3.2 Strain energy
Strain energy is the energy stored by a system under deformation. When a force is applied to a plate, it deforms and stores strain energy. If we take a small element from the plate that has been deformed, we can use stress (σ) to represent the force in the material and strain (ε) to represent the displacement of the material. When load is removed the strain energy or potential energy is given by:
This equation is for three dimensions (for entire volume), by expressing the volume of plate as area of plate times the thickness of plate, where thickness is constant. We can rewrite the equation 3.1 as:
Where
ε – Strain in the element of plate
σ – Stress in the element of plate
t – Thickness of plate (Constant)
A – Area of plate
Equation 3.2 is the potential energy equation which we will use to develop the stiffness matrix, to replace the stress and strain because both stress and strain are unknown.
FEA Elements
Take a thin plate; divide it into finite triangular elements as shown in figure 3.1. Each triangle is an element. Each element has three nodes, which it shares with other elements. The external forces are applied at the nodes. The elements and nodes are used to approximate shape of the object and to calculate the displacement of points inside the boundary of the object.
3.4 Two dimensional stress-strain relationship
According to Hook’s Law (Relationship given by Robert Hook, an English Mathematician in 1678)
Where
σ – Stress
E – Young’s Modulus
ε – Strain
Equation 3.3 is for one dimensional, for two dimensional equation consider a two dimensional element as shown in figure 3.2.
These stresses can be used to write the strain equations
Where
σ – Axial stress
Ε – Axial strain
τ – Shear stress
λ – Shear strain
ν – Poisson’s ratio
E – Young’s modulus
Now we will use the equation 3.4, 3.5, 3.6 to solve stress. Take equation 3.4 solve it for
Substituting the value of from equation 3.7 in equation 3.5 and solve it for
Take equation 3.5 to solve for
Put the value of from equation 3.9 to equation 3.4. And solve it for
From equation 3.6 find the value of
Write the equation 3.10, 3.8, 3.11 in vector form as
Or equation 3.12 can be written as
Where
Use equation 3.13 to rewrite equation 3.2. So that
Becomes
Stress is eliminated in this equation, now next step is to eliminate strain and develop stiffness matrix.
3.5 Two Dimensional Triangular Elements
Take a triangular element from the problem. The nodes of this element will have displacement in x and y direction, number them as shown in figure 3.3 and figure 3.4
We can write the local displacement vector for each triangle as:
For whole object the global vectors can be written as
Which includes all rn terms.
3.6 Shape Functions
We need to compute the displacement of nodes as well as inner points of the triangle. To compute the displacement of inner points we will use interpolation technique based on areas.
Consider a triangle with node a,b,c. Fix the node a and b and displace the node c. Draw the new triangle as shown in figure 3.5 over the original triangle by overlapping the node a and b.
In the figure 3.5 base is same but height changes, it means area is the function of height only or the displacement of node c. Area of triangle is given by half the product of base and height, i.e.,
Where
A – Area of triangle
b – Base of triangle
h – Height of triangle
Apply this on all three nodes the displacement of inner point will be computed by summing the displacement due to three triangular nodes. Figure 3.6 shows the inner point divides a triangle into 3 regions. Let Aa, Ab and Ac are the areas of three regions. From equation 3.18 we know that the area of triangle is A. We can see from the diagram that total area of triangle is the sum of areas of three regions, i.e. Aa, Ab and Ac.
Where
are the shape functions.
By taking a=1, b=2, c=3 the displacement of the inner point can be computed with the equation 3.22 and 3.23. The displacement u is in X direction and v is in Y direction.
From equation 3.20 we know that sum of all shape functions is 1. It means they are not independent from each other. By putting a=1, b=2, c=3 equation 3.20 becomes
If we know two shape functions, we can compute the third one.
, and
By substituting the values of N_1,N_2 and N_3 in equation 3.22 and 3.23, we get
From equation 3.26 we get
From equation 3.27 we get
We can use the same shape function to find the coordinates of a point inside a triangle. Let are the vertices of a triangle. are the coordinates of unknown point inside triangle.
After substituting , and , we get
and
The equations 3.32 and 3.33 can be used to compute the shape functions. By knowing the values of , , and (see figure 3.7) we can solve the equation 3.32 and 3.33 for α and β. If we know the displacement at the nodes we can use the same shape functions to compute the displacement for the point at .
3.7 Elementary Solid Mechanics
Consider a small element of a material having u and v displacements across the element, then strain can be written as
Where
– Strain in X direction
– Strain in Y direction
– shear strain
We have equations of u and v in terms of α and β not in x and y. But by using chain rule
By writing equation 3.35 and 3.36 in matrix form we get
Use equation 3.32 and 3.33 to find the derivatives in matrix in equation 3.37 and 3.38
We can write the equations in general form as
By putting the equation 3.39, 3.40, 3.41, 3.42, 3.43, 3.44 in equation 3.38 we get
As we know if
Then
And Jacobian of a matrix is
Also the area of triangle is
By applying the equation 3.47 in 3.45 we get
Equation 3.50 can be written as
From equation 3.28 we can find and
By putting the value of and from equation 3.53 and 3.54 in equation 3.51 and 3.52 we get
Similarly, by using the same process for v we find that
From equation 3.29 we can find and
By putting the values of and from equation 3.60 and 3.61 in equation 3.58 and 3.59 we get
So the strain defined in equation 3.34
Becomes
Equation 3.64 can be simplified by using relationships in x and y terms.
Now add and subtract from RHS
Equation 3.66 can be written as
Now taking the equation 3.64
By rearranging equation 3.66 we get
By writing the equation 3.67 in matrix form we get
Where
G is 3×6 element strain displacement matrix. It relates three strains to six nodal displacements.
Put equation 3.68 in equation 3.15
For single triangle this equation can be written as
Matrices G and H and thickness of plate is constant, so they can be moved outside the integration. The equation 3.71 becomes
Put (Area of single triangle). The equation 3.72 becomes
The stiffness matrix for a triangle can be represented as
By putting equation 3.74 in equation 3.73 we get
Us is potential energy for single triangle.
The individual triangles can be used to compute strain energy over the entire plate. After sum we get
Equation 3.76 can be written as
Where
U – Potential energy
R – Global displacement vector (sum of all local displacement vectors)
K – Global stiffness matrix (sum of all local stiffness matrices)
3.8 Computing the displacement
The first step for computing displacement is called meshing. In this step we discretized the entire plate being studied into small triangles. These triangles are called elements and their vertices are called nodes. All the triangles should be roughly of same size. They must not overlap with each other. They must share vertices. All the triangles that share a side must share two vertices. Meshing can be done by hand or by using finite element software.
In second step we compute the stiffness matrix it is computed by using equation 3.74
Where
ts – Thickness of element
As – area of element
The term x32,y23 and other similar terms can be computed from coordinates of vertices with relationships.
After computing the stiffness matrix (ks) for each element they are summed in global stiffness matrix K. It is a symmetric matrix. When summing the local stiffness matrix, the degree of freedom of nodes are used to determine which row and column of global stiffness is to use.
After computing the global stiffness matrix, it is used to write the equation
Where
K – Global stiffness matrix
R – Displacement vector
F – Force vector
In next step we apply the constraints to fix the plate in space. It reduces the size of the problem.
In final step we solve the problem to find displacement of triangular nodes. Gaussian elimination or some other technique can be used to solve the problem.